..
A MATHEMATICAL M:lDEL FOR THE ENERGY
AND PROTEIN METABOLISM OF HOMEOTHERMS
by
Kenneth Falter
..
Institute of Statistics
Mimeograph Series No. 81;
Raleigh - April 1972
iv
TABLE OF CONTENTS
.
,
v
.
Page
LIST OF TABLES .
vi
LIST OF FIGURES
viii
1.
INTRODUCTION
1
2.
PHILOSOPHY OF MODELING
3
3.
BACKGROUND AND KEY LITERATURE
9
3 1
3.2
3.3
3.4
0
4.
Roles of Feed Constituents
Definition of Physical Compartments
Current Feeding Systems
Mathematical Approaches . . . .
DEVELOPMENT OF THE MATHEMATICAL MODEL
4.1
4.2
Preliminaries..
.
Conceptual Framework
4.2.1
4.2.2
4.3
4.4
4.5
5.
Nitrogen, Energy and Heat
Further Subdivisions of Nitrogen and
Energy
23
26
27
34
35
37
The Mathematical Model
50
54
A Model for a Growing Steer.
Initial Estimates . .
5.2.1
5.2.2
5.2.3
5.3
5.4
21
Notation
The Flow Laws
TESTING THE MODEL
5.1
5.2
14
16
21
23
Discussion of the Total Compartment Model
Development of Flow Laws
4.4.1
4.4.2
9
9
Evaluation of Constants
Evaluation of Initial Conditions
Initial Estimates of Parameter Values
The Goodness of Fit Criterion
The Iterative Estimation Procedure
6.
RESULTS AND DISCUSSION.
.
7.
CONCLUSIONS AND RECOMMENDATIONS
.
55
62
62
66
69
79
84
86
102
v
TABLE OF CONTENTS ( continued)
Page
8.
LIST OF REFERENCES
9.
APPENDICES
9.1
9.2
9.3
...
.··
.··
·
9.3.3
9.3.4
9. 7
·
·
·· ·
9.3.2
9.6
105
Flow Laws for Milk Production
Flow Laws for Heat Loss
Evaluation of Constants
9.3.1
9.4
9.5
...• ···
·• ··
Evaluation of P
B
Evaluation of f B, f S
Eva1uatio~ of w ' W
Bn
s
Evaluation of r
s
."
108
112
118
118
·
119
120
121
Evaluation of Initial Conditions for the Pools
Evaluation of c
d,o
Evaluation of Kp,B
•
Evaluation of Weights for the Goodness of Fit
Criterion •
.
·
·
123
125
· ··.·....
...·
108
..
.. ..
128
··.
130
•
..
'"
vi
LIST OF TABLES
Page
4.1
Flow laws for the compartment model
4.2
Differential equations for the compartment model
53
5.1
Flow laws for testing the model . • . • . . •
58
5.2
Differential equations for testing the model
61
5.3
Input data for testing the model
63
5.4
Constants used to test the model
67
5.5
Initial pool concentrations and values Of. A.B.an.d cd.'o
used in testing the model
10
5.6
Values of b
76
5.7
Initial estimates of parameter values •
78
5.8
Nitrogen and energy balance variables.
79
5.9
Experimental results
82
BW
.
51
' D, and K . .
BW
5.10 Weights for the goodness of fit calculation.
6.1
....
87
Variance-covariance and correlation matrix of final parameter
values . • . . .
. • . . . • • • . . • • •
88
6.3
Parameter values used for sensitivity test
90
6.4
Summary of
r
values by feeding level and run
92
6.5
Summary of
r values
r
values, degrees of freedom and mean square
. . . • • . . • • . .
94
Mean square error in the raw data, due to the model and
fraction of variation accounted for by the model
95
6.7
Model results
96
6.8
Weighted residuals of final simulation run
9.1
Simplified composition of energy-containing substances in
dry skeletal muscle • . . • . • . . • . . . . . . . . •
6.2
..
Final parameter values - averages and coefficients of
variation . .
. . . ..
..•.
84
6.6
,
.
97
118
vii
LIST OF TABLES (continued)
Page
9.2
Summary of pool concentration data
125
9.3
Values of heat production, dry matter fed and fat energy
balance
. . . . . • • .
126
9.4
Heat production values for zero fat energy balance
127
9.5
Regression constants and values of cd
128
9.6
Values of s
9.7
Values of wi' xi' a, b, and s; for each factor
,0
..
p
131
132
..
viii
LIST OF FIGURES
Page
2.1
Iterative model-building
3.1
Structural and energetic roles of feed nutrients
10
3.2
Physical compartments . . . •
11
4.1
Physico-chemical compartments.
25
4.2
A compartment model . .
28
4.3
Flow law from stores to P
6.1
plot of K P versus
B,
6.2
Weighted residuals versus feeding level (FL) for final
simulation run . • . .
~
7
se
, B by
as a function of fatness
45
breed • • . . . . • . . • •
93
98
9.1
Graph of milk production rate •
111
9.2
Possible forms of A ,A versus body temperature
l 2
115
9.3
Possible form of K'
versus body temperature
Pse,Ph
116
9.4
Possible form of
9.5
Graph of f(x) = xl (x + Of)
9.6
Graph of s versus feeding level and fitted regression
p
,
1 l.nes . . . . . . . . . . . . . . . . . . . . . . . .
~ se,Ph
versus body temperature
........
.
117
122
133
e.
.,
1.
INTRODUCTION
Since the early 1800's, much effort has gone into .evaluating feeds
for animal functions such as maintenance, growth, fattening and other
production.
A number of feeding standards and systems have resulted.
It was early realized that values based solely on feed content were
inadequate, hence the concept of nutrient availability was applied.
These standards have served as useful guides.
Starting with these
standards, feeding regimens are empirically adjusted to take into
.'
j
account economic factors and factors known to affect feed assessment
and animal production.
Many such factors are not handled explicitly
in the standards.
With the advancement of nutritional knowledge, increasing emphasis
has been given to the metabolic processes involved in digestion and
utilization of feed.
Efforts have also been made to formulate
matical equations to describe these processes.
Blaxter
~
mathe~
al. (1956)
derived a two compartment model for the passage of undigested feed
through the digestive tract.
Blaxter and Mitchell (1948) used a
statistical procedure for estimating true digestibility of proteins.
Lucas and Smart (1959) successfully applied this approach to calculating true digestibility of 11 feed components in two types of forage.
Blaxter (1962b) proposed a new system for assessing energy values
of feeds which adjusts the metabolizable energy of a ration for the
plane-of-nutrition effect.
Lucas (1964) developed a model to handle
aspects of digestion'and absorption.
metabolism in a mathematical way.
Waldo (1968) discussed nitrogen
In a recent lecture,
2
Lucas
1
has evaluated and reviewed these modeling efforts, and, build-
ing on these, he has outlined the beginnings of a model to handle the
joint metabolism of nitrogen and energy.
The aims of this dissertation are two-fold.
First, a conceptual
framework will be developed within which the energy and protein
metabolism of homeotherms may be handled.
Second, mathematical formu-
lations will be developed which describe, in a mechanistic sense, the
transport of material from the feed through the metabolic processes
involved in the digestion and utilization of the feed by the animal .
••
This model will formally handle some of the considerations used in
the practical application of the feeding standards.
Using experimental data from the literature, facets of the model
will be tested.
Further testing needed and kinds of data required for
this will be discussed, as will scientific implications.
1
Lucas, H. L. 1968. Mathematical Models in Animal Nutrition.
International Summer School on Biomathematics and Data Processing in
Animal Exper iments. Elsinore, Den.mark.
3
2,
..
PHILOSOPHY OF MODELING
A model may be thought of as a representation, or analog, of some
real object or process (Hawthorne, 1964),
Models may be categorized
according to the type of material of which they are constructed,
Kleiber (1961) described and illustrated a hydraulic scheme, or model,
for energy utilization,
Such a model would be constructed from glass-
ware, water, weights, balls J ro1lers J strings, gears and springs,
Energy utilization also can be modeled by ilsing abstract symbols
in mathematical equations,
for energy utilization,
Such a model would be a mathematical model
Mathematical models may be classified accord-
ing to their purpose as empirical models or rational models,
Empirical models are mainly used to describe or to summarize a
body of data,
They are attempts to fit curves or relations to data in
order to make predictions.
that it fits the data
o
The reason for using an empirical model is
Predictions are restricted to the domain of
values over which the data were collected,
The parameters of the model
and their dimensions, or units, are usually not interpretable in terms
of the system which generated the data
o
This class of models includes
most statistical regression models,
Rational models are used to explain the behavior of a system in
a mechanistic sense,
They are developed by first defining a conceptual
or abstract framework for the system under study,
This framework
describes or embodies the major e1em.ents of the structure and behavior
of the system,
This framework is then translated into a set of mathe-
matical expressions which comprise the model,
The parameters and vari-
ables in these expressions are identified in terms of elements of the
4
system being modeled}
~'~'}
they make sense in terms of the system.
The solution of this model should behave like the system does over a
wide domain of values.
Lucas (1964) stated that models based on as
much rationality as possible lead to the best predictability.
A
general framework for conceptualizing and mathematizing a problem to be
modeled was presented in the context of grasslands problems (Lucas)
1960).
The remarks hold for mathematical modeling in general.
How well a rational mathematical model fits experimental data or
predicts results depends on how well the mathematical expressions
represent the system} and on how close the parameter estimates are to
the true values.
The form and the complexity of the mathematical model depend somewhat on the modeler and on his level of competence in mathematics and
in biology.
We may consider that rational models lie somewhere on a
continuum with extremes at points A and B.
A ...- - - - - - - - -__1 B
Point A represents a purely biological formulation which is
elaborate} very specific and quite extensive.
system} with great precision and detail.
It represents but one
Such formulations are usually
difficult to handle} and are either incapable of analytical solution or
require an exhorbitant amount of time to solve.
An example of such a
formulation might be a model developed to express the biochemical reactions involved in ruminant digestion and fermentation (Baldwin et a1.)
1970).
Forty-five biochemical reactions were defined} leading to 40
simultaneous nonlinear differential equations.
These were solved
.
~
5
numerically by a digital computer and the results presented in tables
..
and graphs.
The point A may be labeled nearly complete reality .
At the other extreme, point B represents a purely abstract
formulation which is quite elegant and represents many systems in
general, but no one system in particular.
An example (Kalman et a1.,
1969) is the representation of a dynamical system by an octup1e (T, X,
r, _,
U, 0, Y,
11> where T is a time set, X is a state set, U
i~
a set
of input values, 0 is a set of input functions, Y is a set of output
values,
tion and
r
is a set of output functions, _ is a
~
is a readout map.
state~transition
func-
Though this is a far cry from biology,
by making certain assumptions about the structure of the model, a body
of theory can be developed.
Point B may be labeled complete abstrac-
tion.
We want to develop or construct a mathematical model lying between
points A and B,
!.~.,
a tradeoff.
The model will have two general
characteristics (Hawthorne, 1964):
(a) Similarity - in some sense, our model is like the real thing.
By ignoring relatively unimportant details, it is practical to
construct and test, or experiment with, our model; and
(b) Nonidentity - the model is not the real ;hing.
than~comp1ete
copy of reality.
It is a 1ess-
Results of testing or experi-
menting with the model may be false.
The model should be.as abstract as possible, but still related to
the biology.
ing senses:
The model may be intractable in one or more of the follow-
6
(a) Mathematically - we are unable to formulate the model at the
desired level of abstraction;
•
(b) Analytically - we are unable to solve the mathematics we have
formula ted;
(c) Experimentally - we either are unable to experiment with the
system modeled or are unable to collect data pertinent to the
model.
These intractabilities are handled by making assumptions or
simplifications to accommodate them.
Possible accommodations to the
above would be:
(a) Diminish the degree of abstraction;
(b) Simplify the formulation or obtain an approximate solution;
(c) Simplify the model to one that is tractable or design new
experiments or experimental techniques.
These accommodations must be consciously made.
does not work}
~'~'}
Then} if the model
it does not behave like the system it purports to
represent} the assumptions or simplifications can be critically reexamined.
Once a model is formulated} it must be tested.
It is set up to
represent a particular situation for which experimental results are
available) is solved} and the results from the model are compared to
the experimental results.
If the model results do not compare favor-
ably with the experimental results) we are then involved in an
iterative process (Figure 2.1).
We must apply our technical knowledge}
of the model and of the process being modeled} to the situation and
modify the model (re-formu1ate) and/or modify parameter values
4
7
Initialization
..
Initial parameter values
(a) from literature
(b) derived estimates
Iteration
....
"-
\
\
\
Test
model
Modify model
and/or
modify parameter
values
Compare with
experimental
results
Not
Acceptable
Acceptable
I
I
I
I
Technical
knowledge
Figure 201
Model valid
Iterative model-building
-_ .... ,.
/
8
(re-estimate).
This cycle continues until the model results compare
favorably with the experimental results over a wide range of experimental situation~.
•
Then it can be judged a valid model.
then be extended to a broader range of experimental situations or to
additional species.
A
The model may
'.
It then must be tested again, as shown by the
dotted line in Figure 2.1, and again enters the iterative cycle.
The model is then used to learn about the system which was
modeled.
It may be used to evaluate the prediction error in order to
estimate the confidence in the predictions.
In those cases in which
there are insufficient data to completely test or completely elucidate
the model, the model will indicate the need for further experimentation and point out which data need to be collected.
The model is also
useful in checking out the assumptions made in developing the model.
In cases in which more than one model appears to be valid, examination
of the results of the competing models will indicate for which input
values their results differ and hence define the critical experiment
needed to select one of the models.
The model may be used to predict performance under conditions not
previously explored experimentally, or it may be used to investigate
performance under conditions not previously considered in order to
identify probable fruitful areas of experimentation.
In the case of
systems which are expensive or difficult to experiment with, it may be
used to characterize the behavior of such systems.
.
9
3.
3.1
A.
BACKGROUND AND KEY LITERATURE
Roles of Feed Constituents
One of the empirical adjustments made when using feeding standards
.,.
is for the fact that protein may have an energetic role as well as a
structural one.
As part of the abstraction process in the development
of a conceptual framework, only two roles of feed constituents will
be considered, namely structural and energetic (Figure 3.1).
be assumed that supplies of regulatory nutrients
(~.~.,
It will
vitamins) are
adequate.
As shown in the figure, the two roles are not mutually exclusive.
Minerals may be considered as almost purely structural.
Proteins have
a dual role, but usually function as structural materials.
Carbohy-
drates and fats are predominantly energetic materials.
3.2
Definition of Physical Compartments
It is important, in developing a conceptual framework, that
definitions be such as to simplify the ensuing mathematical development
of the model.
Often, certain "well-known" concepts or terms are
redefined to provide working definitions which are to be used in the
context of the conceptual framework.
Physical compartments, derived
from Lucas,l which are used to describe animal nutrition and metabolism
are presented in Figure 3.2.
Descriptions and working definitions as
used throughout this dissertation follow.
1
Lucas, H. L. 1968. Mathematical Models in Animal Nutrition.
International Summer School on Biomathematics and Data Processing in
Animal Experiments. Elsinore, Denmark.
10
.
~
Structuro-energetic
Almost
purely
structural
Usually mostly
structural
Predominantly
energetic
minerals
proteins
some lipids
carbohydra tes
fats
Body Proper
Energy Supply
External Production
~.~.
milk
•
Figure 3.1
Structural and energetic roles of feed nutrients
e
..
e
• 4'
~
Fecal Components
Heat Prod.
Food Gas
Feed
Gut
Excre-Fecal Urine Gut Body
Heat
Body Body
Ext.
Cone. Loss Residue Products tion Loss Loss Ferm Process Loss Pool Wear Proper Stores Prod
(P)
(BW)
(B)
(S)
(U)
(F)
(H) ~
(GH)
(BH)
(GP)
(FE)
(C)
(V) (R)
,....--.
~
Ll
i
~
t
>
~
[3<
,
(M)
)I
......--.
,II
e
.
,
B
B~,-
<
P
FE
.-
....
v
,
,
--G
'~~~~
BH
,~~ _'~
"'"
~
I
T
r
r
~
GJ
G
Figure 3.2
Physical compartments
"'
EJ
I-'
I-'
12
Food consumed (C) is partitioned as a result of digestive processes into combustible gas loss (V), a residue in the feces (R), heat
...
of fermentation (GH) and nutrients absorbed into the pool (F).
The pool (F) represents materials in the circulating and interchanging fluids of the body,
or~
..
in the case of heat, the total heat
content of the body.
External production (M) includes such things as living offspring,
milk, eggs, wool, fur and work.
The body proper (B) is the muscle, skeleton, vital organs, and
other tissues, exclusive of adipose.
That is, it comprises the
systems or structure for existence, growth, fat storage and utilization, external production and reproduction.
The energy stores (8), which are here differentiated from and
exclusive of the body proper (B), are mainly adipose tissue,
~.~.,
the depot fat.
It is very important, for purposes of developing the model, to
differentiate between the body proper and the stores.
The body proper
consists of a mixture of structural and energetic materials.
stores are considered to be energetic material only.
The
Any structural
components usually associated with the depot fat are defined as being
part of the body proper.
stores are different.
The functions of the body proper and the
In addition, there are constraints on the
maximum size of the body proper, but virtually none on the stores.
Different mathematical formulations are developed for the flow of
material into and out of these compartments.
The abstraction process
involves grouping together those items which can be mathematically
..
13
handled in a similar fashion.
..
Since the body proper and the stores
cannot be handled similarly, they must be carefully defined and
differentiated.
For brevity, the term, body, may be used in place of
the term, body proper.
In addition to the feed residue, the feces (F) contain metabolic
waste products, or an excretion component (FE), and a gut product
component (GP), both of which have the pool as their source.
The gut
product component, associated with digestion of food, includes tissue
debris from abrasions of the walls of the digestive tract, mucus and
materials secreted in digestive juices.
The urine waste products (U) and the heat loss (H) are eliminated
from the pool.
The pool is the proximate source of materials for nonfermentative
digestive processes, for body building, for energy stores and for
external production.
The body proper and the stores are constantly
interchanging materials with the pool.
The net balance is an in-
crease, maintenance or decrease of body proper and/or of fat stores.
Associated with the various aspects of processing pool materials
is wear and tear on the body (BW), which is analogous to the gut wear
resulting from digestion of food.
Also related to these processes are
heats of reaction, or body heat (BH).
(GH) supply the heat pool.
..
The body heat and the gut heat
Materials oxidized to drive the heat-
producing reactions are supplied by the pool •
Figure 3.2 generally follows the traditional framework for energy
metabolism.
However, certain sub-compartments, which are not usually
identified, have been distinguished.
These are the three fecal
14
components (R, GP, and FE), the two heat production components (GH and
BH) and the distinction between the backflow of material from body to
pool and the body wear (BW).
This representation, which is quite general, may be applied to
proteins and to almost any other nutrient by deleting certain compartments.
For example, by excluding gas loss (V), heat production (GH,
BH), heat loss (H) and stores (S), the remainder is a framework for
protein metabolism.
3.3
Current Feeding Systems
Feed evaluation systems, reviewed and discussed by Kriss (1931)
and Maynard and Loos1i (1962), may be expressed in terms of the physical compartments shown in Figure 3.2.
The digestible nutrients (DN) system states the apparently
digestible amount of each nutrient per unit of feed.
If C represents
the amount of a given nutrient consumed, F the amount in the feces,
and W the weight of the feed, then digestible nutrients may be
expressed as
(3. 1)
DN
= (C-F)/W.
If the digestible nutrient calculation were done individually
for crude protein, ether extract, nitrogen-free extract and crude
fiber, and the TDN values for these nutrients labelled CP, EE, NFE and
CF, respectively, then total digestible nutrients (TDN) , in terms of
energy, may be expressed as
(3.2)
TDN
= CP + 2.25EE + NFE + CF
where EE has 2.25 the energy
v~1ue
of the others, per unit weight.
..
'.
15
MetaboLizable energy (ME)
when measured directly in an energy
J
balance trial" is computed as t.he groe's energy of the feed (C) minus
the energy losses in feces
."
(3.3)
ME
C
=
F
~
V
(F)~
=
combustible gases (V) and urine (U), or
u.
Net energy (NE), when measured directly in an energy balance
trial, is computed as the metabolizable energy less the heat increment.
In terms of Figure 3.2" we have
(3.4)
NE = C
=
F
=
V- U
=
H.
The digestible nutrient system considers only fecal loss, ignoring
the digestive gas, urine and heat losses.
The fecal loss (F) includes
gut products (GP), which represent secretions resulting from the
digestive process, and fecal excretion (FE), which represents the
metabolic waste products.
In compu.ting total digestible nutrients" it is assumed that all
nutrients are to be used strictly for energetic purposes, although
the TUN system specifies levels of digestible protein to satisfy
structural needs.
Insofar as protein is used for structural purposes,
TDN underestimates the ene.rgy supply.
Metabolizable energy accounts for the losses in metabolism except
gut fermentation heat
108S
(GH)
~
\\Jhich is a digestive loss, and body
process heat production (BH) ,9 which is a loss of metabolism.
Net energy brings in the concept of the heat increment, but this
....
quantity can be affected by the state of: the animal and its
ment.
environ~
16
3.4
Mathematical Approaches
Lucas 2 has stressed the importance of taking energetic and
structural materials jointly into account in a proper way.
McMeekan
(1940a, 1940b, 1940c) for example, in reporting his classical experiments with pigs demonstrated this point.
He showed that by varying
the total energy intake and the protein to energy ratio in the feeds
at different stages of the pig's development, he could affect greatly
the composition of the carcass relative to skeleton, muscle, skin and
fat.
Some of the investigators who have contributed to improvements
in the interpretation of studies on digestion and utilization of feed
are discussed below.
They have tried to better explain the mechanisms
involved in the passage of food through the digestive tract.
Schneider (1935) concluded from an investigation of the metabolic
fecal nitrogen in the feces of the rat that there were two components.
One, a constant amount that is probably of excretory origin, was found
to be related to body surface and also to endogenous urinary nitrogen.
This component is represented by the fecal excretion compartment (FE)
in Figure 3.2.
The second component, which was found to be propor-
tional to food intake, is represented as gut products (GP).
The urine
loss (U) includes the endogenous urinary nitrogen.
Blaxter and Mitchell (1948) incorporated the metabolic fecal
nitrogen into their expression for calculating protein requirements.
.....
17
They also used a statistical procedure for estimating true digestibility of feed proteins.
Lucas and Smart (1959) applied Blaxter and Mitchell's procedure
to calculating true digestibility of 11 feed components including
ash, crude protein, crude fiber and nitrogen-free extract in two
types of forage.
(3.5)
The equation used is
y=a+~
in which y is the apparently digestible amount of the feed component,
as a percent of feed dry matter; x is the amount of the feed component
fed, as a percent of feed dry matter;
~
is the true digestibility
coefficient (as a fraction); and a is the intercept, representing
fecal matter other than undigested feed residues.
The term a includes
the secretions of the body due to digestion, represented by gut
products (GP) and excretions due to breakdown of the body, included
in the fecal excretion (FE) in Figure 3.2.
Blaxter et al. (1956), investigating the digestibility of food by
sheep, used mathematical analysis to estimate diurnal variation of
feces production and the length of the preliminary period necessary in
digestion trials.
They also concluded that digestibility of feed
could be predicted by its passage through the gut.
The digestibility
process was assumed to involve two compartments, rumen and abomasum,
with a time lag accounting for action of the duodenum.
A quantitative theory relating the apparent digestibility of
•
nutrients in feeds to the composition of feeds and feces has been
18
developed by Lucas 3
The theory is based on several postulates relating
to description of the feed in physi.ca 1 ;lnd chemh:al terms.
Blaxter (1962a, ppo
295~296)
...
stated~
Firstly what is needed is a method wh~reby the productive
performance. of an individt:1al can be predicted wit.h some
precision from a knowledge of the quantities of different
foods which make up its ration, and of the conditions
under which i.t is kept. • . • The second consideration
is that any such method mu.st be capable of assigning to a
particular food a nutritional worth in a particular set
of circumstances.
In line with this statement, Blaxter (1962b) proposed a sys tern,
called performance prediction, for assessing energy value of feeds.
This system accounts for the plane-af-nutrition effect in calculating
the true metabolizable energy of a ration o
It then calculates the
energy availahle for production as the difference between the true
metabolizable energy of the ration and the metabolizable energy required to maintain the animal.
The method involves assuming a produc-
tion requirement to be met, estimating a ration to satisfy this
requirement, and then calculating the production that the ration will
support.
If this value does not agree with the assumed production
value, the ration is adjusted and the results recalculated o
This
method is a step in the right direction; it takes into account some
factors contributing to non-additivity of individual feed values.
It
does not, however, encompass the interaction of protein and energy
intakes or their ultimate use for growth, fat storage or external
production.
3
Lucas, H. L. 1960. Relations Between Apparent Digestibility and
the Composition of Feed and Feces. Mimeographed Report, Department of
Statistics, North Carolina State Ur..iversity at Raleigh"
•
19
Lucas (1964), building on his previous work in digestibility
..
studies, proposed a model to handle digestion and absorption.
This
model distinguishes digestion from absorption, accounts for the gas
loss and considers the synthesis and degradation by micro-organisms,
in the ruminant, during digestion.
The model introduces the use of
stochastic elements to account for experimental and other errors.
Lucas showed how the stochastic model may be used to evaluate the
predictivity of various chemical fractions in feeds.
Blaxter (1966), in his discussion of the feeding of cows and
the partition of feed in the maximization of the economic return from
milk, recognized that the milk yield, as a function of feeding level,
follows a law of diminishing returns.
The strong diminishing returns
phenomenon is the result of looking at only one output of the animal.
If one considers the partitioning of the feed among the body proper,
the stores and external production, the sum of these values is almost
linearly related to feeding level, but still reflect diminishing
returns to some extent.
The diminishing returns for one output
apparently are explained by a change in the partitioning of the feed
by the animal as feeding level increases.
Waldo (1968; p. 270) in a review of nitrogen metabolism in the
ruminant
stated~
It should be possible to describe nitrogen metabolism in
terms of pool sizes and concentrations, the order of
rates of transfer or reaction, and the magnitude of these
rates. These concepts can aid us in understanding the
relationships between many processes.
Waldo looked only at part of the whole picture.
Byoignoring energy
metabolism, he omitted the importance of the dual structural-energetic
role of protein.
20
In a recent lecture, Lucas
4
dis c us sed the various feed
evaluation methods and efforts in modeling metabolic processes involved
in digestion and in utilization of feed.
Taking these efforts into
account, he has outlined the beginnings of a model to account for the
combined metabolism of nitrogen and of energy.
4Lucas, 1968, 0p. cit.
.
"
21
4.
DEVELOPMENT OF THE MATHEMATICAL MODEL
4.1
Preliminaries
A mathemati.cal model for the partition of nitrogenous and
non~
nitrogenolls energetic materials in the bomeothenrr is to be developed.
It will characterize, mathemat:lcal1y, how the animal digests, absorbs,
transports, synthesizes, catabolizes, interchanges and excretes these
rna ted.a1s.
The model will extend both the concepts embodied in the
feeding standards and the modeling efforts reviewed in Chapter 3.
For a given feed, or input, the model will trace the flow of this
input through the physical compartments of the animal, through its
chemical transformations to other materials, and to its final disposition, or use, by the animaL
The mathematical model which defines the
flows of this material will take into account the physiological state
(~'12"
age, stage of growth, stage of gestation, stage of lactation)
of the animal and the relative amounts of protein and other energetic
material in the feed.
At the first level of abstraction, all materials not classified
in Figure 3.1 as structuro-energetic are ignoredo
Nitrogenous
materials inclu.de protein, free amino acids, urea? ammonia,
creatine, and creatinineo
Ncm~nitrogenou.s
energetic materials include
starch, glucose, fiber, fatty acids, lipids, depot fat, and methane.
For simplicity of notation, the term Henergetic materials" will be
used to denote and include all non-nitrogenous energetic materials.
Although heat is a form of energy, it is considered separately from
energetic materials.
22
Each nitrogenous material, in addition to its nitrogen content,
also contains an amount of energy which is characterized by its heat
of combustion.
The materials can thus be characterized by a binary
system of notation,
(N,E)
where N
= gms of nitrogen per gram of material, and E = kca1 of energy
per gram of material
o
According to figures from B1axter and Rook
(1953), the figures for body protein and depot fat are
body
prote~n
depot fat
(0016, 50322)
(0·00, 9.367)
This notation is easily generalized to handle additional types of
material.
To handle a substance like carbon, a ternary notation is
~.~o,
used,
(N,E,C)
where C
above.
gms of carbon per gram of material and Nand E are as defined
The above examples
become~
body protein (0016, 50322, 0.525)
depot fat
(0.00, 90367, 00765)
Note that both nitrogenous and energetic materials have an
associated energy value.
Hence, both materials can be characterized
by their heat of combustiono
Then, for nitrogenous material, the
kcal of heat of combustion can be converted to grams of nitrogen;
~o~.,
5.322 kca1 of body protein is equivalent to 0.16 grams of
nitrogen.
Thus, each kca1 of body protein is equivalent to
23
0.16/5.322
00301
grams of nitrogen.
'
..
..-
The first step in developing the model, namely the definition of
those physic.al compartments necessary to describe the flow of nitrogenous and energetic materials in the homeotherm, has been discussed
previously in Section 302.
4.2
4.2.1
Conceptual Framework
Nitrogen, Energy and Heat
A physical subdivision of the animal is not adequate to handle
the partitioning of several types of feed materials.
The classifica-
tion of feed materials as either nitrogenous or energetic has been
discussed.
These materials start out together in the feed, but are
handled differently in the various physical compartments.
of starch, an energetic material, some of
in the mouth.
In the case
it is completely digested
This digested starch is more readily absorbed than is
the protein in the feed.
The stores consist solely of energetic
materials, thus no nitrogenous material flows from the pool to the
stores.
The physical compartments must be subdivided into chemical
compartments to ha!ldle nitrogenous and energetic materials.
Heat is produced from reactions involving both types of materials
and is dissipated differently than either y hence a third chemical
compartment
ffitlSt
be included fer hea t.
A further complication is that nitrogenous materials can be
converted to energetic compounds.
Amino acids are deaminated to
keto=acids with the concomitant production of urea and the liberation
24
of heat.
Thus, the nitrogenous and energetic compartments must be
interconnected and the energetic must be connected to the heat production compartment.
Figure 4.1 illustrates a general physico-chemical
compartment model.
The physical compartments are labeled as in Figure 3.2 and the
following subscripts used to denote chemical compartmentation:
n - nitrogenous material
e - energetic material
h - heat.
The nitrogen portion of the food consumed, fecal components,
fecal and urine loss and the pool are shown in Figure 4.1.
C
n
Compartment
represents both protein and other nitrogenous substances expressed
as "protein equivalents".
cases.
The term protein will be used for both
There are no gaseous emissions containing nitrogen, hence this
compartment appears only in the energy part of the figure.
The stores
represent depot fat and have no nitrogen content, hence they appear
only as energy.
The path from C to GH represents the heat of digese
tion and, in the ruminant, fermentation.
The path from P
e
to BH
represents the total heat dissipated from all reactions in the body
other than the heat of digestion.
Since heat is a form of energy, it
does not appear in the nitrogenous part of the figure.
The path from
P to P represents the conversion of nitrogenous material to energetic
n
e
material.
Several compartments are not subscripted.
ment (V) and the stores (S) involve energy only.
The gas loss compartSince there is no
.. '
e
(
--
6
_ Fecal Components
Heat Prod,
Food Gas
Food
Gut
Excre- Fecal Urine Gut
Body
Heat
Cons Loss Residue Products tion
Loss Loss Ferm Process Loss
(C)
(V)
(R)
(GP)
(FE)
(F)
(U)
(GH)
(BH)
(H)
I I)
Cn
.
e
~
Body Body
Ext.
Pool Wear Proper Stores Prod
(P)
(BW)
(B)
(3)
(M)
~
i
'f
GJ
~
IGP n \
(
5]<
'
~)
<.
I I
>~
J
EJ
PDD
,
~.- - - - «
I-€J
i
,1
I
1GP e\
<I
<
Pe
EJ~<-,
I
..
>~ ~J
~ ~~ J ~
y
X IH~5J
I
c:J
<
Figure 4.1
Physico-chemical compartments
~EJ
Legend - subscripts
n - nitrogen
e - non-nitrogenous
energy
h - heat
N
\J1
26
corresponding nitrogenous compartment, the subscript would be
extraneous.
The heat production compartments (GH, BR) and heat loss
(R) have an H in their designation"
To add a subscript would un-
necessarily complicate the notation,
Eody proper (B), body wear (BW)
and external production (M) are unsubscripted for a different reason.
Each of these compartments is considered to represent a well-defined
mixture of nitrogenous and energetic materials.
They flow together
in fairly constant proportions to form body proper and may break down
in similar proportions as body wear and back flow from body to pool.
External production also is formed from a flow of nitrogenous and
energetic material in proportion determined by the particular product.
4.2.2
Further Subdivisions of Nitrogen and Energy
The subdivision of the physical compartments into nitrogenous,
energetic and heat components still does not account for all complicating factors"
The energetic material in the food includes a variety of
substances such as polysaccharides
saccharides, lipids and others.
(~.~.,
starch, cellulose), mono-
These differ in rate of digestibility,
in proportion digested and in end products of digestion.
tion of carbohydrates prOVides the gut heat, GR.
related to the digestion of carbohydrates.
consis~
The fermenta-
The gas loss, V, is
The gut products, GP ,
e
in large part, of lipid material rather than carbohydrates.
To handle these digestive features properly, energy is further
subdivided into carbohydrate and fat components, denoted by subscripts
c and f, respectively,
27
Then, compartment C
repre.sents a mixture of carbohydrates fed,
c
-.
and an average digestibility value is used.
Once in the pool, the
end products of carbohydrate and of lipid digestion are not
distinguished from one dnot1:xer, but are designated as P
."
se
.
The food residue and gut product components of the feces also
are subdivided to correspond to their originating material,
Cf' R
f
and GP
f
for fats and Cc and R for carbohydrates.
c
products represented by GP
c
~.~.,
The gut
are not strictly from carbohydrates, but
rather comprise all gut products which are not from ether extract.
Two categories of pool materials must be distinguished, useful
materials and waste products.
The first comprises metabolizable
useful materials such as amino acids, glucose and fatty acids.
These
compounds are either digested or hydrolyzed forms from the feed or they
are products from the backflow of materials from the body proper and
stores to the pool"
These are available for structure and for energy
for body formation, fat storage and external production.
They will be
designated as simple compounds and labeled as P
for the
sn
and P
se
nitrogenoills and energetic pool compartments, respecti.vely.
The second category is waste products.
These result from body
wear and from chemical degradation of food materials.
usually excreted by the animal.
compounds and labeled as P
d :l
These are
They will be designated as degraded
and P
de"
These refinements are shown in Figure 4.2.
4.3
Discussion of the Total Compartment Model
Figure 4.2 contains
~:he
proposed compartment model.
contain a compartment for the dfge.s tive tract.
It does not
The reason for this is
Fecal Components
Food Gas
Food
Gut
Excre- Fecal Urine
Cons Loss Residue Products tion
Loss Loss
(C)
(V)
(R)
(GP)
(FE)
(F)
(U)
EJ)
•
r;I
~~ -<.: r;;l
~
n
~«~
T-
r:t)
«n
~Fnl
Heat
Gut
Ferm
(GH)
Prod
Pool
Body Heat
Proc Loss Degraded Simple
(BR) (H)
(P)
~ IP sn
<
""'I
I')
(
~
)
P
~)
~
~
I
I
~
@-<:
Figure 4.2
e
.,
i
<~
ru::l..
I
L2.I
Body Body
Ext.
Wear Proper Stores Prod
(BW)
(B)
(S)
(M)
Ph
se
)
<
,
~rsl
<LJ
Legend - subscripts
n - nitrogenous
e - non-nitrogenous
energy
h - heat
f - fats
c - carbohydrates
N
A compartment model
e
00
,0
,
•
e
29
that we will concern ourselves with a controlled type of feeding in
which after a few days on a constant amount of a standard diet, a
'.
nonruminant will reach a state of dynamic equilibrium. The rate of
passage of food at a given point in the digestive tract will be rela-
.'
tively constant and the partitioning of the food into an absorbed
portion, fecal residue J etc. will be characteristic of the food and
the animal (Blaxter et al., 1956; Maynard and 1o0sli, 1962).
ruminant, a longer time period
reached.
i~
For a
necessary for equilibrium to be
We will later consider a 28 day nitrogen and energy balance
trial and so the assumption of equilibrium will be a realistic one.
Thus we can concern ourselves with the partitioning of the food into
absorbed, fecal residue, gas loss and gut fermentation heat portions,
and ignore the details of digestion.
Compartment C
n
represents the protein or nitrogenous material
The path from C to P
represents the digestion of protein and
n
sn
fed.
the absorption of the resultant amino acids into the pool.
nitrogen (F ) comes from three sources.
n
protein, is shown by the path from C
n
Fecal
Feed residue, or undigested
to R.
n
A second source is gut
products (gut wear and secretory materials) shown as the path from P
sn
to GP
n
(Lucas, 1964, p.
376)~
. these are associated with and/or are necessary for
the digestion of feeds (~.~'J mucus and constituents of
the digestive juices).
Schneider (1935) refers to this as the digestive fraction of the
..
metabolic fecal nitrogen.
matter consumed.
P
dn
It is proportional to the quantity of dry
The third source of fecal nitrogen, the flow from
to FEn' may be considered as excretory materials (Lucas, 1964,
30
p. 376):
. waste products of metabolism or excesses in the
body proper for which the gut is one of the paths of
e1imina tion.
..
Schneider (1935) refers to this as the constant fraction of the
metabolic fecal nitrogen.
'.
According to Schneider's data on rats and
swine, this constant fraction is proportional to the metabolic body
size,
!.~.,
body weight to the three-fourths power.
The major part
of the fecal nitrogen is usually accounted for by the undigested feed
residue and the excretory fraction.
If the dry matter consumed and
nitrogen consumed are low, then the gut products fraction may be a
significant proportion of the fecal nitrogen,
Excretion of urinary
nitrogen is shown as the flow from P
to U.
n
dn
This includes what is
usually termed endogenous urinary nitrogen (Schneider, 1935; Maynard
and Loos1i, 1962) as well as all other waste nitrogen.
The feeding of lipids is denoted as compartment C .
f
The diges-
tion of these materials is shown as the path from C to P .
f
se
Un-
digested "fats" flow from C to Rf , and lipids secreted into the
f
feces are shown as the flow from P
se
to GP .
f
The feeding of polysaccharides, including fibrous material, is
shown as C.
c
Cc to Pse
C
c
P
se
Digestion of these materials is shown as the path from
Undigested "carbohydrates" are indicated by the flow from
to R , and carbohydrates secreted into the feces by the flow from
c
to GP •
c
Energetic materials in the urine are shown as a flow from P
U.
e
de
Two other losses are associated with the digestion of carbo-
hydrates, especially with herbivores.
The volatile gas loss,
~.~.,
to
31
methane, is shown as a flow from C
c
to V, and heat production from gut
fermentation is shown as a flow from C
c
to GR.
According to Schoenheimer (1942), lipids of fat depots constantly
undergo synthesis, interconversion and degradation.
Fat and fatty
."
acids are steadily and rapidly regenerated.
depicted by the paths from P
se
These processes are
to S and from S to P
se
, the former
representing the synthesis and the latter the degradation of the
depot fat.
Interconversion of lipids is ignored.
The synthesis of nitrogenous and energetic materials for external
production is shown as the paths from Psn and Pse to M.
We assume that,
for a given species and product, the product has a fairly constant
ratio of energy to nitrogen.
This has been shown for milk of various
breeds of cattle (Overman and Gaines, 1933).
Sufficient amounts of
nitrogenous and energetic material will react to form the product, to
provide energy to drive the reaction (and be dissipated as heat) and
to provide for any waste products.
The synthesis of the body proper for growth and for replacement of
cells, which are constantly breaking down, is shown by the paths from
P
sn
and P
se
to B.
As for production, we postulate a constant energy to
nitrogen ratio for the body proper.
There is a certain amount of fat
associated with the protein in the organs of the body.
fers from depot fat in its function.
This fat
dif~
It will be considered as part
of the total fat in the body, but its formation will be related to the
formation of protein.
The combination of this protein and fat
comprisesthe formation of the body, B.
Schoenheimer (1942) states that
proteins of the body, like lipids of the fat depots, are also in a
32
dynamic state.
Thus proteins are constant.ly being degraded to amino
acids as shown by the path from B to P
synthesized from amino acids.
sn
, and are constantly being
In order to maintain the constant
energy to nitrogen ratio in the body, the breakdown of each unit of
protein will be accompanied by a concomitant breakdown of fat from the
body and this is shown as the path from B to P
se
0
For every reaction or flow of material in the animal, there is a
certain amount of wear and tear on the systemo
This wear and tear is
represented as a breakdown of the body, B, since it is the body which
includes the physiological systems in which the reactions take place.
The wear and tear, BW, resulting from each reaction or flow, is proportional to the flow.
body wear terms·.
The total body wear is the sum of the individual
The nitrogenous and energetic materials are broken
down into simple energetic (P
degraded energetic (P
de
se
)' degraded nitrogenous (P
) materials, and into heat.
dn
), and
The total break=
down, BW, is represented as
BW
where:
~o~ofo
1. 1. 1.
~i
the amount of body wear per unit of the
f.
the i
1.
th
cth
L
flow
flow
A fasting animal will oxidize its tissues to produce heat to keep
warm.
This will entail body weaL
A certain amount of nitrogenous
and energetic material in the feed will spare the oxidation of the
tissues and the body wear (Blaxter, 1962a) and replace the gut wear
resulting from the digestion and metabolism of the feed itself.
This amount is the maintenance level of feeding.
33
Every reaction, or flow of material from one comparbment to
another, produces a heat of reaction.
The energy which is converted
to this heat is supplied by the pool, P
se
•
The amount of heat is
proportional to the amount of material transported, synthesized or
catabolized.
For an exothermic reaction, the heat evolved is handled as a
transfer of energy from the energy pool, P
BH.
,to the heat production,
s~
For an endothermic reaction, the heat absorbed is handled as a
transfer of energy from P
se
to the compartment receiving the product
of the reaction.
The total heat generated from all reactions will thus be the
sum, over all flows, of the product of the flow and the proportionality
factor reflecting the heat of reaction for that flow,
BH
where:
= ~.~.f.
~
~
~
BH
= total
~i
= the
heat production factor for the
f.
= the
~
~
~.~.,
heat production from body processes
.th
.th
1
flow
flow or reaction.
In addition to the heat production and body wear accompanying
every reaction, there also is the production of by-products or waste
products from every reaction.
These waste products flow to P
dn
or P
de
and are then excreted, or in the case of the ruminant, may be utilized
by the rumen microbes.
For each flow, there will be an amount of ni.trogenous and energetic waste per unit of flow.
constants
V
and
~
These amounts are denoted by the
for nitrogenous and energetic waste, respectively, in
34
the pool.
The total amounts of these waste products may be formulated
as:
P
P
where:
dn
de
waste
L. 'V .. f.
waste
LS.f.
~
~ 1.
~
~
~
= the i th flow
f.
~
'V.,s.
~
~
= the amount of nitrogenous and energetic waste which
flow to P
'V. , S.
~
~
o
dn
and P : respectively, per unit of f
de
i
for those f. which produce no waste.
1.
The heat production from all reactions, denoted by compartment BH,
combined with the heat production from gut fermentation, GH, forms the
heat pool, Ph.
The dissipation of heat is shown as the flow from Ph to
H.
The model as described herein is quite general and includes the
major paths or flows of nitrogen and energy.
Conceptually, extension
of the model to handle ruminant digestion is not difficult.
Compart-
ments and flows for the digestive tract and for the ruminal microorganisms would have to be added.
4.4
Development: of Flow Laws
For each of the paths developed in Figure 4.2, we will now
proceed to develop a flow law or mathematical equation.
The simplest
mathematical formulation consistent with nutritional facts and principIes will be proposed.
These formulations are differentials that
express the flow of material per unit of time or the rate of flow of
material.
35
These flow laws are then combLnedt:0' form dlfferential equations
for seven compartments in the rr:odeL
-.
.-
:J.8JId21y
J
and Pdeo
Bo S
,
P sn J P se~
J
The rate of change in a compartment eqtlals the sum of the
flows into it minus the sum of the fl'JWS
The systEm of differer.tial
OUit
oftt •
eq~]at::'onst:hlis
derived ccmstitClltes the
mathema tical model.
40401
Notation
In Section 401, we discussed the fact r:hat both nitrogenous and
energetic materials could be characterized by their heat of combustion
o
Then from the binary notation J a corresponding vallLi.e in grams of
ni trogen cO\Jld be associa ted with each emit: of nea t for the ni trogenollls
compounds.
Hence? the basic tinit of flow will be the kiloca1orie J
kca10
The arnmmt of kcal in any compartment at time t will be denoted
by Illpper case letters for the compartment designation followed by
lower case subscripts where appropriate,
~o~oJ
ni trogenous rna terial in the pool at t:Lme t is P
the kcal of simple
,and the total kcal
sn·
of body proper is Bo
The rate of change, or derivat:ivt y of the amoUlnt of kcal in any
compartment is denoted by lower caae letters for the compartment
followed by lowe.r case subscripts where appropriate.
the rate of change of kcal in F
change of body is bo
sn
For the abuiJe J
is denoted by Pen and the rate of
The units are kcal/timeo
For the flow from one compart:nent to anether, the ccmpartment
from which the material c'riginates
:,s
designated as a derivative.
compartment to which the material is flowing
~s
designated as a
The
36
subscript.
This subscript consists of an upper case letter or letters
for the compartment followed by lower case letters where appropriate.
For example, the flow from P
sn
to P
se
would be designated as p
It may be read as the rate of flow of amino acids to Obketo acids,
flow from body proper (B) to body wear (BW) is designated b
BW
•
sn,Pse
The
'
We have discussed coefficients for waste products in the pool
(\I and
s),
for heat (T]) and for body breakdown (P).
The heat coeffi-
cient for Psn, Pse ' written as T1 Psn,Pse , represents the kca1 of heat
generated per kca1 of flow from P
sn
to P
per unit of time.
se
The firs t
subscript on T1 denotes the originating compartment and the second
subscript, the destination.
The same notation applies for \I,
S,
and
p.
The flow laws are characterized by a "rate" constant denoted by
*
a Greek kappa with a superscripted asterisk, K.
These are denoted in
the same fashion as the waste and heat coefficients discussed above.
Each flow has related to it a certain amount of breakdown and the
production of waste products and heat,
P
se
The heat will be supplied from
and the body breakdown will be deducted from B.
The waste produced
must be subtracted from the gross flow to produce a net flow of
material,
This is incorporated into the rate constant.
For example, consider the flow from P
loss from P
sn
sn
to P
se
The rate of
is shown as
-1<
Psn,Pse
The function ; will be defined in the next section,
for the waste produced in the pool are
The terms
".
37
..
and
,.
The net flow to P
p
se
sn,Pse
from. P
sn
is thus
* pse) (1
(psn
,
\)
Psn,Pse
where
K:
Pne
\)
Psn,Pse
-
)
~
~PsnJPse
In general, to simplify matters, the flow laws will be developed
directly as net flows, ignoring the waste, heat and body breakdown
terms.
However, for those reactions where these terms playa major
role, they will be developed explicitly,
not necessarily imply that they are zero.
Ignoring these terms does
For testing thEi! model and
for predicting, these terms must be estimated.
Additional parameters, and constants and exceptions to the above,
are defined as needed in the subsections to follow.
4,4.2
•
The Flow Laws
According to the notaticm. dee,ned above, the amount of protein
fed per unit time (i.e., kcal/day) will be designated c ,
n
38
The protein fed is either digested or defecated.
There is a
well-known effect of feeding level on digestibility (B1axter et a1.,
1956; Forbes et a1., 1928; 1930).
bi1ity decreases.
As feeding level increases, digesti-
...
We define the true digestibility coefficient of
0.
protein, d*, as if there were no feeding level effect.
n
The effect
of feeding level on digestibility is handled as follows.
For a given diet, define the amount of dry matter fed per day
which results in zero energy balance as cd
feeding level of unity.
,0
.
This corresponds to a
Then the feeding level, FL, for a given ration
would be the actual dry matter fed, cd' divided by cd,o' or
Then we may define the feeding level effect as
where 0 < a
FL
< 1, b
FL
< 0 and 0 < f
FL
< 1.
The digestibility coeffi-
cient of protein, corrected for feeding level, is then
Thus the flow law for the digestion of protein is
(4.1a)
c
= (d ) (c )
n,Psn
n
n
and the flow law for the passage of undigested residue to the feces
(R) is given by
n
(4.1b)
c
= (1 - d ) (c ) •
n,Rn..
n
n
39
The digestion of fats and carbohydrates is handled similarly to
protein.
".
We define true digestibility coefficients, d* and dc'
* for
f
fats and carbohydrates.
Using the same function of feeding level as
above, we have that
The digestion of fats, or passage from Cf to P ' is given by
se
(4.2a)
where c
f
is the amount of fat fed per day.
The undigested fats flow
to R according to
f
(4.2b)
For carbohydrates, there is a complicating factor.
Of the c
c
kcal of carbohydrates fed per day, (d )(c ) of this is digested.
c
c
The
amount of methane produced (V) and the gut heat production (GH)
usually are taken to be fractions of the digested carbohydrate
(Blaxter, 1962a; Bratzler and Forbes, 1940).
We thus partition the
digested energy among the energy of the methane loss, the gut heat
production and the energy absorbed into the pool.
lost as methane is
~c
If the fraction
and the fraction lost as gut heat is
then the fraction absorbed is (1 - ~c
-
'TbH)·
absorption and food residue in the feces are
~GH'
Then, the equations for
40
(4.3a)
c
c,Pse
i-L
= (1 -
c
..
(4.3b)
c c,Rc
=
*.
(1 - dc) (cc) .
The amount excreted as methane is
(4.3c)
c
c, V
= (i-L ) (d ) (c )
c
c
c
.
The equation for gut heat production is
(4. 3d)
c
c,GH
Passage of material to the gut products compartments (GP , GP
n
f
and GP ) is proportional to the amount of dry matter fed per day, cd'
c
The proportionality factors differ for protein, fats and carbohydrates
and are denoted by KGpn '
K
and KGpc ' respectively.
GPf
The flow laws
are
(4.4)
psn,GPn
= (KGPr/(C ) .
d
(4.5)
Pse,GPf
(4.6)
pse,GPc = (KGpc ) (cd)
= (K
GPf ) (cd)
In developing the remaining flow laws, we assume that the flux is
a function of the product of the body size, B, and the basic driving
force at any point as determined by the concentrations of reactants.
The path from P
dn
to U and FE represents the passage of
n
n
excretory nitrogen (Lucas, 1964) or endogenous fecal and urinary
41
nitrogen (Schneider, 1935).
material in P
-.
dn
The concentration of the nitrogenous
may be expressed as
and the product of this term and body size, B, is P
dn
.
Thus the flux
is a function of P •
dn
Schneider (1935) found in rats that the ratio of endogenous fecal
to endogenous urinary nitrogen was approximately constant over a wide
range of nitrogen intakes.
nitrogen excreted from P
to FEn as (1 - fUn).
dn
Thus, we define the fraction of the total
which goes to Un as fUn and that which goes
If the rate constant for flow from P
dn
is Kpdn '
then the flow laws are given by
(4.7a)
Pdn,Un
(4.7b)
P dn FEn = (1 - f Un ) (KPdn ) (P dn )
,
•
The total rate of flow of nitrogenous material as fecal loss
is expressed as the sum of the rates of flow of the three fecal
components.
Similarly, for the passage of energetic waste material from P '
de
we define the rate constant K
and the fraction passing to U as
e
Pde
f
Ue
.
Then the flow laws for energetic waste are given by
(4.8a)
(4.8b)
The total rate of fecal loss of energetic material is expressed
as the sum of the rates of the
fiv~
fecal components.
42
Next, consider the flow of material from P
sion of amino acids to energetic material.
sn
to P
se
, the conver-
We must consider the
.~
relationship between the amounts of material in these pool compartments.
The nutritive ratio for a feed is defined as the ratio of digestib1e protein to the sum of digestible fats and carbohydrates (Maynard
and Loos1i, 1962).
We modify this concept and define the nutritive
ratio for the pool (NR) as the ratio of the amino acids (i.e., digested
protein) in the pool, P
sn
, to the simple energetic materials
digested carbohydrates and fat) in the pool, P
NR
se
(~.~.,
Thus,
= Psn Ip se
We postulate that for a given animal, in a given state, there is an
optimal nutritive ratio, NR.
o
For example, a mature, non-producing
animal may have a given optimal ratio for maintenance.
growing animal, the optimal ratio would differ.
Since the production
of milk requires different proportions of nitrogen (P
(P
se
If it is a
sn
) and energy
) than body growth or maintenance, there would be a different
optimal ratio for the milking animal.
Fattening requires much energy
and little nitrogen and again the optimal ratio differs.
As is well known, when protein intake of growing animals exceeds
a certain level relative to energy intake, the ratio of nitrogen
retained in the body to nitrogen in the urine decreases.
the pool amino acids (P
have
NR
> NRo
sn
Thus, when
) are high relative to pool energy (P
se
), we
43
and the rate of conversion of amino acids to pool energy and urea
This action tends to reduce NR to its optimal value, NR .
a
increases.
When amino acids are low relative to pool energy,
NR
< NRo
and conversion of amino acids to pool energy and urea decreases,
tending to increase NR to its optimal value.
A driving force defined as the ratio of the concentration of
amino acids (P
this behavior.
sn
IB) to the concentration of pool energy (P
se
IB) gives
Then the flow law for the flow of amino acids out of
"1<
the pool, p sn, .p se , is given bv
. the product of the driving force, the
body size and a rate constant or
(4.9a)
This reaction provides urea as a by-product (Dukes, 1955) as well as
heat.
The amount of urea formed (kca1) per kca1 of amino acids
deaminated is vpne .
(4.9b)
Thus the formation of urea is given by
Psn,Pdn
The amount of pool energy formed is the difference between (4.9a) and
(4.9b) or
(4.9c)
p
sn,Pse
44
where
If the amount of body heat produced per kcal of amino acids
".
deamina ted is g1 ven by Tlpne' then the flow
0
f hea t from Ps e to BH
for this reac tion is given by
··k
(1lpne)(psn,pse) .
The flow of energetic material from P
se
product of the pool energy concentration (P
a rate constant.
se
to 8 is defined as the
/B), the body size (B) and
Thus,
(4.10)
According to Schoenheimer (1942);, depot fat is constantly being
degraded to fatty acids and these are constantly being synthesized to
If an animal is fed insufficient energy, the net effect
depot fat.
is a decrease in depot fat.
the flow from 8 to P
se
We postulate two factors which control
,body size and fatness.
The body has the
machinery for the conversion of depot fat to energetic compounds in
the pooL hence, the flow is proportfonal to Bo
the ratio of fat stores to body, 8/Bo
We define fatness as
The flow from 8 to P
se
will be
reduced as fatness is decreased and wi.ll approach a maximum value as
fatness increases.
We formulate the effect. of fatness as a hyperbolic
relationship and the flow is then defined by
(4011)
s
Pse
-
(K:~ p) (B) [ (S/B) / (S/B + r)] .
'"'"
Graphically, this flow is represented in Figure 4.30
45
-.
• 0
- - - - -
~--=--------'--'-
- ....
S/B
r
Figure 4.3
Flow law from stores to P
The paths from P
sn
and P
se
se
as a function of fatness
to B represent the synthesis of body
proper from the nitrogenous and energetic pools.
We will first derive
the flow law for body growth and then partition this growth into
nitrogenous and energetic contributions.
The driving force behind
body growth is considered to be the product of the two pool concentrations, (P
P
sn se
2
/B ).
taken into account.
Y =
Q' -
(~)
The fact that body size is limited must also be
Brody (1945) characterizes growth by
(e
-yt
)
where y is body weight, t is age from conception,
con3tants and
Q'
~
represents the maximum body weight.
and yare suitable
The true relation-
ship appears to be a logistic or S-shaped curve) but for ages after
sexual maturity, Brody's formulation is adequate.
We will represent this phenomenon, not in terms of age of the
animal, but in terms of a maximum growth potential or body size,
denoted by A .
B
In addition to the driving force, the rate of body
46
growth will be proportional to the remaining growth potential of the
animal, the difference between the maximum body size, A , and the
B
current body size, B, or (AB-B).
Multiplying the driving force and
the term for remaining growth potential by the body size .• B, and a
-.
rate constant, Kp B' gives the flow law for body growth,
,
(4.12)
(Kp B) (P
,
::: (Kp B) (P
,
p
sn se
sn
) (P
2
/B ) (A - B) (B)
B
se
) (A - B) /B
B
0
We have previously discussed the idea of the body being a we1ldefined mixture of nitrogenous and energetic material.
If the amounts
of nitrogenous and energetic material in the body are denoted by B
n
and B , respectively, then the total kca1 of body proper, B, is
e
related to these quantities by the relationship
(4.13)
B
+
B
n
B
e
and the ratio of energetic to nitrogenous material, P , is defined by.
B
(4.14)
B /B
e
n
From (4.14), we have
(4. 15)
and substituting into (4.13) gives
(4.16)
B
47
or
(4017)
-,
.-
Substituting (4.17) into (4.15) gives
(4.18)
Thus. given the amoGCnt of body proper, BJ we use (4.17) and (4.18)
to partition this into its nitrogenolls and energetic components
0
We
assume that: the flow of material into the body, given by (4.12), is
similarly partitioned; hence, the flows of nitrogenous and energetic
material into the body are denoted by Pn Band Pe B' respectively, and
,
)
are expressed by
(4. 19a)
P
"- [1/(1 + P ) ](P )
B
B
llJB
(4. 19b)
where PB is given by equation (4.12)0
According to Schoenheimer (1942), the body protein and amino
acids are in a d)'namic state just as are the fat stores and fatty
acidso
We define a flow law for the degradation of the body which
is proportional to the body size, B, or
(4020)
In order to maintain the ratio P within the body, this degradation or
B
backf10w must be partitioned in the same manner as was the synthesiso
Denoting the backflow to P
have
sn
and P
se
by b p and b , respectively, we
n
Pe
48
(4.21a)
(4,21b)
b
Pe
In addition to the degradaticn of a part of the body tissues to amino
acids and simple energetic forms" some of the body breakdown is wear
and tear on the system,. or body wear, due to carrying out all the
The tctal body wear (BW) may be represented as
previous reactions,
a sum of breakdown terms from all the reactions,
Then the flow law
for breakdown of the body proper due to body wear is
(4.22a)
h
~o
where
1
.BW
-, L.
,l
~
.f .
1
1
and f,. are as defined in Section 4,3.
1.
This material is then
partitioned am..::'ng Pdn: P se and Pde·
The nitrogen of the body protein goes to P
(!:.~.,
urea and other sub3tances)0
P
dn
dn
as a waste product
does not ccmtaLn so much
energy per gram of nitrogen as does body protein, hence the additional
energy goes either to P -' .' or L) P
ue'
se
where it can be utilized,
For
each kcal of body broken dc:)",m as body wear,we define the following
partition:
The following constraint alsc holds:
Thus the parti,tion equations are
49
-.
."
(4. 22b)
b
(4.22c)
b
(4. 22d)
b
BW,Pdn
BW,Pse
BW,Pde
(\)BW) (b BW)
(n
BW
) (b
BW
)
(~BW) (b BW)
.
Flow laws for external production are not discussed here.
The model
was tested against data from steers, whose production is encompassed
by body growth and addition to fat stores.
However, the development
of equations for the production of milk are found in Appendix 9.1.
As for body wear, the heat production from body processes is
accumulated as a sum of heat production terms from each body process.
As discussed in Section 4.3, we denote this by
(4.23)
P se, BH == I:1 T].~ L1
o
where T]. and f. have been previously defined.
1
1
If we denote total heat production as HP, then the rate of heat
production, hp, is the sum of the gut fermentation and body process
heat production, or
(4.24)
hp
c
c, GH
+ P
se,BH
The heat production flows into the heat pool (Ph)'
dissipated, as shown by a flow from Ph to H.
This heat is then
The data used to test
the model were collected at a temperature in the range of thermoneutrality.
A mechanism and flow laws for body temperature regulation
under varying environmental temperatures are presented in Appendix 9.2.
50
4.5
The Mathematical Model
The flow laws developed above are the elements of the differential
equations which form the mathematical model.
These flow laws, numbered
as in Section 4.4, are listed in Table 4.1.
The differential equations
derived therefrom, one for each compartment in Figure 4.2, are listed
in Table 4.2.
The differential equation for each compartment is
defined as the sum of the flows into the compartment minus the sum of
the flows out of it.
The compartment values, as functions of time,
are evaluated by integrating these differential equations, subject to
appropriate initial conditions.
The development thus far is represented by the light bulb in
Figure 2.1.
model.
We have completed our assumption or derivation of a
Subsequent sections will be concerned with deriving initial
parameter values and with the iteration part of model-building.
51
Table 4.1
..
Flow laws for the compartment model
(4.la)
c
(4.lb)
c
(4.2a)
c
n,Psn
n,Rn
=
(l-d n) (c n )
f,Pse
(4.2b)
(4.3a)
c
(4.3b)
c
(4.3c)
c
(4.3d)
c
(4.4)
psn,GPn
(l-IJ.
c,Pse
c,Rc
=
c
-11GH) (d c )
(c )
c
(l-d ) (c )
c
c
c,V
c,GH
(4.5)
(4.6)
(4.7a)
pse,GPc
Pdn,un
(4.7b)
(4.8a)
(4.8b)
(4.9a)
Pde,FEe
*
Psn,Pse
(4.9b)
(4.9c)
(4.10)
Psn,Pse
= (Kp s) (P )
,
se
continued
52
Table 4.1 (continued)
(4011)
sPse
(4. 12)
PB
(4. 19a)
Pn B
(40 19b)
Pe B
-
(4.20)
b
= (K
(4. 21a)
b
(4.2Ib)
b
(4.22a)
b
(4. 22b)
b
(4.22c)
b
(4. 22d)
b
(4023)
(4024)
(K:
P
, p) (B) [ (SiB) I (S/B+r) ]
= (Kp B) (P ) (P ) (AB-B) IB
,
sn
se
[11 ( 1 +P ) ] (P )
,
,
S
B
B
[ PBI (1 +P ) ] (P )
B
B
, p) (B)
B
pn
= [l/(l+P )] (b p )
pe
= [PBI ( 1 +P ) ] (b p )
B
BW
BW,Pdn
BW,Pse
BW,Pde
Pse,BH
hp
B
= 2:,p.f.
~
~
~
:=
(\lEW) (b
:=
(~) (b
)
BW
BW
)
= (~BW) (b )
BW
:=
2:.
:=
c
~
'fl. f.
~
c,GH
~
+ P
se,BH
.-.
53
Table 4.2
Differential equations for the compartment model
ok
Pn,M + b pn - Psn,Pse
(4.26)
+b
Pe
+b
BW,Pse
-P
se,S
+s
Pse
-P
se,BH
(4.27)
Psn,Pdn + bBw,Pdn - Pdn,FEn - Pdn,Un +
(4.28)
b
(4.29)
hp
(4.30)
b
(4.31)
s
(4.32)
f
(4.33)
u
(4.34)
f
(4.35)
u
(4.36)
v
+p
sn,Pse
-c
c,GH
Li~ifia
BW,Pde
s
Pse
n
n
Pdn , Un
e
c f ,. Rf + c c,Rc + Pse,GP f + P se,GPc + Pd e,FEe
e
= Pde,Ue
aSummation is over those fluxes producing nitrogenous waste in
addition to P
Pd and b
which are shown explicitly. Such terms
sn, n
BW ,Pd
n
must appear as losses against the proper compartments in order to
balance the equations.
bSummation is over those fluxes producing energetic waste in
addition to bBw,Pde which is shown explicitly. Such terms must appear
as losses against the proper compartments in order to balance the
equations.
54
TESTING THE MODEL
We test: the m<o,dEd <;dth twc' purposes in mind.
First? to
e~am:ine
the results in a qualitative sense to see if their pattern is reasonable} and seccl1ld" to derive the best estimates for each parameter in
the model.
FrQIT: our kn':jwledge of animal DrJtrition.. the pattern of compartment
values as, a fClnction of time is generally known.
If the model results
are qualitatively rea",:onable., we have some assurance that no unreasonable assumptions were made in deriving the model and that no major
factors have been omitted.
Pessimistically;, it might be considered
that a combination of unreasonable assumptions and major factors
omitted might have equal and opposite effects on the results and cancel
out.
This view is rejected on the grounds that the model was derived
from nutritioaal concepts,. and it is cmlikely that such a cancelling
Given th<at the resr::dts are qualitati.vely reasonable<J we then must
exami.ne them quantitatIvely.
This i.nvolves deriving a best set of
estimates (lif th.e parameters.
"Best lY means that over a series of
situations;, for which experimental results are available., the output
for the particular model
mental data.
is closest, in some sense) to the experi=
Having obtained this best set, the degree of closeness
is evaluated.
The remainder of this section IS cO>:'.ce:r.ned 'Nith a description of
the meldel tel be testEd, the data
LO
be "-ised; the initial estimates of
the parameters} selection of constar:ts-, and the i.terative method used
to arri.ve at the final parameter estimates.
'.
55
5.1
A Model for a Growing Steer
Data available to test the model shown in Figure 4.2 were
ed under a variety of conditions)
l.~.)
from
steers~
collect~
from dry cows)
from milking cows J from animals on controlled feeding regimens and from
"
those on ad 1i.bitum feedi.ng.
model in its entirety.
No one set of data pe.rmits testing the
A set of data were found? however J that permit
testing of key features of the model except for external production
and body temperature regulation.
Two experiments on growing steers (Forbes et al.) 1928; 1930)
contain data on feed composition and consumption) body weight and
nitrogen and energy balance.
experiment.
Two steers were involved in each
In the first experiment the steers were Aberdeen=Angus
identified as steers 36 and 47.
In the second experiment, they were
Shorthorns identified as steers 57 and 60.
They were fed mixtures of
alfalfa hay and corn meal at feeding levels ranging from one=half to
three times maintenance and hay alone at the maintenance level.
The
maintenance level as defined in Forbes et al. (1928; 1930) is only
approximate as zero energy balance was not attained.
It is used below
as defined in their paper.
Each diet was fed for a 30-day period and observations were made
during the last 20 days.
During the period of observation J the animal
presumably reached a state of dynamic equilibriurn J where the rate of
passage of food at a given point in the digestive tract was almost
constant, and the partitioning of the food into an absorbed portion)
fecal residue) etc., had effectively stabilized.
56
We thus reduce the model to a simpler version which still retains
the critical features of the joint nitrogen and energy metabolism.
That is, the external production compartment from Figure 4.2 is
eliminated.
The energetic material in the gut products component of
the feces is similar to ether extract.
Thus, all energetic gut
products are represented by the flow from P to GP and compartment
se
f
GP
c
is eliminated.
Since the values of the
~.,
1.
the coefficients for heat production
from body processes, are not known, and the data were not sufficient
to estimate them, the handling of heat production had to be simplified.
The heat produced from gut fermentation will still be calculated in
terms of digested carbohydrates as given by equation (4.3d).
The
heat loss per day will be taken as the amount of heat dissipation
measured in the calorimeter, say D, or
D .
=
Then the amount of heat production per day from body processes will be
taken as the difference between the heat dissipated and the gut
fermentation heat, or
=
D - c c,GH
Thus, the total heat production, hp, equals the heat dissipation,
since
hp
D
and the heat pool remains at a constant level.
...
57
~iJ
The values of the
known nor estimable.
the body wear coefficient.s, also were not
Hence J the degradation of body as body wear, bBw>'
is assumed to be proportional to the heat dissipation.9 DO' and the
proportionality factor will be estimated in the testing process.
The nitrogen flow will be explicitly followed using the binary
notation for nitrogenous compounds.
Each compound has an associated
heat of combustion value (kcal/gm nitrogen).
Dividing each flow law
for nitrogenous material (kcal) by this heat of combustion value gives
the appropriate flow law in terms of grams of nitrogen.
These flow
laws will be designated by prefixing the energy flow law by
subscripted lower case n.
~
The heat of combustion value will be denoted
by a lower case h, prefixed by a subscripted n and suffixed by a
subscript to denote the material to which the value applie.s.
A heat
of combustion value wi.thout the prefix will have units of. kcal/gm
material.
For example,
h
n Bn
denotes kcal equivalent of body protein
per gm body nitrogen, whereas h
protein.
Bn
denotes kcal body
p~otein
per gm body
With the above explanations, the flow laws in Table 5.1
represent the model to he tested.
Values of \) and
S, the waste.
production coefficients.9 will be assl'JIIled to be zero except where
explicitly shown.
The differential equations derived from t.hese flow laws are given
in Table 5.2..
Testing the model involves solving this system of
differential equations and comparing the results obtained to the
nitrogen and energy balance tables given in Forbes et
A computer program (IBM J 1968) for the
360~Model
to numerically solve this system of equat:i.ons.
~!:
(192.8, 1930).
75 computer was used
The RungeooKutta
58
Table 5.1
F1c", laws for te.sting the model
(5.1a)
(5.lh)
(5. Ie)
(5. Id)
c
C
n n,Psn
c
n,Rn
c
n nyRn
-
n,Psn/ nh Cn
(1- d ) (c. )
n
= c
n
/ h
n,Rn' n Rn
(5.2a)
c f, Pse
(d f ) (c f)-' (d~) [a fL +~b fL) (F L) ] (c f)
(5.2b)
c
(l-d ) (c )
f
f
(5.3a)
c
(5.3b)
c-(l-d)(c)
f .'
Rf
cyPse
(1=1J, =i1 )(d )(c)::.: (l-IJ. -i1H)(d*)[afL+(bfL)(FL)](C)
c ''CH
c
c.
'
c ''G
c
'
c
c,Rc
c
c
(5.3c)
(5.4)
(5.5a)
(5. 5b)
(5.6)
(5. 780)
(5.7b)
(5.7c)
(5.7d)
n P dn, FEn
:'" p
I h
dn, FEnl n FEn
(5.8a)
(5.8b)
continued
59
Table 5.1 (continued)
(5.9a)
(5.9b)
(5.9c)
2,
P .
(5.9d)
.-
(. KPn.e ) (P snlI p se') (B).
(Kp~s)(Pse)
(5.10)
.... (K . ) (B)[(S/B)/(S/B + r)]
S op
(5.11)
(5.l2a)
/ h .
snJPdnl a-Pdn
PH
.. (K p B)(P )(P )(AB=B)/B
J
so.
se
.
- [1/ (1 + PB) ] (P )
B
(5.l2b)
(5.l2c)
="'
Po. :; B/nhBn
(5.l2d)
(5.13a)
b
(5.13b)
b
(5. Bc)
0.
-
P
Po.
ByP
) (B)
[1/ (l + PB) ](b p )
Po.
b
(Ie
"'" bpn/nh B
(5.l3d)
(5.14)
:=
(5.15)
Pse,BH
(5.16)
hp
(5. Db)
b
D
::; D
=
c
c.,GH
BW.jPdn
continued
60
Table 5.1 (continued)
(5.17c)
b
n BW,Pdn
(5.17d)
b
(5. 17e)
b
BW,Pse
BW,Pde
61
Table 5.2
Differential equations for testing the model
*
Psn,Pse
(5. 18a)
(5. 18b)
n c n,Psn - n P sn,GPn - n P n,B + n b Pn - n P sn,Pdn
(5.19)
= c
+
f,Pse
+b
C
c,Pse
-P
BW,Pse
-
se,S
Pse,GPf - Pe,B + bPe
+s
Pse
-P
se,BH
+p
sn,Pse
(5.20a)
Psn,Pdn + bBW,Pdn
(5.20b)
P
+b
-P
-P
n sn,Pdn
n BW,Pdn
n dn,FEn
n dn,Un
(5.21)
bBw,Pde - Pde,FEc - Pde,Ue
(5.22)
hp
(5.23a)
(5. 23b)
Pdn,Un
b
n
(5.24)
s
(5.25a)
f
(5. 25b)
Pdn,FEn
b
b
n Pn
s
n
c
n,Rn
+ P
b
n BW,Pdn
Pse
sn,GPn
+ P
dn,FEn
f
n n
(5.20a)
u
(5.26b)
u
n n
(5.27)
f
(5.28)
u
(5.29)
v
n
e
nPdn,Un
- cf,Rf + cc,Rc + Pse,GPf + Pde,FEc
e
c
c,V
62
variable step numerical integration method was used.
The program,
entitled Continuous System Modeling Program (CSMP) was run over enough
time periods for the system to approach equilibrium, as we have assumed
the steer does.
The final values were then used to compare with the
experimental data.
5.2
Initial Estimates
Use of the CSMP computer program requires that values of all
constants and parameters and initial values for all compartments of
the model be provided.
Parameter estimates are refined by an iterative
procedure until an optimal set of parameter values is derived.
5.2.1
Evaluation of Constants
For each experiment (Forbes et al., 1928; 1930), data are provided
on the steers' body weight, dry matter fed (gms), nitrogen fed (gms),
ether extract fed (gms), carbohydrate fed (gms), total energy fed
(kcal) and total heat production (kcal).
Constants are needed to
convert the data in grams dry matter into units compatible with the
model, kcal and grams of nitrogen.
The given data (Forbes et a1.
J
1928; 1930) are summarized in Table 5.3.
The first constant evaluated is the ratio of energy to nitrogen
in the body, P (Appendix 9.3).
B
for the flow law sp
se
Next, a constant, r , is evaluated
s
, equation (5.11) (Appendix 9.3).
Then, factors
to convert body weight into kcal of body, B, and kcal of stores, S,
are computed.
9.3) •
These are denoted f
B
and fS' respectively (Appendix
We also must be able to convert kcal of body protein, B , and
kcal of fat, S and Be' to body weight.
n
We define constants w
and w '
S
Bn
'0
63
Table 5.3
Input data for testi.ng the model
Steer~
Feeding
Leve1 a
Body Size
(kg)
Dry Hatt.er
Feed Values .fB!!.ls)
Ether
Nitrogen Extract
hydrates
Heat
Production
(kca1)
Carbo~
360.5
471.2
1.0
481.2
1.0 (hay) 499.9
'490.2
1.5
482.9
2.0
1,885
3,762
5,763
5.,353
7,037
39.5
78.8
145.6
11.5.9
152.2
50.7
101.1
86.8
148.1
194.6
1,484.0
2,960.6
4,225,9
4,186.4
5,503.7
8,156.0
9,839.7
11,635.0
11,8.54. 1
13,888.1
470.5
1.0
1.0 (hay)
1.5
2.0
474.8
484.8
499.0
494.6
486.2
1;>863
3,790
5,771
5,617
7,384
39.0
79.4
145.8
121.5
159.8
50.0
101.9
87.0
155.3
204.2
1,466.1
2,983.3
4,231. 8
4,392.8
5,774.9
7,754.5
9,383.0
11,254.6
11,696.9
13,536.3
600.5
LO
1.0 (hay)
1.5
2.0
2.5
3.0
381.0
310.9
412.1
332.9
3.58.4
39108
426.6
1,681
2,828
4,983
4,237
5,704
7.,520
9,489
32.9
55.6
117.1
83.3
112.2
148.0
186.8
46.3
77.5
70.5
117.2
158.0
207,9
267.9
1,336.7
2,9 250 . 6
3,748.6
3,370.8
4,538.6
5,982.1
7,544.9
7,476.0
7,252.9
9,790. 1
8,82L 4
11,156.9
13,976.4
16,133.1
570.5
LO
1.0 (hay)
1.5
2.0
2.5
398.3
359.6
425.6
384.3
403.1
443.7
1.,700
3,085
5.,125
4,612
6,233
8,057
33.3
60.7
121.0
90.8
122.6
158.7
46.8
84.6
72.5
127.7
172.3
227.3
1,351. 9
2,455.4
3,854.6
3,669.5
4,958.3
6,405.6
7,939.0
7,908.7
9,953.7
9,493.3
11,851. 2
14,408.2
aFeeding level is expressed as a fraction of the lYmaintenancel!
ration.
64
respectively for this (Appendix 9.3).
The intake values (gms) are
converted to kcal using appropriate heat of combustion values.
In order to simplify matters, we assume that the heats of combustion of nitrogenous matter (kcal/gm N) in feed, body, Rn~ GP
n
and P
sn
2:.~.,
are equal,
h
n Cn
~
h
n Bn
= nh Rn = nh GPn = nh Psn = 34.2
where the value is taken from Forbes et ale (1928; 1930).
Assuming that, on the average, body protein contains 16 percent
nitrogen, then the heat of combustion of body protein (kca1/gm protein)
is
h
Bn
=
(h
n Bn
)(.16)
=
5.472 .
The heat of combustion value for ether extract (Maynard and Lcos1i,
1962) is taken as
Since the total energy fed is given (Forbes et aL, 1928; 1930)
and we can calculate the energy fed as nitrogen and as ether extract
from the constants given above, we can solve for the heat of combustion
of Cc from the following
equation~
Thus,
he c
= [(total fed)
~ (c n )( n h n ) C
(cf.)(hCf)J/c
..
c •
Substituting feed values from Table 5.3 and heat of combustion
values derived above into this equation leads to
6.5
h
CC
"'. 4.51
for mixed diets of alfalfa hay and corn meal, and for a diet of hay
alone, a value of
he c "' 4.74
The heat of combustion for methane (Forbes et: ale, 1928; 1930) is taken
as
For U
n
J
we use the value given by Forbes et a1. (1928; 1930) in units
--
of kcal/gm nitrogen;
h
n Un
= 7.45 •
The nitrogenous waste material flowing to P
as a result of the
dn
deamination of amino acids is urea.
Its heat of combustion is 5.414
kcal/gm nitrogen (Blaxter? 1962a) which is lower than the value for U .
n
Since the heat of combustion of urinary nitrogenous compounds is 7.45
kca1/gm and the heat of combusticm of the urea nitrogen in the urine is
5.414 kca1/gm, the nitrogenous waste resulting from body breakdown must
have a heat of combustion higher than 7.45.
To simplify handling these
flows, we will assume that the heats of combustion
material flowing into and out of P
equals 7045.
dn
h
n Un
~
()\f
all nitrogenous
Hence
7,,45
In the deaminaticm of a.mino acids .• all nitrogen from the amino
acids flows to Pdn'
Thus?
66
1 gram n Psn - 1 gram n Pdn
for this reaction.
In terms of energy,
."
h
kcal in P
sn
n p sn
.,
or
1 kcal in P
sn
- ( h pd / h p ) kcal in P
•
n
n n sn
dn
Thus
The factor v
BW
is derived similarly.
In the process of body
breakdown, all nitrogen from the body flows to P '
dn
1 gram n B-1
gram n P n .
n
d
One gram of
B is equivalent to h
kcal of B and this is equivalent
n n
n Bn
n
to (1 + PB) (nhBn) kcal of total body, B.
equivalent to nhPdn kcal of P •
dn
Also, 1 gram of
n
P
is
dn
Thus
or
and thus
.151 •
Constants are summarized in Table 5.4.
5.2.2
Evaluation of Initial Conditions
Initial conditions for all compartments, except B, S, and the four
pools, are taken as zero.
For B, the body weight of the animal is
multiplied by the factor f •
B
Using a prefixed subscript z to denote
67
Table 5.4
Constant.s used
Constant
Valu.e
tCI
test the model
Description
PB
.44
ratio of energy to nitrogen in the body y B /B
e n
r
.034
constant in flow law s
s
f
f
B
S
wBn
W
s
Pse
(Appendix 9.3)
1.423
kcal B per gram body weight (Appendix 9.3)
.895
kcal S per gram body weight (Appendix 9.3)
.868
grams
.107
grams body per kcal fat (S + B ) (Appendix 9.3)
e
fat~free
body per kcal B (Appendix 9.3)
n
h
n Cn
34.2
kcal feed protein per gram feed nitrogen
h
n Bn
34.2
kcal body protein per gram body nitrogen
h
n Rn
34.2
kcal fecal food residue (protein) per gram nitrogen
h
n GPn
34.2
kcal fecal gut wear per gram nitrogen
h
n Psn
34.2
kcal pool amino acids per gram pool nitrogen
h
h
h
h
Bn
Cf
Cc
V
{
5.472
kcal body protein per gram body protein
9.4
kcal feed ether extract per gram feed ether extract
4.51
4.74
kcal feed carbohydrate per gram feed carbohydrate
for mixed diet and hay diet, respectively
13.3
kcal methane per gram methane
h
n Un
7.45
kcal uri.ne per gram urine nitrogen
h
n Pdn
7.45
kcal pool material per gram pool nitrogen
h
n FEn
7.45
kcal fecal excretion per gram fecal nitrogen
\)Pne
.218
kcal to P
\)BW
.151
kcal to P
dn
dn
per kcal amino acid deaminated
per kcal body broken down
68
i.nitial (CJr zerc time) c,:m:'IitLc'(l and w'B
fOl
body weight;, we have
For grams of nitrogen at zero time~ we mlist convert zB to kcal of
-,
protein)
B whEre
z n
and then divide by t.he heat of combustion to give
For the initial condition on stores y we multiply body weight by f "
S
Initial conditions for t1:le pools are derived in Appendix 9.4.
Initial
values in terms of grams of nitrogen are derived by dividing by the
appropriate heat of combustion.? i. e ••
<-=
-
,/
P / h
z sn nPsn
z Pdn/ n h Pdn
There are two values which must be derived for each experiment
simulated.
The flow law for transport from pool to body requires a
maximum body size in kcal." A.B.
(Shorthorn and
Aberdeen~Angus)y
For each of the breeds involved
a maximJrn bod:v size in grams j w
.?
Bmax
is assumed.
We then take
"B= (fB ) ("rBmax ) •
For steers 36 and 47 J the AberdeenoAngus ~ we assum.e an average breed
value of 650 kg for the maxim;;.m I:>:'d)' sile,? and for steers 57 and 60"
69
the Shorthorns, an average breed value of 750 kg.
AB = (1.423)(650,000) =
Thus,
924,950 kca1
for steers 36 and 47, and
AB = (1.423)(750,000) = 1:067,250 kcal
for steers 57 and 60.
The second required value for each experiment is the feeding level
used in evaluating the parameter d.
n
This value is the ratio of dry
matter fed, cd' to a base value denoted cd,o.
estimate of that
The base value is an
amount of dry matter which must be fed to yield zero
energy balance for the steer.
For each steer, linear regression was
used to relate heat production to the balance of fat energy and heat
production to dry matter fed (Appendix 9.5).
The predicted value of
dry matter for zero fat energy balance was taken as cd
,0
•
Initial
conditions are summarized in Table 5.5.
5.2.3
Initial Estimates of Parameter Values
Initial estimates of parameters are derived in various ways.
Data
from many sources are used and- judgments are made to derive these
values.
(5.30)
There are 20 parameters to be estimated.
From equation (5.la),
dn = (d*)[f(FL)J
n
We will arbitrarily assume that d*
n
=1
and what we call a
actually represent (d~)(aFL) and similarly for b
(5.2a) and (5.3a),
FL
.
FL
will
From equations
70
Table 5 • .5
Initial pool concentrations and values of A and cd
used
in testing the model
.B
,,0
Term
Value
Description
z Psn/ z B
.00103
concentration of simple nitrogenous
material in the pool
z Pdn/ z B
.000665
concentration of degraded nitrogenous
material in the pool
z P se / z B
.00514
concentration of simple energetic
material in the pool
.00331
concentration of degraded energetic
material in the pool
P
/
z de z
B
AB-Shorthorn
924,950 }
maximum body size, kcal
AB-Aberdeen-Angus
c
c
c
c
1,067,250
d,o
-
Steer 36
d,o
-
Steer 47
d,o
-
Steer 57
2,558
d,o
-
Steer 60
2,313
3,976
grams of dry matter fed which
correspond to zero fat energy balance
71
(5.31)
(5.32)
(d;) [ f (FL)
d
c
J
(d:> [f (FL) ]
We will denote d
c
as the digestibility coefficient for carbohydrates
for a mixed diet of corn meal and alfalfa hay.
For the diet of hay
alone) this coeffici.ent is denoted as dc(hay).
The equations for flow of secretory material from the pool to the
feces, (.5.5a) and (5.6) involve parameters K
and KGPf"
GPn
for these paramete.rs are kcal/gm since cd is in gm!day.
f
GPf
The units
If f
GPn
and
represent the fraction of the dry matter fed which is secreted
into the feces as grams of crude protein and of ether extract,
respectively, then
(5.33)
K:
GPn ::::; (fGPn) (h Bn)
and
(5.34)
We define the parameter IJ.
c
kcal of digested carbohydrate.
We define a new
parameter~
as the kcal of methane produce.d per
In terms of f ,
c
f ' as the grams of methane produced
V
per gram carbohydrate digested for the mixed diet" and fV(hay) as
the corresponding term for the diet of hay only.
Then to determine ,...c·
II. 9
grams of methane and the gram of digested carbohydrate must be con=
verted to kcaL
(5.35a)
(5. 35b)
Thus,
72
Combining those parameters frciffi TablE .solwhie!l are not: functions
of other parameter::> and a'::din,g in tLc' :;e defined in equations (5030)
.9
through (5035) gives the f,,110w'ing 2.0 parameters:
'*
*,';
f Cpn " fUn' KpneJ d c .\' d c (hay).\' de fv;, fV(hay)
Kp ,? S,o
KS :' p,?
Kp J
B~ KB~, p" KBw-" and
,1
a
' b
K
FU FU pdnJ
' f '
f CPe " K:pde ) 'Ue"'
nBW o
The concept of digestion being affected by feeding level was not
originally incorp"jrated, into the modeL
parameter valu.es "rere
that time.
ES timated
When it was added, the
by examining the model output up to
Values for each steer differed slightly, but the average
initial value;5were a
.'" 09465 and bFL'c
FL
~.01630o
Based on previou.s work of Lucas and Smart (1959)" we estimated
initial values of f
CPn
',;0(
.04 and f
CPe
'" 0016.
The parameter K
was estimated from the ni.trogen and the energy
Pdn
balance data for Steer 47 ted at maintenance (Forbes et a1.
The urinary excretion was 4709 gros ni.trogen/day.
.
com'b ustl,on
f
0,
192.8)
0
Using a heat of
.
""f' '7,O~J
1<; k
. t, "
t'h"
urine
c.,
.ca"1/gm nl.,_,r,",gen."
,1.8
35608 kcal/day.
J
IS
. 'I.en.t t :0
eqlllVa
According to Schneider (l935), endogenous fecal
nitrogen in the rat is abou,t 1/6:)£ the u.rinary nitrogeno
Using this
value for steers." fecal nitrogen is about 59047 kcal/day for a total
of 416 03 kcal/dayo
Summing equations (507a) and (501c) and solving for
K
gives
Pdn
We will use the initial condition
zP
dn
P'
z dn
LJr Pd,n0
The initi,al, pool value
equals the product of the initial pool concentration and the
initial body SiZE, or
B) ( ZB)
z P d n ". (,P
z d n,/ z'
'
0
73
From Table 5.5, the first term equals .000665.
-.
From Section 5.2.2, we
have
From Table 5.3, w equals 484.8 kg for Steer 47 at maintenance and from
B
Table 5.4, f
z
equals 1.423.
B
B
Hence,
= (484,800)(1.423) = 689,870
and hence,
zP
dn
= (.000665)(689,870) = 458.76 .
Thus
Kpdn
= 416.3/458.76 = .90745 .
Now, total urine energy was 768.3 kca1/day.
Subtracting the con-
tribution of nitrogen leaves 411.5 representing Pde,.Ue'
Assuming no
flow from P
to FE leads to
de
c
Applying the same reasoning as above, we have that
z
P
- 2283 kca1 and
de
thus,
= 411.5/2283
K
Pde
From the above
discussion~
~ .18024
note that we have also partitioned the flow
from P
between urine and feces and
dn
fUn
f
Ue
6/7 "" . 8571
=
1.00
The digestibility factors, like a
FL
and b
in the equation for
FL
feeding level effect, were estimated by examination of the model
74
results at the time these parameters were added to the model.
Thus,
d* - .8541
c
.0
d~ = 1.00 .
..
It was recognized that digestibility of the carbohydrates in the hay
only diet would be lower than for the mixed diets and so a different
value of d* was used for this case,
c
d~(hay) = .7606 .
The factor for methane production, f ' was estimated from data in
V
Forbes et al. (1928; 1930).
For each steer at each feeding level, the
grams of carbohydrate (crude fiber plus NFE) digested per day and the
grams of methane produced per day were recorded.
The ratio of
carbohydrate digested to methane produced gave a value for each steer
at each feeding level and the average of these 23 values was
f
V
= .0469 .
Again, for hay, methane production is a higher fraction of digested
carbohydrates due to the higher proportion of fiber in the hay.
For
hay, the value is
f
V(hay)
=.0527.
Flatt et al. (1965) and Blaxter (1966) mention cows losing 16-20
and 20.1 megacalories per day during early lactation.
Using this as
an indication of the maximum amount of fat stores which may be
depleted in a day by the steers, we assumed
sp se = 21,000 kcal/day •
75
Assuming the fatness to be high enough so that equation (5.11) can be
approximated by
'.
s
"
Pse
=:
then
Using the average body size of the four steers in Forbes et a1.
(1928; 1930) of 611,890 kca1, we have
K
S,
P
=:
.03432 •
For Steer 47 at maintenance, or more accurat.e1y, at near maintenance, the fat energy added is 689 kca1/day and the protein (B )
n
The fat associated with B is
added is 208. 7 kca1/day.
n
thus depot fat added is 689
=
92 : : : 597 kca1/day.
From the differential
equation for stores, equation (5.24), and equation (5.10)
s
c
597 -- (Kp S) (3546) - 21,000 •
,
Thus
The derivation of
Ie
P,
B
=:
~
JB
is given in Appendix 9.6.
.001619 •
From equation (5.17a) we see that
'"
"BW
-- b BW/n •
The result is
76
FrQ~ ~he
data on steers under fasting conditions (Forbes!!
!l.,
1928;
193Q; Forbes and Kriss, 1932), the values of bBW and D are taken
(Table 5.6) and a value of K
BW
average
v~lue
BW
Table
S.~
The
•
Values of bBW ' D, aod K
BW
,I
b
Steer
60
K
D
BW
1,857.442
1,645.842
1,725.245
1,901. 270
36
47
57
BW
7,651
7,396
7,482
6,904
.24277
.22253
.23058
.27539
Average
.24282
From equations (5.l3a) and (5.23a) we have
and
FrQm above, for Steer 47, protein added per day is 208.7 kcal and the
f~t
associated with this is 92 kcal/day, or about 300 kcal/day of B
From Table 5.6, b BW = 1646.
Substituting values already derived for Kp B and initial pool conis ad4ed.
Thus, we take b
= 300.
,
centrattons into (5.l2a) and taking
and
ll:
B for Steer 47 as 689,870 kcal
As as 1,067,250 kcal gives PB = 2231.14.
bp
= 285.14
.'
~
is
= .24282
K
for each steer.
i~ c~l~ulated
Solving for bp gives
77
and
K P = .0004177 •
B,
'0
The parameter
TI
BW
was introduced into the model after several
simulations had been run.
TI
BW
= .621
Examination of the data led to the estimate
•
From equations (5.9a) and (5.l8a) we have
(5.36)
Based on data from Steer 47 fed at near maintenance, we assume that
p
sn
= O.
From
equat~ons
(5.l2b) and (5.l3b), we have
Pn , B = (1/1.44) (2231.14) = 1549.40
and
bpn = (1/1.44)(285.14) = 198.01 •
Averaging the true digestibility coefficients of protein for cattle,
sheep and goats (Lucas and Smart, 1959) leads to a value of dn of
.94, and from Table 5.3, n c n
= 79.4
and from Table 5.4, n hen
= 34.2,
thus
=
(.94)(34.2)(79.4)
= 2552.55
.
From equations (5.5a) and (5.33) and Tables 5.3, 5.4 and 5.7 J
= (.04)(5.472)(3790) = 829.56
Substituting into equation (5.36),
"
78
Table 5.7
Initial estimates of parameter values
Parameter
a
b
f
FL
FL
GPn
Value
.9465,
-.01630
.04
KPdn
.90745
fUn
.8571
~ne
.002688
d*
c
.8541
d*
.7606
c(hay)
d*
f
f
f
f
v
V(hay)
GPe,
~de
1.0
.0469
.0527
.016
.18024
l.0
6.090
.03432
.001619
.0004177
.24282
.621
"
79
2552.55 - 829.56 - 1549.40 + 198.01
= 371. 6
and substituting for the remaining values leads to
*
IL-
-Pne
=.002688.
The initial estimates of parameter
values are listed in Table
5.7.
5.3
The Goodness of Fit Criterion
The goodness of fit of the model to the experimental data will be
based on the nitrogen and energy balances.
The nitrogen and energy
balance variables for the steer data and the corresponding terms from
the model are matched in Table 5.8.
Table 5.8
Nitrogen and energy balance variables
Item
a
Terms in Model
c
n n
Nitrogen - fed
Energy
- in urine
u
n n
-
r + ngpn + fe
n n
n n
in feces
- in protein
- fed
(and pools)
- in urine
- in feces
- in methane
- in protein
- in fat (and pools)
b + nPsn + nPdn
n n
c
u
r
n
n
n
+ c
c
+ u
+ r
+ c
f
e
f
+ r
c
+gp
n
+gp
f
+ fe
v
b (= b
n
+ b )
e
s + Psn + Pdn + Pse + P
de
aUnits for nitrogen are grams/day and for energy are kca1/day.
e
80
e
The amount of nitrogen or energy fed per day must equal the
amount excreted per day plus the amount being added to the body,
stores, or pools per day.
This balance is expressed as follows:
(5.37)
c = u + r + ngpn + fe + b + nPsn + nPdn
n n
n n
n n
n n
n n
(5.38)
c
n
+ c
c
+ c
f
= u
n
+ u
+ r
e
+ v + b
+ r
n
+ b
n
e
+ r
f
+ fe
c + gpn + gPf
e
+ s + Psn + Pdn + P
se + Pde
These two equations are not independent since each term in (5.37)
is related to a corresponding term in (5.38) by its heat of combustion.
Recall that
n h Cn
= nh Rn = nh GPn = nh Bn = nh Psn
-- 34 • 2
7.45 .
If we multiply (5.37) by
(5.39)
c
n
= un
+ ( nh Cn
n
h
Cn
' the result is
n'hUn ) (u n) + r n + gp n + (h
n Cn - nh FEn) (fe n )
Subtracting (5.39) from (5.38) gives
+
(5.40)
U
e
-
(h
h
) (fe )
n Cn
n FEn
n
and rearranging terms and substituting for the heats of combustion
gives
(5.41)
+ fe e + v + b e +
S
+ Pse + P e •
d
'\
81
The terms in pare.ntheses represent energy, which originally was in a
nitrogenous compound.. and which has been added to the
energy from the deamination of amino acids.
an energy balance in the system due to
non~nitrogenous
Equation (5.41) represents
non~nitrogenous
compounds.
We can assume that the right-hand sides of equations (5.37) and
(5.41) are constant.
For equation (5.37) then, if we match the values
of urine and fecal output per day) we will also match the value of
nitrogen remaining in the animal per day (body plus pools).
Similarly,
for equation (5.41)) if the model output values match the experimental
values for urine, feces and methane excreted per day, they will also
match the amount retained per day (stores plus body plus pools).
Hence, the goodness of fit criterion will be based on a comparison of
u, f , u , f and v with the corresponding experimental results.
n n nne
e
The experimental results are given in Table 5.9, with the above five
terms labeled Urine N, Fecal N, Urine E, Fecal E and Methane E,
respectively.
The goodness of fit criterion is in the form of a sum of squares
of deviations between model results and experimental results for each
entry in Table 5.9.
A glance at the table indicates, however, that if
each model value were, say, five percent different from the experimental value, the difference in fecal energy would dominate the sum of
squares.
To overcome this a set of weighting factors are derived to
equalize the variances of the factors.
level, as well as among factors.
The variances vary over feeding
The values for Steers 36 and 47 are
in good agreement with each other, as are the values for Steers 57 and
60.
However, there is a difference between the two grou.ps of steers.
82
Table 5.9
Experimental results
a
FLSteer
a
Urine N
Fecal N
a
Urine E
0.536
41. 9
11. 4
253.845
1,936.62
831. 3
47
41. 3
11. 7
229.015
1,,862.16
827.3
57
39.7
10 • .5
216.735
I J 712.90
744.6
60
39.1
11. 6
215.305
1.• 731. 48
737.9
1.036
47.1
23.0
386.305
3,679.40
I J 481. 2
47
47.9
23.7
353.345
3 J .588.46
l,449.2
57
47.1
17.9
304.535
3,027.02
1,271. 7
60
40.4
17.8
280.720
2,852.64
1,060.8
1.0 (hay)36
89.0
48.2
597.850
9,632.46
1,656.0
47
89.5
47.6
576.525
9 J 616.88
l,678.7
57
78.7
41. 3
568.78.5
8.. 492.54
lJ593.3
60
72.6
42.2
531. 730
8 J 559.26
1,465.2
1.536
70.9
39.5
473.495
5.,317.90
1,841. 5
47
74.2
40.6
444.010
5.. 551.78
1,880.2
57
44.9
30.4
423.995
4 J 443.92
1,781. 4
60
39.8
30.0
356.290
40'439.90
lJ482.5
2.036
83.4
57.6
512.2'70
7,824.38
2,16804
47
81.1
58.1
497.505
8,155008
2,444.6
57
60.3
43.7
53.5.565
6,534.76
2,335.2
60
49.7
41. 8
492 0035
6,,054.84
2 J 085.7
2.557
79.3
60.3
627.015
8,8Il.54
2,930.3
60
66.5
58.5
612.975
8,336.30
20'518.0
3.060
88.4
76.3
6'17.020
I1 349.04
3}O74.5
a
,"
Fecal E
j
Methane E
FL::: feeding level as a fraction of th.e I1 ma :i.ntenance ii ration,
N = nitrogen (gros/day), E =. energy (kcal/day).
83
This is not unexpected as the two groups of steers correspond to two
breeds and two experiments.
Weighting factors are calculated by the following method.
each
For
factor~
(1) Calculate the variance between experimental results for
Steers 36 and 47.
Call this si;
(2) Calculate the variance between results for Steers 57 and 60,
2
and call this s2'
2
(3) Pool the variances to give s .
p"
(4) Calculate the pooled standard deviation, s p , as the square
root of s 2 .
p"
(5) Calculate a regression line of s
p
versus feeding level;
(6) Using the regression line from (5), calculate predicted
standard deviations, s*, for each factor at each feeding level.
p
To calculate the goodness of fit for each steer J we proceed as
follows:
(1) For each feeding level, and for each of the five factors in
Table 5.9, calculate the difference between the model result and the
experimental result.
Call this deviation 8';
(2) Divide 8' by the appropriate weight to give 8 (= 8'/weight);
(3) Calculate the sum of squares,
5 values,
r,
as the sum of squares of the
i.~o,
r
= L:8
2
0
The computations of the weights are given in Appendix 9.7 and the
weights are summarized in Table 5.10.
84
Weights for the goodness of fit calculation
Table 5.10
.--
Feeding
Level
Urine N
Fecal N
.4780
.3347
19.63
0.5
Urine E
Fecal E
Methane E
8.597
7.864
1.0
2.248
.5174
20.86
79.70
76.61
1. 0 (hay)
4.019
.7001
20.86
79.70
76.61
1.5
4.019
.7001
22.09
168.6
145.4
2.0
5. 789
.8828
23.32
257.5
214.1
2.5
7.559
1.066
24.54
346.4
282.8
3.0
9.330
1. 248
25.77
435.2
351. 6
5.4
The Iterative Estimation Procedure
The iterative estimation procedure involves the method of steepest
descent;
i.~.,
given a vector of parameter values,
~,
a corresponding
sum of squares, r, and the vector of derivatives, dr/d~, we determine
that direction in
~-space
which results in the greatest reduction in r.
We then move a certain distance in this direction to a new set of
parameter values, re-evaluate r and the derivative and repeat the
process.
Let
~j = an nx1 vector of parameter values following the
-n
j
r
th
i tera tion,
j
= sum
j
= vector
L
-n
:::
(L~)
~
of squares corresponding to ~j
=
of derivatives, one value for each parameter
orJ./o~~
~
The iterative procedure is to take
i
=
1,2, ••. ,n •
85
where k is a scalar denoting the distance along the L v'ector which we
move.
'.
We cannot estimate the L vector analytically as the system of
differential equations cannot be solved analytically.
derivati.ves are approximated by finite differences.
Therefore, the
We define the
following:
roj = sum
of squares corresponding to ~j
llj
=a
ll~
-~
- the
r~
= sum of squares based on
1
= an nxl vector of l's •
n
~
-n
diagonal matrix of increments on
.th
~
column of II
j
i
~
= 1,2, ... ,n
~j + ll~
-~
Then
As
~j+1
~j becomes smaller, the size of the elements (o~) of II
~
and of k are reduced in order to converge closer to the minimum value
of
r
and the optimal set of parameter values,
~.
86
6.
RESULTS AND DISCUSSION
The model was fitted for each steer.
For each steer a series of
iterations were carried out using the method of steepest descent.
A
point was reached where further reduction in the swn of squares, f,
appeared virtually impossible.
The parameter values corresponding to the final runs are given in
Table 6.1.
The parameters are also averaged over the four steers and
the coefficient of variation given.
Examination of the coefficients of
variation reveals that all but, five of the parameters are remarkably
coq,sistent.
Except for K;ne~ f Gpe '
Kp Band K p all but one of
S'p:J
B,
K
the remaining coefficients are less than 3 per cent and that for f
is 8.06 per cent.
GPn
For the five listed above, the breed averages are
also given (Table 6.1).
Steers 36 and 47 are
Aberdee.n~Angus
and
Steers 57 and 60 are Shorthorns.
In addition to estimating the values of the parameters, we are
also interested in estimating their variances and covariances.
These
are necessary in estimating the variance of the compartment values or
of the derivatives of the compartment values.
variance matrix is given (Table 6.2).
A correlation and co-
The underlined elements are the
diagonals of the matrix and contain the variances of the parameters.
The upper right part of the matrix contains the correlation coefficients
and the lower left part contains the covariances.
The variances and covariances are coded values.
Alongside the
row headings and below the column headings are numbers in parenthese,s.
These stand for negative powers of 10.
To decode the ijth covariance,
say s*(1j) to the actual covariance;, s(ij), where Pi is the number in
.0<
.,
e
Table 6.1
..
..
e
Steer 47
1.00200
Steer 57
.974097
Steer 60
,978085
Average over
Four Steers
,985531
eVa
=.0395800
=.0408788
=,0405472
=,0405610
1, 76
.0232201
.0215682
,0202752
.0223574
8.06
,438200
.438200
.438199
.438199
b
.870704
.861498
,887197
,887290
.876672
~ne
.00722799
.0106301
.0100345
.00975440
.00941175
.855239
.855398
.861397
.864024
.859014
.51
.656320
.661565
.665774
.635601
.654815
2.04
.993897
.993993
.996972
.35
1.0
1.0
1.46
15.95 ,00892904
.0488357
.0494198
,0482809
.0516488
.0495463
2.98
f VChay )
,0548425
.0554984
.0542194
.0580016
,0556405
2.98
f
,0505173
.0513194
.0394395
.0390366
.0450782
.0863266
.0863291
.0864183
.0864159
.0863725
.06
.965594
.965198
.961197
.961283
.963318
,25
GPe
K
Pde
fUe
1.5
Kp S
J
1.5
1.5
1.5
1,5
By Breed
Steers
Steers
36 J 47
57, 60
1,26
fUn
V
14.98 .0509184
.00989445
.0392380
0.0
KSJ P
Kp B
J
K
.0378494
,0431698
.0615897
.0625355
.0512861
.000129204
.000101692
.0000435954
.0000282780
.0000756924 62.98 .000115448 .0000359367
.000430977
.000285667
.000985627
.000774249
.000619130
K
BW
n BW
0197585
.191300
.197197
.196884
.195742
1. 52
.620603
0624098
.613297
.613401
0617850
.87
BJP
e
Final parameter values = averages and coefficients of variation
Parameter Steer 36
a
.987943
FL
b
=,0412381
FL
f
.0243660
GPn
.438196
I1>dn
d*
c
d*
c(hay)
d*
f
f
1
24.64 .0405096
.0620626
51.50 .000358322 .000879938
00
a eV
b
= coefficient
ev < 001.
of variation (per cent).
......
Table 6 02
Variance=covariance and correlation matrix of final parameter values
Parameter
(Code)
(2)
a
FL
b
(2)
1.,')44
(3)
068
FL
f
-.03
=008
.67
=.58
(3)
f Gpo
(3)
1.453
=00972
30246
K .
Pdn
f fn
U
(6)
=7.336
08982
=1.984
(2)
=L546
=04681
=1.850
08269
=L513
=.0.560
=20894
01855
4.020
K;'(
Pne
d*
.1709
(3)
.5073
f
Un
(2)
.97
aFt
b
pL
.6018
Ie
Pdn
GPn
(3)
.65
(6)
3.583
~-
.4418
2.797
K*
Pne
a
d*
.09
c
(2)
=.81
051
077
=.80
(3)
006
.25
=056
=097
.56
,92
.18
.98
.40
017
=0 4 6
L 631
.08
.92
=.39
=.96
2.253
,36
.86
=037
~.57
=097
01579
-~
(2)
L 326
d*
f
f
(2)
.3817
.0619
2.357
03044
=04267
=01953
=.1488
(3)
=.1661
00263
=3.407
.2466
.5055
04434
-.3553
f
(3)
= • .559?
.8851
=5.520
.8308
7512
.1383
40561
=.0848
=08538
V
f
V(hay)
GPe
(2'>
o
~de
f
Ue
(4)
=.5574
(3)
2.528
.3164
P
(2)
1.226
=00661
~;B
(4)
.4258
=.0039
(3)
=03764
=01047
KS
.'
IeB.'p
K
BW
(3)
TT
(2)
BW
=3.128
06491
=2.017
01884
16.12
1. 319
.5189
=20001
= 1. 002
50371
.34
.4854
01222
=5.619
~.1636
310
=.5510
L 197
=.5238
=.8355
=.3285
=02894
1.046
.2359
04669
.6251
.3025
.2189
=.6940
=01802
10416
1.437
08466
=.5554
=04259
.1992
07926
.51'78
01937
1.494
=2.868
=1. 219
.2346
L 703
=2.356
=8.327
(2)
=.18
d~
c (hay)
f
(2)
088
=04440
.3364
d*
c(hay)
.26
(2)
c
.".
d*
=20839
=02188
= 1. 566
=1.021
~6.
30197
08390
=1. 782
=04349
.9908
.5366
=05184
=04047
=.2034
08153
.1602
.3896
. 00953
.1189
=01498
=01056
20795
=.6880
=2. 713
=00920
00098
=.2204
=30321
=.5249
08190
.1817
continued
e
.
\
..
-
,
.•
00
00
e
e
••
..
e
~
T
tit
Table 6.2 (continued)
Parameter
f
V(hay)
f
K:
Pde
(4)
-.87
fUe
(3)
.85
KS, P
(2)
=. 78
Kp :; B
(4)
KB:; P
(3)
=.95
K
BW
(3)
=.85
'TT
(2)
~.09
~.09
(3)
.25
.25
.29
=.23
.18
-.07
=.01
=.46
=.95
.49
f'
(3)
=062
091
=.92
093
=.96
.98
=0 74
=.23
.82
K
Pdn
fUn
K*
Pne
(6)
009
=062
. 009
=.41
.48
=.52
.59
=.62
033
"".50
=.21
(2)
027
.27
=097
.95
=.93
.89
=085
.96
.74
(3)
.20
020
=.32
.39
=.43
,52
=.56
.12
=.61
=.ll
d*
c
d*
(2)
055
055
=.97
096
=.96
,96
=097
.85
.46
=.93
.38
=.34
.33
=.35
.43
=.18
=.28
.38
=.98
.96
=.95
~.50
.96
a
FL
b
FL
~GPn
V
(3)
f
GPe
(2)
.90
(Code)
(3)
.72
BW
(2)
.97
-LO
c(hay)
d*
(2)
=.95
=.95
(2)
=.32
=.32
1.0
f
(3)
2.181
1.0
=.34
.31
~.32
.38
=.48
.03
-.01
-.26
(3)
7.348
8.251
=.34
.31
~.32
.38
=.48
.03
-.01
=.26
(2)
=.3407
=1. 148
.99
~.98
.95
~.94
=.54
.98
(4)
.2380
.99
~.96
.94
.49
=.96
~.99
.97
~.93
=044
094
=.99
.89
.35
=.90
=.83
~028
.86
f
f
V
V(hay)
f
GPe
K
Pde
fne
(3)
=1. 150
.8018
.4561
=.3470
=30874
1. 612
~LO
=1.0
.2657
=1. 238
LO
-LO
50785
KS:; P
Kp :; B
(2)
0 7 063
2.379
=.8327
.6430
=3.020
(4)
=03346
=1. 127
.3069
=.2371
1.117
KB:; P
K
BW
(3)
00128
.0£.29
=.2036
.1554
(3)
-.0626
=02108
(2)
=.2113
=07119
'TT
BW
=L093
.3551
=.7160
07480
=3.189
-.2664
L224
10597
=.5980
.2273
03585
=.1262
.1017
065
=.96
=03924
.6150
8.850
=.71
.2223
-.1653
=1. 146
1.304
~.
6155
aparameter Kp S is omitted as its variance and covariances equal zero.
:;
.2904
00
\0
90
. t h e numb er 1n
.
parent h eses next to t h e 1. th row h ea d'1ng and p. 1S
J
paren th eses below the J.th co 1umn h ea d'1ng, th en
.. ) = s*(~J')
x 10
s ( 1J
....
-(p.+p.)
1
J
For example, the covariance between
= 2)
(power
KP , B
(power
= 4)
and fUn
is given by
= -.5184
x 10-(4+2)
= -.5184
x 10- 6 •
These values are tabulated to show their estimates based on the
limited number of trials we investigated.
Each correlation coefficient
and covariance is based on four observations.
The mean for each factor
has been estimated, hence these values have only two degrees of freedome
Similarly, the variances have three degrees of freedom.
No
inferences or hypothesis tests are called for or warranted.
A small study was made to evaluate the sensitivity of the model
to averaging parameters over all four steers and for the five parameters previously shown to have large coefficients of variation, over
breed.
Seven trials were run, with the parameters varied as shown in
Table 6.3.
Table 6.3
Run
Parameter values used for sensitivity test
KP , B'
KB,p' KS, P
KPne' f GPe
Others
1
Individual
Individual
Individual
2
Individual
Individual
Averaged
3
Individual
Breed average
Averaged
4
Individual
Averaged
Averaged
5
Breed average
Breed average
Averaged
6
Breed average
Averaged
Averaged
7
Averaged
Averaged
Averaged
91
For each steer and for each ru.n;> values of the sum of squares of
.
residuals) f J are tabu.lat.ed for each feeding level and the total given
for each steer (Table 6.4).
Looking at the total
Steer 60.
one.
r
values;> run one is superior for all but
For Steer .36, runs two and five are almost as good as run
For Steer 47;> runs four;> six and seven were fairly close. to run
one and for Steer 60, runs four and six were far superior to run one.
We obviously did not reach the minimum
r
value for Steer 60 when
fi.tting that steer individually, for substituting the values of run
four for those of run one gave great improvement.
It is probable that
the minimum value has not been attained for any of the steers.
forced to halt the search for the minimum value of
r
We were
due to precision
problems in the computer program.
There also appear to be compensating factors in the model.
For
instance, Kp Band K
can be varied quite a bit) without changing
,
B;>P
the sum of squares very much.
If one is lowered, the other can also
be lowered and the pool and body sizes maintained relatively constant.
The covariance between the two is negative, but this is due to breed
differences.
A plot of the values (Figure 6.1) indicates positive
correlation within breed.
We believe that
~
:I
Sand KS p act similarly.
;>
iterations done, Kp S remained constant.
;>
However) over the
It is possible that the rates
of breakdown of body and stores p K p and K p;> respectively y are
B;>
S)
measures of the rate at which all reactions occllr.o ..!..~. y they act H.ke
clocks in the system.
Fix.ing values of these two parameters for all.
steers and then evaluating animal to animal and breed to breed variation in the other parameters may be very meaningfuL
92
Table 6.4
SteerFLa
36
0.5
r
Summary of
values by feeding level and run
2
3
Run
4
5
6
146.03
90.19
104.41
247.52
96.10
234.32
243.88
1.0
115.79
80.27
99.15
77.67
94.92
73.26
76.95
1.5
35.77
44.97
42.74
27.97
43.15
28.27
27.94
2.0
60.97
101. 64
92.32
85.56
93.41
86.54
84.90
104.05
145.56
138.00
83.32
140.46
85.33
85.79
462.61
462.63
476.62
522.04
468.04
507.72
519.46
0.5
93.62
151.42
140.52
101.15
138.31
102.40
106.38
1.0
85.84
128.69
113.36
78.91
114.96
80.60
84.04
1.5
36.40
61.50
59.99
39.10
59.93
39.07
39.02
2.0
37.20
52.10
55.10
43.34
55.04
43.32
42.24
1.0H
Total
81.77
151. 86
153.30
83.86
152.16
82.96
83.22
334.83
545.57
522.27
346.36
520.40
348.35
354.90
0.5
124.72
217.18
225.33
105.22
228.34
102.11
137.48
1.0
54.47
48.20
47.79
63.34
48.52
63.62
65.24
1.5
38.44
28.66
27.92
43.41
31.06
46.24
40.48
2.0
30.22
22.77
22.22
34.64
24.44
36.64
32.88
43.62
126.88
33.39
134.84
32.76
133. 70
47.34
207.32
34.92
124.77
49.30
45.62
197.37
224.44
Total
418.35
485.04
489.72
501.27
493.05
495.28
546.14
60 0 • 5
224.72
297.76
286.49
75.04
290.07
87.23
163.36
1.0
12.96
15.10
15.40
23.93
14.35
23.38
26.58
1.5
27.10
15.47
15.83
23.10
14.14
21. 72
22.11
2.0
24.97
18.88
19.38
28.98
17.10
26.94
25.77
2.5
33.07
29.38
29.99
42.38
27.52
40.12
38.52
3.0
41.46
48.01
48.79
62.82
44.84
59.07
54.59
160.99
68.52
69.08
116.47
78.34
127.04
153.16
525.27
493.12
484.96
372.72
486.36
385.50
484.09
b
1.0H
Total
1
7
47
57
2.5
1.0H
1.0H
Total
a
FL
b
H
= feeding
= hay.
level as a fraction of the "maintenance" ration.
93
Aberdeen-Angus
.00100
.00080
.00060
/
Shorthorn
.00040
.00020
o 0.'----L:----I.:----l-:----':---:-7--7-=---:;-&-:-~
.2
.4
.6
.8
1.0
1.2
1.4
Figure 6.1
4
Kp , B (x 10 )
Plot of K p versus Kp B by breed
B
,
,
At any rate, discussion of the goodness of fit will be based on
the final runs with individual parameter values (run one).
r
value shown in Table 6.4 is based on a simulation run in which 20
parameters were estimated.
r
Each total
Thus the degrees of freedom associated with
is the difference between the total number of observations and 20.
There are five observations per feeding level, hence for Steers 36 and
47,
r
is based on five degrees of freedom; for Steer 57, 10 degrees of
freedom; and for Steer 60, 15 degrees of freedom.
To put the
r
values
on an equal footing, divide by the degrees of freedom, giving a mean
square
r
value (Table 6.5).
94
Table 6.5
a df
Sunnnary of
r valu.es
r
va1ues J degrees of freedom and mean square
a
Steer
r
36
462.61
.5
92 • .522
47
334.83
5
66.966
57
418~
35
10
41. 835
60
525.27
15
35.018
df
Mean Square
degrees of freedom.
To evaluate the goodness of fit J we must compare
squares of the raw data.
r
to the sum of
The experimental results (Table 5.9) and the
weights associated with them (Table 5.10) are used.
Dividing the
results by the weights gives a new tab1e J like Table 5.9 J but with
values of approximately equal variance.
replications and Urine NJ Fecal NJ etc.
Considering feeding levels as
J
as factors, an analysis of
variance is performed on the weighted variables.
We are interested in
the error term which represents the sum of squares within each factor,
corrected for the mean of that factor.
If
~L
denotes the number of
feeding levels for a steer J then in the analysis of variance J the error
term has
(5)(~L~1)
degrees of freedom.
Carrying out the analysis of
variance and dividing the sum of squares for the error term by the
degrees of freedom gives the mean square for error for the raw data.
It is this number which is compared to the mean square error for the
model.
The ratio of the mean square error from the model to that for
the data indicates how much of the variation in the data has been
accounted for by the model results (Table 6.6).
95
Table 6.6
Mean square error in the raw data, due to the model and
fraction of variation accounted for by the model
Steer
MSE Data
MSE Model
% Accounted For
36
1974.015
92.522
95.31
47
1830.065
66.966
96.34
57
1257.152
41. 835
96.67
60
1292.113
35.018
97.29
Thus even though the best fits may not have been obtained, the
model sti.ll accounted for over 95 per cent of the variation in each
steer's data.
The model results for each steer have been tabulated (Table 6.7)
against feeding level, for each factor.
From these and the
experi~
mental results (Table 5.9), weighted residuals for each steer have
been tabulated (Table 6.8) and plotted (Figure 6.2) against feeding
level, for each factor.
For completeness of presentation, weighted
residuals are also given for protein (B ) and stores (S), even though
n.
these do not enter into the calculation of f.
The urine N residuals indi.cate a change in urine excretion at the
0.5 and 1.0 feeding levels.
The model has averaged out the excretion
patterns and predicts low at the lower feeding levels and high at the
upper levels.
The fecal N residuals are not
residuals for the hay diet are low.
trend.
consi~tent
among steers.
All
Steer 36 seems to show a linear
96
Table 6.7
a
FLSteer
Model resuits
a
Urine N
,~-
Fecal
... N
~----_
~
a
E
.Urine
-.-"--="-----
Fe,.:al E
-~
Methane E
0.536
41. 2.
14. 7
356. 040
1.,945.55
798.3
47
40.4
14.1
322..051
1)836.61
808.9
5'7
39.2
12,.6
3.'55 . .548
l.~
683.07
708.1
60
38.8
12..1
;334.307
1,615.46
7.52.6
1.036
55.5
26.9
428.362
4~171.42
1,560.6
47
.56.8
26.4
389.442
4.,066.02
1,610.3
57
41.9
20.5
352.927
3) 349.70
1,256.0
60
40.2
18.3
323.038
2,980.17
1,240.3
1. 0 (hay)36
87.0
45.8
505.118
10,271. 98
1,,877.7
47
90.6
45.0
466.027
9.,190.48
1,418.8
57
66.6
39.2
442.897
9 y 168.25
1,651. 1
60
71,3
38.1
43.5.190
9~355.95
1)637.2
1.536
71.0
39.2
515.425
6,193. '74
2,168.3
47
76.3
40.4
484.612
6.369.19
1,321. 1
57
52.4
31.6
423.024
5,.394.36
1,827.5
60
51.0
28.2
392.404
4;828.06
1,808.2
2.036
83.8
52.8
602. 786
8.'))2.61
2 1 797.2
47
89.9
54.6
561. 406
8 ;1865.37
2 J 987.9
5'7
66.0
45.3
527.586
7.842.89
2,398.4
60
65.2
40.5
495.995
7,.011003
2,365.4
2.557
82.5
62.8
641. 30 '7
10,935.72
2? 995.2
60
82.7
57.1
621. 220
10) 0,'59.80
3,9° 04 • 6
3.060
98.2
78.4
716.743
13.,831.32
3,635.0
a
FL
feeding level, N
nitrogen (gm.s/day)y E
energy (kcal/day).
•
4
97
Weighted residuals cf final simulation run
Table 608
•
Steer
36
47
57
60
a
FL
a
Urine N
Fecal N
----, - - - -
Prine E
Fecal E
--~--
CH
E
4
---,
P!'iJL
E
Fat E
----
20
=5.49
.68
6.25
L04
~12.5l
=049
-4 044
8002
2.89
2078
-1.29
=.36
1090
5.19
2025
.06
-L83
.07
-So 41
3.88
2,83
2094
2010
-2.09
00)
-1.98
7,28
4.74
-2.97
"2.. 33
",3 011
,47
LO
3095
.5. 18
]099
2. 010
-110 7S
=052
LOR
.27
~3067
1.73
,-50 30
5035
3.39
091
-L83
1.5
053
~022
1.83
4.84
3.03
-1. 34
-,1. 65
200
.49
-3092
2.74
2076
2..54
024
-L 79
005
-L 11
6032
7007
-3 047
·,4064
=3035
.09
LO
-2059
5.09
2.32
4 005
-.20
3.26
·,1.11
LOH
- 3. 00
03
8048
075
9.41
-2.
0.5
,~L49
9.90
5021
1.04
1.0
30 73
7.60
2002
LOH
~o
49
-3.36
1.5
.03
200
,~4.
n
1.5
L87
-3000
1 &' ,J
'''0:
-.04
]064
032
-5.85
200
.98
1.80
-.34
.5.08
030
-3064
-L45
.,1. 27
205
.42
2.38
.58
60 n
023
-2.28
-2.17
0.5
'~. 56
L42
6006
-13.50
1.87
~.41
025
LO
-.07
089
2.03
1.60
2.34
'~.
29
-088
LOH
=.31
-5087
=.46
2024
3056
-2.84
L5
2.78
-2.'52
1.63
2030
2024
~6.
200
2067
~L52
017
3071
1.31
,~7
.02
~L06
205
2.15
'~o
79
034
4098
I. 72
=6010
-]. 79
300
L05
10 70
].54
So 70
1.59
-3094
-2032
FL
:=:;
l'J
~"6.
1000
30
feeding level J LOR -= hay diet at ma intenance leveL
c> •
85
4+
Urine N
4
LEGEND
3+
X
2+
-1
-2
-3
!
Ii
8+
6-1-
c
01
l~
A
D
2
I
2~
•I
I
3
~
FL
~ymbol
36
•
47
~
57
0
60
X
C
0
8
Fecal N
•
6
•
A
c
e
X
I
~
!
1
Figure.602
.
..
0
0
JI
i
2
lH
l~
~
X
X
t
~
X
A
2
't
I
Urine E
C
•
4
=4+
e
A
•
-2+
-6
,
X
C
~
4-+.
2+
~
~
+
lOI
A
)(
Steer
X
C
1+
oI
X
X
A
•
D
h
I .. FL
3
Ax
0
X
-2
•
A
~
1
II
4
l~Q
X
~
2~
3
-4
-6t
A
c
-8
\0
00
Weighted residuals versus feeding level (FL) for final simulation run
e
l'
~
e
e
~
.
X.
10+
•
A
4..1.
CI
2 4-
o
=2
I
+
4
Fecal E
0
8+
6+
•
6-
iii
A
0
0
2
X-
X
I
~
X
I
1
,IH
I
,
,
1~
2
2~
' .. FL
-2
)(
-4
=41
0
I
p
AD
-5 40-
-10
*
•
2
+
X
~
•
A.
X
)(
0
~
1H
0
1~
[]
2
X
Q
2~
FL
3
I
A
I
.c
t
Fat E
1
.,
f
1H
1~
~
•
i"
2
0
X
I
e
~
X
,~
!
~
FL
0
X
-3
1
IH
1:.
I
I
Figure 6.2 (continued)
C
1~
X
X
-1
-2
•
A
~
3
+
•
-15
Protein E
~
A
!
t!!.
=1
-3
0
e
3
[]
': !
•
•
0
~
D
X
1
~
.,
Methane E
A
3
X
•
..
e
I'
•
b.
0
..
A
2
L
3
X
lJ
A
0
~
2~
X
0
)(
-0
-0
100
Urine E shows higr" predicU::ms at the lower fe.eding leve.ls and a
low prediction for the hay.
Since urine E resu.lts from body
break·~
down" we possibly must improve the formulatiom of that flc\N' lawo
The fecal E residuals are all high except for the 0.5
level.
fe~ding
The hay values are higher than. the rest and t1:'.ere :s:eems to be
a linear trend with feeding leveL
The methane E residuals are all high, but are small in magnitude
compared to the others.
steers.
The 0.5 feeding level is low for three
A slight adjustment in the parameter f
V
would seem. to be all
that is needed,. assuming that d*J which controls carbohydrate diges=
c
tion, is correct.
The protein E residuals seem. to have less patte.rn than
but are larger in magnitude.
Improvement in
the~~t~Rel·s,
t~ie body~relat:ed
param=
eters will probably correct this.
The fat E residuals are negative for all but the 0.5 feeding
level.
They trend downward as feeding level increases.
Improvement
in the stores-related parameters shou.ld solve this.
Trends in the res idua Is seem to indicate tha t there is an effE'c t
of feeding level on the compartment whose value is being predicted.
The fonuulation for di.gestibility, which decreasE's digestibility as
feeding levE,l increases.? is an attempt to handle this problem.
parameters f'Jr this formula;, a
ment.
FL
The
and b
prcbably need SOlme imprcn7e=
FV
When they are at their optimum values, most of the linear trends
in the residuals should disappear.
If not J then some additional flew
laws may need to be modified as was the digestibility fC?ImlJla.
101
Even thcugh the optimum parame.ter valoes have not been cttained y
and there might be imprcv'ement in some cf the f10,,1 laws" it is
apparent from th,e high percentage of variation in the experimental
results accounted for that the model
wo~ld
be lisehd in predicting
animal performance over wide ranges of protein and energy intakes and
their ratio.
If desired9 an economic framework can be easily imposed.
W2
7.
CONCLUSIONS AND RECOMMENDATIONS
A conceptual framework defining the physical and chemical compartments necessary for handling protein and energy metabolism of
homeotherms has been developed.
A mathematical
tion, has been developed in this framework.
mode~or
representa-
This framework and model
treat the energy and protein metabolism as an input-output system.
They trace the flow of feed components (input) through the physical
compartments, through chemical transformations and ultimately to the
final uses by the homeotherm (output) such as body gain, fat storage
Or external production.
The model takes account of the physiological
state of the homeotherm, of its maximum capacities for growth and
production, and also of feed composition, the amount of feed and the
energy to protein ratio in the feed.
The model deals jointly with
the structural and energetic needs of the homeotherm and with the
structural and energetic roles of the feed constituents.
The model was tested against experimental data from the literature.
These data are from direct calorimetry studies on four steers
which were fed mixtures of hay and corn meal at feeding levels ranging
from one-half to three times maintenance and hay alone at the maintenance level.
The model explained more than 95 percent of the
variation in the data.
Although the patterns in the plots of residuals from the fit
indicate that the optimum set of parameter values was not attained,
they were not such as to lead to questioning of the model; hence the
conceptual framework proposed and the model developed are valid.
W3
With some improvements, and extension to milking cows, this
model can be used to study the partitioning of nitrogen and energy
among the various body processes for animals on a controlled feeding
regimen, whose intake is measured accurately.
Further work on
factors affecting consumption of feed is necessary in order to extend
the model to the case of ad libitum feeding.
The model should be tested over a broader range of experimental
situations,
~.~.,
milking cows, mature cows, calves, and on other
species such as rats, chickens, sheep and goats.
Data on pool concentrations and fatness would simplify future
testing of the model and make the model output more accurate.
A mathematical analysis of the system would be useful.
Identification of stationary points and prediction of the long range
behavior of the system under constant input would be useful.
The handling of digestibility coefficients should be generalized.
Two coefficients were identified for carbohydrates; one for the mixed
diet and one for the hay.
similarly identified.
The methane production coefficients were
A more general approach would be to define
coefficients for each type of material,
1.~.,
protein, carbohydrate
and ether extract, which would apply to the hay as well as to the
grain, and treat the inputs separately.
They would then be combined
in the same proportions as they appear in the diets.
The search for the optimum set of parameter values should be
pursued.
More precision must be introduced into the computer program
and more judgment is called for also.
should be fixed and held constant.
The coefficients K
This will allow
B,P and KS, P
~
, Band ~S~
,
W4
vary and indicate animal to animal and breed to breed variation.
This should also improve the behavior of the steepest descent method
as the compensating effect of these pairs of parameters will be removed.
A combination of more precision in the steepest descent method
and more intuition would probably lead to this optimum set.of values.
After testing the model over a broader range of situations, an
attempt should be made to estimate the heat
(~)
coefficients as described earlier.
(~
and the body breakdown
105
8.
LIST OF REFERENCES
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W. B.
Baldwin, R. L., H. L. Lucas and R. Cabrera. 1970. Energetic
relationships in the formation and utilization of fermentation
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of Digestion and Metabolism in the Ruminant. Proceedings of the
Third International Symposium, Cambridge, England. Oriel Press,
Newcast1e.Upon-Tyne, England.
B1axter, K. L. 1962a. The Energy Metabolism of Ruminants.
Thomas, Springfield, Illinois.
Charles C.
B1axter, K. L. 1962b. Progress in assessing the energy value of
feeding-stuffs for ruminants. Journal of the Royal Agricultural
Society of England 123:7-21.
B1axter, K. L. 1966. The feeding of dairy cows for optimal production. The George Scott Robertson Memorial Lecture. Queen's
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B1axter, K. L., N. McC. Graham and F. W. Wainman. 1956. Some
observations on the digestibility of food by sheep, and on related problems. British Journal of Nutrition 10:69-91.
B1axter, K. L. and H. H. Mitchell. 1948. The factorization of the
protein requirements of ruminants and of the protein values of
feeds, with particular reference to the significance of the
metabolic fecal nitrogen. Journal of Animal Science 7:351-372.
B1axter, K. L. and J. A. F. Rook. 1953. The heat of combustion of
the tissues of cattle in relation to their chemical composition.
British Journal of Nutrition 7:83-91.
Bratz1er, J. W. and E. W. Forbes. 1940. The estimation of methane
production by cattle. The Journal of Nutrition 19:611-613.
Brody, S. 1945. Bioenergetics and Growth.
Corporation, New York, N. Y.
Reinhold Publishing
Crampton, E. W. and L. E. Lloyd. 1959. Fundamentals of Nutrition.
W. H. Freeman and Company, San Francisco, California.
Dukes, H. H. 1943.
tion, revised.
The Physiology of Domestic Animals. Fifth EdiComstock Publishing Company, Inc., Ithaca,
N. Y.
Dukes, H. H. 1955. The Physiology of Domestic Animals. Seventh
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106
Flatt: W. P., L. A. Moore., N. W. Hooven and R. D. Plowman. 1965.
Energy metabolism studies with a high producing lactating dairy
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Forbes, E. B., W. W. Braman and M. Kriss. 1928. The energy metabolism
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the energy metabolism of cattle in relation to the plane of
nutrition. Journal of Agricultural Research 40~37-78.
Forbes, E. B. and M. Kriss. 1932. The analysis of the curve of heat
production in relation to the plane of nutrition. Journal of
Nutrition 5~183=187.
Hawthorne, G. B., Jr. 1964. Digital simulation and modeling.
Datamation, October~25-29.
IBM Corporation. 1968. System/360 Continuous System Modeling Program
(360A-CX-16X). User's Manual No. H20-0367-2, New York, N. Y.
Kalman, R. E., P. L. Fa1b and M. A. Arbib. 1969. Topics in Mathematical System Theory. McGraw-Hill Book Co.: New York, N. Y.
Kleiber, M. 1961. The Fire of Life.
New York, N. Y.
John Wiley and Sons, Inc.,
Kriss, M. 1931. A comparison of feeding standards for dairy cows,
with especial reference to energy requirements~ editorial review.
Journal of Nutrition 4~14l~161.
Lucas, H. L., Jr. 1960. Theory and mathematics in grassland problems,
pp. 732-736. Proceedings of the Eighth International Grassland
Congress, Reading, Berkshire.1 England.
Lucas, H. L., Jr. 1964. Stochastic elements in biological models;
their sources and significance: pp. 355-385. In John Gur1and
(ed.), Stochastic Models in Biology and Medicine. The University
of Wisconsin Press, Madison.
Lucas, H. L., Jr. and W. W. G. Smart, Jr. 1959. Chemical composition
and digestibility of forages. Proceedings of the Sixteenth
Southern Pasture and Forage Crop Improvement Conference, State
College, Mississippi.
Maynard, L. A. and J. K. LoasH. 1962. Animal Nutrition. Fifth
Edition. McGraw-Hill Book Company,. Inc., New York, N. Y.
McMeekan, C. P. 1940a. GroW"th and development in the pig, with
special reference to carcass quality characters. I. Journal of
Agricultural Science 30;276-343.
107
McMeekan» C. P. 1940b. Growth and developmer.t in the pig., with
special reference bJ carcass quality characters. II. The
influence of the plane. of nutrition on growth and developmen t.
Journal of Agricultural Science 30~387=436.
McMeekan, C. P. 1940c. Growth and development in the pig, with
special reference to carcass quality characters. III. The
effect of the plane of nutrition on the form and composition of
the bacon pig. Journal of Agricultu.ral Science 30~511=569.
Overman, O. R. and W. 1. Gaines. 1933. Mi1k~energy formulas for
various breeds of cattle. Journal of Agricultural Research
46~ 1109-11200
Reid, J. T., G. Ho Wellington and Ho 00 Dunno 1955. Some relationships among the major chemical components of the bovine body and
their application to nutritional investigationso Journal of
Dairy Science 38:1344-1359.
Schneider, B. Ho 19350 The subdivision of the metabolic nitrogen in
the feces of the rat, swine and man. Journal of Biological
Chemistry 109: 249~278.
Schoenheimer~
R. 1.9420 The Dynamic State of Body Constituents.
Harvard University Press J Cambridge. Massachusetts.
Waldo J D. R. 1968. Nitrogen metabolism in the ruminant.
Dairy Science 51:265-275.
Journal of
108
9.
9.1
APPENDICES
Flow Laws for Milk Production
Milk production is represented in Figure 4.2 by the paths from
compartments P
sn
and P
to M.
se
The flow laws for milk production are
derived somewhat differently than the other flow laws derived in
Section 4.4.
The quantity of interest is not milk production, M, but
the milk production rate~ denoted
Pw
and having units of kca1/time.
The notation PM refers to the total milk production arising as a
result of the flow of materials from both the nitrogen and energy
pools, P
sn
and P
se
We postulate that the rate of change of the milk
production rate is proportional to two factors:
(a) the rate of change of the pool concentrations, and
(b) the difference between the "genetic maximum" milk
production rate for the cow and the current rate.
Since nitrogen and energy are needed jointly for milk production,
we consider the product of the two pool concentrations as the driving
force, and the rate of change of the pool concentrations is thus given
by
where d denotes derivative.
Brody (1945, p. 703) says the following abou.t the life curve of
lactation:
The rise in milk production up to seven or eight years
in dairy cattle parallels in shape J although it lags
in time, the rise in body weight..
109
In addition to the life curve of lactation, we must consider the
lactation-period curve of lactation.
maximum and then declines.
This curve rapidly rises to a
According to Brody (1945, p. 703):
• • • it appears that the decline in milk production
following the attainment of the maximum yield at the
prime of life or the prime of the lactation period is
exponen tia 1 .
We must postulate a genetic maximum production curve which takes these
features into account.
For each lactation period, we can define a maximum milk production
curve, per unit of body size, asa difference of two exponentials
(Brody, 1945),
Then, for the first seven or eight lactations, multiplying this by
body size will give the maximum milk production curve,
~B.
Then the
difference between the genetic maximum milk production rate and the
current rate is given by (\iB - PM).
be
Kp ,M and
Letting the proportionality factor
representing the rate of change of milk production rate by
d(PM)' our differential equation is:
Exact solution of this equation requires an appropriate initial condition.
P
se
of
We postulate that a certain minimum concentration of P
is necessary to "drive" the milk production.
~
also affects this minimum value.
sn
and
However, the value
A large value of
~, ~.~.,
a
large potential to produce, requires a lower concentration to "drive"
the production.
If
~
is an appropriate constant, then we define the
initial condition as follows:
the rate of milk production, PM equals 0
110
2
when (P sn Pse/B ) ::: ~/A,
or:
'M
(9.2)
Solving the differential equation (9.1), subject to the initial
condition (9.2), yields the differential equation for milk production
rate,
We consider milk to be a fairly
we11~defined
mixture of nitrog-
enous and energetic material and denote the nitrogenous material,
casein, by Mn and the energetic material,
e.~.,
~~
lactose, by M.
e
!.~.,
Then
the total amount of milk produced, M, is related to these quantities
by:
(9.4)
M ::: ~
+ Me
and the ratio of energetic to nitrogenous material,
(9.5)
0.. :::
'M.
~.
is defined by:
M /M •
e n
From (9.5) we have
(9.6)
M
e = (PM)(M)
n
and substituting into (9.4) gives
(9. 7)
or
(9.8)
•
111
Substituting (9.8) into (9.6) gives
(9.9)
Thus, given the amount of milk, M, we use (9.8) and (9.9) to
."
partition this into its nitrogenous and energetic components.
We
assume that the flow of material to form milk, given by (9.3), is
partitioned similarly.
Denoting the flows of nitrogenous and energetic
material for milk production as p
n,
M and p
e,M'
respectively, the flow
laws are:
(9. lOa)
(9. lOb)
For a given value of
~,
the graph of the milk production rate,
PM' as defined by equation (9.3) is shown in Figure 9.1.
~B
---------------
p
~
~
Figure 9.1
p
sn se
B
Graph of milk production rate
2
112
9.2
Flow Laws for Heat Loss
Heat in the heat pool, Ph' which cannot be used by the animal to
keep warm, is dissipated as shown by the flow from Ph to H, the
compartment for heat loss.
Four factors in heat loss are radiation, conduction, convection
and water evaporation.
area.
Consider the heat loss per unit of surface
According to Kleiber (1961), radiation is proportional to
(Ti - T~) where Tl ,T
2
are the temperatures of the sender and receiver
of the radiation in degrees Kelvin.
temperature, (T
For small differences in
4 - T4) may be approximated by (T
l
2
l
- T ) and
2
radia tion ex 6T
where 6T
= Tskin
- T
, the difference between skin temperature of
air
the animal and air temperature, in of or °C.
Heat loss due to con-
duction is proportional to 6T, or
conduction ex 6T
Heat loss by convection is proportional to air velocity (v) and
temperature difference, or
convection ex (v) (6T)
For a given experimental situation, we assume that the air velocity
is constant and write
convection ex 6T .
Heat loss by water evaporation is proportional to air velocity and
the difference in vapor pressure at the surface and in the air, t:NP,
or,
113
evaporation a (a + bv) (6VP)
where a and b are suitable constants.
Again assuming that air velocity
is constant, we have
evaporation a 6VP •
It is reasonable to assume that at the evaporating surface (lungs,
skin), the vapor pressure is equivalent to 100 per cent relative
humidity.
Denoting this as VP
and the vapor pressure of air as
lOO
VP • we write
a'
evaporation a (VP
lOO
- VP a ) •
Now, surface area is proportional to body size to the two-thirds
power, and kcal of body (B) is proportional to body size; hence,
Z 3
surface area is proportional to B / •
Combining the four factors above and multiplying by surface area
leads to the equation for heat lossJ
(9.11)
where Al is an emission factor for the animal and A
Z
is a vaporization
factor and each are functions of the body temperature.
The product
Z
A B / 3 may be considered an "effective" surface area for heat transfer
l
Z 3
and A B /
Z
as an "effective" surface area for evaporation.
In order
to prevent heat from flowing into the animal, we will not allow (6T)
or (VP
lOO
=
VP ) to be negative.
a
Thus, the notation in equation (9.11)
might be more correctly written as
(6T)
=:
max (0 , 6T)
114
However, for simplicity, it will be left as shown.
Al and A must be
Z
defined to satisfy the following conditions:
(a) when the animal is cold .. Al and A
Z
are minimized,
(b) when the body temperature starts to rise above some optimal
value J a small increase in water evaporation occurs, but conduction,
radiation and convection will handle most of the dissipation of the
heat, and
(c) further above the optimal temperature, conduction, radiation
and convection reach their maximum levels and water evaporation
increases to dissipate the extra heat.
In order to satisfy the above J we first define body temperature
as T •
B
Body temperature will be proportional to body heat divided by
body mass.
We take Ph' the kcal of heat in the heat pool, as our
measure of body heat.
B is proportional to body mass, hence,
We define the optimal body temperature for the animal as T.
o
In
order to simplify the initial formulations, we also assume that the
body, skin and rectal temperature of the animal are equal and use T
B
to denote them.
No attempt will be made to derive formulae for Al and
AZJ but a graph (Figure 9.Z) will illustrate possible forms which
satisfy points (a) - (c).
The dissipation of heat requires energy to drive the reaction.
This energy is supplied by P
P
se
to BH and then to Ph.
se
and results in a transfer of heat from
ObviouslYJ the amount of heat thus trans-
ferred must be less than the amount of heat dissipated, else the
animal will not be able to keep cool.
This is what happens at very
115
J-------+-------------"~TB
o
Figure 902
Possible forms of A ,A versus body temperature
l 2
high environmental temperatures when the body is unable to dissipate
its heat without working very hard (producing much heat) and thus it
produces more than it loseso
Kleiber (1961, p. 162)
states~
Under these latter circumstances) the body temperature
rises, as also does the metabolic rate) because the
cellular processes are now uncontrolled and operate
according to Van't Hoff's law. If this positive feed~
back continues J it becomes a fatal vicious cycle.
We will not include a fatal point i.n the model1 but wi.11 note that if
the heat produced in dissipating heat exceeds the heat dissipated, the
animal is in trouble.
(9. 12)
pY
se,Ph
::0;
This energy transfer may be formulated as
K'
Pse,Ph
where K;seyPb is a 'function of body temperature, T •
B
This function,
in shape, will resemble the sum of Al and A of equation (9.11),
Z
except that it will not level out at higher temperatures as Al and A
2
116
do.
In magni.tude: it will be quite a bit less than Ph H except at
J
high body temperatures.
A graphical representation of a possible form
of K:~se,Ph is shown in Figure 9.3.
K'
Ps€;) Ph
T
Figure 9.3
o
Possible form of Ki
versus body temperature
Pse,Ph
When body
temperature falls below T0 J the animal becomes cold.
.
To warm u.p it increases its metabolic rate? or generates more heat.
The animal oxidizes energy from P
se
to Ph to raise the body temperature.
(9.13)
_
to provide heat to BH and thence
We formulate this as
l("
Pse, Ph
where K~se:,Ph is a function of body temperature which is high at low
temperatures and minimal above T.
o
Figure 9.4.
A possible form is shown in
117
Ie"
Pse,Ph
TB
I------~T---------------"~
o
Figure 9.4
Possible form of Ie"
versus body temperature
Pse,Ph
In Table 4.2, the differential equations for the comparbment model
were given.
p
The differential equation for the heat pool (4.29) was
h
= hp - P
h,H
where hp represented the rate of heat production from both gut fermentation and body processes, and Ph H represents dissipation of heat or
.,
heat loss.
We must now augment hp by the heat produced in dissipating
heat and in warming up.
Thus, we may
write~
(9.14)
where pI Ph and p" Ph have been defined above and are functions of
se,
se,
body temperature.
(9.11).
The flow law for Ph H has been given as equation
,
118
9.3
9.3.1
Evaluation of Constants
Evaluation of P
B
Combining data from several sources (Crampton and Lloyd, 1959;
Maynard and Loosli, 1962; Blaxter and Rook, 1953) yields the following
table:
Table 9.1
Simplified composition of energy-containing substances in
dry skeletal muscle
Component
Fraction
kcal/gm
of Component
.8
5.322
4.2576
.2
9.367
1. 8734
. a
P rote~n
Fat
b
kcal/gm
c
of Dry Tissue
aMixed, deposited material containing, on the average, 16 per cent
nitrogen.
b
.
Conta~ns
no
.
n~trogen.
CEqual to fraction times kcal/gm of component.
Carbohydrate is omitted since it comprises less than 1 per cent of
the total body.
Now, one gram of body, B, consists of 4.2576 kcal B
and 1.8734 kcal B.
e
(9.15)
n
Therefore,
PB
=e
B /B
= 1.8734/402576
n
.44 •
Some useful relationships derived from (9.15) are
and
119
or
(9.16)
9.3.2
Evaluation of f , f
B S
Given the body weight of a steer, we want to determine those
fractions of the total which can be taken as depot fat and as body.
We then will convert these fractions to kca1.
Let w designate body weight in grams.
B
We assume an average
fatness of 14.2 per cent (Reid et a1., 1955).
fat (grams)
= .142
Thus,
w
B
and by subtraction, the remainder is fat-free body, therefore
fat-free body (grams)
= .858 wB .
The fat is partitioned between stores, S, and body, B , with a heat of
e
combustion of 9.367 kca1/gm.
(9.17)
S + Be
=
Thus,
(9.367) (.142) (w B)
The average protein content of the
= 1.33
fat~free
wB .
body is 21.64 per cent
(Reid et a1., 1955), hence
protein (grams)
=
(.2164) (fat-free body)
=
(.2164) (.858) (w )
B
and using a heat of combustion value of 5.322 kca1/gm for protein, the
value of B is
n
B
n
From (9.15),
=
(5.322)(.2164)(.858) (w )
B
.988 w •
B
120
B = P oB = (.44)(.988) (w ) = .435 ow
B
e
B n
B
and since
then,
B
= .988 owB +
.435 ow
B
= 1.423 owB
•
Therefore,
(9. 18)
f
B
= 1.423
•
Also, from (9.17),
and therefore,
(9.19)
9.3.3
fS
= .895
•
Evaluation of wBn'
W
s
Given the energy content of the
want to calculate the body weight.
body,~
and of the stores, S, we
We use B to determine the weight
n
of the fat-free body and Band S to determine it for the fat.
e
kca1 of S or B corresponds to
e
1/9.367
grams of fat.
(9.20)
= .107
Hence
wS =·107.
One kca1 of B corresponds to
n
1/5.322
= .1879
One
121
grams of protein.
.
Assuming that protein comprises 21.64 per cent of the
fat-free body, the .1879 grams of protein imply
.1879/.2164
= .8683
grams of body weight.
(9.21)
w
Bn
Hence
= .868
and
= .868·Bn
9.3.4
+ .107· (S + B)
•
e
Evaluation of r s
The flow law for s
Pse
(9.22)
f(x) = x/ (x +
where x
SiB and
Ol
contains a term of the form
Ol)
= rs•
The graph of this function is a hyperbola
(Figure 9.5) with asymptotes at x
condition, we can evaluate
Ol.
= -Ol and x =
For a given initial
00.
Suppose that the condition is
Then
f
o
= x / (x
and solving for
Thus to evaluate
+
0
0
Ol
gives
Ol
Ol)
in (9.22), we must specify x
=
(S/B)
and f .
0 0 0
It
was decided that when the amount of fat in the stores, S, equalled the
amount of fat in the body, B , of an average animal (i.e., one with
e
14.2 per cent fat in its body), then f(x) = 0.9.
122
f(x)
1
f
------------
o
----...----..,..----.. . .
---------------3~x
Figure 9.5
Graph of f(x) = x/(x +
01)
From Section 9.3.2 above, one gram of body weight contains 1.423
kca1 (B) of which .988 kca1 is in Band .435 kca1 is in B.
n
Be /B
= .435/1.423 = .3057
e
•
Thus, the initial condition is that when
SiB = .3057 ,
f
o
=.9.
Then,
(9.23)
rs
= 01 = (.3057) (1
- .9)/.9
= .3057/9 = .034
.
Thus,
123
9.4
Evaluation of Initial Conditions for the Pools
Dukes (1955) gives normal ranges of several chemical constituents
•
of the blood of mature domestic animals.
are (in mg per 100 ml whole blood)
For the cow, pertinent values
~
Amino acid nitrogen
Total non-protein nitrogen
Urea nitrogen
Sugar
Lactic Acid
4
20
6
40
- 8
- 40
- 27
- 70
5 - 20
Dukes (1943) gives a value of .0567 gms fat per 100 gms blood.
Taking the specific gravity of whole blood as 1.052 for cattle (Dukes,
1955), this is equivalent to
(.0567) (1.052) = .0596
grams of fat per 100 ml whole blood.
For P , we consider amino acid nitrogen.
sn
Taking the midpoint of
the range as an average figure, we have 6 mg amino acid nitrogen per
100 ml whole blood.
Assuming amino acids to contain 16 per cent
nitrogen, we have
6/.16
=
37.5
mg amino acids or .0375 gm amino acids per 100 ml whole blood.
For P , consider the average of the total NPN values to be 30 mg
dn
NPN per 100 mI.
Subtracting the amino acid nitrogen value of 6 mg per
100 ml leaves 24 mg of nitrogen per 100 ml in P
mately 16.5 mg per 100 ml is urea nitrogen.
nitrogen to mg of P
present in urea.
nitrogen.
dn
dn
•
Of this, approxi-
Thus to convert the mg of
material, we assume that all the nitrogen is
Urea consists of approximately 45.75 per cent
Thus we have
124
24/.4575- 52.46
mg of Pdn or .0525
gms
The energy pooL,
Pdn per 100 ml whole blood.
P
se-• will contain substances like sugar, volatile
fatty acids, lactic acid and fat.
Taking average figures for sugar and
lactic acid as was done above.? and using the sugar value to estimate
the fatty acid content gives
55 + 55 + 12.5 - 122.5
mg or .122 gms non-fat P
se
per 100 ml whole blood.
From above, we
have .0596 gms fat pe.r 100 ml whole blood.
Given the above va1ues J we must now convert to kcal of pool
materials per kcal of B to give initial pool concentrations.
conversion factors will be used.
Several
Albritton (1952) gives a value of
85 gms water per 100 ml whole blood for cattle.
Reid et al. (1955)
provide average concentrations in the fat-free body of water as .7291
and protein as .2164.
per .2164 gms protein.
Hence, we consider the ratio. 7291 gms water
The heat of combustion of protein is taken as
5.322 kcal/gm (Blaxter and Rook, 1953) and the factor 1. 44 kcal B
per kcal B was derived in Section 9.3.
n
If k denotes the concentration of materials in blood (gms/lOO ml
whole blood) and h denotes the heat of combustion of these materials
(kcal/gm), then the conversion formula is~
kca1 material _ ( k gros material) (100 ml whole blood) ( 07291 gIn water)
kcal B
100 ml whole blood'
85 gm water
.2164 gm protein
1 kcal B
(1 gm protein
n )(h kcal material
5.322 kca1 B ) (1.44 kca1 B
1 gm material) •
n
125
Multi.plying the terms out and cancelling units give the concentrations
as
•
materi.al
(0051'72) (k') (h)
-kcal kcal
B
-0
. .
.
'.
-
0
If we consider Pdn as mostly urea J then we may take h "" 2.450
in the blood, we. will take h ,= 9000
we
For fat
Summarizing the above (Table 9.2),
have~
Table 9.2
Summ.ary of pool concentration data
k (gros/IOO ml blood)
Pool
h (kcal/gm)
Concentration
00375
5.322
.00103
P
dn
.0525
2.45
.000665
fat
P
se other
00596
9.0
0122
3.75
.00277 }
.00514
.0023'7
P
sn
Pde
.00331
No infonnation was available for P
satisfy the following
P
de
de
., so it was evaluated to
ratio~
concentrat.ion
P
dn
concentration
-co
P
concentration
se
905
P
concentration
sn
Evaluation of Cd
,0
For each steer J data were given on heat producti.on, dry matter
fed and fat energy balance per dayo
We wish to derive a value of dry
matter fed which corresponds to zero fat ene.rgy balance.
given in Table 9.30
The data are
126
Table 9.3
Values of heat production, dry matter fed and fat energy
balance
a
D
Steer 36
a
DM
EB
0.5
8,155.8
1,885
-3,082.8
1.0
9,839.7
3,762
1.0 (hay)
11,635.0
1.5
2.0
Feeding
Level
Feeding
Level
Steer 47
a
DM
EB
7,754.5
1,863
-2,625.8
-15.2
9,382.8
3,790
688.8
5,763
-685.0
11,254.6
5,771
210.9
11,854.1
5,353
2,377.9
11,692.9
5,617
3,288.0
13,888.1
7,037
4,431.8
13,536.3
7,839
5,598.4
D
Steer 57
DM
EB
D
D
Steer 60
DM
EB
0.5
7,939.1
1,700
-3,098.0
7,476.2
1,681
-2,720.8
1.0
7,908.7
3,085
530.1
7,252.9
2,828
472.2
1.0 (hay)
9,953.7
5,155
474.1
9,790.1
5,013
89.3
1.5
9,493.3
4,612
2,761. 7
8,821.4
4,237
2,240.1
2.0
11,851. 2
6,233
3,987.2
11,156.9
5,704
3,371.6
2.5
14,408.2
8,057
6,071.5
13,976.4
7,520
4,967.6
16,133.1
9,489
7,171.6
b
3.0
EB
aD = heat production (kca1/day), DM = dry matter fed (gms/day),
energy balance (kca1/day).
= fat
bNo observations for Steer 57 at this feeding level.
127
The method used was to estimate that value of heat production
correspondi.ng to zero fat energy balance..
•
•
Linear regression was used
with heat production as the dependent variable and energy balance as
the i.ndependent variable.
The values for the hay feedings were omitted
from this regression for all steers, and for Steers 57 and 60, the 0.5
feeding level data were also omitted.
The heat production values thus
derived are given in Table 9.4.
Table 9.4
Heat production values for zero fat energy balance
Steer
Heat Production (kcal/day)
36
47
57
10,228.3
9,351. 8
6,892.5
60
6,348.4
Then, a regression line was fit, for each steer, relating heat
production (dependent variable) to dry matter fed (independent
variable).
Using the regression line, the value of dry matter fed
which corresponded to the heat production value of Table 9.4 was
calculated.
If we represent the regression line by
y = a + bx
where y = heat production and x ._. dry matter fed, and if Yo is the
value in Table 9.4, then
c
d,o
= x0 =
(y
The values of a, band c
0
~ a)/b
dJo
thus derived are given in Table 9.5.
128
Table 9.5
Regression constants and values of cd
Steer
a
b
36
5,937.9
5,845.6
3,478.2
3,136.6
1.07900
.98064
1. 33462
1. 38861
47
57
60
;0
c
d,o
•
3,976
3,575
2,558
2,313
Evaluation of K B
P2
9.6
The flow law for body growth, equation (5.12a) is
Brody (1945) considers body growth to follow an exponential law.
This
holds fairly well after birth, but starting at conception; the logistic
law seems applicable.
The term in the second bracket of the above
equation is a logistic type term.
If we denote the term in the first
bracket by K' and use w (grams) instead of B (kca1), we may represent
B
body growth by
Note that we are temporarily ignoring body breakdown, b
BW
and bp •
Dividing both side.s of this equation by w leads to the following
B
equivalent form,
d (In w ) / d t = (K') (A
B
w
- w )
B
where K
1
=:
(K I) (A ) and K
2
w
=:
-
B
K' •
B
Constants K1 and K are estimated from the data in Brody (1945)
2
on Holstein growth, pages 571-572.
The derivative is estimated by
129
differences, hence,
•
or
where t , t are values of time at which observations are made.
1
2
From
data ranging over values of w from 660 to 1760 kg, a rough estimate
B
of
K'
is
K'
=
,0001446 (kg=months)
-1
,
Converting units leads to
K: ' -
,48182 x 10 =8 ( gram= days) - 1
and converting grams B to kca1 (1 gram = 1. 423 kca1) gives
K'
=
,68563 x 10=8 (kcal=days)=l ,
Now
P
(K
P,B
) (
P
sn) (~)
B
B
Substituting initial values of pool concentrations gives
K
- .001295 (kca1=days)=1 ,
P,B
In order to compensate for body breakdown, h
ignored, we mus t increase
'1>
BW
and b
B to sus taiD the growtho
p
which were
If we arbi=
"
trari1y increase the above value by one=fourth, the initial estimate
will be taken as
Kp B
J
.~.
,001619 .
l~
9.7
Evaluation of Weights for the Goodness of Fit Criterion
The evaluation of weights follows the procedure outlined in
Section 5.3 and uses the data in Table 5.9.
If for any factor, we let x
36
' x
47
' x
57
and x
60
denote the value
given in Table 5.9 for the steer indicated in the subscript, then the
variance between Steers 36 and 47 is given by
and between Steers 57 and 60 by
The pooled variance is then
or
s
and s
p
2
p
2
is the square root of s .
P
For feeding level 2.5, there are no
observations on Steers 36 and 47, hence
The values of s
p
When regressing s
are given in Table 9.6.
p
on feeding level for each of the factors, we
must recognize two points.
First, for feeding levels 0.5 through 2.0,
the pooled variances are based on two sets of two values each, and
hence, have two degrees of freedom.
For feeding level 2.5, we have
one set of two values and hence, one degree of freedom.
Thus, we must
fit a weighted regression line with weights equal to the degrees of
131
Values of s
Table 9.6
•
I
Feeding
Level
a
Urine N
0.5
1.0
1.0 (hay)
1.5
2.0
2.5
.4242
3.6718
3.0602
3.0372
5.6135
9.0509
aN
freedom.
= nitrogen,
E
p
:;::
Fecal N
a
Urine E
Fecal E
Methane E
.5700
.3535
.5408
.5852
.9823
1.2727
12.4355
20.3317
21. 3765
36.9233
22.9829
9.9277
38.3715
98.3342
34.2563
116.9572
291. 4128
340.2880
3.9016
106.6569
65.0478
150.6974
186.1025
291. 5401
energy.
The second point is that for the hay diet, the amount of
nitrogen fed is close to that of feeding level 1.5 for the mixed diets.
Hence, for Urine N and Fecal N, the hay data will be treated as feeding
level 1.5 in determining the regression line and in predicting the
value of s * to be used for the weight.
p
I f we d enote t h e ~· th f ee d'~ng 1 eve 1 b y x.,
~
freedom) by w.~ and the value of s p by
y~j
~
.
. h t (d egrees
~ts we~g
0
f
then the regression line is
y:;::a+bx
and by least squares theory,
- (y.)] / [Dl. (x. - x)
- 2]
b :;:: [Dl. (x. - x)
.~~
~
a
=y
-
~
bi = (Dl.y.
.
~
~
~
.~~
~
bDl.x.)
. ~ ~ /Dl.
. ~
~
~
Using the values of x., a and b, we derive the predicted values,
~
s * which are used as weights for the goodness of fit criterion.
p'
values of w., Xi' a, band s * are summarized in Table 9. 7.
~
p
observed (Table 9.6) and predicted (Table 9.7) values of s
The
The
p
and the
132
regression line for each factor are given in Figure 9.6.
no observed value of s
p
There was
for feeding level 3.0 as there was only one
observation per factor, that for Steer 60.
Using the regression line,
however, a predicted value is calculated.
Table 9.7
Feeding
Level
\
Values of wi' xi' a, b, and s *p for each factor
w.
~
s*
p
x.
~
Urine N
.4780
Fecal N
0.5
2
0 . .5
1.0
2
1.0
2.248
.5174
1.0 (hay)
2
1.5
4.019
.7001
1.5
2
1.5
4.019
.7001
2.0
2
2,0
5.789
.8828
2.5
1
2.5
7.5.59
1.066
9.330
1. 248
3.0
Feeding
Level
•
w.
~
.3347
a'
-1. 2924
.1520
b:
3.5407
.36541
s*
p
X.
~
Urine E
Fecal E
Methane E
0.5
2
0.5
19.63
1.0
2
1.0
20.86
79.70
76.61
1. 0 (hay)
2
1.0
20.86
79.70
76.61
1.5
2
1.5
22.09
168.6
145.4
2.0
2
2.0
23.32
257.5
214.1
2.5
1
2. .5
24.54
346.4
282.8
25. 77
435.2
351. 6
a:
18.399
- 98.065
- 61. 5684
b·
2.458
177.77
137.49
3.0
8.597
7.864
,.
~
e
••
I:
p
~
~
~
I
1. 6~
-
Fecal N
•
I
4
,
0
Urine E
40t20
S
o
t
300f
•
•
!
,
I
I
I
4001-
s
./
100L
v'
0.5
I
Fecal E
300
Methane E
p
01
0
p
0
200f
s
s
•
•
I
,
•
I
I
P
e
Urine N
]
s
..
..
e
-1
200
P
100
I
,
,
1.0 1.5
2.0
Feeding Level
2.5
I
3.0
~
I
/'
•
0
0
0.5
1.0 1.5
2.0
Feeding Level
2.5
3.0
t-'
Figure 9.6
Graph of s
p.
versus feeding level and fitted regression lines
I.J-l
I.J-l