Comparative evaluation of mathematical functions to describe growth and efficiency
of phosphorus utilization in growing pigs
E. Kebreab, M. Schulin-Zeuthen, S. Lopez, J. Soler, R. S. Dias, C. F. M. de Lange and J.
France
J Anim Sci 2007.85:2498-2507.
doi: 10.2527/jas.2006-738 originally published online Jun 12, 2007;
The online version of this article, along with updated information and services, is located on
the World Wide Web at:
http://jas.fass.org/cgi/content/full/85/10/2498
www.asas.org
Downloaded from jas.fass.org at Serials/Acq. Dept., Library on August 13, 2009.
Comparative evaluation of mathematical functions to describe growth
and efficiency of phosphorus utilization in growing pigs1
E. Kebreab,*†2 M. Schulin-Zeuthen,* S. Lopez,‡ J. Soler,§ R. S. Dias,*
C. F. M. de Lange,* and J. France*
*Centre for Nutrition Modelling, Department of Animal and Poultry Science, University of Guelph, Guelph,
N1G 2W1, Canada; †National Centre for Livestock and Environment, Department of Animal Science,
University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada; ‡Departmento Produccion Animal,
Universidad de León, 24071 Leon, Spain; and §IRTA, Centre de Control Porcı́, 17121 Monells, Girona, Spain
ABSTRACT: Success of pig production depends on
maximizing return over feed costs and addressing potential nutrient pollution to the environment. Mathematical modeling has been used to describe many important aspects of inputs and outputs of pork production. This study was undertaken to compare 4
mathematical functions for the best fit in terms of describing specific data sets on pig growth and, in a separate experiment, to compare these 4 functions for describing of P utilization for growth. Two data sets with
growth data were used to conduct growth analysis and
another data set was used for P efficiency analysis. All
data sets were constructed from independent trials that
measured BW, age, and intake. Four growth functions
representing diminishing returns (monomolecular),
sigmoidal with a fixed point of inflection (Gompertz),
and sigmoidal with a variable point of inflection (Richards and von Bertalanffy) were used. Meta-analysis of
the data was conducted to identify the most appropriate
functions for growth and P utilization. Based on Bayes-
ian information criteria, the Richards equation described the BW vs. age data best. The additional parameter of the Richards equation was necessary because
the data required a lower point of inflection (138 d)
than the Gompertz, with a fixed point of inflexion at
1/e times the final BW (189 d), could accommodate.
Lack of flexibility in the Gompertz equation was a limitation to accurate prediction. The monomolecular equation was best at determining efficiencies of P utilization
for BW gain compared with the sigmoidal functions.
The parameter estimate for the rate constant in all
functions decreased as available P intake increased.
Average efficiencies during different stages of growth
were calculated and offer insight into targeting stages
where high feed (nutrient) input is required and when
adjustments are needed to accommodate the loss of
efficiency and the reduction of potential pollution problems. It is recommended that the Richards and monomolecular equations be included in future growth and
nutrient efficiency analyses.
Key words: function, growth, phosphorus, pig
©2007 American Society of Animal Science. All rights reserved.
INTRODUCTION
Traditionally, growth functions have been used to
describe increase in BW over time. One of the outputs
of generating growth curves is to have the ability to
predict accurately the time at which pigs will be ready
for market. A useful growth function should describe
data well and contain biologically and physically mean-
1
This research was undertaken, in part, thanks to funding from
the Canada Research Chairs Program.
2
Corresponding author: [email protected]
Received November 7, 2006.
Accepted June 6, 2007.
J. Anim. Sci. 2007. 85:2498–2507
doi:10.2527/jas.2006-738
ingful parameters (France et al., 1996). Protein accretion (Ferguson et al., 1994), BW (Bridges et al., 1992),
and mineral deposition (Mahan and Shields, 1998) are
among attributes to be quantified using growth functions. In swine nutrition, Schinckel and de Lange (1996)
identified protein accretion, partitioning of energy,
maintenance requirement, and feed intake as primary
requirements in developing growth models. Growth
functions can be grouped into 3 categories: those that
represent diminishing returns behavior (e.g., monomolecular), sigmoidal behavior with a fixed point of inflection (e.g., Gompertz, logistic), and those encompassing
sigmoidal behavior with a flexible point of inflection
(e.g., Richards, von Bertalanffy). The flexible functions
are often generalized models that encompass simpler
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Growth and phosphorus intake analysis in pigs
Table 1. Data sources used in the study
Data
set
Source
1
Schulin-Zeuthen, (2000);
Schulin-Zeuthen and Danfær (2001)
2
Jondreville et al. (2004);
Tibau et al. (2002)
3
Swidersky et al. (2003)
Growth
period, d
Genotype
Diet,1 % of as-fed
911
Birth to 1,007
(Landrace × Yorkshire)
× Duroc
209
68 to 210
1,330
70 to 155
(Large White × Landrace)
× Pietrain or
(Large White × Landrace)
× Large White
Yorkshire
DM 89.9, dCP 14.3, NE 9.26,
P 0.586, aP 0.378
DM 89.6, dCP 13.1, NE 9.11,
P 0.551, aP 0.317
DM 88.9, dCP 11.9, NE 8.96,
P 0.517, aP 0.256
DM 88.2, dCP 12.7, NE 8.49,
P 0.507, aP 0.241
DM 89.0, dCP 11.2, NE 8.18,
P 0.497, aP 0.244
DM 88.8, dCP 10.7, NE 7.87,
P 0.458, aP 0.224
DM 88.4, dCP 9.56, NE 7.87,
P 0.450, aP 0.224
DM 89.6, CP 20.0, NE 9.55,
Ca 1.07, P 0.845, aP 0.406
DM 87.3, CP 18.5, NE 9.21,
Ca 1.02, P 0.817, aP 0.378
DM 88.3, dCP 13.2, Ca 0.348,
P 0.377, aP 0.122
DM 87.2, dCP 13.5, Ca 0.521,
P 0.436, aP 0.183
DM 87.1, dCP 13.2, Ca 0.685,
P 0.493, aP 0.240
DM 88.9, dCP 13.1, Ca 0.858,
P 0.553, aP 0.301
Number
of pigs
1
CP = 6.25 × N (NRC, 1998), dCP = digestible protein, NE (MJ/kg) = 7.72 Scandinavian Feed Units (Andersen and Just, 1983), NE = 0.75
digestible energy (Andersen and Just, 1983), and aP = available phosphorus.
models for particular values of certain additional parameters. Although the Gompertz has been applied
widely, other functions have been used to describe
growth in nonruminant animals (e.g., Darmani Kuhi
et al., 2003b).
Another application of growth curves is to relate BW
to cumulative nutrient intake. It is useful to examine
intake in order to assess genetic improvements (Bermejo et al., 2003) or to increase return over feed costs.
Phosphorus is an essential nutrient that has received
attention for environmental reasons such as limited
supply and pollution of ponds and streams (SchulinZeuthen et al., 2005). Therefore, optimizing of P intake
and understanding factors affecting P utilization in pigs
have environmental and economic benefits. Growth
functions can be used to determine efficiency of nutrient
utilization (Darmani Kuhi et al., 2003a), which is the
derivative of the relationship between BW and cumulative nutrient intake.
The aim of this study was to evaluate 4 candidate
mathematical functions for best fit in describing specific
data sets on pig growth, and to evaluate these 4 functions for describing of P utilization for growth in a separate experiment.
MATERIALS AND METHODS
Animal Care and Use Committee approval was not
obtained for this study because the data were obtained
from existing databases (Table 1).
Data Sources
A total of 2,450 observations from 102 pigs originating from 3 independent data sets (Table 1) were used
in this study. All 3 data sets had data on individual
pigs and were derived from experiments conducted in
Denmark, Spain, and Canada. Separate analyses were
undertaken to relate BW to time and BW to cumulative
P intake. Data sets 1 and 2 were used for growth analysis, and data set 3 for P efficiency analysis.
Growth Analysis. Two data sets with a total of 1,120
observations from 48 pigs were used to compare 4 equations to describe growth in pigs. Data set 1 consisted
of birth to maturity records from 5 castrated, 5 male,
and 5 female pigs. The pigs were crossbred ¹⁄₄ Danish
Landrace, ¹⁄₄ Yorkshire, and ¹⁄₂ Duroc. Individual BW
was recorded weekly and feed intake twice weekly. All
pigs received 7 diets (based on barley and soybean)
during their life span to maximize growth based on
Danish standards (Table 1). The pigs were raised between 1996 and 2001 (Schulin-Zeuthen, 2000; SchulinZeuthen and Danfær, 2001).
Data set 2 consisted of data collected from 33 Spanish
grower-finisher pigs (25 to 140 kg) of 2 sexes (females
and castrated males) with BW and feed intake recorded
every third week. The pigs were crossbred ¹⁄₄ Large
White, ¹⁄₄ Landrace, and ¹⁄₂ Large White or Pietrain.
They received a grower and a finisher diet, based on
corn (32.6 and 28.5%), soybean meal (24.5 and 21%),
barley (10 and 15%), and wheat (10 and 10%) for grower
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2500
Kebreab et al.
and finisher diets, respectively (Tibau et al., 2002; Jondreville et al., 2004; Table 1).
Phosphorus Efficiency Analysis. Data set 3 with
1,330 observations was used to test the fit of 4 functions
to describe efficiency of P utilization in growing pigs.
An experiment using 54 Yorkshire pigs (30 males and
24 females), with an initial BW of 25 kg conducted at
the University of Guelph (Swidersky et al., 2003), was
used to construct data set 3. The pigs were randomly
assigned to 1 of the 4 dietary P levels. The treatments
were low, medium, recommended, and high available
P intakes corresponding to 52, 78, 105 and 130%, respectively, of the NRC (1998) requirements for grower
and finisher pigs. Pigs had ad libitum access to feed and
feed intake was recorded automatically. For individual
pigs housed in groups, 2 diets were prepared for grower
and finisher phases, based on corn (73.1 and 80%) and
soybean meal (24.0 and 17.3%; Swidersky et al., 2003;
Table 1). The data set was selected because of the wide
range of P intakes. The 4 functions can also describe Ca
efficiency, as Ca was varied in tandem with P because of
their close association; however, the diets were formulated to be first-limiting in P.
Mathematical Models
Growth Analysis. The growth equations chosen in
the study represent diminishing returns behavior
(monomolecular), sigmoidal with a fixed point of inflection (Gompertz), and sigmoidal with a variable point of
inflection (von Bertalanffy and Richards) as described
in Chapter 5 of Thornley and France (2007).
The monomolecular equation, the simplest equation
used in this study, was used in the following form with
3 parameters:
W = Wf − (Wf − W0)e−kt,
where W is BW (kg) at age t (d), W0 is initial BW at
birth (t = 0), Wf is asymptotic BW reached as t → ∞, and
k is a rate constant (d−1). The monomolecular equation
describes the progress of a simple, irreversible firstorder reaction.
The Gompertz equation can be derived by assuming
substrate (feed) is nonlimiting, the quantity of growth
machinery is proportional to W, and effectiveness of
the growth machinery decays exponentially with time
according to a decay constant k (d−1). The Gompertz
equation was used in the following form:
W=
⎡⎛
W0 exp⎢ ⎜ln
⎣⎝
⎤
Wf ⎞
⎟(1 − e−kt)⎥.
W0 ⎠
⎦
Inflection in this sigmoidal growth function occurs at
W = Wf/e.
In the von Bertalanffy equation, the assumptions are
that the substrate is nonlimiting, and the growth process is the difference between anabolism and catabolism. The form of the von Bertalanffy used in this
study was
W = [Wfn − (Wfn − W0n)e−kt]1/n,
where n (dimensionless, with bounds 0 < n ≤ 1/3) and
k (d−1) are parameters. Inflection occurs at W = (1 −
n)1/nWf.
The Richards equation is an empirical construct and
therefore does not have the underlying biological basis
of the previous models. However, it belongs in the same
group of classic growth models, and its flexibility, due
to its shape parameter n (dimensionless), makes it a
generalized alternative to other equations (e.g., those
with fixed inflection points such as the Gompertz). The
Richards equation was used in the following form:
W=
(W0n
W0Wf
,
+ (Wfn − W0n)e−kt)1/n
where k (d−1) is a positive constant, and n is ≥ −1 to
ensure physiological growth rates as W → 0. Inflection
occurs, provided n is > −1, at W = Wf [1/(n + 1)]1/n.
Phosphorus Efficiency Analysis. Growth equations
can also be used to obtain parameters describing conversion of intake (e.g., feed, P, or other nutrient) to
animal products such as BW. Data set 3 contains data
with different levels of P and Ca intake but similar
intake of other nutrients, so the equations can be tested
for their ability to determine values of efficiency of P
utilization. To describe efficiency, BW needs to be expressed in terms of intake. In this analysis BW, W (kg)
was regressed against cumulative intake of available
P, P (g), and the models described above, with age replaced by cumulative intake, were evaluated for their
ability to determine the efficiency of P utilization.
The instantaneous efficiency of P utilization for gain,
kP (kg of BW/g of P), is given by differentiating the W
vs. P relationship:
kP =
dW
,
dP
and the average efficiency of utilization of P for gain
(kP) between 2 P intake levels (P1 and P2) is
P2
P2
∫
∫ kPdP
kP =
P1
P2 − P 1
=
P1
W2
dW
dP
dP
P2 − P 1
∫ dW
=
W1
P2 − P1
=
(W2 − W1)
.
(P2 − P1)
Statistical Procedures
Growth Analysis. The 4 models described were fitted to the 2 data sets combined using the nonlinear
mixed procedure (PROC NLMIXED, Littell et al., 1996;
SAS Inst. Inc., Cary, NC). Mixed model analysis was
chosen because the data were gathered from various
studies; therefore, it was necessary to consider analyzing not only fixed effects of the dependent variable but
also random effects (because the studies represent a
random sample of a larger population of studies). For
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Growth and phosphorus intake analysis in pigs
this analysis, a nonlinear mixed procedure based on
Craig and Schinckel (2001) and Schinckel and Craig
(2002) was employed. Forward selection was used to
find the optimum model with lowest value of Bayesian
information criterion (BIC; Leonard and Hsu, 2001).
Trial was coded as a random effect, and between-pig
variability was modeled by introducing a parameter (c)
to vary the mature BW, which was also a random effect
(Craig and Schinckel, 2001). For example, for the monomolecular equation, the model fitted was W = (c + ␮) ×
[Wf − (Wf − W0)e−kt], where ␮ is the random effect of
each pig. Introducing sex as a fixed source of variation
in the model did not improve the performance of the
models because most of the variability associated with
differences in sex had been explained by the betweenpig variability parameter.
Phosphorus Efficiency Analysis. Nonlinear mixed
model analysis was also used in determining efficiency
of P utilization. First, the equations were fitted to the
data, and then derivatives of the growth curves were
calculated. In fitting the equations, the fixed effects of
available P intake level and sex were considered. The
NLMIXED procedure of SAS does not support directly
the concept of an independent discrete (class) variable,
so the analysis was performed using dummy variables
to represent treatments, sex, or both. Each parameter
was considered individually at first, and then a combination of parameters was used. Model comparison using
an F-test showed that the rate parameter (k) was significantly affected by treatment (P < 0.01) and inclusion
of other parameters did not improve the overall model
performance. Therefore, the model with separate k values but a common W0, Wf, and n was fitted.
Distribution of random effects was assumed to be
normal and the dual quasi-Newton technique was used
for optimization with adaptive Gaussian quadrature as
the integration method. Performance of the models was
evaluated using the significance level of the parameters
estimated, the variance of error estimate, and its approximate SE. Comparison of models was based on BIC,
which are model-order selection criteria centered on
parsimony and imposing a penalty on more complicated
models for inclusion of additional parameters. Bayesian
information criterion combines the maximum likelihood (data fitting) and the choice of model by penalizing
the (log) maximum likelihood with a term related to
model complexity, as follows: BIC = −2 log(Ĵ) + Klog(N),
where Ĵ is the maximum likelihood, K is number of
free parameters in the model, and N is sample size. A
smaller numerical value of BIC indicates a better fit
when comparing models.
Both analyses were performed with the assumption
that variance distribution for the fixed factor was normal. Random effects were assumed to be normally distributed.
RESULTS
Growth Analysis
In general, all the equations fitted BW vs. age data
well below approximately 350 kg (Figure 1). The nonlin-
2501
ear analysis of the pooled data sets showed that W0
estimates were close to zero for all of the models compared. The estimate of W0 from the monomolecular
equation was greatest compared with the other equations. The von Bertalanffy equation estimated W0 to be
decreased compared with the other equations (Table 2).
Estimates of Wf were within a 0.07-kg range for the
sigmoidal equations. More importantly, the random effect multiplier based on BW, (c), was also within a narrow range. Due to the introduction of a random effect
of between-pig variation, the parameter estimate of mature pig BW (Wf) as described above is not absolute.
Therefore, to aid comparison between models, the estimated mature BW was calculated by multiplying Wf
and c (Table 2) with the monomolecular and Gompertz
equations showing greatest and the lowest estimates,
respectively. The rate constant parameter estimates (k)
ranged between 0.0028 and 0.0067, and were significantly different from zero for all models. Finally, the
parameter n estimates were −0.40 and 0.33 for Richards
and von Bertalanffy equations, respectively.
Comparison of models based on BIC suggested an
advantage in using a more complex model (Richards)
compared with simpler models such as the Gompertz
or monomolecular. The von Bertalanffy equation also
appeared to improve predictions (Figure 1) compared
with the monomolecular equation. However, the addition of an extra parameter in the von Bertalanffy equation did not result in significant improvement in goodness-of-fit with this model compared with the Gompertz
equation, as reflected by their similar BIC values.
Phosphorus Efficiency Analysis
Cumulative available dietary P intake was calculated
from feed intake measurements and corresponding
available P content of the diet. For a given BW, the
amount of available P consumed up to that particular
time was entered in the database.
Fitting the equations to plots of BW vs. cumulative
available dietary P intake (Figure 2) showed that the
parameter estimate for initial BW at zero P intake
ranged from 19.2 for the monomolecular equation to
28.7 kg for Richards equation (Table 3). Final BW estimates ranged from 131.9 (Richards) to 193.3 kg (monomolecular). All initial and final BW estimates for all 4
models were significant parameters in the model. Rate
constant estimates ranged from 0.0009 to 0.0082; n
parameter estimates were −0.87 and 0.33 for the Richards and von Bertalanffy equations, respectively.
The model with separate rate constant values for
each treatment category significantly improved on the
general model in all equations compared. However,
comparisons of the models using t-tests of parameter
values failed to reject the null hypothesis that BW gain
was not significantly different between sexes (P > 0.05).
Animals fed high levels of available P showed the lowest
estimates of rate constants, which increased as the lev-
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Kebreab et al.
Figure 1. Relationship between BW (kg) and age (d) in data sets 1 and 2. Symbols represent observed values and
the lines are regression lines fitted using the different functions.
els of available P decreased. In all models, the differences in the estimate of rate constants among dietary
P levels were significant (Table 3).
Comparison of models based on BIC did not show
any advantage in using a more complex equation than
the monomolecular (Table 3). The 3 sigmoidal equations
did not do as well as the monomolecular equation, but
their BIC values were within a narrow range. The Richards equation was marginally improved compared with
the other sigmoidal equations, and its parameter esti-
mates were close to those of the monomolecular equation, with a point of inflection at a very small BW over
the range of cumulative P intake. The Gompertz and
von Bertalanffy equations were the least suitable for
analyzing efficiency of P utilization.
Calculations of maximum efficiency of P utilization
showed wide variation between models. Maximum efficiency was at a considerably lower BW for Richards
(8.0 kg) than for von Bertalanffy (56.5 kg) and Gompertz
(57.9 kg) equations. The monomolecular equation as-
Table 2. Parameter estimates and SE obtained by nonlinear regression of BW vs. age (data sets 1 and 2)
Monomolecular
Item1
Estimate
W0, kg
0.05
19.5
Wf, kg
c
21.9
Wf × c, kg
427
k, d−1
0.0028
n
σ
226
BIC
8,872
Calculated growth trait
t*
NA
W*
NA
Gompertz
SE
P-value
0.009
0.45
0.501
0.01
<0.001
0.0001
<0.001
9.97
0.021
Estimate
0.026
1.01
378
382
0.0067
190
8,744
189
139
Richards
von Bertalanffy
SE
P-value
Estimate
SE
P-value
Estimate
SE
P-value
0.001
0.016
0.021
<0.001
<0.001
0.001
0.015
0.022
<0.01
<0.001
0.001
<0.001
<0.001
8.26
<0.001
0.0001
0.091
31.2
<0.01
<0.01
<0.001
0.008
1.08
360
389
0.0054
0.33
190
8,745
0.0008
0.0176
0.014
0.0001
0.030
1.06
372
394
0.0047
−0.40
191
8,726
0.0001
0.032
31.6
<0.001
<0.01
<0.001
138
91.7
158
110
1
W0 is initial BW at t = 0, Wf is asymptotic mature BW, c a parameter to estimate between-pig variability of mature BW, k is a rate constant,
n is a shape parameter influencing point of inflection, σ is SE of model (= √variance (σ2) of error), BIC (Bayesian information criterion) is a
measure of regression fit, and t* and W* are age and BW at the inflection point, respectively.
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Growth and phosphorus intake analysis in pigs
Figure 2. Relationship between BW (kg) and cumulative available phosphorus intake (g) in Data set 3. Symbols
represent observed values (circle, square, triangle, and inverted triangle correspond to treatments with available P
intakes of 52, 78, 105, and 130% of the NRC (1998) recommendation, respectively), and the lines are regression lines
fitted using the 4 candidate equations.
sumes the efficiency is continuously decreasing until
an asymptote is reached, when efficiency becomes zero.
The growth curve was divided into 4 phases as shown
in Figure 3. The lag phase was defined as time from
the initial BW estimate until the animal attains 10%
of its final BW, the accelerating phase continues from
the end of lag phase until maximum growth rate (point
of inflection) is achieved, then the animal goes into the
Table 3. Parameter estimates and SE obtained by nonlinear regression of BW vs. cumulative P intake (data set 3)
Monomolecular
Item
1
Estimate
SE
W0, kg
19.2
2.14
Wf, kg
193
5.26
k1, g−1
0.0038 0.00015
k2, g−1
0.0025 0.00010
k3, g−1
0.0019 0.00008
k4, g−1
0.0013 0.00005
n
σ
20.5
0.34
BIC
7,557
Calculated growth trait
t1*
NA
t2*
NA
t3*
NA
t4*
NA
W1*
NA
Gompertz
P-value
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
Estimate
27.5
157
0.0082
0.0055
0.0041
0.0027
25.6
7,667
67.9
101
136
206
57.9
Richards
von Bertalanffy
SE
P-value
Estimate
SE
P-value
Estimate
SE
P-value
0.34
3.04
0.00022
0.00015
0.00011
0.00007
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
0.97
7.26
0.00017
0.00016
0.00010
0.00008
0.005
0.21
<0.001
<0.001
<0.001
<0.001
0.25
28.7
132
0.0031
0.0020
0.0014
0.0009
−0.87
25.3
7,666
27.0
178
0.0060
0.0040
0.0030
0.0020
0.33
25.9
7,671
0.36
4.71
0.00021
0.00014
0.00011
0.0007
0.006
0.64
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
<0.001
12.5
17.3
23.4
34.1
8.0
52.8
84.7
113
170
56.5
1
W0 is initial BW at t = 0, Wf is asymptotic mature BW, k is a rate constant, n is a shape parameter influencing point of inflection, σ is SE
of model (= √variance (σ2) of error), BIC (Bayesian Information Criterion) is a measure of regression fit, and t* and W* are available P intakes
(g) and BW (kg) at the inflection points, respectively. Subscript numbers for k and t denote treatments with P levels of 52, 78, 105, and 130%
of the NRC (1998), respectively.
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Kebreab et al.
Figure 3. Growth stages for a representative sigmoidal curve and graphical representation of instantaneous efficiency
for the monomolecular equation. W0 is initial BW at time, t, = 0, Wf is final BW when t → ∞, Wl is BW at the end of
the lag phase (calculated as W0 + 0.1Wf), W* is BW at inflection, Wd = 0.9Wf, and kP is instantaneous efficiency of
phosphorus utilization for growth.
deceleration phase until it reaches 90% of its mature
BW, and finally into the stationary phase that continues
until the approach of asymptotic final BW. Figure 3
also depicts graphically instantaneous efficiency of P
utilization taking the monomolecular equation as an
example.
Average efficiency of P utilization for gain in the
stages described above is given in Table 4. The calculations were made by taking into account the significant
differences in the rate constant. Averages from 25 to
100 kg, 25 to 50 kg, and 50 to 100 kg (BW intervals
covering the growing-finishing periods) are also given
because these could have implications for commercial
production systems. As expected, the greatest average
BW gain and efficiency of P utilization for the monomolecular equation was during the first phase, whereas
for the Gompertz and von Bertalanffy equations, greatest BW gain was during the acceleration phase. For
all the equations, P efficiency approaches zero in the
stationary phase.
DISCUSSION
Weight gain has been described using growth equations, and in nonruminants, the Gompertz has traditionally been the model of choice (Wellock et al., 2004).
In this study, a simpler model (monomolecular) and
others without the limitation of a fixed point of inflection (which occurs at 1/e times the final BW in the
Gompertz), such as Richards and von Bertalanffy equations, were evaluated. The analyses showed that, in
general, all 4 growth equations fitted the data reasonably well (Figures 1 and 2).
Performance of the equations in describing the data
sets was evaluated using BIC, which suggested differences in model choice for growth in BW and efficiency
of P utilization. A notable difference in parameter estimates in both sets of analyses was for W0. Estimates
of W0 were lower in the growth analysis because the
data contained information on birth weights whereas
birth weight data were not available for the P efficiency
analysis. The W0 estimates were within the range of
0.042 and 9.5 kg in the first and second analyses, respectively. The Wf estimates in growth analysis have to be
looked at in conjunction with the c parameter, which
was within 18 kg for the sigmoidal curves. The introduction of random effect of mature BW increased the goodness-of-fit (lower BIC value) compared with a generalized model with only experiment as the random element
in the model. It has also improved the approximate SE
of the parameter estimates.
Comparison of BIC values for the growth analysis
revealed that the Richards equation, perhaps due to
the broad limits on its additional parameter, was superior to the other equations. As parameter n approaches
particular values, the Richards equation encompasses
other simpler models, such as the monomolecular (n =
−1), Gompertz (n = 0), and logistic (n = 1). Table 2
shows that n estimate was about halfway between the
monomolecular and Gompertz equations. The flexibility
of the Richards equation increased the accuracy of prediction, and the need for an additional parameter was
justified because the shape of the curve could not be
described as well by the monomolecular (because it is
not a sigmoidal equation) or the Gompertz (due to its
fixed inflection point).
Four phases were arbitrarily assigned to evaluate
BW gain (Figure 3), and the average efficiency of BW
gain with time within the phases was calculated. The
values based on generalized sigmoidal equations were
close to each other but different from those based on
the monomolecular equation. The importance of this
kind of calculation is to identify periods of greatest BW
gain so that inputs can be adjusted accordingly. More
crucially, by recognizing the rate of decline in growth
during the stationary phase, more informed decisions
can be made about the impact of varying slaughter
weight on nutrient use and income over diet costs. From
Downloaded from jas.fass.org at Serials/Acq. Dept., Library on August 13, 2009.
2505
Growth and phosphorus intake analysis in pigs
Table 4. Calculated average efficiency of available P utilization in various stages of growth
based on parameter estimates for the 4 candidate equations1
Average efficiency,2 kg of BW gain/g of available P intake
Equation
k1
Monomolecular
Gompertz
Richards
von Bertalanffy
k4
Monomolecular
Gompertz
Richards
von Bertalanffy
k(W
k(W –W*)
k(W*–W
0.62
0.43
0.50
0.45
0.53
0.47
0.47
0.47
0.22
0.31
0.32
0.28
0.014
0.035
0.039
0.020
0.59
0.43
0.39
0.45
0.44
0.44
0.45
0.44
0.48
0.44
0.43
0.44
0.21
0.14
0.29
0.15
0.18
0.16
0.16
0.16
0.074
0.10
0.11
0.095
0.005
0.008
0.013
0.007
0.20
0.14
0.13
0.15
0.15
0.15
0.15
0.15
0.17
0.15
0.14
0.15
0
–W )
l
l
d
)
k(W
d
–W )
f
k(W
25
–W )
50
k(W
50
–W
100
)
k(W
25
–W
100
)
1
Two of the 4 rate constants (k) for treatments with the greatest (k1) and lowest (k4) available P intake
levels are shown for comparison of efficiencies.
2
W0 is initial BW at time t = 0, Wf is final BW when t → ∞, Wl is BW at end of lag phase (calculated as
0.1Wf + W0), W* is BW at inflection, Wd = 0.9Wf, and k denotes average efficiency. For the monomolecular
and for Richards equations (for P intake only), W* was calculated as (Wf − Wl)/2 because of the absence of
the inflection point for the former and low value for the latter equation. The subscripted numbers appearing
in the last 3 column headings are BW in kilograms.
the Richards equation, it can be calculated that the
pigs were expected to gain nearly 1 kg/d during the
accelerating phase, which goes down by about 40% in
the next phase; by the time they reach the stationary
phase, the animals were 97% less efficient in gaining
BW. The data also show that there was uniformity up
to the end of the accelerating phase (d 300 of age); then,
as animals matured, there were greater differences
among animals due to the BW gains being mainly attributable to fat deposition.
Apart from describing the pattern of growth, nonlinear equations can be used to investigate efficiency of
utilization of individual nutrients for BW gain. Usually,
mineral requirements in pigs are estimated by empirical methods based on feeding a range of dietary concentrations of each mineral element and examining the
response in terms of certain performance (BW gain) or
physiological (mineral balance) attributes to assess the
most adequate dietary concentration (i.e., the one resulting in maximum response). In general, as P intake
is increased, there is an increase in BW gain, feed intake, and feed efficiency (gain to feed ratio); this relationship is curvilinear, such that maximum response is
reached at a certain level of P intake, which is variable
depending upon animal characteristics (genotype, sex)
and the growing stage (Ketaren et al., 1993; Ekpe et
al., 2002; Hastad et al., 2004). The response is less
pronounced as the animal grows; hence in animals of
greater BW the level of P intake required to reach the
maximum response is lower (Ketaren et al., 1993; Hastad et al., 2004). From this approach, requirements
have been mostly expressed as percentage of the mineral in the diet (or alternatively per unit of energy in the
diet; ARC, 1981). However, in order to assess optimal
requirements for a given growth rate, it is important
to express requirements of P (and any other mineral)
directly in relation to protein and fat gain, but this
goal cannot be achieved with the information currently
available in the literature.
Alternatively, the use of nonlinear equations to describe the relationship between BW and cumulative
available P intake can be used to estimate the efficiency
of P utilization in terms of BW gain per unit of available
P intake at any stage of growth, provided that other
nutrients are not limiting expression of nutrient retention rates. Using this approach, 4 nonlinear equations
were examined. The Richards equation seemed to yield
as good a fit as the Gompertz, given the almost identical
BIC values. However, in this case, the estimate of parameter n was close to −1 (Table 3), approaching diminishing returns behavior without a point of inflection,
so that the simple monomolecular equation was adequate to describe the relationship between BW and cumulative available P intake in pigs, which is confirmed
by the lowest BIC value of the model. Most of the rate
parameter estimates of the Richards and monomolecular equations were not significantly different from each
other, which further proves the flexibility of the Richards equation and the suitability of the monomolecular
equation to fit these profiles, without the need for an
additional parameter. The same equation (Richards)
can be used for both growth analysis and efficiency
of nutrient utilization, and can yield different growth
patterns for each of the 4 dietary P levels. Therefore,
only 1 equation is required for all levels to assess the
optimum level.
Calculations for efficiency of P utilization based on
the best performing model (monomolecular equation)
showed that the average efficiency is greatest during
the 2 initial phases and slows down as the pig’s cumulative consumption of available P increases. By the time
the pig reaches the stationary phase, efficiency approaches zero. This trend is the same in pigs fed low
or high levels of P (Table 4). The steady decline in
Downloaded from jas.fass.org at Serials/Acq. Dept., Library on August 13, 2009.
2506
Kebreab et al.
efficiency of P utilization observed as a pig grows agrees
with the lower P requirements recommended (ARC,
1981; NRC, 1998) for different growth phases (decreasing from weaning to 100 kg) and can be attributed to
a number of reasons. The amount of P required for
maintenance is greater in larger pigs, and thus relatively less P is used for growth (ARC, 1981; NRC, 1998).
Also, it is important to examine variations in P deposition in the BW gain. Considering the principles of skeletal tissue growth, there is a steady and progressive
increase in the rate of P accumulation per unit of fatfree BW from birth to slaughtering weight (Mahan and
Shields, 1998). However, given the exponential increase
in fat deposition in adipose tissue, the rate of increment
of P content in empty BW is inversely related to BW,
as demonstrated by Mudd et al. (1969) in trials using
carcass analysis, reporting values of 8.1, 5.3, and 1.6 g
of P/kg BW gain at 20, 50, and 90 kg of empty BW,
respectively. Our analysis also shows that as BW increases, the proportion of P retained decreases. The
ARC (1981) suggested that the rate of storage of Ca and
P (g/kg of BW gain) with growth could be represented by
a linear fall between birth and 50 kg of BW (with a
linear relationship between that rate and BW) and a
constant rate thereafter. Calculations of efficiency of P
utilization as d(BW)/d(P intake) using nonlinear methods takes into account the relationship between BW
and P as described by different classical growth equations. It is noteworthy that the levels of Ca and P resulting in maximum growth rate and efficiency of gain
are not necessarily adequate for maximum growth mineralization (ARC, 1981). To achieve maximum deposition of Ca and P, and increase the bone mineral content,
a greater mineral supply would be required (NRC, 1998;
Hastad et al., 2004). The approach suggested herein
could be used not only to quantify the trend in efficiency
of P utilization throughout the life span of the animal,
but also to examine differences between genotypes with
different lean growth rate (Bertram et al., 1994) or
the effects of sex (comparison among gilts, barrows, or
boars; Thomas and Kornegay, 1981; Ekpe et al., 2002).
This information could also be used to estimate P requirements in pigs at various stages of the life cycle,
adjusting of P input levels during the finishing stages
of pig production.
Nonlinear mixed analysis can give reliable parameter
estimates and is a useful tool when analyzing data from
multiple studies. Although the Gompertz equation has
been used extensively, the Richards equation was superior in describing growth over time and is recommended
for use in data analysis in nonruminant animals. The
extra parameter in the model improved the predictions
and therefore was required as indicated by lower BIC
values. The nonlinear relationship between BW and
cumulative available P intake was best described by
the simpler monomolecular equation. Calculations of
average or instantaneous efficiency of P utilization can
be used to predict the animals’ response to changes in
intake and the potential to reduce excess P excretion
to the environment. It is recommended that future studies on growth and nutrient intake analyses consider
other models such as the monomolecular and flexible
equations such as the Richards as well as the
Gompertz.
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References
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