Pulsatile Pressure and Flow Through
Distensible Vessels
By Victor L. Streeter, Sc.D., W. Ford Keitzer, M.D., and David F. Bohr, M.D.
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• The problems of characterizing pulsatile
patterns of pressure and flow in the arterial
system are intriguing and complex. Ejection
of blood from the left ventricle initiates nonlinear transients in pressure and flow at the
root of the aorta. These transients initiate
complex pulse patterns that are propagated
throughout the arterial tree. Some of the factors that must be reckoned with in an analysis
of these patterns are itemized in the list that
follows: 1) The force initiating the transients
is itself complex; the velocity of ventricular
ejection increases rapidly with the opening
of the aortic valve, then declines slowly to
reach a negative nadir with the closure of the
aortic valve. 2) The distensibility of the walls
of the arteries receiving this positive increment of pressure and flow has an important
influence on pulse patterns. This physical
property of the arterial wall is also responsible for changes in configuration and velocity
of the transients as they pass over the arterial tree. The pulsatile patterns are further
distorted by 3) frictional losses of energy
with both positive and negative flow of the
blood, and by the 4) branching and tapering
architecture of the arterial tree. 5) The resistances to forward motion of blood through
the distal arteriolar beds also have their effects on the contours of the arterial pulses
observed upstream.
From the Departments of Civil Engineering, Surgery and Physiology, University of Michigan, Ann
Arbor, Michigan.
Supported in part by the National Institutes of
Health, U. S. Public Health Service through project
Xo. HE07443-01.
The experimental work was completed while Dr.
Keitzer was supported in part by projects B-2290C1
and H-4260C3 from the U. S. Public Health Service
and the Veterans Administration Hospital, Ann
Arbor, Michigan.
Received for publication February 25, 1963.
Circulation Research, Volume XIII, July 19G3
Both experimental and theoretical methods
have been used to analyze and to quantify the
influence of these factors on the arterial pulse
pattern. The experimental approach has profited substantially from new instrumentation
which permits a recording of the transients
of both pressure and flow with a high degree
of fidelity at various levels of the arterial
tree.1"4 The theoretical approach employs
established mathematical relations between
known physical parameters which permit the
prediction of pressure and flow at specific
points in the arterial tree and specific times
in the cardiac cycle. If an adequate mathematical expression were available to describe
these relations, the fit of these predictions to
the actual measured values would then become a valuable tool both for assessing the
accuracy of the selected values for the physical parameters and for evaluating the significance of the various factors that influence
these transients in the arterial tree.
The theoretical analysis of these transients
then has two requirements: 1) realistic values
for factors that influence the transients, and
2) a mathematical statement of the interrelationship of these factors which will define
pressure and flow with respect to time and
position in the arterial tree. Approximations
of the required quantitative values for the
physical factors involved can be found in the
literature. However, one of the more unyielding problems has been the development of a
mathematical expression for the interrelation
of these factors in pulsatile flow in a distensible vessel. Owing to the difficulties encountered, greatly simplifying assumptions have
been made, such as laminar flow, linear frictional resistance, and steady oscillatory
flow.1- 2 The resulting equations, despite the
restrictive assumptions, are very complicated
STREETER, KEITZER, BOHR
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and do not lend themselves to ease of computation when practical boundary conditions are
introduced.
The current study presents a mathematical
approach which permits a more realistic solution of specific equations describing these
interrelations. Specific values for pressure
and flow with respect to time and position in
the arterial tree can be computed. This approach starts with two established equations
dealing with pressure and flow transients and
the physical factors that influence them.
These simultaneous equations are: 1) the continuity equation, which equates the net influx
of blood entering a small segment of the arterial tree with the increase of volume of that
segment (fig. 2), and 2) the momentum equation (Newton's second law), which equates
the forward force acting on this segment of
blood with the backward force plus the force
exerted by the arterial wall and the force required to overcome friction in the artery (fig.
3). The inclusion in this equation of the
statement for friction makes it a nonlinear,
partial differential equation which has not
been solved satisfactorily by previous methods. The critical aspect of the current approach is the application of the method of
characteristics which permits the solution of
these two simultaneous equations. Specific
values can be computed for the unknown
functions of pressure and velocity along characteristic lines relating the independent variables of time and distance (figs. 4 and 5).
With the aid of a high speed computer these
unknowns have been determined for frequent
periods of time (l/400th of a physiological
pulse cycle) and for short segments along the
arterial tree (1 cm).
In developing this approach we were not
aware that in 1958 Lambert5 applied similar
principles to this problem.* Although he employed the method of characteristics, the use
of a graphic solution instead of a computer,
and the inadequate consideration of physical
•We are indebted to the Editorial Board of Circulation Research for calling this publication to our
attention.
conditions in the circulatory system limited
the usefulness of his attempt and led to unrealistic results.
The problems of reflections and their interpretation in an analysis of unsteady flow become less significant with this approach. A
pressure pulse is transmitted through the vessel at a speed that depends upon the tube
properties and the pressure within the tube.
As the pressure varies both with distance
along the tube and with time, the speed of
the pulse wave changes continuously with the
independent variables, distance along the tube
and time. For any change in speed of the
pulse wave, reflections are set up which move
in the reverse direction, and the transmitted
wave is affected as well. As reflections also
occur at boundaries, i.e., entrances, exits,
branches and obstructions, previous methods
of keeping track of all these reflections become hopelessly complicated.
The method outlined here takes all these
reflections into account automatically, not by
keeping track of them separately, but by satisfying all of the basic equations at closely
spaced sections along the vessel at frequent
time intervals, and by satisfying the boundary equations. Owing to the relative ease of
handling the computations for interior sections and for boundaries, solutions may be
programmed to simulate conditions in branching arteries with satisfactory accuracy.
The theory of characteristics, which applies
to the solution of hyperbolic partial differential equations, first gained prominence in solution of supersonic flow problems by Courant
and Friedrichs.6 These methods were extended to applications of free-surface flow
cases later, by Stoker.7 More recently they
have been applied to waterhammer situations
by Lai,s Streeter and Lai,9 and Streeter,10 in
which nonlinear terms for wall expansion and
for wall friction have been retained in the
equations. This paper presents an extension
of the theory to flexible vessels, tapering vessels, and vessels with distributed outflow
along their length, together with in vivo experimental confirmation of the method.
Circulation Re-eareh, Volume XIII, July 1963
PULSATILE PRESSURE AND FLOW
Theoretical Methods and Results
In this section the mathematical and physical relationships for flow through flexible
tubes are derived, including equations for
tapering vessels with distributed outflow.
ii
I
T+dT
PD + d(PD)
I
A. DERIVATION OF BASIC EQUATIONS
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The assumptions required for analytical
handling of pulsatile flow are first discussed,
followed by development of elasticity relations and the continuity and momentum equations for vessels of constant initial diameter.
From these relationships the characteristic
equations are obtained and finite difference
methods applied to develop the equations for
the method of specified time intervals. After
discussion of boundary conditions, an example is presented.
1. Basic Assumptions
The basic assumptions required in developing the working equations are:
a. One-dimensional flow; the velocity at a
cross section is given by the average velocity
at the section at a given instant.
b. The vessel walls are elastic, with a Poisson ratio of 0.5, and they are tethered; hence
the volume of elastic vessel wall per unit
length remains constant.
c. The fluid density is constant. Compressibility of blood is small compared with expansion of the vessel walls under increased
pressure, and
d. Pressure losses due to wall friction may
be expressed as proportional to some power
of the velocity at a cross section.
2. Elasticity Relationships
Although arteries have viscoelastic properties, their effect seems to be minor and previous investigators11 have concluded that the
stress-strain curve may be considered linear
without introducing appreciable error.
If D is the inside diameter of the vessel,
and h the effective wall thickness, then as a
consequence of the constant volume of elastic
wall material resulting from the assumption
of Poisson's ratio of 0.50,
Dh = Doho
(1)
By using the method of analysis for stresses
within the walls of thin-walled vessels, at
Circulation Research. Volume XUT, July 1903
T+dT
FIGURE 1
Relation between tensile force change in ioa.ll dT
and pressure change dP.
pressure P, figure 1, the tensile force per unit
length of tube is given by
2T = PD
since the force component on the curved half
section of the tube is equal to the pressure
times the projected area of the curved surface
(Ref. 14, p. 48, 52). Now, considering an
increment of pressure dP, the equilibrium
equation, from figure 1, is
2(T + dT) = PD + d(PD)
as the projected area also increases by dD.
By taking the difference of the last two equations, and after solving for dT,
, T _ d(PD)
Since the diameter change, on a percentage
basis, is much less than the pressure change,
the term PdD/2 is neglected in expanding
the right-hand side, leaving
dT = ^ dP
(2)
By dividing through by the wall thickness the
change in tensile stress dS (force per unit
area) is obtained
,a
dS =
dT
DdP
IT = ~W
Now, by dividing through by the elastic modulus of the vessel wall, T, the unit strain is
obtained, which is the change in length per
unit length caused by dP. Since circumference changes are proportional to diameter
changes, the unit strain is dD/D,
dD
D
J3 dP
2h Y
(3)
STREETER, KEITZER, BOHR
After use of equation 1 to eliminate h, and
after separating variables
dP =
(4)
Integrating
By use of equations 1 and 7
ho D_
DJ A_
Ao
h Do
After equating expressions 8 a n d 9
a' = a? - gH
(5)
in which the condition has been used that D
= D o when P = 0. By multiplying and dividing the right-hand side by 7r/4 to introduce
the vessel cross sectional area A, and correspondingly Ao,
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A_
1
(6)
PD0
1 hoY
This form of equation permits the vessel to
blow up when PD 0 = h0Y. For the normal
pulse pressure range where linear elastic relationships are assumed, the pressure is below
this limiting value. Peterson, Jensen and
Parnell 11 have demonstrated that for the
physiological pressure pulse the stress: strain
relationship in the wall (Young's modulus)
can be considered to be constant. Burton 12
and Bergel13 have, however, emphasized the
physiological importance of a pressure dependence of Young's modulus in the higher
pressure range. Here adventitial connective
tissue adds support to 'the wall structure. In
considering larger pressure transients such as
occur in exercise it may be appropriate to
use a nonlinear elastic relationship, as Lambert 5 has done.
The pressure P has been expressed as force
per unit area (dynes/cm 2 ). It is customary
to express it in terms of the height of a liquid
column (the fluid flowing). These are related
by P = pgH in which p is the mass densitj7
(g/ml), g is gravity (980 cm/sec 2 ), H is the
pressure in height of fluid flowing (cm). Two
additional substitutions are made, let
hoY
Then
1
=
A.
=
D:
1 - gH/a:
(3)
(9)
(10)
" a " is the speed of t h e pressure pulse in the
vessel. This statement, t h a t " a " decreases
with a n increase in head, is t r u e if Y o u n g ' s
modulus remains constant. 0
^Relation Between Pulse Velocity and Arterial
Pressure: Equation 10, indicating that an increase
in arterial pressure causes a decrease in pulse velocity,
is at odds with recent reckoning,* and early experimental evidence.17' u King,19 by assigning ' ' elastomeric" properties to the artery wall, developed an
equation which indicates that pulse velocity is a
positive function of arterial pressure. The fact of
the matter is that if the increase in pressure is
accompanied by an increase in artery circumference
with no change in wall stiffness, then pulse velocity
decreases; conversely, if, with the increase in pressure, the wall gets stiffer without an increase in
circumference, then pulse velocity increases. I t is
probable that either or both of these changes occur
in the artery, depending on the cause and duration of the increase in arterial pressure. Thus, in
the context of our current information, a speculative ramification aimed at exploring the issue of
whether pulse velocity increases or decreases with
an increase in arterial pressure must consider the
specific circumstances under which the pressure increases. The following speculations are illustrative:
1) In an experimental situation in which the active
tension of vascular smooth muscle does not change
and the intraluminal pressure is increased, it has
been well established that the artery wall gets stiffer
as it balloons out." This stiffness would be responsible
for an increase in pulse velocity which might overcompensate for the decrease in pulse velocity effected
by the increase in circumference. Such a relation
between intraluminal pressure and pulse velocity has
been observed in isolated vessels17 and in man.18
2) In exercise the magnitude and steepness of
the pulse pressure increases while the pressure in
the arterial system just prior to the systolic upstroke, i.e., diastolic pressure, is not much changed.
It is important to note that there is in exercise a
several fold increase in the concentration of circulating catecholamines. Since even during rest the
blood concentration of these constrictors is effective
in contracting smooth muscle of the aorta,20 increase
in catecholamines that occurs with exercise would be
expected to cause an increase in active tension in
the wall of the aorta. This would make the artery
wall stiffer and increase the pulse velocity by a
Circulation Research, Volume XIII, July 1963
PULSATILE PRESSURE AND FLOW
From equation 9
D =
(11)
The speed of the pressure pulse changes both
with time t and with distance x along the vessel, and whenever " a " changes reflections are
produced.
The partial derivatives of A with respect
to the two independent variables, time t and
distance x are needed later and result from
equations 8 to 11 (a variable subscript x or t
represents partial differentiation with respect
to that variable, i.e., Ax = 3A/3x).
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Ao
Ax
Ao
a1
2aB'
ax,
A,
2 a;
a1
mechanism quite unrelated to a change in intraluminal
pressure. This stiffness would make the wall less
distensible. In this stiffened condition the pressure
dependent increase in circumference which is necessary
to effect a further increase in stiffness would be
small. Therefore, in this artery, stiffened by active
contraction of vascular smooth muscle, there should
be less of a tendency for Young's modulus to decrease
or increase with a given increase in arterial pressure. Pulse velocity would be less influenced by
arterial pressure when smooth muscle tone is high
than when it is low. In exercise this relationship is
complicated by the fact that the vessel is further
stiffened by an increased steepness of the pressure
pulse. Here the viscous component of wall stiffness
is proportionately greater.
3) The increase in arterial pressure associated
with chronic hypertension is accompanied by an
increase in stiffness of the arterial wall (Bohr,
D. F., unpublished observation). This is due primarily
to an increase in the non-contractile structural components of the wall. The contribution of active tension
from smooth muscle to the total tension in these
stiff hypertensive vessels has not been quantified.
Again, since these vessels are less distensible the
change in circumference for a given change in pressure must be quite small, tending to keep the stressstrain relationship relatively constant over a given
pressure range. Changes in circumference and stiffness
would be small and therefore the pressure dependent
variation in pulse velocity might not be great.
Although it is relatively simple to introduce a
pressure dependent Young's modulus into the program currently described, the quantitative nature of
this dependence is not known and it is quite likely
that the character of the dependence will vary
depending on the basis for the pressure change.
Circulation Research, Volume XIII, July 1003
PAV+(/>AV)xdX
-Jdxl—
FIGURE 2
Material balance showing mass per unit time entering and leaving an eleviental volume.
In these equations Ao is considered to be constant. Then from equation 10
2a . ax = — gHx
2a . at = — gHt
Now, by eliminating ax, at, and Ao from the
last three sets of equations
Ax _ gHx
At _ gHt
A
a1
A
a1
Also, from the relation P =pgH
Px = PgHx
,..„,.
(13)
These elastic relationships are used in the
continuity and momentum equations that are
now developed.
3. Continuity Equation
The continuity equation is a material balance for a small segment of vessel, which
states that the net mass inflow per unit time
is just equal to the time rate of increase of
mass within the segment, figure 2.
If V is the average velocity at the entrance
to the element, the rate of mass inflow is
pAV, and the rate of mass outflow is pAV +
(pAV)xdx. The net mass inflow per unit time
is then —(pAV)xdx and must just equal the
time rate of increase of mass within the segment (pAdx) t . Equating these expressions
- (PAV)xdx = (pAdx)t
(14)
p is constant for all practical purposes when
considering flow in a distensible vessel. A and
V are dependent variables. After expanding
equation 14, remembering that x is independent of t, and then dividing through by the
mass of the fluid segment, p Adx,
A
Vx+Ai=0
(15)
STREETER, KEITZER, BOHR
TIME RATE OF INCREASE
OF MOMENTUM
MOMENTUM IN
/>AV 2
Force acting on segment of fluid (solid lines), and momentum relationships (dash lines).
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which is the continuity equation and must
hold throughout the vessel.
4. Momentum Equation
The momentum equation when applied in
the x-direction to the fluid in a segmental volume, figure 3, is a statement that the resultant
x-component of force on the segment of fluid
is just equal to the net efflux of x-momentum
plus the time rate of increase of x-momentum
within the segment. The resultant force component, figure 3, is
(pAV2)xdx, with a net efflux of (PAV2)xdx.
The time rate of increase of momentum within
the segment is given by (pAdxV)t. After
combining the force and momentum terms,
- PxAdx - TOlrDdx = (pAV'),dx + (PAdxV)t
By expanding the partial derivatives, then
dividing through by the mass of the segment
pAdx, and after replacing Px by pgHx,
gH x
^ V « + 2VV,
A
Vt +V A -i = 0
A
F = PA + (P + Px H£) A,dx - [PA + (PA)xdx] The first term is due to pressure within the
fluid acting over the cross section at x, the
second term is the force component in the xdirection due to the tube wall pushing against
the fluid (zero for constant A, as Axdx is the
increase in cross section in the length dx).
The term in brackets is the force pushing
against the element on the distal side. The
action of fluid friction at the tube wall is
given by the product of shear stress T0 at the
wall and area of wall surface 7rDdx. This force
is assumed to act wholly in the x-direction.
By expanding the expression for F it becomes
F = - P,Adx -
+ PXAX
Since the term with square of dx becomes of
a higher order of smallness as dx approaches
zero, it may be dropped from the equation.
The momentum influx at x is pAV 2 and
the momentum efflux at x + dx is pAY 2 +
may be written in
The wall shear stress
the form
TO =
k
2
(17)
For established, steady laminar flow k =
16/R, with R the Reynolds number VDp//i,
in which /x is the fluid viscosity (poise). For
turbulent flow k = f/4, with f the commonly
used Darcy-Weisbach friction factor.14 By
inserting equation 17 into equation 16,
using f,
T V*
2V
vAi = o
(18)
This is the momentum equation for flow
through a distensible vessel. By multiplying
equation 15 by V and subtracting it from
Circulation Research, Volume XIII, July 1968
PULSATILE PRESSURE AND FLOW
equation
sults*
18, substantial
simplification
gHx+VVx+Vt+^-=0
re-
These expressions must be the same,
equating them and solving for X
By
(19)
and
Equations 15 and 19 contain the continuity
and momentum principles. After substituting
the elastic relationships given by equations
12 into equation 15 it becomes, upon simplification,
VHX + H t + — Vx = 0
(24)
x =
Xow, by restricting the applicability of
equation 21 to those characteristic lines for
which equations 22 and 23 are satisfied, it
may be written in the simple form
(20)
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Equations 19 and 20 are used to develop the
final equations.
5. Development of Characteristic Equations
By calling equation 19 Lx and equation 20
Lo, they may be combined linearly using an
unknown multiplier X, as follows:
By applying X = +g/a to equations 22 and 25
gdHdVfV»
a dt
dt
2D
(26)
dx
(27)
L = L, + XL, = x[Hx(f + V) + HJ + [VX(V + — ) + Vt] + ^
X
e
If two distinct values of X are taken, two
equations result which contain the momentum
and continuity principles. The theory of
characteristics determines two special values
of X which result in great simplification of
the equations. To review some fundamental
relations in calculus, if H = H(x, t) and V =
V(x, t ) , then the total derivatives of H and
V with respect to t are
dH
dt
dV
dt
dx
dt
dx
dt
wherein these relationships H and V are
pressure and velocity of a particle as it moves
(x becomes a function of t ) . By examining
equation 21 it is seen that the first bracket
contains d H / d t if
X'
dt
(22)
and the second bracket contains dV/dt if
TT
-
Ail
LI A
(23)
*By writing the V2 term in the friction expression
as V V , it will reverse sign if the flow reverses,
and hence act in the proper direction at all times.
Circulation Iraearch, Volume XIII, July 1903
2D
=0
(21)
and by applying X = — g/a to the same equations
a
dH + dv + fz:
dt ^ dt
dt ^ 2D
2D
dt
dx
(28)
(29)
Equation 26, a total differential equation, is
valid only along a line defined by equation
27, if a plot is made on an x — t plane, as
shown in figure 4. If C + in the figure represents the characteristic line defined by dx/dt
= V + a and passing through a point R
where V, H, x and t are known, then equation
26 may be used as one relation between pressure H and velocity V along this line. Similarly, equation 28 is valid only along a line
defined by equation 29, figure 4. Xow with V
and H known at the known points R and S,
four equations are available at the intersection P of the two characteristic curves, for
computing V, H, x and t.
6. Method of Specified Time Intervals
For computation purposes, equations 26 to
29 are written as finite difference equations.
10
STREETER, KEITZER, BOHR
t
The V2 friction term has been replaced bv
V|V|.
VR, En, Vs, and II S are now to be computed from the equations, Avith reference to
figure 5, by linear interpolation. The mesh
(values of Ax, At) is assumed to be fine
enough so that the slopes of the characteristic
lines are given adequately by evaluating V +
a and V — a at C. From figure 5
c_
VR — V
— xA
Vc — VA
FIGURE 4
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Characteristic lines C + and C— drawn through
points R and S respectively, where V and H are
known. Their intersection P is a point where x,
t, V', and H may be determined from the equations.
XC
-
xA
By remembering that x P =
31
Xc -
XR
(34)
AX
, from equation
= (V + a)c At
Then
For use with a high-speed digital computer,
the theory of characteristics method may be
extended to a system in which the time and
distance intervals are preselected.15 This is
called the method of specified time intervals,
and entails interpolation of values of V and
H at unknown points R and S, figure 5, such
that P occurs at equally spaced distances Ax
along the vessel for equal time increments. In
figure 5, consider that V and H have been
computed for the first two rows at the equally
spaced sections. Values of V and H are to be
computed next for point P at time tP, which
is t c + At, and values of V and H are known
at A, B, and C.
Equations 26 to 29 are written as finite difference equations, for points R and P on C +
and for points S and P on C_, with At = tP
—tK = tp — ts. The mesh (Ax, At) is assumed
to be so fine that the velocity in the friction
term may be evaluated at known conditions
at C. Also aR and as are replaced by ar, as
well as DR and D s by D c .
— (Hp — H R ) + Vp — VR H
ac
(31)
V c | At
!
2 D c
(32)
XP
XA
= AX — (xC -
XR)
= Ax - (V + a) C At
(35)
For convenience the grid mesh ratio At/Ax is
called ©. Substituting into equation 35, the
first of the following four equations is obtained,
VR = V
(Vc - VA) - 0(V + a)c) (36)
H B = HA + (Hc - HA)
- 0 (V + a)c)
(37)
V8 = VB + (Vo - VB) ( + 0 (V - a)c)
(38)
H8 = HB + (Hc - HB)
+ 0(V - a)c) (39)
The last three equations are found in a manner similar to that used in obtaining equation
36. In
0 =
At
Ax
(40)
At must be selected so that R and S lie within
the reach defined by points A and B.
With the interpolated values of V R , H R ,
V s , H s known, equations 30 and 32 may now
be solved for H P and V P ,
HR
+ Ha
£2
(VR
- V8)
(41)
2D
xP - xR = (V + a) c At
rr\ , v
v ,
— Us) + Vp — Vs H
-
HP =
At
(30)
- g
XR
- x 8 = (V - a ) c At
(33)
+
2^
(HR
-
Hs)
fcVc 1 Vc | At
(42)
Equations 36 through 41 permit all interior
points of the grid to be computed, i.e., all
points where corresponding points, A, B, and
C are known. At the end points an additional
Circulation Research. Volume XIII, July 19G3
11
PULSATILE PRESSURE AND FLOW
t
C.
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FIGURE 5
•By specifying At and Ax, which locates point P, values of V and R at B and S may
be found by interpolation from values at A, C, amd B.
condition is needed to solve for VP and
HP, since only one of the equations 30 and 32
is available at a given boundary. This additional condition is called the boundary condition.
7. Boundary Conditions
Boundary conditions may take on many
forms. Some examples are given here to illustrate procedures in developing them. At the
proximal end of the vessel equation 32 applies, figure 6,
VP = V8 + £ (HP - Hs) - fcVc 1 Vc | At
a
2DC
(43)
in which VP and H P are the unknowns, as
Vs and H s are given by the interpolation
equations 38 and 39. If a known pulse inflow
QP into the vessel is known at this instant,
then a second equation becomes available, as
follows:
Qp = VP I
(44)
DP'
= ^
lo
*
By substituting equation 45 into equation 44
to eliminate DP, and after solving forVP,
Vp
=
"
= k, (H P - H B ) m = VP -A
4
gHp)
Circulation Research, Volume XIII, July 1063
DP1
(47)
equation 30, solved for VP is
VP = V R - JL (H P - H R )
-
fcVc | V c | At
2DC
(48)
By use of equations 10 and 11, equation 47
may be solved for VP in terms of H P
VP =
But from equations 10 and 11
DP1
which is the second relationship required.
With the pulse inflow given for each time increment, VP and HP at the proximal end may
be computed progressively as the rest of the
solution proceeds. Another example is to have
the pressure pulse specified as a function of
time at x = 0. In this case the known H P is
inserted into equation 43 and VP may be computed.
At the distal end, figure 7, the outflow majr
be expressed in terms of the pressure difference between the vessel H P and the terminal
bed HB as
4k,
™
t
(H P - H B ) m (a o s - gH P )
(49)
A trial solution may be required, depending
upon the value of exponent m in the terminal
bed resistance relationship. Care must be
taken in applying a terminal bed Q — II relation. Within the vessel proximal to the bed,
there is no simple Q — H relation, primarily
12
STREETER, KEITZER, BOHR
o
B
FIGURE 6
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
Proximal boundary relations. One relation between
V and H at P is obtained from values of V
and H at S.
due to the complex reflections. If a segment
of an arterial system is to be analyzed, the
boundary conditions should generally be expressed as H versus t, Q versus t, or VP
versus t.
For known pressure as a function of time,
VP is found directly from equation 48.
8. Example
To illustrate the application of the characteristics theory to a flow situation, a known
pulse flow is injected into a distensible vessel,
with a linear terminal bed at the distal end.
The computer program, figure 8, is in the
MAD (Michigan Algorithmic Decoder) language and an IBM 709 is used in performing
the calculations. The pulse flow from the
heart of a dog, measured at the ascending
aorta by the Square "Wave electromagnetic
flowmeter,* has been expressed by a series of
empirical formulas. The average flow is 35.65
ml/sec over the 0.4 sec pulse cycle, with a
maximum inflow of 160 ml/sec. A friction
factor f = 0.3 was used and a time increment
of At = 0.004 sec taken with twenty equal
reaches of Ax = 2.5 cm. The pressure H, velocity V, diameter D and flow rate Q were calculated, and values printed out for every
0.008 sec time interval and at 5 cm distances
along the tube. The terminal bed pressure
•Carolina Medical Electronics, Inc., TVinston-Ralem,
Xortli Carolina,
FIGURE 7
Distal boundary relations.
was taken as 100 mm Hg, and the problem
was initiated by first setting up a steady flow
equal to the average flow (QAVE).
The sequence of main calculations in the
program are as follows (fig. 8) :
1. Calculation of steady state problem,
storing of H, V, D, and Q for the 21
sections 2.5 cm apart for time t = O.
2. Calculation of interior points, statements 45 through 54, for the next time
increment.
3. Calculation of the proximal boundary
condition, statements 55 through 67.
4. Calculation of the distal boundary condition equations, statements 68 through
78.
5. Print out of every second set of results,
incrementation of T and U, and check
on determination of end of solution.
The program was run through three pulse
cycles. The end of the second and third pulses
showed rather close agreement, indicating
that steady oscillatory flow has almost been
established. A gross check on continuitj' was
made as a measure of the accuracy of the finite difference method. The total inflow in
the third pulse is 35.65 X 0.4 = 14.26 ml. The
total outflow is 14.34 ml, and the volume of
fluid within the tube has decreased by 0.29
ml. This indicates an error in continuity of
about 1.5%, which could be further reduced
by taking shorter reaches and smaller time
increments, or by going to a method of calculation0 having second order accuracy.
Circttlation RmrarcJi, Volume XIII. July Wfi,l
PULSATILE PRESSURE AND FLOW
13
PULSATILE FLOW THROUGH A DISTENSIBLE IUBE
DIMENST(iN""v"('20),VPI2OI.HI2O),HPI2OI ,01201 ,01 20) .AAI 20)
INTEGER I.U.N.P.KK
,
P R T N T COHHtHltl
""'
GIVEN OATA FOR PROBLEM*
READ D A T A _
.
PRINT
2.P.KK
PRELIMINARY
«uul
.002
»GJ3
«UQ4
CONSIDERATIONS
PPT".32«PTIME
" pfTi"."i*»pTiHE
OM-160.
A0-S6"RTV(Vi"fHVCk7t"lfHbiD0) I
•0011
•oo«
*Olp
•o"ll
" "
A.SQRT.IAq»A0-G»SG»HO)
m-
_
'ui'ooVio'/A'
«0l*
"
* "•ifis
VP-QAVE/J.785*«D»D)
>V71"i"d
_
"
«U16
'i'iY'
flfjflfl MifUfM-9i.il.6-H
HI I )«HO-UHF«I
_
•pi'O
oi'ii'6
"
OIII-OAVE
"A"AVI"V-S'(5KT;TAO'«AO-G»S"G«H(I »)
K-QAVE/1H0-HB1
K-QAVE/1H0-H
T--S-»DELT
"
'i'dii"
«022
'iZi'i'
«Q?»
"
.k;r....
A
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
Cl-K/l.78S*»D0»D0«A0«*OI
C2-Cl«(*0»Ap»C»SG»HB)
"l'i''€V*Q»'io
"" "
•
>027
'026
i~6i<t~
"
VZH'JZr'l'H,
. ••«??
ALN'bUKI . IAU*AO-l»«ili«HINl )
PRINT COMHENTtO
HEAOSICM H G ) t
VELOCI I IESI CM/
2S"ECT;
oiVMETERS(CM)",' "ANOFLOWICC/SECli
PRINI COMMENT»0
TIME
FLOW
X/L.0
.1
.2
i
^"3
;V""
75
.6 ' ' .7
.8
'.9
IV
3t
2).HI
16),HI18),H(20)
,H(,"v("12
10) ,H(
•HI
1<I
" P PRINT
" R T N T " FFORMAT
" I J R " M A YBl.T
' ¥ 2 V V,0,H(0)
V 6 ) 7 V V 2 I,H(21
, V I*). ,HvC()6 ),H(6I
, vI a ,M(8I
I'.vYidi
>. 12)
vc'i*
1,Vi
216),V(18),V(20)
•Oil
*O32
«O32
«0i3
'iOii'
»013
«034
»o3S
«035
y;D"(OT.6"(2f.dr*).016>. 0(8). Diiov.oi i2i.o( iVivev
"
216),U(181tU(20)
'
VECTOR""VALUES'Bli»lH0,F8.3.F9.2',S2',3H H«,llFb.'3«»
VECTOR VALUES R2-»1H , S 2 0 . 2 H V - , 11F8. 3*«*
VECTOR VALUES B 3 - H H . S 2 0 . 2 H 0 - , 11F8. 3«»
WH£N"gVE'R"u7CV(l",fR*ANSFER
ioiii"
'Oil
~'~iji~6
•OH
«0«0
10 Al
"CALCULATION OF TNTE'RIOR POINTS
THROUGH A5.F0R I " 1 , 1 , I . T . N
HR-HI 1-1 )»(H( I ) - H ( 1-1) ) . | l . - T H . f V I I I » » A U I I )
VR-VI1-1)»iV(Y)-V(I-l))•(l.-TH»(V(I)*AAI1)I)
VS-VJ1*1)»(V(1)-Vt1»1))»I1.*TH«(V(I)-AAIII)I
• H 4 " " H ( ' l T i V r H r r f r f T l ) ( l T ( ( I ) A A ( I )))
'
'
2T/I2..OID)
AA(1 jiSOR"T."(A0i*6-G."SC.HPIIIl
OilI-00'AO/AAI I )
U( 1 ) - . 7 8 S * « 0 ( I >.Ul I )«V>11 I '
.Oil
.0S2
.053
•0',*
CALCULATION OF PROXIMAL BOUNDARTTBNDnToFT^
WHENEVER r.L.O.,0-OAVE
WHENEVER T . G E . 0 . . " A N 0 7 T . L E . P M r , 0 ' l j M . T / ( I M * E X P ' . I T / I M - l . ) l
WHENEVER r.G.PNI.»NO.T.lE.PPr .O-OHM/ITH.EXP. ( T / I H - 1 . ) ) - . i 6 « 0
'2M.(T-PMT»/lPPr-PM"rf"
"
J
'VCVSuTlr."r»6»A0'-C"»'sBiHJ' "
" "
VS-V(1)*(V-V(I)|.(l.»TMi(V-»CII
"
0"OO.AO/SORT.(AI
iOO
<0<>6
"»0*T
"
y
«0S»
«0'J6"
.0">7
»(J-i7~
«U60
'i'o'bl"
«U62
V
_
DISTAL BOUNDARY CONDITION
yEE-yR«G.SG*HR/ALN r F.y(N)..ABS.V(NI*0ELT/(2.*0(N))
CS«G'iSG/ACN
"
C6-ICVC2I/C3
_
m-
Hf>
.!M"C6/2.-S0RT.(C6.C6M.-C7)
wl»rN»VvElE"c5VHP"iNJ
'
ACN«SORT.(AO«AO-i;.SG.HP(NII
"blN"JV|)bVA0"7A'C"N
""A'6
"
""
v(i).ypii)
M(T)VHp'(iT
TO' A3
Figure 8
Circulation Research, Volume XIII, Jidy 10G3
-
(part 1)
•P_7*_
""".075
.076
.C~77
m
__ _»OBP
idol
"
WHENEVER U/KK.KK.E.U.TRANSFER
r«"A*N"S'FE"ft"ftf'A<
«p/0
iO7l
*082
i'083
(See legend pape 14)
STREETEE, KEITZER, BOHR
14
- ...GIVEN
DAIA
.. FOR PROBLEM
. .-. ?_.*..5.I>P_OOqOE__06,t
10.000000.
0 «
980.000000,
SG »
UELT - 4 . 0 0 0 O Q 0 E - 0 3 ,
PUHE • .
FLOW
X/L-
'.000
.0
HO -
.1
13.11.?
7.863
"1.760
19.128
13.713
12.899
12.962
13.083
4.672
1.758
11.344
13.093
1.083
1.759
2.630
3. 803
5.125
H= """l4Vf0'J" T3.458
12.948
V»
39.743
18.362
1.591
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
Di
1.824"" 1.780 . 1.751
0- 103.879 45.673 3.831
H=>
V-
14.531
14.047
49.33V "JV.381
14.931
TTTfT
160.06
14.772
48.927
14.517
T4".60"3
1.844
92.424
15.06 7 15.001
14.877
57.605 50.660 3J9.903
1.881 1.876 "1.868
160.000 140.041 109.327
15.182 15.179 15.127
56.4IV 50.487 42.366
1.888
1.888
1.885
0« 158.010 141.377 118.190
HV"
"U-
15.280
54.184
r.~895
15.316
49.269
lT8"98"
13.446
.401
1.779
.998
13.533
4.542
1.784
11.352
13.823
4.91~6
1.801
13.297
-.154
1.770
-.380
13.540
-1.161
15.412
*8.06l
T.96'4
136.911
13.553
-6.044
1.785
- 1 5 . 125
•300000,
N •
?L
300.
K K =•
2
15.494
42.935
l. oooooi:
.7
14.138
12.038
1.820
31.325
14.364
12.946
1.834
34.214
14.4^6
14.033
1.843
37.428
14.096
14.309
14,442
14.536
15.116
" 6r453" "* "9".029"""T2V6"22
12.510
1.618
16.743
1.831
23.772
I.Oil
31.942
1.842.
39.975
13.793
-.783
1.799
14.027
1.510
1.813
14.21B
5.350
1.825
14.345
9.965
1.8l3
13.999
26.300
14. 390
14. 7/9
1.816
3 9 . 130
-1.991
3.900
13.756
-5.389
13.940
-2.5"63
1.797
-13.669
1.808
-6.580
13.710
-8.809
1.794
-22.274
14.09B
14.210 14. 2S3
2.121
8.015
14. 4S1
1.810
5.500
1.825
20.958'
13.B43 "13.958
14.044
-5.621
-.420 6.379
1.802
-14.340
1.809
-1.080
1.814
16.493
14.028
1.817
36.361
14.570 14.009 13.610 13.587 13.624 13.646
23.704
1.939 -12.0_17 -12-162 -8.488 -2_.7_70
1.848 1.812" 1.788" 1.787
1.789
63.548
5.002 -30.188 -31.006 -21.343
13.661
4.625
1. 791
11.6b I
13.674
J2.9B0
1.792
32.746
14.502
14.941
31.446 15.210
1.843
1.8 72
86.553' 40.583
13.46B
4.545
13.471
12.426
14.871
26.047
1.867
71.330
101.431
15.425
42.915
1.905
122.370
15.264
42.536
1.894
119.858
13.888 13.554
-3.545 -11.313
13.503
-8.421
14.302
13.688 13.400
10.333 ">.154 -6.794
1.830"
1.79Y V."/76"
27.191 -10.489 -16.B35
14.662 14.000
23.801" 9.307
8
(part
13.325
-1.773
\"Tii'
-4.371
1.812
1.778
23.994
-1.852
14.889
33.984
1.868
93.185
14.343
24.347
1.833
64.250
13.650
11.306
1.791
2B.476
14.588
36.540
1.849
98.0U4~
13.9U0 13.332
26.253' "~14. 109
1.811
1.772
67.589 3 4 . 8 0 7
15.001
41.148
1.876
113.744
13.2d9
4.8b5
'fY
T3".2oT
f. 7 V6
29.202
13.427 13.190
-.746
-.129
1./64
1.701
1.854
64.222
15.416
44.175
Flgurt
13.484
-2.595
1.805 1.785 1.782 1.781 1.780
1.7B0
30.932
-9.070 -28.315 -21.007 -6.465
11.313
14.266
15.266 15.040 14.722
45.489 46.240 44.981 39.174
1.857 "1.828
1.894 1.879
f.905
1.906
1.911 1.910 1.905
1.894
117.721 120.312 123.059 125.883 128.195 128.181 121.886 102.836
f5.498
41.964
so.ooopog
KHO •
13.802
l*-034
1JL-568
13.550^ J 3 - * ? * _ 1 3 . 7 3 8 _13._B06_ ±3. 8 W
57125 13.527
12.526 -8.132 -12.697 -11.097 -7.590 -2.077
1.803 1.805
1.814 1.786 1.785 1.791
1.7961.800
13.341 34.5^8
31.591 -20.373 -31.771 -27.957 -19.228 -5.285
15.478
15.476
15.395
15.215
*5.716
43.475
41.353
38.732
1.909
1.909
1.903 ' 1.891
130.871 124.443 117.655 108.741
H15.436
15.505 J 5 . 5 0 2
V44.671
43.833
43.242
0»
1.906
1.911
1.911
"u"-T2T."482"T2*5"."737 124.021
1.784
-2.903
13.384
13.558
-9.899
-9.090
1.785
1.775
- 2 2 . 5 0 0 -24.782
H15.358 15.415
15.411
15.318 15.095
V- 51.303 47:573 43.567 39.271 33.548
l)«
1.901 1-905 1.904
1.8981.882
0- T4'5.567 135.552 124.100 111.094 93.368
HV«
'00«
.6
13.275
' 4.748
1.769
11.6 70
15.299
15.176
43.332
36.230
1.897' 1 . 8 8 8
0» 152.860 139.357 122.410
.5
13.846
11.390
1.802
29.06?
13.134
13.339
-2.881
-4.875
1 . 7 7 41.761
1.773
796 -7.017- - 1 2 . 0 3 2
H~S 14.760 14.468 14.004
V» 54-790 44.065 24.657
D-" T;"8"60" T."B41 1.812
0" 148.865 117.297 63.581
-
L •
_
AND FLOW(CC/SfcC)
13.289
13.543
9.600
10.716
' l."770
1.785
23.619
26.803
13.366
10.433
U«
1.845 1.815
"0"i" 1 3 1 . 8 9 8 "fifl.^lS 2 5 .
148.87
.4
1. 300000,
3 5. 610UUO,
P =
01AMETERS(CM),
.3
13.009
5.573
'1.754
13.469
J.752
9.164
F >
.400000,
12.959
2.891
1.751
6.964
1.748
OAVE •
13.600000,
12.945
.000
1.751
.000
1.794
61.359
103.88"
.2
00 <
14.000000,
VELOCITIESICM/SEL),
"r<V26T — "JITS*"
.016'
.100000,
HB «
HEAOSICMHG),
TIME
^
THICK -
13.119
1 1.4^2
1.760
-.316
13.162
1.763
10.776
13.656
2 7._549
1.791"
69.414
12.999
6.193
ll"754"
14.957
12.900
11.009
12.9^8
8.925
12.8/3
10.6b6
1.751
r"
1 . 74 I
13.0V6
15.9B4
1. 759
38.844
12.873
10.6d6
1.74 7
25.607
2)
FIGURE 8
Computer program and sample of calvuhttions for pulse flow into a flexible tube. Pulse
time 0.4 sec, unstressed tube i.d. — 1.3 an, h — 0.1 cm, Y = 5 X 106 dynes/cm*.
Problem teas started as a steady state floic irith flow of 35.65 ml/sec and head of 14
cm Mg at proximal end. Friction factor f = 0.3 and losses varying as square of the
velocity. The calculations are printed out for the third pulse after steady floic, with
values given for each 0.008 sec time interval (except .160 to .360 sec).
Circulation RatcarcJi, Volume XIII, July 1963
15
PULSATILE PRESSURE AND FLOW
95. 15
15.257
' 33". *86
15.458
15.453
40.119
15.376
42.558
15.235
45.195
15.035
47.817
49.912
14.781
14.*49
13.966
13.431
50.529 ~Ka.6l>~2 "toU'lV ~267690~
J..8.94
1.908 1.907
1.902
1.892 1.878 1.861
l,«40
1.811 !«_/_?$
95.153 114.678 121.604 128.410 134.445 138.312 137.489 127.740 103.589
66.269
112
14.900
15.314
15.382
15.309
15.179
15.005
14.797
14.549
14.236
21.867
36.025
42.089
45.947
49.574
52.501
54.100
53.496
48.837
1.869
1.898
1.902
1.897
1.888 1.876 1.862
1.846
1.826
60.010 101.884 119.630 129.900 138.821 145.168 147.373 143.201 127.934
25.25
14.485
9.476
1.842
25.252
128
14.971
15.243
15.220
15.107
14.959
14.790
26.831
40.230
46.356
50.728
54.168
56.279
1.874
1.893
1.891
1.883 1.873 .1.862
74.006 113.185 130.193 141.301 149.286 153.232
"
.144
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
,152
.160
.00
,360
.00
14.3U6
39.295
1.834
103.791
14.930
15.089
15.021
14.903
14.771
14.656
14.648
14.801
33.612
45.628
51.233
54.999
57.331
57.222
51.515
36.964
1.871
1.882
1.877 1.869 1.861
1.853
1.853 1.B63
92.440 126.930 141.825 150.970 155.894 154.335 1J8.860 100.722
14.521
24.715
1.844
66.032
14.821
14.912
14.841
14.757
14.745
14.903
40.711
50.651
55.024
57.123
55.345
46.689
1.864 1.870 1.865 1.860 1.859
1.869
111.092 139.118 150.361 155.169 150.211 128.151
15.171
32.113*
1.88H
89.ttd7
14.053
14.555
1.815
37.658
14.450
14.700
14.769
14.769
14.891
15.1/3
32.732
46.535
53.872
55.139
50.627
39.861
1.840
1.856 1.861
1.861
1.869
1.88C
87.019 125.900 146.466 149.910 138.848 111.567
15.4'JU
27.417
1.907
78.325
14.020
14.388
14.641
14.825
15.083
23.014
38.955
49.252
50.696
43.6B7
1.813
1.836
1.852
1.864
1.8fa2
59.410 103.121 132.698 i38.379 121.4B4
32.988
1 . 904
23.8/5
L-J!? 9
93.953
69.158
15.582
27.232
1.917
78.5/0
15.747
21.403
" 1.92 9
62.530
13.815
.000
13.799
5.199
_J_. 8JH
.000
1.800
13.223
13.774
10.237
1.798
25.995
13.679
.000
1.792
.000
13.674
5.56 7
1.792
14.044
13.654
10.400
1.791
26.201
13.698
16.326
1.794
41.250
14.026
14.456
14.881
15.2/7
35.793
28.922
40.102 42.365
1.868
1.895
1.813
1.840
74.691 106.661 116.103 100.950
13.507
7.545
1.782
18.826
13.601
15.155
1.788
38.047
13.709
22.433
1.794
56.723
13.794
28.587
1.799
72.690
13.840
32.857
1.802
83.803
13.857
34.743
1.803
88.716
13.8/4
33.861
1.804
86.562
13.913
29.862
1.806
76.536'
13.960
22.823
1.8U9
58.680
13.9/1
13.704
1.610
35.3H9
13.338
6.033
1.773
13.535
19.440
1.784
48.596
13.652
25.834
1.791
65.077
13.753
30.596
1.79 7
77.5U7
13.841
32.738
1.802
83.503
'if. 934"
31.724
1.808
81.422
14.042
27.678
1.814
71.557
"147138
14.890
13.421
12.558
1.777
31.161
21.464.
1.H2U
55.8t>5
147169'
14.248
1.822
3 7. l">3
J3.199
' ~."doo~
4.891
1.764
1.765
~7o66~ 11.965
13.264
10.333
1.768
25.382
13.370
16.378
1.775
40.506
13.506
22.351
1.782
55.766
13.659
2 7.02 7
1.791
68.111
13.821
29.159
1 .801
74.276
13.991
28.147
1.811
72.518
14.154
24.527
1.821
63.891
A* - 2 7 3
19.6^8
.1-82 9
bl.oid
14.310
14.5H9
1.831
3H.414
13.084
4.094
'" 1 . 7 5 8
9.942
13.133
8.552
1.761
20.830
13.227
13.465
1.766
32.997
13.374
18.385
1.775
45.481
13.570
22.302
1.7U6
55.878
13.799
24.108
1.800
61.323
14.035
23.387
1.814
60.435
14.36/
14.235
20.85B
17.7/0
1.834
1.826"
46.903
54.637
I4.3)b
14.741
I. 837
39.197
12.987
3.503
L-.75 3
.boo" 8.453
13.029
7.075
1.755
17.120
13. 118
10.702
1.760
26.046
13.275
14.114
1.769
34.692
13.502
16.729
1.782
41.729
13.778
17.988
1.798
45.691
14.054
17.872
1.815
46.244
14.273
16.968
1.82*
44.564
14.4U2
15.901
1.017
42.113
1 4 . 4 39
14.8 K
1.6.4 V
3 9 . b/.l
12.952
5.644
1.751
13.592
13.049
7.919
1.756
19.187
13.217
9.622
1.766
23.565
13.463
10.726
1.760
26.686
13.757
11.432
1.797
28.997
14.043
12.121
1.814
31.338
14.26b
13.012
1.82H
34.15">
14.396
13.967
1 4 . 4 16
14.Bo6
l.BJ'/
39.536
.368
13.17.7
13.069
.000
1.758
.000
,392
14.604
14.434
56.477 C J 2 . 5 4 O
1.850
l.Uiv
151.768 139.530
13.881
35.952
1.805
91.951
12.975
— V6W
12 '."8 94"
.000
1.748
.000
12.907
2.939
1.749
7.058
Figure
8
B. EQUATIONS FOR TAPERED DISTENSIBLE
VESSEL WITH DISTRIBUTED OUTFLOW
Since the vascular system is so complex,
and several computer statements are generally required for any boundary condition, a
special program has been developed for flow
through a vessel having its unstressed diameter varying with length along the tube, figCirculation Research, Volume XIII, July 1003
(part
15.409
15.634
1.83O
37.046
3)
ure 9, and with flow leaving the vessel in a
distributed manner. Such a vessel could represent the aorta, with branches of various
sizes along its length. The outflow along the
vessel is set up as a flow per unit length of
vessel, with rate proportional to the head difference inside and outside the tube.
16
STEEETER, KEITZER, BOHR
with Ao a function of x. Then
A
V A
V A
ao
A
"V
FIGURE 9
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
If q represents the outfloAV per unit length,
then
q = k, (H - HB)
(50)
in which k, has an assigned value for each of
the N sections of the vessel. The continuity
equation, equation 14, now has an extra term
- (pAV)xdx - pqdx = (pAdx)t
-a.' -gH
(51)
2a • a, = - g H x
Hence
Ax
, &[x
Aox
vVP
H
_
VR
q.
=
A*.-
(52)
A
+ VB
,
g
m
+ — (H, HR
+ Hs
+
p
^
8)
- Va
By assuming that the fluid leaving through
the walls has its axial momentum reduced by
contact with the branch, the momentum equation becomes
(
gH x
(53)
(54)
VP = V 8 + -£• (H P - H 8 ) a
which simplifies to
a
in which a is the rate of change of unstressed
vessel area per unit length. A t / A is obtained
as before.
Following the previous procedures in developing the finite difference equations, the two
controlling equations become (the interpolation equations VR, H R , V S , H S are unchanged) :
V P = V R - -i- (H P - H R ) - IYi
A_t
A
a
and
Tapered vessel with distributed outfloiu along the
walls. Floiv per unit length through i-th reach
shoivn. HB is terminal bed pressure.
+
a1
But
a1
y Ax_ y
A + x
°A
(55)
By addition, and then by subtraction, the two
equations yield values of VP and H P at interior points:
fVc 1 —Vc I A t
(56)
2aCaVc At _ 2a c 'qAt"|
Ao
Aoao
(57)
I
The boundary conditions are handled exactly as before, except that either equation
54 or equation 55 is used, depending upon
whether it is a right-end or a left-end bound-
- PxAdx - ro,rDdx = (pAV')x dx + (pAVdx)t + pqVdx
After reducing this equation in a manner
similar to equations 18 and 19, one obtains
equation 19 as before.
In order to take the tapering effect into
account Ax is evaluated as follows:
A = Ao^a3
ary, respectively. For actual computation
equation 50 is used to eliminate q from the
working equations. As very small time increments are generally used (0.001 sec), it is a
satisfactory approximation to allow H to
equal H c in equation 50, although this is not
absolutely necessary. With this type of distributed outflow (equation 50), each branch is
Circulation Research, Volume XIII, July 196S
17
PULSATILE PRESSURE AND FLOW
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
TIME . SEC.
FIGURE 10
Pressure-time graphs for two cross sections of the femoral artery of a dog. Distance
between sections 9.4 cm, proximal diameter 0.316 cm, distal diameter 0.228 cm. Branch
arteries between the two sections are ligated.
treated as if it were a terminal bed, i.e., a
definite relation between pressure H and flow
q is established for each branch.
Comparison of Measured Flow in a
Tapering Vessel with Flow Computed
from Pressure-Time Data
To compare flow calculated from experimentally measured pressures with measured
flow, an in vivo experiment was performed.
Pressure and flow data were obtained from
the femoral artery of a dog. Pressure was determined by inserting two short needles in
the artery at points 9.4 cm apart. Small
branches between were ligated at the vessel
wall. Suitable catheters and fittings were connected between the needles and two Sanborn
manometers. Flow was measured simultaneously by means of an electromagnetic flow
probe placed immediately upstream from the
proximal pressure needle. Data were recorded
on a four-channel Sanborn (350) system at a
chart speed of 100 mm/sec. Figure 10 illustrates the measured proximal and distal pressures obtained for one cycle.
Circulation Research, Volume XIII, July 1968
The flow recorded for the same cycle is
shown by the solid line in figure 11, and the
flow calculated by using the two measured
pressures as boundary conditions is shown by
the dashed line. Pressure-time values were
read off the strip chart for each 0.01 sec. By
parabolic interpolation these data were replaced by values for 0.001 sec intervals over
the pulse cycle. The 0.001 sec intervals afforded
a finer mesh for computation of flow by the
method of specified time intervals already described. During the experiment the diameter
of the artery and the thickness of the wall
were measured with calipers at the two pressure sites. For the computation the diameter
of the artery was assumed to decrease linearly
along seven equal reaches of the 9.4 cm length.
The flow was assumed to be steady initially
at about average value, with the head (pressure) constant along the artery. After two
cycles have been computed the calculated flow
pulse has almost achieved its steady oscillatory character.
The calculated flow does not coincide ex-
18
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
19
PULSATILE PRESSURE AND FLOW
ring in pulsatile flow will be asjinmetrical;
that is, the losses where the flow is in a backward direction will be greater than those
where the flow is forward.
In making the calculations for the illustration here the friction term was expressed
f
D 1
OT^ V c I V c I ~ . By a gradient method, val-
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
ues for f and n were obtained that gave the
best fit (least square method) with the flowmeter data. These results yielded values for
f of about 0.4 and for n about 2.0. These
values for f and n are not great enough to
express the increased losses when the flow
reverses. If this hypothesis is correct, this
asymmetrical discrepancy between the calculated and recorded backward flows would be
correctable by finding appropriately larger
values for f and n when the sign of the
friction changes with reversal of the flow.
Summary
The basic differential equations for elastic
wall material and for continuity and momentum are derived, including fluid frictional resistance of the wall of the tubes, based on
one-dimensional flow. These partial differential equations are transformed into four ordinary differential equations using the theorj'
of characteristics. Then difference equations
are developed and by an interpolation method
(method of specified time intervals) equations
are obtained for computation of velocity and
pressure at equally-spaced sections along the
vessel at specified equal time intervals. The
equations are first applied to a flexible tube
of initial constant diameter, with a pulse flow
taken from in vivo experiments.
Equations are then developed for tapering
tubes with distributed outflow along their
lengths (to simulate branches). Pressure-time
data from femoral artery measurements are
then used to compute flow through the artery,
and the results are compared with electromagnetic flowmeter data. Computed flow is
greater than measured flow if a friction factor
for laminar flow is used. Frictional losses of
energy in the normal pulsatile flow in the
femoral artery are similar to those of turbulent flow.
Circulation Research, Volume XIII, July 1963
Acknowledgment
The authors are indebted to the University of
Michigan Computing Center for use of its digital
computer.
References
1. WOMERSLET, J. E.: An Elastic Tube Theory of
Pulse Transmission and Oscillatory Flow in
Mammalian Arteries. WADC Technical Beport
Tr 56-614, January, 1957.
2. MCDONALD, D. A.: Blood Flow Through Arteries,
Baltimore, Williams & Wilkins Co., 1960.
3. WARNER, H. R.: Use of analogue computers in
the study of control mechanisms in the circulation. Federation Proc. 21: 87, 1962.
4. BEMINGTON, J. W.: Contour changes of the
aortic pulse during propagation. Am. J.
Physiol. 199: 331, 1960.
5. LAMBERT, J. W.: On the nonlinearities of fluid
flow in nonrigid tubes. J. Franklin Inst. 266:
83, 1958.
6. COURANT, B., AND FRIEDRICHS, K. O.:
Super-
sonic Flow and Shock Waves, New York,
Interscience Publishers, 1948.
7. STOKER, J. J.: The formation of breakers and
bores. Communications on Applied Mathematics,
1: no. 1, 1948.
8. LAI, C.: A study of waterhammer including
effect of hydraulic losses. Doctoral Dissertation, University of Michigan, 1961.
9. STREETER, V. L., AND LAI, C.: Waterhammer
analysis including fluid friction. J. Hydraulics
Division, Proc. ASCE, Paper 3135, HT3, May,
1962, pp. 79-112.
10. STREETER, V. L.: Waterhammer analysis with
nonlinear frictional resistance. Conference on
Hydraulics and Fluid Mechanics, University
of Western Australia. In press, presented
December, 1962.
11. PETERSON, L. H., JENSEN, R. E., AND PARNELL,
12.
13.
14.
15.
16.
J.: Mechanical properties of arteries in vivo.
Circulation Res. 8: 622, 1960.
BURTON, A. C.: Relation of structure to function
of the tissues of the wall of blood vessels.
Physiol. Rev. 34: 619, 1954.
BERGEL, D. H.: The static elastic properties of
the arterial wall. J. Physiol. 156: 445, 1961.
STREETER, V. L.: Fluid Mechanics, ed. 3, Xew
York, McGraw-Hill Book Company Inc., 1962,
pp. 210-222.
LISTER, M.: The numerical solution of hyperbolic
partial differential equations by the methods
of characteristics, in Mathematical Methods
for Digital Computers, Xew York, John Wiley
and Sons, 1960.
RODBARD, S.: Physical principles of liquid flow,
in Cardiovascular Functions, A. Luisada, ed.,
Xew York, McGraw-Hill Book Company Inc.,
1962, pp. 2-139.
20
17.
STREETEE, KEITZEE, BOHR
BRAMWELL, J.
C,
DOWNING, A.
C,
AND HILL,
A. V.: Effect of pressure on the extensibility
of the human artery. Heart 10: 289, 1923.
18.
BRAJIWELL,
J.
C,
MCDOWALL, R.
J.
S.,
AND
MCSWINEY, B. A.: Variation of arterial elasticity with blood pressure in man. Royal Soe.
Proc. Series B 94: 450, 1923.
19. KING, A. J.: Waves in elastic tubes: Velocity
of the pulse wave in large arteries. J. Appl.
Phys. 18: 595, 1947.
20.
BOHR, D. F., GOULET, P. L., AND TAQUINI, A.
C,
JR. : Direct tension recording from smooth
muscle of resistance vessels from various organs. Angiology 12: 478, 1961.
Appendix
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
Symbols
a = speed of pressure pulse through vessel
ao = speed of pressure pulse through vessel at zero
pressure
A = area of vessel cross section, point on x-t plot
Ao = area of vessel cross section at zero pressure
B = point on x-t plot
C = designation of characteristic curve, point on
x-t plot
D = diameter of vessel
Do = diameter of vessel at pressure zero
f = Darcy-Weisbach friction factor
F = force on fluid element
g — acceleration due to gravity
h •=. thickness of vessel wall
h0 = thickness of vessel wall at pressure zero
H = pressure expressed in length of column of fluid
flowing
k = constant
L = label for a differential equation
m = exponent
P = pressure, or point to be computed
q = outflow per unit length
Q = flow through vessel
R = known point on characteristic curve
R = Reynolds number
S = tensile stress in vessel wall; known point on
characteristic curve
t = time
T = tensile force per unit length in tube wall
V = velocity in vessel
x = distance along vessel
Y = elastic modulus of vessel wall
a = rate of change of unstressed vessel area per
unit length
6 = ratio At/Ax
X = undetermined multiplier
fi = dynamic viscosity
p = density of fluid
To = frictional wall shear stress
Notation for Michigan Algorithmic
Decoder (MAD) Program
A
= speed of pressure pulse at pressure HO
AC
= speed of pressure pulse at pressure H
AO
= speed of pressure pulse at zero pressure
ACM = speed of pressure pulse at pressure H ( N )
Cl, C2, C3, C4, C5, C6, C7 = constants
D
= diameter
DELT = time increment
DHF
= steady state pressure drop in reach Ax
F
= friction factor
G
= gravity
H
= pressure, from previous calculation
HO
= initial pressure
HB
= bed pressure
HP
= pressure to be calculated
HR
= interpolated head
HS
= interpolated head
I
= integer
K
= constant in terminal bed relation
KK
= constant
N
= number of reaches
P
= constant
PMT, PPT, PTT = constants in computing pulse
P T I M E = pulse period
Q
= flow
QAVE = steady-state flow
QM
= maximum pulse flow
RHO
= density
SG
= specific gravity of mercury in terms of
fluid flowing
T
= time
TM
= constant in computing pulse
TJ
= integer
V
= velocity, from previous calculation
VP
= velocity to be calculated
VR
= interpolated velocity
VS
= interpolated velocity
Circulation Research,
Volume. XIII, July 19G3
Pulsatile Pressure and Flow Through Distensible Vessels
VICTOR L. STREETER, W. FORD KEITZER and DAVID F. BOHR
Downloaded from http://circres.ahajournals.org/ by guest on June 18, 2017
Circ Res. 1963;13:3-20
doi: 10.1161/01.RES.13.1.3
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Copyright © 1963 American Heart Association, Inc. All rights reserved.
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