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A simple mathematical model of muscle-induced ejection flows
MATHEMATICAL
BIOSCIENCES
A Simple Mathematical Ejection Flows
13, 335-359
(1972)
335
Model of Muscle-Induced
A. HELFGOTT AND E. 0. TUCK Department of Applied Mathematics,
University
of Adelaide
AND
R. J. CRAIG AND P. S. HETZEL Cardio-Pulmonary Investigational Unit, Royal Adelaide South Australia
Communicated
Hospital,
by F. Grodins
ABSTRACT Problems involving expulsion of fluid from a container due to the action of muscles in the walls of the container, are analysed by use of a highly simplified model. The resulting inhomogeneous linear wave equation is solved for a particular class of muscle contraction time histories, enabling prediction of period and total volume of outflow and exit velocity pulse shapes. Applications in physiological contexts are indicated.
1. INTRODUCTION
There are a number of important physiological situations in which a fluid is caused to move due to the action of muscles in the walls of a container. Of primary importance is the expulsion of blood from the left and right ventricles of the heart. Other examples include ureteric transport, uterine contraction, urine expulsion from the bladder, bowel motility, ejaculation from the male reproductive system, and, in the more general biological field, squid-like propulsive mechanisms. In this paper the action of the left ventricle will be examined whenever a specific example is required. However, the same general principles apply to other similar muscle-induced ejection flow systems. At first sight if we wish to construct a theoretical model for these processes, we are confronted with a problem of overwhelming complexity. Little is known (although much has been written!) about the way muscles really work, and they, after all, are responsible for the whole flow. Even if we knew enough about the mechanical properties of individual muscles, we should still be faced with problems associated with the complex distribution and orientation of muscle fibres in the walls of the container, and with Copyright
0 1972 by American Elsevier Publishing Company, Inc.
336
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND P. S. HETZEL
the intricate sequence of electrochemical events which programs their contractions. (See MacCallum [I], Streeter et al. [2] and Durrer et al [3] for the left-ventricular situation.) From the fluid point of view we should be concerned with the nonNewtonian viscous properties of physiological fluids and the fact that they are moving in three dimensions in a vessel of complicated and timevarying geometry. Finally there is the question of exit conditions at the outlet from the container. If we have in mind applications to the left ventricle, we should attempt to model the effect of the aorta and, indeed, of the whole arterial system, on flow conditions in the ventricle itself. The approach to be adopted in the present work is of simplifying the problem to the greatest extent consistent with retention of features which are thought to be significant in determining expulsion characteristics. The essential assumptions to be made are outlined in the following section; further non-essential assumptions are also made later in order to simplify the mathematical analysis of special cases.
2.
RANGE
OF INVESTIGATION
AND
ASSUMPTIONS
A recent paper on muscles by Apter and Graessley [5] provides up to the present time perhaps the most satisfactory macroscopic mechanical model for muscle behavior. This subject presently attracts considerable attention, and there seems little doubt that better models will be developed in the future. However, the muscle model in [5] is already too complicated to put into the present framework, and we propose here a linear muscle model consisting of a linear elastic spring and a contractile element coupled in series. This simplified muscle model is discussed in some detail in the next section. Much work has been done in blood rheology (reviewed by Fung 161). However, we take the liberty of neglecting this aspect altogether, assuming an inviscid incompressible fluid. This is a good approximation for leftventricular expulsion, which is at a relatively high Reynolds’ number, and may also be justified in some of the other applications mentioned. Further, we neglect gross geometry variations in the container by assuming it to be a long cylindrical tube in which the flow takes place essentially in one direction only, namely along the axis of the tube. (Fig. la.) The exterior and interior surfaces of the tube are tethered so as to prevent longitudinal motion, so that the tube wall can move only in a radial direction. Longitudinal stresses in the tube wall are neglected, and so is the inertia of the tube. One-dimensional flows of this kind where the container walls possess passive elastic or viscoelastic properties have been used to model arterial
MUSCLE INDUCED
EJECTION
331
FLOWS
blood flow (see review by Rudinger [7]), and a characteristic of this approach is that there results a set of hyperbolic partial differential equations, similar to those of gas dynamics. The (passive) elastic properties of the wall provide energy storage, which leads to a wave-like behavior
#‘-------r
i
p(X,t)
5= CONSTANT
I
4
ClRCUMFEREhlAL
MUSCLE
FIG. I. (a) Geometrical arrangement of the muscle tube, (b) excitation of circumferential muscles at speed V.
(“hydro-elastic” magnitude
waves)
with
a speed
(for
small
amplitude
waves)
of
(1) where E is the Young’s modulus, 6 the thickness and yg the radius of the container wall, while p is the fluid density. This wave phenomenon with speed c given by (1) is also apparent in the present model of active flow induced by the living energy-generating walls of the muscle container, in which we take into account the forcing effect of the muscles. Next, we turn to the electrical events which activate the muscle tube. Muscles are turned on (activated) by an electrical voltage signal (activation
338
A. HELFGOTT,
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
potential) which depolarizes the membrane surrounding them (Fig. 2). This in turn leads to electrochemical activities, resulting in subsequent muscle contraction. We shall not discuss the electrochemical mechanism triggered by the activation potential but instead, in Sec. 3, a procedure is described for modelling the net result of electrochemical events. In the present analysis it is assumed that every circumferential muscle fibre along the tube has a “switch” that is turned on by an electrical excitation signal travelling at speed V from the closed end to the open end of the tube (Fig. lb.) The turning on of the “switch” indicates the start of electrochemical activities, followed by contraction of the muscle fibre at location x.
FIG. 2. (a) Schematic representation of the linear muscle model, (b) pre-determined program of individual muscle activation.
This leads to the definition meters :
of two important
non-dimensional
para-
i. = V/c
(2)
/I = CT/L,
(3)
and where T (Fig. 3) is a measure of the time taken for an individual muscle to contract fully, while L is the length of the tube. In the present work we investigate the influence of such parameters as 1. and p on the flow pattern and the performance of the muscle tube. In the analysis, i plays a role similar to the Mach number in flight through the atmosphere, and p is the ratio of Tto the time needed for the “hydroelastic” wave to travel the whole length of the tube.
MUSCLE INDUCED
EJECTION
339
FLOWS
Analogue models of the complete cardiovascular system have been formulated (see [4]), but the modelling of the actual pumping mechanism has been of minor interest in these studies. On the contrary, in the present work this is the crucial point, and we choose to neglect the effect of the
t+$.
bm
of arrival
potential
d Activation-
at X
FIG. 3. (a), (b) and (c) Various idealized individual muscle activation programs, (d) wave of activation travelling at speed V, (e) three-dimensional representation of a pressure excitation P(x, t) based on the activation wave of Fig. 3(d).
arterial system on the ejection process, replacing the true exit conditions at the open end of the tube by a constant pressure condition. This pressure Pexit is of course not necessarily identifiable with the external or ambient pressure pa outside the wall of the muscle tube (Fig. la). For example in
A. HELFGOTT,
340
E. 0. TUCK,
R. J. CRAIG
the case of the left ventricle, Pexit is the aortic pressure, same as the pericardial pressure pa. 3.
THE LINEAR
MUSCLE
AND P. S. HETZEL
which is not the
MODEL
In the present work we adopt the simplest possible linear muscle model. From the physical point of view, it consists of a linear elastic spring (SE) of spring constant k per unit cross section area, coupled in series to a contractile element (CE) (Fig. 2a). In the literature it is possible to find many suggestions for muscle models, consisting of various combinations of springs, dashpots, and contractile elements. We have avoided unnecessary complications by choosing a simple model which nevertheless, as we shall see, possesses some of the most important properties of muscle. It is easy to see how this model produces tension or does mechanical work. For example, if we hold the muscle at constant length I and allow the contractile element to shorten, the spring will be stretched, thus generating tension in the system. On the other hand, if we attach a weight W to the muscle model and allow the contractile element to shorten by the amount Al, work WA1 will be done, while the spring remains at constant length. These two extreme examples are known as “isometric” and “isotonic” contractions, respectively. It is evident that the contractile element is essentially an agency capable of shortening. The actual force developed in the model depends mainly on what we couple to the contractile element, e.g. on the spring, in the isometric experiment. In order to proceed with the analysis of model properties and the derivation of the governing equations, it would be desirable to have answers to two important questions, namely: (a) How is contraction initiated? (b) How does muscle contract? The answer to the first question is known in general. The application of an activation potential to the membrane surrounding a muscle fibre (Fig. 2a) triggers a very involved set of electrochemical events that eventually leads to mechanical contraction. However, regarding the second question, no final answer is known, and we choose to model only the net result of all electrochemical events in the following way. We start with the following c I z y z, y0
= = = = = =
definitions
(Fig. 2a)
stress in the muscle fibre, length of muscle fibre, length of the contractile element, length of the series elastic spring, the value of z for a completely relaxed (non-activated) the value of y for an unloaded muscle (G = 0).
muscle, and
MUSCLE INDUCED
EJECTION FLOWS
341
We now suppose that at time t = t,, an activation signal (or a train of such signals) is applied to the membrane. The result is that the length z(t) of the CE begins to decrease, and we make the assumption that z(t) is a giuerz function of time, in particular not depending on the instantaneous muscle load a(t). This could be achieved mechanically by a rack-and-pinion system, in which the pinion has a pre-determined time-dependent angular position, as in Fig. 2b. The length 10r of an unloaded and relaxed muscle is In any other state the length fibre are
ZO,= Yo + z,* I and the “external”
strain
(4) E of the muscle
z=y+z
(5)
and
respectively. For a muscle of unit cross-sectional given by the stress in the spring
area the stress o at any instant
o = kAy = k(y - y,),
is (7)
and, from (4) and (5) we have Ay = y - yo = (f - Z,,) + (z, - z). (8) This means that the elongation of the spring is given by the change in length of the whole muscle fibre plus the shortening of the contractile element. This can also be obtained from inspection of Fig. 2. Finally, from (6), (7): (8) we get o = k(l - lo,) + k(z,. - z) = E.2 + .f(f),
(9)
where E = kl,, is Young’s
modulus,
(10)
and
(11) f(t) = 0, - 4t)l. is precisely the tension (stress) developed in the muscle during isometric (constant length) contraction. It is a given arbitrary function, determined by z(t). The stress-strain relation (constitutive law) for muscle given by (9) is an inhomogeneous linear Hooke’s law. In some cases, as we shall see later, it is more convenient to define the “external” muscle strain a* as The functionf(t)
A. HELFGOTT,
342
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
1,
EE, = E r
( The relation
between
E and
1 . >
-
Or
(13)
is derived from (6), (12) and (13), thus
E*
I 0’s E = E, + A&* = E, + - + I &*. 1Or (E ) Substituting
(14)
(14) into (9) we get Ao = d - gs = (E +
b,)E*
+
f(t).
(15) In order to make use of (9) or (15) in the generation of muscle-induced ejection flows, we must make some assumptions about the form of the forcing termf(t) which determines the pattern of flow. In the following work we leave,f(t) as a general function of time as far as possible. When we do want to specify f( t) it is appropriate to take the simplest possible forms, consistent with the simplified model (9), but representing the essential features of muscle excitation. Three elementary forms off(t) will be mentioned. Case 1.
The simplest
excitation
function
f(t) where H(t) is Heaviside’s That is
of all is (Fig. 3a)
= f,H(t),
(16) and f. is a given constant.
unit step function 0
t
f(t)= i f. t > 0.
(17)
This corresponds to a situation where at time t = 0 the muscle contracts instantaneously to a maximum level, producing a stress increase fo. For t > 0 the muscle remains fully contracted (tetanized). Case 2. The muscle is allowed to complete contraction shortening T (Fig. 3b) and then remains tetanized. 0, S(t)
=
I
t
<
0,
fO;,
0
<
t
<
f 02
t
>
T.
in a finite time of
T,
(18)
This reduces to (17) as T -+ 0. Case 3. The muscle is allowed to contract, and then relax, as shown in Fig. 3c.
remain
tetanized
for a while
343
MUSCLE INDUCED EJECTION FLOWS
t < 0,
’ 0,
O
f(l)=
fcb
T
T, -
foT2
-
t T,’
T,,
(19)
7.1 < t d 7.2, t > T,.
0,
All these cases have some degree of validity in various applications, e.g. in cardiac muscle a mechanism similar to (19) causes the muscle to contract and relax. More general models are discussed by Helfgott [IO]. From (11) we obtain
the velocity V CE
of the contractile
_;dfw
=dL(t)= dt
k
dt ’
which is also a given function. It was mentioned tension development in an isometric contraction. l-l,=-
$-f(t)
element,
s
already Further,
= - %f(t)
(20)
that f(t) is the from (I 5),
(21)
is the amount of shortening in an isotonic (constant load) contraction. The above is not only a grossly simplified representation of what happens in actual muscles, but falls far short of mathematical models which have already been postulated in the vast physiological literature on this subject. However, it appears that for our purposes the above will suffice, and it certainly has the advantage of analytical simplicity. The model (9) does represent two important aspects of muscle behavior, namely, energy production (shortening element) and energy storage (spring), but does not appear to include either the dynamic stress-velocity property of muscles (Hill [8]) or the dynamic dependence of isometric tension development {f(t)} on Z, (or initial load), because such circuit elements as a dashpot and non-linear elasticity are missing. However, these effects can if required be incorporated to a certain extent in a linear model performing small amplitude motions about an equilibrium condition. This is done [e.g. with f(t) given by (IS)] by allowing the parameters E, T, and f0 to vary according to the equilibrium values I, and 0,. For example, the velocity of the contractile element v
=
CE
_
r.-df(t)
k
dt
= -- “f-0 kT
for
0 < t < T,
344
A. HELFGOTT,
E. 0. TUCK,
made to depend in an arbitrary ment of.f,, k, and T.
R. J. CRAIG
AND P. S. HETZEL
way on 1, and o’sby suitable
adjust-
4. THE LINEAR FLOW MODEL The vessel in which flow takes place is assumed (See Fig. 1) to be a thin-walled circular cylinder composed of muscles arranged circumferentially. The cylinder wall contracts during muscle excitation, but we shall assume that such displacements are small perturbations of the rest state, and likewise induce flow velocities which are small. Thus, if the equilibrium radius of the cylinder is r0 = ZJ27r we define the perturbation to this radius as 1-l a = a(x, 1) = 2 2n so that the instantaneous radius aneous cross-section area is
w
’
is r = r,, + a = 112~~and the instant-
S = 77(ro + ~2)~. If u = u(x, t) is the fluid velocity, (continuity)
we obtain
(23)
for conservation
(A%), + s* = 0, and for conservation
of momentum
of mass (24)
(Euler)
1 u, + UU, = --p,, P
(25)
where p(x, t) = pi(X, t) - Pexit is the rise in hydrodynamic pressure over the constant exit pressure Pexit and p the fluid density (assumed constant). Equations (24) and (25) assume that the fluid is inviscid and incompressible, and that the flow is effectively one-dimensional along the cylinder, so that the lateral velocity components may be neglected compared with U. However, at this stage we have not yet assumed small disturbances, and the above equations may be used in a more general non-linear study of ejection processes. (See Helfgott [lo].) If we do now assume small displacements of the tube walls and correspondingly small values for U, we have from (23) and (24) nr&,
+ 2nrou,
= 0,
or a, = -+r+4,,
(26)
1 u, = --px.
(27)
and from (29,
P
MUSCLE INDUCED
345
EJECTION FLOWS
Equations (26) and (27) are the basic flow equations, supplemented by a relationship between p and a obtained properties. 5.
THE INHOMOGENEOUS
WAVE
and need to be from the muscle
EQUATION
We now suppose that the walls of the tube are of small thickness 6. This, together with the assumptions made in Sec. 2, leads to a simple static equilibrium relation between the hoop stress (stress along the circumferential muscle fibre) and the pressure difference across the wall of the tube (excess internal pressure).
Before the tube starts contracting, p = 0, and the tube is pre-stressed the pressure difference Api = pexit - pa to a level given by
From (15), (28) and (29), neglecting (e.g. pa), we obtain Aa = r~ - a, = ?p
by
terms of second order of magnitude
+ 2
= (E + a,)&* + j-(x, t),
(30)
where f can vary from one muscle fibre to the other along the tube. With s* = a I’0’
(31)
we obtain p = % 4
+ P,
(32)
where P = :f(x,
t) = P(x, t),
(33)
is the forcing pressure. Equation (32), which plays a role analogous to the equation of state in gas dynamics (a’ being analogous to the density variation in that case), is a simple linear relationship between pressure and radius changes, with a given forcing pressure P (analogous to a heat energy source in a burning gas). Notice that there is no direct effect of the pre-stress 0, of the tube on Eq. (32); this follows from the linearization and may not be strictly correct for all physiological situations, such as the function of the left ventricle. 23
346
A. HELFGOTT, E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
Using (32) to eliminate
a(~, t) from (26), we have r;
i.e.,
Either giving
pt = P, - %,. (34) 2ro u or p may now be eliminated from the pair of Eqs. (34) and (27), Ptt - C2Pxx = Pm
(35)
and 1 2 % - c u,, = -- PL where Ed c2 = __ 2P7-0’ Equations (35) and (36) are inhomogeneous linear one-dimensional wave equations, the wave speed c being defined by (1). This speed is simply the speed at which hydro-elastic waves travel along the wall of the tube in the absence of muscle excitation. The forcing terms on the right of these equations control the flow, and represent the total influence of muscle activity. The problem having been reduced to such a familiar partial differential equation, we can call upon a great variety of well established mathematical techniques and solutions. For example, such equations occur in the theory of transmission lines (Jaeger and Newstead [9]) and we may re-interpret transmission-line solutions in terms of the present problem. However, in the present paper we treat only a sample problem for which the solution is obvious by inspection. 6.
BOUNDARY
AND INITIAL CONDITIONS
In order to model accurately the real flow situation it is in principle as important to supply the correct boundary and initial conditions as it is to use the correct flow equations. However, once again we take the liberty of grossly simplifying these conditions, which in the case of boundary conditions would be extremely complicated non-linear relationships between the pressure and velocity time histories at the ends of the tube. We now assume that the tube extends from x = 0 to x = L, and that the left end is closed, i.e. u=O
at
x=0,
(37)
347
MUSCLE INDUCED EJECTION FLOWS
while the right end is held at the constant pi = Pexit =
exit pressure Pexit, i.e.
C0IlSt.p
(38)
or
p = pi - P&t = 0 at X = L. (39 In view of the Euler Eq. (27), the condition (37) on the velocity is equivalent to the condition px=O at x=0 (40) on the pressure. More accurate representations of the real physiological boundary conditions are of course easy to envisage and to formulate for any specific problem. For example, in situations such as left ventricular ejection we may be unhappy about the constant pressure condition (38), and indeed there is no doubt we can do a much better job of modelling the impedance offered by the aorta and the whole arterial system to flow from the aortic valve. However, this would be contrary to the spirit of the present paper, and is left for future work. (See Helfgott [lo].) As far as initial conditions are concerned, we start the process from an equilibrium state, in which the fluid is at rest at the exit pressure, i.e. at t=O u(x, 0) = 0, (41) and P(X, 0) = pi(X, 0) - &it = 0. (42) Again, by use of the continuity equation (26) and muscle equation (32) we may eliminate the velocity, obtaining initial conditions on the pressure alone, replacing (41) by pr(?c, 0) = P,(x, 0). (43) The use of a state of rest as an initial state may also be questioned in applications to the left-ventricular ejection, which is preceded by an isovolumetric phase in which the pressure builds up, with the aortic valve closed. However, little internal movement occurs in this phase and if we use the pressure at the instant of opening of the valve as our exit pressure, the above initial conditions remain appropriate. Finally, a terminating condition ending the flow process is required. Clearly the object of the muscle-walled container is to discharge or transport its content of biological fluids from one location to another. Hence, only the case of out&ev from the container is important and will be treated here. From the mathematical point of view, we allow the solution to proceed as long as u(L, t) > 0 for 0 < t < t,, (40 but consider the problem terminated the first time since t = 0, ve have
at the instant
u(L, t,) = 0.
t = t, > 0 at which, for (45)
348
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND
P. S. HETZEL
In biological systems where an actual valve is present condition (45) models its closure reasonably well, e.g., aortic valve closure in the left ventricle. In other cases, we consider that the flow model is realistic only for 0 < t < t,, and must look to a more accurate model for subsequent events. 7. THE SOLUTION FOR A TRAVELLING WAVE OF EXCITATION Certain final assumptions which have to be made before we can obtain solutions to (35), (36) concern the program of electrochemical activation of the entire muscle tube. Clearly, many kinds of muscle-induced flows can be generated, depending on the choice of the forcing function P(x, t). In the present work we shall only be concerned with muscle tubes in which all muscles are identical to each other, i.e., the individual muscle activity law ,f(t) is the same for all muscles. Further, the solution to be presented below is for the case P(X, t) = F(t - x/V),
(46)
which determines the x-dependence of P(x, t). This corresponds to a wave of muscle activity, starting at the closed end x = 0 at t = 0, and moving toward the open end x = L at uniform speed V. The function H(t) = Sf(t)/r,, is identified with P(0, t) or in fact at any fixed x with the individual muscle activity law. This means that every muscle is identically activated, but is stimulated at an instant t = t,, = x/V which differs from its neighbor. Note that it has not yet been necessary to specifyf(t), and hence F(t) remains an arbitrary function. Geometrical representation of (46) yields cylindrical surfaces as shown in Figs. 3d and 3e for,f(t) given by (18) or (19). The interesting possibility of examining analytically pathological defects in muscle performance by allowing more general dependence on x in the linear model is left for future work. For the specific case of the heart, muscle damage, together with defects in the conducting system, all leading to changes in the normal P(x, t), are treated numerically in some detail by the non-linear model of Helfgott [lo]. The exact solution of the wave equation (35) for the pressure difference p(x, t), subject to the boundary conditions (39) and (40) and initial conditions (42) and (43) is
PC% t>=
gJ [F(t -
-j5
x/V) _p
(
t-X-_c
L
L
c
r; )
(,+.y_L._b
+p” (
c
c
,+x2-4 c
c
v> v1
MUSCLE INDUCED
EJECTION
349
FLOWS
+p
(I_T_3L_$ 1 (1+“_5L_; 1 p ( > -,(t+g$ ’ -f5
c
---
VC
p-2 -
(9
.
c
.
.
.
c
.
.
.
.
.
.
c
.
>
.
.
t-”
[
c
+;_g
+P(r+;_!c)
+ . . . . . . . . . . . .
(47) 1 This solution is obtained by Laplace transform methods in Appendix A; however it has a simple physical explanation, and its correctness may be established by direct verification, as follows. The term in &t - x/V) contributes a particular integral of the inhomogeneous wave equation (35), whereas all other terms are hydro-elastic waves, i.e. solutions of the homogeneous equation, so that certainly (35) is satisfied. The middle column of terms represents hydro-elastic waves travelling to the right while the last column represents left-travelling waves. Each pair of terms is obtained from the pair above it by reversing sign and delaying by the interval 2L/c for passage of one hydro-elastic wave back and forward, and this process continues indefinitely, generating an infinite series solution. The initial conditions (42) and (43), are satisfied in the present example because the argument of all P” functions is negative at t = 0. The boundary condition (39) is clearly satisfied by horizontal cancellation of terms, while the boundary condition (40) is satisfied by diagonal cancellation of terms, apart from the first term inside each bracket. These terms themselves combine to satisfy (40). Physically, equation (47) indicates that, starting at x = 0 at time t = 0, two waves proceed down the tube toward x = L, one of which is the imposed muscle excitation signal at speed V and the other the hydroelastic wave at speed c, generated instantaneously by reflection at the closed end x = 0. The net effect is a pulse of positive pressure and velocity travelling toward x = L. When the leading edge of this pulse reaches x = L, it is reflected as a hydro-elastic wave with negative pressure, but still positive fluid velocity. The reflections from both original waves arrive back at the closed end x = 0 at a time L/c after they reached the open end. Thus, the wave which started as a muscle excitation (first square bracket of (47)) arrives back at t = L/V -t LJc, while that which started as a hydro-elastic wave (second
350
A. HELFGOTT, E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
square bracket) arrives back at t = 2Llc. These again by the closed end, and travel back as waves velocity to the open end again, and so on. The expression for the velocity U(X, t) is easily by solving the wave equation (36) or from the p(x, t), using the Euler equation (27). We find u(x, t) =
waves are now reflected of negative pressure and obtained, either directly, solution for the pressure
,(v:_c2)p(++~) +P(t_f_!+o+pL) _,(,_g+(t+;-!&) + . . . . . . . , . . 1
+
V2
pc( v2 - 2)
C
-p”
(,_x_L_4 ) (t+,“-“-” 1 ( ) (,+x_?E_4 1 c
+F
-
t-;_+;
c
v
+p”
V
-p
c
c
. . . . . . . . . .
.
v>
(48)
Notice from the first two terms of (48) that, irrespective of which signal travels faster (i.e. of whether V < c or V > c), the first velocity pulse is positive, because of the multiplying factor k’//lp(V - c’). We are particularly interested in the flow through the outlet at x = L, i.e. from (48)
MUSCLE INDUCED
351
EJECTION FLOWS
where 1 = V/c is the ratio
of the excitation speed to the hydro-elastic wave speed. In fact, although formally (49) is an infinite series, we shall observe in the following section that u(L, T) necessarily takes negative values at times earlier than the smaller of 5L/c and L/V + 4L/c. Hence we use the solution only at times when the first four terms of (49) are sufficient. Physically, this simply means that the process is terminated by arrival of a negative velocity signal after only one reflection from the closed end, and we discontinue the solution before further reflections take place. 8.
OUTFLOW TIMES
We now investigate analytically the time t, of termination of outflow at x = L. To keep this analysis tractable, we consider only the particular case when the muscle activation law is given by (18) i.e., the muscle becomes tetanized (a state of sustained contraction) at t = T. Hence, the forcing pressure signal is (0, P(t) =
df(t) __
zz
r.
I
Pot -
t < 0, o
<
t
T’
\POT t >
<
”
T
(50)
T,
where PO = Gfo/ro. The more complicated signal (19) including relaxation (Fig. 3c) will be considered in future work. although analytic determination of t, is then unlikely. The determination of the outflow time t, from (49) subject to (50) is quite straight forward and only the results will be presented here. The time t, depends on the two parameters 3, = V/c and ,LL= CT/L and we can identify six regions of the (%, p) parameter plane. A sample calculation is given (for region III) in Appendix B. Region I
The arguments of the first four terms in (49) lie between 0 and t = t,, while the arguments of all subsequent terms are less than or equal to zero. In this case we have
Tat
C&/L = 4 + l/A,
(51)
and the region is defined by either i > 1,/l > 4,
(52)
2 < 1, /l > 3 + 1/n.
(53)
or This region (and Regions V, VI to follow) is one in which conditions are clearly unfavorable for optimum expulsion, since some muscles near x = L have not yet fully contracted when the process is terminated.
352
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND P. S. HETZEL
Region ZZ
The first term in (49) has argument greater than T(“saturated”), the next three have arguments between 0 and T, and all others have argument less than zero. Now C&/L = 2 + &f - (1 - +)/It, (54) and the region is defined by %> This is a more favourable at t = t,.
1,
2
situation,
(55) since all muscles are fully contracted
Region ZZZ The first two terms of (49) are saturated, the third term has argument less than T, while the fourth and all subsequent terms have
argument
less than zero, i.e., the fourth
wave has yet to arrive. Now
ctJL = 2 + &ll + (1 - &g/A,
(56)
and the region is defined by /z>l, Equations
,Ll<2.
(56), (57) are derived in Appendix
(57) B.
Region IV
Again the first two terms of (49) have saturated, but now the excitation signal travels slower than the hydro-elastic wave and the first negative wave to arrive is the fourth term; the third and all other waves have yet to arrive at t = t,. Now C&/L = 3 + +/LL(1-
I”),
(58)
and the region is defined by 23i--1
I < 1,
(59)
P<3,;1-1.
Region V The leading hydro-elastic wave [second term of (49)] has saturated, the first and fourth waves have arrived but not saturated, while the third and all other waves have yet to arrive. Now c&/L =
1 + (7 + 2/Jcl)n /I(3 + ;I)
’
and the region is defined by 232-l -___ R/1+1
II
1
a
Region VZ The second wave has saturated, the first, third and fourth waves have all arrived but not yet saturated, and all other waves have yet to arrive. Now 1 + (5 + 2/J)1 - 4L2 ctJL
=
;1(3 - 2)
’
(62)
MUSCLE INDUCED
EJECTION
353
FLOWS
II
FIG. 4. Regions A = V/c, p = CT/L.
of distinct
performance
of the tetanizing
muscle tube, where
5-
L-
x-t.1 ppL.tI
7-
2:
l-
rr-- \,/-
x
-20
\
:’ _!_._.-J!~..__.&Ly’ ’ / : : /---’
/ /
o-/ 0
_-___-___---_ !
/
\
i!
a5
______~
1.0
1.5
2.0
:
2.5
3.0
FIG. 5. Output velocity u(L, t) for various values of X at p = 0.25, in Region III of Fig. 4.
354
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND P. S. HETZEL
and the region is defined by ;1 < 1,
a + 1/n < ,U < 3 + 1,/n.
(63)
These six regions are shown in Fig. 4. Velocity pulse shapes are given in Fig. 5 for the most important Region III, at the representative value p = 0.25. 9.
TOTAL VOLUME OF EXPULSION
One aim of the physiological systems modelled by the present work may be to expel the maximum possible volume of fluid, and we can investigate this phenomenon using the results for the tetanizing muscle tube. The total volume of expulsion may be defined either by Q = nri
fo s0
4-h t) 4
(64)
i.e. the net outflow from the tube, or in terms of the difference between the internal volume of the tube at t = 0 and t = t,. Making use of (49) for u(L, t), we may now obtain the volume Q by elementary, if tedious, integration, for the case considered in Sets. 7 and 8 of a travelling wave of excitation which tetanizes the muscle [Eq. (18)]. Only a few selected results will be mentioned here; the computation for Region III is given in Appendix B. Firstly, we may observe that the case 2 > 1 (i.e., V > c) always produces more total volume than the case 2 < 1. This is reasonably clear on intuitive grounds, since if V < c there is a tendency for the outtlow to cease at a time when the muscles near the exit have not yet fully contracted, as in Regions I, V, and VI of Fig. 4. However, even in Region IV, when this is not so, the value of Q is always less than that for Region III at the same value of p. Secondly, it is found that the faster the muscles contract (i.e. the smaller is cc>the better, as far as volume of expulsion is concerned. Again this agrees with what we should expect intuitively. Thus of the six regions of Fig. 4, Region III (A > 1, 1( < 2) always gives the greatest value of Q, namely (See Appendix B) Q = (2 - $=$
: 0
Note that m,2PoL/pc2 is the total internal volume change which would occur if every muscle contracted isotonically at a zero stress level, i.e. in the absence of any pressure difference across the wall of the tube. The fact that a value of twice this amount is obtainable (at p = 0) is due to the favorable effect of the negative pressure pulse which travels back up the
MUSCLE
INDUCED
EJECTION
FLOWS
355
tube after reflection from the open end, sucking the wall of the tube down further in a passive elastic manner. An additional feature of the result (65) is that it is independent of the parameter i, so long as 2 > 1. Thus any speed V of travel of the excitation signal which exceeds the hydro-elastic wave speed c seems to be equally desirable from the point of view of total volume of expulsion. The effect of 1. in Region III is to change the shape of the output pulse without altering its total volume content, as indicated by Fig. 5. For example, near 1. = 1 (i.e., as V + c), we obtain a very high, short, pulse containing about half of the total volume, followed by a long time period (about 2L/c) of low-output Ilow containing the remaining half of the volume. On the other hand, at very high values of I (i.e. all muscles essentially contracting simultaneously), we obtain a single pulse. It would appear that the true situation (e.g. in left-ventricular ejection) is often somewhere between these extremes. Clearly many questions remain to be answered in a performance analysis of the pumping mechanism, and we shall not pursue this question further here. 10. CONCLUSIONS
A simple model of muscle-induced ejection flows has been developed. The general expression for the forcing pressure function P(x, t), representing the muscle tube’s activation program, enables generation at will of various kinds of ejection flows. In spite of its admitted crudity in so many respects as a representation of actual physiological systems, the model is capable of demonstrating essential features of these flows. The model has the advantage that analytical solutions in terms of the general function P(x, t) exist for the pressure and velocity as a function of time anywhere in the tube. For example, the very important exit velocity u(L, r) is obtained analytically in terms of P(x, t), by solution of a classical wave equation, enabling prediction of outflow duration, total volume ejected, and shape of exit velocity pulse. The example of the “tetanizing muscle tube” treated in Sets. 8 and 9 demonstrates the procedure involved in applying the model to specific physiological systems of known P(x, t). The model must of course not be judged by this actual specification of P(x, t). However, the results obtained for the shape of the exit velocity pulse show that qualitative agreement with physiological systems is possible, e.g., simulation of high velocities at the early stages of the ejection period. Further, parametric conditions for maximum total volume flow can be established, this being a significant step toward a performance analysis or optimization of the ejection process.
356
A. HELFGOTT,
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
Perhaps the most interesting aspect of the present work is the possibility, with only a little more analytical difficulty, of investigating various defects in the performance of the pumping mechanism. To give one example only, it is a relatively simple task to extend the present work to cover the situation in which for some portion of the length of the tube the forcing pressure P(x, t) remains zero for all time, i.e. the corresponding muscles are not excited. The effect would be a ballooning out of the tube over this section, with a subsequent loss in efficiency of pumping. Such situations do, of course, occur in the defective heart. Nothing in the present discussion should be taken as implying that it is not worthwhile constructing more accurate models, including muscle visco-elasticity, relaxation and non-linearity, flow three-dimensionality, non-linearity and viscosity, thick-walled tubes, output impedance, etc. and we have analyzed some of these effects (Helfgott [lo]). However, it must be recognized that, with the inclusion of each additional physical phenomenon, even though the model is thereby made more accurate, our understanding of the important ejection phenomena may be made more difficult. Indeed, perhaps the most important conclusion of all from the present work is the fact that it is even possible to reduce such a complex physiological problem to a tractable mathematical form, without in the process sacrificing essential physiological properties of the systems studied. ACKNOWLEDGMENT This project was supported in part by grants from the National Health and Medical Research Council of Australia, and the Commissioners of Charitable Funds of South Australia.
APPENDIX
A: LAPLACE
Defining
TRANSFORM
SOLUTION
OF THE WAVE
the usual Laplace transform [9] m p(x, t)e-s’ nt = Z[p(x, F(x, s) = s0
t)],
EQUATION
(Al)
etc., we have from (35) sZp(x, s) - c*p.&x, s) = s*qx, where we have already used the initial conditions case (46) we have
s),
(A2) (42), (43). In the special
P(x, s) = e-Xs’“F(s),
(A3)
s2 s2 p,, - p- - - -cie-xs~v~(s)e
(A4)
so that (A2) becomes
MUSCLE INDUCED
357
EJECTION FLOWS
Equation (A4) is an elementary 2nd-order ordinary differential equation with respect to x; its solution subject to the boundary conditions p = 0 at x = L and pX = 0 at x = 0 (obtained by Laplace transforming (39) and (40), respectively) is + P(x, S) = $J($
;(e-sx/c_ X
e-“SIv
_
e
-zsLlc+sx/c)
e-sL/c-sL/v(esx/c
+
+
e-sx/c)
_ 1
[
+
,-WC
1 (A5)
On taking
the inverse
PC&t> =
Laplace transform
[(pt-x)-
of (A5), we obtain
p-1(1
+
e-2sL/c)-lpQ(t)
V
where
1 )
(A6)
,,=E[P(t-f)-P(i-~+5)1
(A7) Expanding
(1 + e-2SL’C)-1 = 1 - e-zSL’c f e-4SLlc - . . ., we have finally
Ax, t> =
&
H ) p” t - ;
- Q”(t)
+P(i-F)-6(1-T)+...].
(A8)
which is easily seen to be identical with the solution when (A7) is used to express 0 in terms of P. APPENDIX
B:
COMPUTATION
OF OUTFLOW
TIME AND VOLUME
Region III is defined in Sec. 8 as the region terms of (49) are saturated at t = t,, i.e. t, -L/V>
(47) given in the text,
T,
FOR REGION
III
in which the first two (Bl)
and t, - L/c > T, the third term is non-zero
but not saturated,
032) i.e.
0 < t, - L/V - 2Llc < T,
(B3)
and the fourth term has yet to arrive, i.e. t, - 3Llc < 0.
04)
358
A. HELFGOTT,
The inequality (B4) guarantees terms of (49) vanish identically Under these circumstances, u(L, t) = which vanishes
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
that the fourth, fifth, and all subsequent at 1 = t,. near t = t,, equation (49) reduces to
1) (2 + 1) - 2 - $ (c-;-F)], t
W
when L t,=F+T+
2L
I-1 -T,21
w-5)
I.e., 1-l ctJL = 1 -I- 2 + L 21 lJ = 2 -t 3j.f + (1 - &.$I, as in (56). From (B6) and (B3) we have A > I, whereas from (B2) we have ,u c 2, confirming (57). The remaining inequalities (Bl) and (B4), are satisfied automatically. The volume Q defined in (64) is computed as follows
as given in equation
(65) of the text.
REFERENCES 1 J. B. MacCallum, W&h Festschri., Johns Hopkins Hospital Reports 9, 307(1900). 2 D. D. Streeter, H. M. Spotnitz, D. P. Pate], J. Ross, and E. H. Sonnenblick, Circulation Res. 24, 339 (1969). 3 D. Durrer, R. Th. Van Dam, G. E. Freud, M. J. Janse, F. L. Meijter, and R. C. Arzbaecher, Circulation 41, 899 (1970).
MUSCLE INDUCED
EJECTION
FLOWS
359
4 E. 0. Attinger, Pulsatile bloodflow. McGraw-Hill, New York, 1964. 5 J. T. Apter and W. W. Graessley, Biophys. J. 10,539 (1970). 6 Y. C. Fung, Applied Mechanics Reviews, 1 (1968). 7 G. Rudinger, “Review of Current Mathematical Methods for the Analysis of Blood Flow”, Biomedical Fluid Mechanics Symposium, A.S.M.E., New York (1966), p. 1. 8 A. V. Hill, J. Physiology (London) 56, 19 (1922). 9 J. C. Jaeger and G. Newstead, An introduction to thelaplace transformation. Methuen and Co. Ltd. (1969). 10 A. Helfgott, “Mathematical Modelling of the Heart”, Ph.D. Thesis, The University of Adelaide (1971, unpublished).
BIOSCIENCES
A Simple Mathematical Ejection Flows
13, 335-359
(1972)
335
Model of Muscle-Induced
A. HELFGOTT AND E. 0. TUCK Department of Applied Mathematics,
University
of Adelaide
AND
R. J. CRAIG AND P. S. HETZEL Cardio-Pulmonary Investigational Unit, Royal Adelaide South Australia
Communicated
Hospital,
by F. Grodins
ABSTRACT Problems involving expulsion of fluid from a container due to the action of muscles in the walls of the container, are analysed by use of a highly simplified model. The resulting inhomogeneous linear wave equation is solved for a particular class of muscle contraction time histories, enabling prediction of period and total volume of outflow and exit velocity pulse shapes. Applications in physiological contexts are indicated.
1. INTRODUCTION
There are a number of important physiological situations in which a fluid is caused to move due to the action of muscles in the walls of a container. Of primary importance is the expulsion of blood from the left and right ventricles of the heart. Other examples include ureteric transport, uterine contraction, urine expulsion from the bladder, bowel motility, ejaculation from the male reproductive system, and, in the more general biological field, squid-like propulsive mechanisms. In this paper the action of the left ventricle will be examined whenever a specific example is required. However, the same general principles apply to other similar muscle-induced ejection flow systems. At first sight if we wish to construct a theoretical model for these processes, we are confronted with a problem of overwhelming complexity. Little is known (although much has been written!) about the way muscles really work, and they, after all, are responsible for the whole flow. Even if we knew enough about the mechanical properties of individual muscles, we should still be faced with problems associated with the complex distribution and orientation of muscle fibres in the walls of the container, and with Copyright
0 1972 by American Elsevier Publishing Company, Inc.
336
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND P. S. HETZEL
the intricate sequence of electrochemical events which programs their contractions. (See MacCallum [I], Streeter et al. [2] and Durrer et al [3] for the left-ventricular situation.) From the fluid point of view we should be concerned with the nonNewtonian viscous properties of physiological fluids and the fact that they are moving in three dimensions in a vessel of complicated and timevarying geometry. Finally there is the question of exit conditions at the outlet from the container. If we have in mind applications to the left ventricle, we should attempt to model the effect of the aorta and, indeed, of the whole arterial system, on flow conditions in the ventricle itself. The approach to be adopted in the present work is of simplifying the problem to the greatest extent consistent with retention of features which are thought to be significant in determining expulsion characteristics. The essential assumptions to be made are outlined in the following section; further non-essential assumptions are also made later in order to simplify the mathematical analysis of special cases.
2.
RANGE
OF INVESTIGATION
AND
ASSUMPTIONS
A recent paper on muscles by Apter and Graessley [5] provides up to the present time perhaps the most satisfactory macroscopic mechanical model for muscle behavior. This subject presently attracts considerable attention, and there seems little doubt that better models will be developed in the future. However, the muscle model in [5] is already too complicated to put into the present framework, and we propose here a linear muscle model consisting of a linear elastic spring and a contractile element coupled in series. This simplified muscle model is discussed in some detail in the next section. Much work has been done in blood rheology (reviewed by Fung 161). However, we take the liberty of neglecting this aspect altogether, assuming an inviscid incompressible fluid. This is a good approximation for leftventricular expulsion, which is at a relatively high Reynolds’ number, and may also be justified in some of the other applications mentioned. Further, we neglect gross geometry variations in the container by assuming it to be a long cylindrical tube in which the flow takes place essentially in one direction only, namely along the axis of the tube. (Fig. la.) The exterior and interior surfaces of the tube are tethered so as to prevent longitudinal motion, so that the tube wall can move only in a radial direction. Longitudinal stresses in the tube wall are neglected, and so is the inertia of the tube. One-dimensional flows of this kind where the container walls possess passive elastic or viscoelastic properties have been used to model arterial
MUSCLE INDUCED
EJECTION
331
FLOWS
blood flow (see review by Rudinger [7]), and a characteristic of this approach is that there results a set of hyperbolic partial differential equations, similar to those of gas dynamics. The (passive) elastic properties of the wall provide energy storage, which leads to a wave-like behavior
#‘-------r
i
p(X,t)
5= CONSTANT
I
4
ClRCUMFEREhlAL
MUSCLE
FIG. I. (a) Geometrical arrangement of the muscle tube, (b) excitation of circumferential muscles at speed V.
(“hydro-elastic” magnitude
waves)
with
a speed
(for
small
amplitude
waves)
of
(1) where E is the Young’s modulus, 6 the thickness and yg the radius of the container wall, while p is the fluid density. This wave phenomenon with speed c given by (1) is also apparent in the present model of active flow induced by the living energy-generating walls of the muscle container, in which we take into account the forcing effect of the muscles. Next, we turn to the electrical events which activate the muscle tube. Muscles are turned on (activated) by an electrical voltage signal (activation
338
A. HELFGOTT,
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
potential) which depolarizes the membrane surrounding them (Fig. 2). This in turn leads to electrochemical activities, resulting in subsequent muscle contraction. We shall not discuss the electrochemical mechanism triggered by the activation potential but instead, in Sec. 3, a procedure is described for modelling the net result of electrochemical events. In the present analysis it is assumed that every circumferential muscle fibre along the tube has a “switch” that is turned on by an electrical excitation signal travelling at speed V from the closed end to the open end of the tube (Fig. lb.) The turning on of the “switch” indicates the start of electrochemical activities, followed by contraction of the muscle fibre at location x.
FIG. 2. (a) Schematic representation of the linear muscle model, (b) pre-determined program of individual muscle activation.
This leads to the definition meters :
of two important
non-dimensional
para-
i. = V/c
(2)
/I = CT/L,
(3)
and where T (Fig. 3) is a measure of the time taken for an individual muscle to contract fully, while L is the length of the tube. In the present work we investigate the influence of such parameters as 1. and p on the flow pattern and the performance of the muscle tube. In the analysis, i plays a role similar to the Mach number in flight through the atmosphere, and p is the ratio of Tto the time needed for the “hydroelastic” wave to travel the whole length of the tube.
MUSCLE INDUCED
EJECTION
339
FLOWS
Analogue models of the complete cardiovascular system have been formulated (see [4]), but the modelling of the actual pumping mechanism has been of minor interest in these studies. On the contrary, in the present work this is the crucial point, and we choose to neglect the effect of the
t+$.
bm
of arrival
potential
d Activation-
at X
FIG. 3. (a), (b) and (c) Various idealized individual muscle activation programs, (d) wave of activation travelling at speed V, (e) three-dimensional representation of a pressure excitation P(x, t) based on the activation wave of Fig. 3(d).
arterial system on the ejection process, replacing the true exit conditions at the open end of the tube by a constant pressure condition. This pressure Pexit is of course not necessarily identifiable with the external or ambient pressure pa outside the wall of the muscle tube (Fig. la). For example in
A. HELFGOTT,
340
E. 0. TUCK,
R. J. CRAIG
the case of the left ventricle, Pexit is the aortic pressure, same as the pericardial pressure pa. 3.
THE LINEAR
MUSCLE
AND P. S. HETZEL
which is not the
MODEL
In the present work we adopt the simplest possible linear muscle model. From the physical point of view, it consists of a linear elastic spring (SE) of spring constant k per unit cross section area, coupled in series to a contractile element (CE) (Fig. 2a). In the literature it is possible to find many suggestions for muscle models, consisting of various combinations of springs, dashpots, and contractile elements. We have avoided unnecessary complications by choosing a simple model which nevertheless, as we shall see, possesses some of the most important properties of muscle. It is easy to see how this model produces tension or does mechanical work. For example, if we hold the muscle at constant length I and allow the contractile element to shorten, the spring will be stretched, thus generating tension in the system. On the other hand, if we attach a weight W to the muscle model and allow the contractile element to shorten by the amount Al, work WA1 will be done, while the spring remains at constant length. These two extreme examples are known as “isometric” and “isotonic” contractions, respectively. It is evident that the contractile element is essentially an agency capable of shortening. The actual force developed in the model depends mainly on what we couple to the contractile element, e.g. on the spring, in the isometric experiment. In order to proceed with the analysis of model properties and the derivation of the governing equations, it would be desirable to have answers to two important questions, namely: (a) How is contraction initiated? (b) How does muscle contract? The answer to the first question is known in general. The application of an activation potential to the membrane surrounding a muscle fibre (Fig. 2a) triggers a very involved set of electrochemical events that eventually leads to mechanical contraction. However, regarding the second question, no final answer is known, and we choose to model only the net result of all electrochemical events in the following way. We start with the following c I z y z, y0
= = = = = =
definitions
(Fig. 2a)
stress in the muscle fibre, length of muscle fibre, length of the contractile element, length of the series elastic spring, the value of z for a completely relaxed (non-activated) the value of y for an unloaded muscle (G = 0).
muscle, and
MUSCLE INDUCED
EJECTION FLOWS
341
We now suppose that at time t = t,, an activation signal (or a train of such signals) is applied to the membrane. The result is that the length z(t) of the CE begins to decrease, and we make the assumption that z(t) is a giuerz function of time, in particular not depending on the instantaneous muscle load a(t). This could be achieved mechanically by a rack-and-pinion system, in which the pinion has a pre-determined time-dependent angular position, as in Fig. 2b. The length 10r of an unloaded and relaxed muscle is In any other state the length fibre are
ZO,= Yo + z,* I and the “external”
strain
(4) E of the muscle
z=y+z
(5)
and
respectively. For a muscle of unit cross-sectional given by the stress in the spring
area the stress o at any instant
o = kAy = k(y - y,),
is (7)
and, from (4) and (5) we have Ay = y - yo = (f - Z,,) + (z, - z). (8) This means that the elongation of the spring is given by the change in length of the whole muscle fibre plus the shortening of the contractile element. This can also be obtained from inspection of Fig. 2. Finally, from (6), (7): (8) we get o = k(l - lo,) + k(z,. - z) = E.2 + .f(f),
(9)
where E = kl,, is Young’s
modulus,
(10)
and
(11) f(t) = 0, - 4t)l. is precisely the tension (stress) developed in the muscle during isometric (constant length) contraction. It is a given arbitrary function, determined by z(t). The stress-strain relation (constitutive law) for muscle given by (9) is an inhomogeneous linear Hooke’s law. In some cases, as we shall see later, it is more convenient to define the “external” muscle strain a* as The functionf(t)
A. HELFGOTT,
342
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
1,
EE, = E r
( The relation
between
E and
1 . >
-
Or
(13)
is derived from (6), (12) and (13), thus
E*
I 0’s E = E, + A&* = E, + - + I &*. 1Or (E ) Substituting
(14)
(14) into (9) we get Ao = d - gs = (E +
b,)E*
+
f(t).
(15) In order to make use of (9) or (15) in the generation of muscle-induced ejection flows, we must make some assumptions about the form of the forcing termf(t) which determines the pattern of flow. In the following work we leave,f(t) as a general function of time as far as possible. When we do want to specify f( t) it is appropriate to take the simplest possible forms, consistent with the simplified model (9), but representing the essential features of muscle excitation. Three elementary forms off(t) will be mentioned. Case 1.
The simplest
excitation
function
f(t) where H(t) is Heaviside’s That is
of all is (Fig. 3a)
= f,H(t),
(16) and f. is a given constant.
unit step function 0
t
f(t)= i f. t > 0.
(17)
This corresponds to a situation where at time t = 0 the muscle contracts instantaneously to a maximum level, producing a stress increase fo. For t > 0 the muscle remains fully contracted (tetanized). Case 2. The muscle is allowed to complete contraction shortening T (Fig. 3b) and then remains tetanized. 0, S(t)
=
I
t
<
0,
fO;,
0
<
t
<
f 02
t
>
T.
in a finite time of
T,
(18)
This reduces to (17) as T -+ 0. Case 3. The muscle is allowed to contract, and then relax, as shown in Fig. 3c.
remain
tetanized
for a while
343
MUSCLE INDUCED EJECTION FLOWS
t < 0,
’ 0,
O
f(l)=
fcb
T
T, -
foT2
-
t T,’
T,,
(19)
7.1 < t d 7.2, t > T,.
0,
All these cases have some degree of validity in various applications, e.g. in cardiac muscle a mechanism similar to (19) causes the muscle to contract and relax. More general models are discussed by Helfgott [IO]. From (11) we obtain
the velocity V CE
of the contractile
_;dfw
=dL(t)= dt
k
dt ’
which is also a given function. It was mentioned tension development in an isometric contraction. l-l,=-
$-f(t)
element,
s
already Further,
= - %f(t)
(20)
that f(t) is the from (I 5),
(21)
is the amount of shortening in an isotonic (constant load) contraction. The above is not only a grossly simplified representation of what happens in actual muscles, but falls far short of mathematical models which have already been postulated in the vast physiological literature on this subject. However, it appears that for our purposes the above will suffice, and it certainly has the advantage of analytical simplicity. The model (9) does represent two important aspects of muscle behavior, namely, energy production (shortening element) and energy storage (spring), but does not appear to include either the dynamic stress-velocity property of muscles (Hill [8]) or the dynamic dependence of isometric tension development {f(t)} on Z, (or initial load), because such circuit elements as a dashpot and non-linear elasticity are missing. However, these effects can if required be incorporated to a certain extent in a linear model performing small amplitude motions about an equilibrium condition. This is done [e.g. with f(t) given by (IS)] by allowing the parameters E, T, and f0 to vary according to the equilibrium values I, and 0,. For example, the velocity of the contractile element v
=
CE
_
r.-df(t)
k
dt
= -- “f-0 kT
for
0 < t < T,
344
A. HELFGOTT,
E. 0. TUCK,
made to depend in an arbitrary ment of.f,, k, and T.
R. J. CRAIG
AND P. S. HETZEL
way on 1, and o’sby suitable
adjust-
4. THE LINEAR FLOW MODEL The vessel in which flow takes place is assumed (See Fig. 1) to be a thin-walled circular cylinder composed of muscles arranged circumferentially. The cylinder wall contracts during muscle excitation, but we shall assume that such displacements are small perturbations of the rest state, and likewise induce flow velocities which are small. Thus, if the equilibrium radius of the cylinder is r0 = ZJ27r we define the perturbation to this radius as 1-l a = a(x, 1) = 2 2n so that the instantaneous radius aneous cross-section area is
w
’
is r = r,, + a = 112~~and the instant-
S = 77(ro + ~2)~. If u = u(x, t) is the fluid velocity, (continuity)
we obtain
(23)
for conservation
(A%), + s* = 0, and for conservation
of momentum
of mass (24)
(Euler)
1 u, + UU, = --p,, P
(25)
where p(x, t) = pi(X, t) - Pexit is the rise in hydrodynamic pressure over the constant exit pressure Pexit and p the fluid density (assumed constant). Equations (24) and (25) assume that the fluid is inviscid and incompressible, and that the flow is effectively one-dimensional along the cylinder, so that the lateral velocity components may be neglected compared with U. However, at this stage we have not yet assumed small disturbances, and the above equations may be used in a more general non-linear study of ejection processes. (See Helfgott [lo].) If we do now assume small displacements of the tube walls and correspondingly small values for U, we have from (23) and (24) nr&,
+ 2nrou,
= 0,
or a, = -+r+4,,
(26)
1 u, = --px.
(27)
and from (29,
P
MUSCLE INDUCED
345
EJECTION FLOWS
Equations (26) and (27) are the basic flow equations, supplemented by a relationship between p and a obtained properties. 5.
THE INHOMOGENEOUS
WAVE
and need to be from the muscle
EQUATION
We now suppose that the walls of the tube are of small thickness 6. This, together with the assumptions made in Sec. 2, leads to a simple static equilibrium relation between the hoop stress (stress along the circumferential muscle fibre) and the pressure difference across the wall of the tube (excess internal pressure).
Before the tube starts contracting, p = 0, and the tube is pre-stressed the pressure difference Api = pexit - pa to a level given by
From (15), (28) and (29), neglecting (e.g. pa), we obtain Aa = r~ - a, = ?p
by
terms of second order of magnitude
+ 2
= (E + a,)&* + j-(x, t),
(30)
where f can vary from one muscle fibre to the other along the tube. With s* = a I’0’
(31)
we obtain p = % 4
+ P,
(32)
where P = :f(x,
t) = P(x, t),
(33)
is the forcing pressure. Equation (32), which plays a role analogous to the equation of state in gas dynamics (a’ being analogous to the density variation in that case), is a simple linear relationship between pressure and radius changes, with a given forcing pressure P (analogous to a heat energy source in a burning gas). Notice that there is no direct effect of the pre-stress 0, of the tube on Eq. (32); this follows from the linearization and may not be strictly correct for all physiological situations, such as the function of the left ventricle. 23
346
A. HELFGOTT, E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
Using (32) to eliminate
a(~, t) from (26), we have r;
i.e.,
Either giving
pt = P, - %,. (34) 2ro u or p may now be eliminated from the pair of Eqs. (34) and (27), Ptt - C2Pxx = Pm
(35)
and 1 2 % - c u,, = -- PL where Ed c2 = __ 2P7-0’ Equations (35) and (36) are inhomogeneous linear one-dimensional wave equations, the wave speed c being defined by (1). This speed is simply the speed at which hydro-elastic waves travel along the wall of the tube in the absence of muscle excitation. The forcing terms on the right of these equations control the flow, and represent the total influence of muscle activity. The problem having been reduced to such a familiar partial differential equation, we can call upon a great variety of well established mathematical techniques and solutions. For example, such equations occur in the theory of transmission lines (Jaeger and Newstead [9]) and we may re-interpret transmission-line solutions in terms of the present problem. However, in the present paper we treat only a sample problem for which the solution is obvious by inspection. 6.
BOUNDARY
AND INITIAL CONDITIONS
In order to model accurately the real flow situation it is in principle as important to supply the correct boundary and initial conditions as it is to use the correct flow equations. However, once again we take the liberty of grossly simplifying these conditions, which in the case of boundary conditions would be extremely complicated non-linear relationships between the pressure and velocity time histories at the ends of the tube. We now assume that the tube extends from x = 0 to x = L, and that the left end is closed, i.e. u=O
at
x=0,
(37)
347
MUSCLE INDUCED EJECTION FLOWS
while the right end is held at the constant pi = Pexit =
exit pressure Pexit, i.e.
C0IlSt.p
(38)
or
p = pi - P&t = 0 at X = L. (39 In view of the Euler Eq. (27), the condition (37) on the velocity is equivalent to the condition px=O at x=0 (40) on the pressure. More accurate representations of the real physiological boundary conditions are of course easy to envisage and to formulate for any specific problem. For example, in situations such as left ventricular ejection we may be unhappy about the constant pressure condition (38), and indeed there is no doubt we can do a much better job of modelling the impedance offered by the aorta and the whole arterial system to flow from the aortic valve. However, this would be contrary to the spirit of the present paper, and is left for future work. (See Helfgott [lo].) As far as initial conditions are concerned, we start the process from an equilibrium state, in which the fluid is at rest at the exit pressure, i.e. at t=O u(x, 0) = 0, (41) and P(X, 0) = pi(X, 0) - &it = 0. (42) Again, by use of the continuity equation (26) and muscle equation (32) we may eliminate the velocity, obtaining initial conditions on the pressure alone, replacing (41) by pr(?c, 0) = P,(x, 0). (43) The use of a state of rest as an initial state may also be questioned in applications to the left-ventricular ejection, which is preceded by an isovolumetric phase in which the pressure builds up, with the aortic valve closed. However, little internal movement occurs in this phase and if we use the pressure at the instant of opening of the valve as our exit pressure, the above initial conditions remain appropriate. Finally, a terminating condition ending the flow process is required. Clearly the object of the muscle-walled container is to discharge or transport its content of biological fluids from one location to another. Hence, only the case of out&ev from the container is important and will be treated here. From the mathematical point of view, we allow the solution to proceed as long as u(L, t) > 0 for 0 < t < t,, (40 but consider the problem terminated the first time since t = 0, ve have
at the instant
u(L, t,) = 0.
t = t, > 0 at which, for (45)
348
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND
P. S. HETZEL
In biological systems where an actual valve is present condition (45) models its closure reasonably well, e.g., aortic valve closure in the left ventricle. In other cases, we consider that the flow model is realistic only for 0 < t < t,, and must look to a more accurate model for subsequent events. 7. THE SOLUTION FOR A TRAVELLING WAVE OF EXCITATION Certain final assumptions which have to be made before we can obtain solutions to (35), (36) concern the program of electrochemical activation of the entire muscle tube. Clearly, many kinds of muscle-induced flows can be generated, depending on the choice of the forcing function P(x, t). In the present work we shall only be concerned with muscle tubes in which all muscles are identical to each other, i.e., the individual muscle activity law ,f(t) is the same for all muscles. Further, the solution to be presented below is for the case P(X, t) = F(t - x/V),
(46)
which determines the x-dependence of P(x, t). This corresponds to a wave of muscle activity, starting at the closed end x = 0 at t = 0, and moving toward the open end x = L at uniform speed V. The function H(t) = Sf(t)/r,, is identified with P(0, t) or in fact at any fixed x with the individual muscle activity law. This means that every muscle is identically activated, but is stimulated at an instant t = t,, = x/V which differs from its neighbor. Note that it has not yet been necessary to specifyf(t), and hence F(t) remains an arbitrary function. Geometrical representation of (46) yields cylindrical surfaces as shown in Figs. 3d and 3e for,f(t) given by (18) or (19). The interesting possibility of examining analytically pathological defects in muscle performance by allowing more general dependence on x in the linear model is left for future work. For the specific case of the heart, muscle damage, together with defects in the conducting system, all leading to changes in the normal P(x, t), are treated numerically in some detail by the non-linear model of Helfgott [lo]. The exact solution of the wave equation (35) for the pressure difference p(x, t), subject to the boundary conditions (39) and (40) and initial conditions (42) and (43) is
PC% t>=
gJ [F(t -
-j5
x/V) _p
(
t-X-_c
L
L
c
r; )
(,+.y_L._b
+p” (
c
c
,+x2-4 c
c
v> v1
MUSCLE INDUCED
EJECTION
349
FLOWS
+p
(I_T_3L_$ 1 (1+“_5L_; 1 p ( > -,(t+g$ ’ -f5
c
---
VC
p-2 -
(9
.
c
.
.
.
c
.
.
.
.
.
.
c
.
>
.
.
t-”
[
c
+;_g
+P(r+;_!c)
+ . . . . . . . . . . . .
(47) 1 This solution is obtained by Laplace transform methods in Appendix A; however it has a simple physical explanation, and its correctness may be established by direct verification, as follows. The term in &t - x/V) contributes a particular integral of the inhomogeneous wave equation (35), whereas all other terms are hydro-elastic waves, i.e. solutions of the homogeneous equation, so that certainly (35) is satisfied. The middle column of terms represents hydro-elastic waves travelling to the right while the last column represents left-travelling waves. Each pair of terms is obtained from the pair above it by reversing sign and delaying by the interval 2L/c for passage of one hydro-elastic wave back and forward, and this process continues indefinitely, generating an infinite series solution. The initial conditions (42) and (43), are satisfied in the present example because the argument of all P” functions is negative at t = 0. The boundary condition (39) is clearly satisfied by horizontal cancellation of terms, while the boundary condition (40) is satisfied by diagonal cancellation of terms, apart from the first term inside each bracket. These terms themselves combine to satisfy (40). Physically, equation (47) indicates that, starting at x = 0 at time t = 0, two waves proceed down the tube toward x = L, one of which is the imposed muscle excitation signal at speed V and the other the hydroelastic wave at speed c, generated instantaneously by reflection at the closed end x = 0. The net effect is a pulse of positive pressure and velocity travelling toward x = L. When the leading edge of this pulse reaches x = L, it is reflected as a hydro-elastic wave with negative pressure, but still positive fluid velocity. The reflections from both original waves arrive back at the closed end x = 0 at a time L/c after they reached the open end. Thus, the wave which started as a muscle excitation (first square bracket of (47)) arrives back at t = L/V -t LJc, while that which started as a hydro-elastic wave (second
350
A. HELFGOTT, E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
square bracket) arrives back at t = 2Llc. These again by the closed end, and travel back as waves velocity to the open end again, and so on. The expression for the velocity U(X, t) is easily by solving the wave equation (36) or from the p(x, t), using the Euler equation (27). We find u(x, t) =
waves are now reflected of negative pressure and obtained, either directly, solution for the pressure
,(v:_c2)p(++~) +P(t_f_!+o+pL) _,(,_g+(t+;-!&) + . . . . . . . , . . 1
+
V2
pc( v2 - 2)
C
-p”
(,_x_L_4 ) (t+,“-“-” 1 ( ) (,+x_?E_4 1 c
+F
-
t-;_+;
c
v
+p”
V
-p
c
c
. . . . . . . . . .
.
v>
(48)
Notice from the first two terms of (48) that, irrespective of which signal travels faster (i.e. of whether V < c or V > c), the first velocity pulse is positive, because of the multiplying factor k’//lp(V - c’). We are particularly interested in the flow through the outlet at x = L, i.e. from (48)
MUSCLE INDUCED
351
EJECTION FLOWS
where 1 = V/c is the ratio
of the excitation speed to the hydro-elastic wave speed. In fact, although formally (49) is an infinite series, we shall observe in the following section that u(L, T) necessarily takes negative values at times earlier than the smaller of 5L/c and L/V + 4L/c. Hence we use the solution only at times when the first four terms of (49) are sufficient. Physically, this simply means that the process is terminated by arrival of a negative velocity signal after only one reflection from the closed end, and we discontinue the solution before further reflections take place. 8.
OUTFLOW TIMES
We now investigate analytically the time t, of termination of outflow at x = L. To keep this analysis tractable, we consider only the particular case when the muscle activation law is given by (18) i.e., the muscle becomes tetanized (a state of sustained contraction) at t = T. Hence, the forcing pressure signal is (0, P(t) =
df(t) __
zz
r.
I
Pot -
t < 0, o
<
t
T’
\POT t >
<
”
T
(50)
T,
where PO = Gfo/ro. The more complicated signal (19) including relaxation (Fig. 3c) will be considered in future work. although analytic determination of t, is then unlikely. The determination of the outflow time t, from (49) subject to (50) is quite straight forward and only the results will be presented here. The time t, depends on the two parameters 3, = V/c and ,LL= CT/L and we can identify six regions of the (%, p) parameter plane. A sample calculation is given (for region III) in Appendix B. Region I
The arguments of the first four terms in (49) lie between 0 and t = t,, while the arguments of all subsequent terms are less than or equal to zero. In this case we have
Tat
C&/L = 4 + l/A,
(51)
and the region is defined by either i > 1,/l > 4,
(52)
2 < 1, /l > 3 + 1/n.
(53)
or This region (and Regions V, VI to follow) is one in which conditions are clearly unfavorable for optimum expulsion, since some muscles near x = L have not yet fully contracted when the process is terminated.
352
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND P. S. HETZEL
Region ZZ
The first term in (49) has argument greater than T(“saturated”), the next three have arguments between 0 and T, and all others have argument less than zero. Now C&/L = 2 + &f - (1 - +)/It, (54) and the region is defined by %> This is a more favourable at t = t,.
1,
2
situation,
(55) since all muscles are fully contracted
Region ZZZ The first two terms of (49) are saturated, the third term has argument less than T, while the fourth and all subsequent terms have
argument
less than zero, i.e., the fourth
wave has yet to arrive. Now
ctJL = 2 + &ll + (1 - &g/A,
(56)
and the region is defined by /z>l, Equations
,Ll<2.
(56), (57) are derived in Appendix
(57) B.
Region IV
Again the first two terms of (49) have saturated, but now the excitation signal travels slower than the hydro-elastic wave and the first negative wave to arrive is the fourth term; the third and all other waves have yet to arrive at t = t,. Now C&/L = 3 + +/LL(1-
I”),
(58)
and the region is defined by 23i--1
I < 1,
(59)
P<3,;1-1.
Region V The leading hydro-elastic wave [second term of (49)] has saturated, the first and fourth waves have arrived but not saturated, while the third and all other waves have yet to arrive. Now c&/L =
1 + (7 + 2/Jcl)n /I(3 + ;I)
’
and the region is defined by 232-l -___ R/1+1
II
1
a
Region VZ The second wave has saturated, the first, third and fourth waves have all arrived but not yet saturated, and all other waves have yet to arrive. Now 1 + (5 + 2/J)1 - 4L2 ctJL
=
;1(3 - 2)
’
(62)
MUSCLE INDUCED
EJECTION
353
FLOWS
II
FIG. 4. Regions A = V/c, p = CT/L.
of distinct
performance
of the tetanizing
muscle tube, where
5-
L-
x-t.1 ppL.tI
7-
2:
l-
rr-- \,/-
x
-20
\
:’ _!_._.-J!~..__.&Ly’ ’ / : : /---’
/ /
o-/ 0
_-___-___---_ !
/
\
i!
a5
______~
1.0
1.5
2.0
:
2.5
3.0
FIG. 5. Output velocity u(L, t) for various values of X at p = 0.25, in Region III of Fig. 4.
354
A. HELFGOTT,
E. 0. TUCK,
R. J. CRAIG
AND P. S. HETZEL
and the region is defined by ;1 < 1,
a + 1/n < ,U < 3 + 1,/n.
(63)
These six regions are shown in Fig. 4. Velocity pulse shapes are given in Fig. 5 for the most important Region III, at the representative value p = 0.25. 9.
TOTAL VOLUME OF EXPULSION
One aim of the physiological systems modelled by the present work may be to expel the maximum possible volume of fluid, and we can investigate this phenomenon using the results for the tetanizing muscle tube. The total volume of expulsion may be defined either by Q = nri
fo s0
4-h t) 4
(64)
i.e. the net outflow from the tube, or in terms of the difference between the internal volume of the tube at t = 0 and t = t,. Making use of (49) for u(L, t), we may now obtain the volume Q by elementary, if tedious, integration, for the case considered in Sets. 7 and 8 of a travelling wave of excitation which tetanizes the muscle [Eq. (18)]. Only a few selected results will be mentioned here; the computation for Region III is given in Appendix B. Firstly, we may observe that the case 2 > 1 (i.e., V > c) always produces more total volume than the case 2 < 1. This is reasonably clear on intuitive grounds, since if V < c there is a tendency for the outtlow to cease at a time when the muscles near the exit have not yet fully contracted, as in Regions I, V, and VI of Fig. 4. However, even in Region IV, when this is not so, the value of Q is always less than that for Region III at the same value of p. Secondly, it is found that the faster the muscles contract (i.e. the smaller is cc>the better, as far as volume of expulsion is concerned. Again this agrees with what we should expect intuitively. Thus of the six regions of Fig. 4, Region III (A > 1, 1( < 2) always gives the greatest value of Q, namely (See Appendix B) Q = (2 - $=$
: 0
Note that m,2PoL/pc2 is the total internal volume change which would occur if every muscle contracted isotonically at a zero stress level, i.e. in the absence of any pressure difference across the wall of the tube. The fact that a value of twice this amount is obtainable (at p = 0) is due to the favorable effect of the negative pressure pulse which travels back up the
MUSCLE
INDUCED
EJECTION
FLOWS
355
tube after reflection from the open end, sucking the wall of the tube down further in a passive elastic manner. An additional feature of the result (65) is that it is independent of the parameter i, so long as 2 > 1. Thus any speed V of travel of the excitation signal which exceeds the hydro-elastic wave speed c seems to be equally desirable from the point of view of total volume of expulsion. The effect of 1. in Region III is to change the shape of the output pulse without altering its total volume content, as indicated by Fig. 5. For example, near 1. = 1 (i.e., as V + c), we obtain a very high, short, pulse containing about half of the total volume, followed by a long time period (about 2L/c) of low-output Ilow containing the remaining half of the volume. On the other hand, at very high values of I (i.e. all muscles essentially contracting simultaneously), we obtain a single pulse. It would appear that the true situation (e.g. in left-ventricular ejection) is often somewhere between these extremes. Clearly many questions remain to be answered in a performance analysis of the pumping mechanism, and we shall not pursue this question further here. 10. CONCLUSIONS
A simple model of muscle-induced ejection flows has been developed. The general expression for the forcing pressure function P(x, t), representing the muscle tube’s activation program, enables generation at will of various kinds of ejection flows. In spite of its admitted crudity in so many respects as a representation of actual physiological systems, the model is capable of demonstrating essential features of these flows. The model has the advantage that analytical solutions in terms of the general function P(x, t) exist for the pressure and velocity as a function of time anywhere in the tube. For example, the very important exit velocity u(L, r) is obtained analytically in terms of P(x, t), by solution of a classical wave equation, enabling prediction of outflow duration, total volume ejected, and shape of exit velocity pulse. The example of the “tetanizing muscle tube” treated in Sets. 8 and 9 demonstrates the procedure involved in applying the model to specific physiological systems of known P(x, t). The model must of course not be judged by this actual specification of P(x, t). However, the results obtained for the shape of the exit velocity pulse show that qualitative agreement with physiological systems is possible, e.g., simulation of high velocities at the early stages of the ejection period. Further, parametric conditions for maximum total volume flow can be established, this being a significant step toward a performance analysis or optimization of the ejection process.
356
A. HELFGOTT,
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
Perhaps the most interesting aspect of the present work is the possibility, with only a little more analytical difficulty, of investigating various defects in the performance of the pumping mechanism. To give one example only, it is a relatively simple task to extend the present work to cover the situation in which for some portion of the length of the tube the forcing pressure P(x, t) remains zero for all time, i.e. the corresponding muscles are not excited. The effect would be a ballooning out of the tube over this section, with a subsequent loss in efficiency of pumping. Such situations do, of course, occur in the defective heart. Nothing in the present discussion should be taken as implying that it is not worthwhile constructing more accurate models, including muscle visco-elasticity, relaxation and non-linearity, flow three-dimensionality, non-linearity and viscosity, thick-walled tubes, output impedance, etc. and we have analyzed some of these effects (Helfgott [lo]). However, it must be recognized that, with the inclusion of each additional physical phenomenon, even though the model is thereby made more accurate, our understanding of the important ejection phenomena may be made more difficult. Indeed, perhaps the most important conclusion of all from the present work is the fact that it is even possible to reduce such a complex physiological problem to a tractable mathematical form, without in the process sacrificing essential physiological properties of the systems studied. ACKNOWLEDGMENT This project was supported in part by grants from the National Health and Medical Research Council of Australia, and the Commissioners of Charitable Funds of South Australia.
APPENDIX
A: LAPLACE
Defining
TRANSFORM
SOLUTION
OF THE WAVE
the usual Laplace transform [9] m p(x, t)e-s’ nt = Z[p(x, F(x, s) = s0
t)],
EQUATION
(Al)
etc., we have from (35) sZp(x, s) - c*p.&x, s) = s*qx, where we have already used the initial conditions case (46) we have
s),
(A2) (42), (43). In the special
P(x, s) = e-Xs’“F(s),
(A3)
s2 s2 p,, - p- - - -cie-xs~v~(s)e
(A4)
so that (A2) becomes
MUSCLE INDUCED
357
EJECTION FLOWS
Equation (A4) is an elementary 2nd-order ordinary differential equation with respect to x; its solution subject to the boundary conditions p = 0 at x = L and pX = 0 at x = 0 (obtained by Laplace transforming (39) and (40), respectively) is + P(x, S) = $J($
;(e-sx/c_ X
e-“SIv
_
e
-zsLlc+sx/c)
e-sL/c-sL/v(esx/c
+
+
e-sx/c)
_ 1
[
+
,-WC
1 (A5)
On taking
the inverse
PC&t> =
Laplace transform
[(pt-x)-
of (A5), we obtain
p-1(1
+
e-2sL/c)-lpQ(t)
V
where
1 )
(A6)
,,=E[P(t-f)-P(i-~+5)1
(A7) Expanding
(1 + e-2SL’C)-1 = 1 - e-zSL’c f e-4SLlc - . . ., we have finally
Ax, t> =
&
H ) p” t - ;
- Q”(t)
+P(i-F)-6(1-T)+...].
(A8)
which is easily seen to be identical with the solution when (A7) is used to express 0 in terms of P. APPENDIX
B:
COMPUTATION
OF OUTFLOW
TIME AND VOLUME
Region III is defined in Sec. 8 as the region terms of (49) are saturated at t = t,, i.e. t, -L/V>
(47) given in the text,
T,
FOR REGION
III
in which the first two (Bl)
and t, - L/c > T, the third term is non-zero
but not saturated,
032) i.e.
0 < t, - L/V - 2Llc < T,
(B3)
and the fourth term has yet to arrive, i.e. t, - 3Llc < 0.
04)
358
A. HELFGOTT,
The inequality (B4) guarantees terms of (49) vanish identically Under these circumstances, u(L, t) = which vanishes
E. 0. TUCK, R. J. CRAIG AND P. S. HETZEL
that the fourth, fifth, and all subsequent at 1 = t,. near t = t,, equation (49) reduces to
1) (2 + 1) - 2 - $ (c-;-F)], t
W
when L t,=F+T+
2L
I-1 -T,21
w-5)
I.e., 1-l ctJL = 1 -I- 2 + L 21 lJ = 2 -t 3j.f + (1 - &.$I, as in (56). From (B6) and (B3) we have A > I, whereas from (B2) we have ,u c 2, confirming (57). The remaining inequalities (Bl) and (B4), are satisfied automatically. The volume Q defined in (64) is computed as follows
as given in equation
(65) of the text.
REFERENCES 1 J. B. MacCallum, W&h Festschri., Johns Hopkins Hospital Reports 9, 307(1900). 2 D. D. Streeter, H. M. Spotnitz, D. P. Pate], J. Ross, and E. H. Sonnenblick, Circulation Res. 24, 339 (1969). 3 D. Durrer, R. Th. Van Dam, G. E. Freud, M. J. Janse, F. L. Meijter, and R. C. Arzbaecher, Circulation 41, 899 (1970).
MUSCLE INDUCED
EJECTION
FLOWS
359
4 E. 0. Attinger, Pulsatile bloodflow. McGraw-Hill, New York, 1964. 5 J. T. Apter and W. W. Graessley, Biophys. J. 10,539 (1970). 6 Y. C. Fung, Applied Mechanics Reviews, 1 (1968). 7 G. Rudinger, “Review of Current Mathematical Methods for the Analysis of Blood Flow”, Biomedical Fluid Mechanics Symposium, A.S.M.E., New York (1966), p. 1. 8 A. V. Hill, J. Physiology (London) 56, 19 (1922). 9 J. C. Jaeger and G. Newstead, An introduction to thelaplace transformation. Methuen and Co. Ltd. (1969). 10 A. Helfgott, “Mathematical Modelling of the Heart”, Ph.D. Thesis, The University of Adelaide (1971, unpublished).