A COMBINED

LEFT VENTRICULAR/SYSTEMIC ARTERIAL MODEL*

PHILIPD. COREY?and RONALD R. WEMPLE Colorado State University. Fort Collins. Colorado X0521. U.S.A. and TERRY J. VANDER WERFF: Cardiovascular Pulmonary Research Laboratory. University of Colorado Medical Center. Denver. Colorado 80220. U.S.A. Abstract-A model mathematically coupling left ventricular performance and systemic arterial dynamics is described. This model demonstrates that these two heretofore separate research areas can be brought into intimate quantitative contact. This model and its more sophisticated successors should provide a better understandmp of the interaction between the heart and the arterial system. particularly when one or both of these is abnormal.

All mathematical representations of the arterial system (e.g. McMahon et al.. 1969; Anliker er al.. 197 1; Wemple and Mockros. 1972) model the heart as having a prescribed cyclic flow or pressure independent of arterial dynamics. Such models are useful for understanding how arterial geometry and wall properties affect the pressure and velocity waveform development, but they do not give information on how pressure and flow in the arterial system affect heart performance. A useful dynamic description of heart muscle performance is provided by a three-dimensional diagram relating muscle tension. length, and contractile element shortening velocity (Braunwald et al.. 1968). These three variables define a surface along which the muscle shortening of systole may be plotted (Fig. 1). Each closed pathway along the surface relating velocity of contractile element shortening. iension, and fiber length represents a particular myocardial contractile state (Brutsaert and Sonnenblick. 1969). This paper combines these two separate research areas by developing a complete mathematical model which incorporates descriptions both of left ventricular performance and of the dynamics of the systemic arterial system. This model can provide the capability for

* Rrcrioed I hfu>. 1974. t Present address: Polymer Intermediates Department, E.I. duPont de Nemours & Co.. Old Hickory. Tennessee. U.S.A. $ Bioengineering Department. University oE Capetown and Groote Schuur Hospital. Cape Town. South Africa.

examining how arterial abnormalities. such as the increased peripheral resistance of hypertension. the increased stiffness associated with arteriosclerosis. and anomalies such as coarctation, affect left ventricular performance. The model can be used to estimate ventricular oxygen consumption from contractile element work (Bretman and Levine, 1964). Another important use is for analysis of the effectiveness of heart assist devices. such as the intraaortic balloon and left heart bypass. in modifying the pressure and-flow generation and contractile element work of a failing heart.

Fig. I. Three-dimensional surface representing tenslon. length and velocity relationship for the heart muscle.

P. D. COREY.R. R. WEMPLEand

IO LEFT

VENTRICULAR

MODEL

The velocity of left ventricular contractile element shortening (I&), the muscle fiber length (1).and the tension in the muscle (T) are interrelated defining the three-dimensional surface shown in Fig. 1 (Braunwald er al., 1968). More complicated mathematical descriptions of the heart’s performance (e.g. Fung, 1970; Ghista and Sandler, 1969; Hanna, 1973) can be put into the model later, but for the present we will develop as simple a description of left ventricular performance as possible. The relation between the velocity of shortening of the contractile element and the muscle tension at a given muscle length is given (see Nomenclature) by Hill (1938): (T + a)v,, = b (Ts - T),

(1)

where Ts is the tension on the curve labelled 3. iye. in the T-l plane in Fig. 1. By setting T = 0 in the above. one obtains a V2 = b T3,

(2)

where the V, is the velocity of contractile element shortening on curve 2, i.e. in the V,-1 plane. It should be noted that a and b are functions of the muscle length. However, many investigators(e.g. Hill, 1938; Brutsaert and Sonnenblick, 1969) have shown that a/T, is approximately a constant for a given muscle. Thus, we can write a = Y&,

ment shortening and fiber length for constant tension can be described by a segment of an ellipse:

(6) Equations (4-6) define a three dimensional surface shown in Fig. 1, characterizing the left ventricle’s performance. These equations can be combined to give an equation relating V,,, T, and 1 at any point on the surface :

X

2

l-l,

[i 10

1,

>

Cl (T_ -

q=

l 1 = Ov

(7)

Tli (7” - 732 - ’

1

=”

(8)

The left ventricle will be assumed to be a thin-walled sphere with contained volume (4/3) rR3 and inner circumference 2 nR. The rate of flow of blood (Q) out of the ventricle and into the aorta is the negative of the rate of change of ventricular volume:

(4) Q = A,V = -4nR’

for a given muscle length. Brutsaert and Sonnenblick’s (1969) experiments on cat papillary muscle suggest that a linear relationship exists between the tension and fiber length at a constant shortening velocity:

T=T,(l-+-‘).

-

where V, is the maximum velocity of contractile element shortening, T, is the maximum tension, and L, is the minimum fiber length. (To maintain consistency of nomenclature. we use ‘m’as the subscript denoting the maximum value. In general, one finds Vmaxinstead of V, and To (or PO) instead of T, in the literature.) Since the inner radius (R) of the left ventricle is proportional to the muscle fiber length, we can rewrite equation (7) as:

(3)

where 7 is a constant. The factor a represents the rate of heat production during the muscular contraction. whereas b represents the absolute rate of energy liberation (Hill, 1938; Katz, 1970). Combination of the above three equations gives (T + i’T,) v,, = yv, cr, - T)

T. .I. VAWDERWERFF

(5)

where I, is the maximum fiber length and the subscript *I’ refers to the curve in the C;,. - T plane. Finally. the results of Sonnenblick (1965) Brutsaert and Sonnenblick (1969) and Sonnenblick et al. (1969) suggest the relationship between the velocity of contractile ele-

f,

where Vis the aortic blood velocity, A, is the (constant) area at the aortic valve, and t is time. The two-element Hill muscle model (1938) gives an equation for the velocity of contractile element shortening (V,,), the velocity of series elastic element shortening (V,,). and the total muscle length shortening veloxity (V,,): I& = v,, + r;,.

(10)

ViJ is simply equal to the negative rate of change in ventricular circumference:

vc/ =

__2KdR dr .

(11)

A combined left ventricular%ystemic arterial model Letting I,,. be the length of the series elastic element. I,..

_

3%’ -

_dl,,. _ dL di___.__=-A. dt

dTidt d T Id/,

dT dt

(12)

The derivative in the denominator can be approximated (Sonnenblick. 19644:Parmley and Sonnenblick. 19671as:

II

CJ”~P+~;-Qt INPUT

CARDIAC

MEAN

FLOW

FLOW,

RATE+

DISTAL

END

TERMINATION

FIN.2. Arterial model. dl= K,,.T. d/s

A combined left ventricular systemic arterial model.

A COMBINED LEFT VENTRICULAR/SYSTEMIC ARTERIAL MODEL* PHILIPD. COREY?and RONALD R. WEMPLE Colorado State University. Fort Collins. Colorado X0521. U...
624KB Sizes 0 Downloads 0 Views