Artificial Organs 20(6):632436, Blackwell Science, Inc., Boston 0 1996 International Society for Artificial Organs

Numerical Simulation of Nonpulsatile Left Ventricular Bypass Norimasa Mitsui, Shintaro Fukunaga, Yoshihiko Koura, Satoru Morita, Hiroshi Hotei, Masafumi Sueshiro, Taijiro Sueda, and Yuichiro Matsuura The First Department of Surgery, Hiroshima University School of Medicine, Hiroshima, Japan

Abstract: A computer simulation was carried out to investigate the influence of nonpulsatile left ventricular assistance on hemodynamics. A simulation circuit was constructed to represent the circulatory system. A source of current was added to denote the nonpulsatile blood pump. The left and right ventricles were replaced by variable compliances. Left heart failure was simulated by decreasing the amount of compliance change of the left ventricle. We introduced a pulsatility indicator (PI) to clarify the pulsatility characteristics in the hemodynamics; this PI was defined as the ratio of the pulse pressure (PP) to

the mean aortic pre$sure (AoP). When nonpulsatile bypass flow increased, the mean AoP, tension time index (TTI). and diastolic pressure time index (DPTI) increased, and cardiac output, PP, and PI decreased. When assisted flow increased with the constant total flow rate, the mean AoP and DPTI changed little; the PP, TTI, and PI decreased, and the endocardia1 viability rate increased. The PI would be helpful in evaluating the effect of pulsatility. Key Words: Computer simulationNonpulsatile blood pumpHemodynamics-Heart failure-Variable compliance-Left ventricular bypass.

Circulatory assist systems using rotary blood pumps have been developed and used successfully for years (1). Development of safer, cheaper, and more reliable rotary blood pump systems is ongoing at various institutions ( 2 4 ) . Artificial circulation with nonpulsatile blood pumps, however, is thought to yield nonphysiologic conditions. A computer simulation of the circulatory system was employed to investigate hemodynamics (5-7). We also have developed a simple numerical simulation circuit and studied the effect on hemodynamics of the total artificial heart, intraaortic balloon pumping (IABP), and biventricular assistance (8-10). In this article, we present the simulation circuit, method of solution, and the influence of the nonpulsatile left ventricular bypass pump on various hemodynamic conditions. METHOD

lar compliance, and inertia of the blood in the vessels are replaced by the symbols for electrical resistance ( R ) , capacitance ( C ) , and inductance ( L ) , respectively. To denote the nonpulsatile left ventricular bypass pump, a source of current is added from the left atrium to the ascending aorta. The pumping function of the left and right ventricles is simulated by changing the capacitance of the variable capacitors denoting each ventricle. The capacitance, C,-C,, represents the compliance of the left ventricle (C,), the ascending aorta (C,), the peripheral arterial vessels (CJ, the systemic vein with right atrium (CJ, the right ventricle (C5),the pulmonary artery (Q,and the pulmonary vein with left atrium (C7),respectively. The inductance LR and L,2 correspond to the inertia of the blood in the aorta (L8) and the pulmonary artery (LI2),respectively. The resistance R8-R12represent the vascular resistance of the aorta (&), arteries of the head and upper extremities ( R J , arteries of the abdomen and lower extremities (R,"), the coronary artery ( R , , ) , and the peripheral vessels of lung ( R , J , respectively. The resistance R,,-R,, represent the resistance of the 4 cardiac valves; their function is expressed by the diodes D,,-D,,.

Circulatory simulation circuit The circulatory simulation circuit is constructed as shown in Fig. 1. The vascular resistance, vascuReceived January 1996. Address correspondence to Dr. Norimasa Mitsui, The First Department of Surgery, Hiroshima University School of Medicine, 1-2-3 Kasumi, Minami-ku, Hiroshima 734, Japan.

632

COMPUTER SIMULATION MODELING

633

FIG. 1. The simulated circulatory circuit designed for the numerical calculation is shown. C, compliance: L, inertia: R , resistance; P, pressure; Q , flow rate: D, diode. Subscripts: 1, left ventricle; 2, ascending aorta; 3, peripheral arterial vessels; 4, systemic vein and right atrium; 5, right ventricle: 6, pulmonary artery; 7, pulmonary vein and left atrium: 8,descending aorta; 9, arteries of head and upper extremities; 10, arteries of abdomen and lower extremities; 11, coronary artery; 12, peripheral vessels of lung; 13, tricuspid valve; 14, pulmonary valve: 15, mitral valve; 16, aortic valve; L, nonpulsatile left ventricular bypass pump.

Governing equations Assume that Ql-Q7 is the flow rate into C1-C7, respectively; and Q8-QI6 is the flow rate through Lg-Ri6, respectively; and PI-P, is the pressure at C,-C',, respectively. Q, is the flow rate of the current source representing the nonpulsatile blood pump connected from the left atrium to the ascending aorta. P,-P,, Q1-QI6, and C , and Cs are treated as functions of time. By relating the pressure and the flow rate into each capacitance, we obtain Eqs. 1-7.

CiP, = JQidt (i = 1-7)

(1-7)

The pressure decrease by the resistance and inductance yields Eqs. 8-16.

PI

- p2 =

R16Q16

P2 - P3 = RgQg

+ Lg d(g:)-

(8) (9)

The conservation of mass at each junction gives Eqs. 17-23.

(17) (18) Q2 = Qi6 + QL - Q8 - Q9 - Q i i (19) Q3 = Q8 - Qio (20) Q4 = Q, + Q I O + Q i i - Q13 (21) QS = Q13 - Q14 (22) Q6 = Q i 4 - Q12 (23) Q7 = Q12 - Q,, - QL Thus, knowing these 23 equations with 23 unknowns, we can solve them using appropriate initial conditions to determine Q1-Q,6 and Pi-P7 as functions of time. Numerical method Before solving the 23 equations, we eliminate Q,Q7 by substituting Eqs. 17-23 with Eqs. 1-7. QI

=

QIS

-

Q16

(24) CiPi = J(Qis - Q 1 6 W (25) C2P2 = J(Q16 + QL - Qa - Q9 - Q i W (26) C3P3 = J(Q8 - Q i o W C4P4 = J(Q, + Q I O + Q I I - Ql3ldf (27) (28) CsP5 = J(Q13 - Ql4)dt (29) C6P6 = J(Qi4 - Q 1 2 W (30) C7P7 = l ( Q n - QIS - QLW Then, we replace the integral and differential terms in Eqs. 9, 15, and 24-30 with the trapezoidal rule Artif Organs, Vol. 20, N o . 6 , 19%

634

N . MITSUI ET AL.

and central difference formulae of the small time increment, At,

g(t) =

dJ(2) + dt

~

+ g") 2

(31) -

(f" '

-

.f")

At

(32) Substituting the relationship in Eqs. 31 and 32 into Eqs. 8-16 and 24-30 with 5ome calculations, we finally obtain the following set of linear equations.

prt 1 I

-

p;" - KI6Q','d1= 0

Computational procedure The flow chart in Fig. 2 outlines the computational procedure. First, all of the information needed to determine the problem, as listed in Table 1 , is set up with the appropriate initial conditions. Second, by starting with the given values for time step t", the coefficient matrix A and column vector b in Eq. 33', are determined. Third, solving Eq. 33 gives us the value of x, i.e., the values of PI-P, and Q,-Q,, for time step tN'I. Fourth, the iteration is repeated until the time step t " + I reaches I cardiac cycle. Fifth, the time step is renewed, and the calculation is repeated until the solution converges with the periodic solution. The values of capacitance C , and C, at systole are given as

+ Cl,,

C,

=

(C,.', - C,,,)exp(-tlAt/lO

-

1)

C,

=

(Cs,d - C,.,)exp( -t/Atll0

-

0.5)

(34)

+ C5,5

135) where el,(,and C5,dare the values of C , and C5 at diastole; and and C5,5 are the asymptotic values of C', and C5 at systole, respectively. To avoid the accumulation of round-off error during the iteration, we kept the blood volume in the whole body constant. The blood volume is calculated by

Initial Conditions

n=n+l

Output hemodynarnics

These equations can be expressed perfunctorily using matrix A and vector x and h as Ax = b. A i - / ; / ' O r p i f l \ , V o / . 20,

N o . 6. I Y Y 6

& FIG. 2. A flow chart is s h o w n for t h e s t e p s in t h e computation.

635

COMPUTER SIMULATION MODELING TABLE 1. Values of the parameter used in the numerical simulation C, C, C, C, R, R,,,

R,, R,: R,, R,,

= = = = = = = = = =

8.0 at diastole to 0.1 at systole 1.0 8.0 at diastole to 0.5 at systole 50 0.5 1.2 10 at diastole and 1,000 at systole 0.12 0.01 0.01

C*

DPTI EVR

0.8 C4 400 C, 3.0 L, 0.01 R, = 3.0

L,, R,, R,, V,

= = = =

= = = =

0.002 0.01 0.01 4,500

~

=

ccr+'p;+' =

(40) (41)

=

TTI relates to the work of the left ventricular muscle and indicates the oxygen consumption of the myocardium. DPTI relates to the coronary blood flow, which indicates the oxygen supply to the myocardium. EVR represents the oxygen balance in the myocardium and indicates the effect of circulatory assistance (13,14). We then introduced the pulsatility indicator (PI) to clarify the pulsatility characteristics in the hemodynamics; the PI is defined as the ratio of the pulse pressure (PP) to the mean aortic pressure (AoP), as follows:

C, compliance (mlirnrn Hg); L , inertia (mrn Hg x sec'irnl); R, resistance (mm Hg x seciml); V,, blood volume in whole body (ml).

(i

(P2 - P4)dt DPTI/TTI

= Jdjast

(36)

1-7)

When v ' ' + ~differs from the prescribed value vo, the values of P t L 1and P;+' are compensated for in every time step using Eqs. 37 and 38:

In the numerical calculation, the cardiac cycle was fixed at 750 ms (80 beatdmin), and the percentage of systole was fixed at 40%. Left heart failure was simulated by decreasing the amount of capacitance change of the left ventricle (by increasing the value of C,.,,).The Gaussian elimination method was used to solve Eq. 33. The results of the numerical calculation in some conditions are shown in Table 2 and Fig. 3. Accordthe cardiac ing to the increase of the value of el,,, output from the left ventricle (COLV), the AoP, TTI, DPTI, and EVR decreased. The left atrial pressure (LAP) also increased but the central venous pressure (CVP) changed M e . When nonpulsatile bypass flow increased, mean AoP, TTI, and DPTI increased, and COLV, LAP, PP, and PI decreased. With constant total flow with assistance, the mean AoP and DPTI were almost the same level as the control; PP, TTI, and PI decreased, and EVR increased. AoP curves in 1 cardiac cycle are shown

where Pf'l and P;" are the modified values of P z + ' and P;+', respectively.

RESULTS Each parameter used for the calculation is shown

in Table I . These parameters were based on our preliminary studies and the data of other investigators ( I I , 12). The tension time index (TTI), diastolic pressure time index (DPTI), and endocardial viability ratio (EVR) also were calculated:

TABLE 2. Result of the numerical simulation ~~~

c1.s (mlimm Hg)

Control mHFO mHFl mHF2 sHF0 sHF1 sHF2 sHF3

0.1

0.4 0.4 0.4 1.2 I .2 1.2 1.2

Bypass Total flow COLV flow Q16 QL Ql6 QL (Limin) (Limin) (Limin)

+

4.7 4.3 3.3 2.3 3.5 2.5 1.5 0.0

0.0 0.0 1.2 2.4 0.0 1.2 2.4 4.8

4.7 4.3 4.5 4.7 3.5 3.7 3.9 4.8

PP CVP LAP AoP(s1dim) P4 P2.s-P2,d P2 (mm Hg) (mm Hg) (mm Hg) (mm Hg)

w

114167189 103162182 104170186 103178189 84/52/68 8416017I 84168175 9219019I

47 41 34 25 32 24 16 2

9 9 9 9 8 8 8 9

10

13 I1 10

IY 18 16 9

TTI

DPTI

EVR

PI

(mm Hg x sec)

( m m H g x sec)

(-)

(-)

27.4 25.4 25.9 26.5 21.5 21.9 22.5 27.3

35.1 31.9 34.1 36.0 25.8 27.9 29.9 36.6

1.28 1.26 1.32 1.36 1.20 1.27 1.33 1.34

0.53 0.50 0.39 0.28 0.46 0.34 0.21 0.02

~~~

COLV, cardiac output of the left ventricle; bypass flow, bypass flow of nonpulsatile left ventricular bypass pump: total flow, sum of COLV and bypass flow; AoP (sldirn), aortic pressure at systoleidiastoleirnean; PP, pulse pressure; CVP, central venous pressure; LAP, left atrial pressure; TTI, tension time index (Eq.39); DPTI, diastolic pressure time index (Eq.40); EVR,endocardial viability ratio (Eq. 41); PI, pulsatility indicator (Eq. 42); Control, normal heart; mHF0, mild heart failure; m H F l , mHF2, mild heart failure with assistance; sHF0, severe heart failure; s H F l , sHF2, sHF3, severe heart failure with assistance. A r t f Organs, Vol. 20, No. 6 , 1996

N . MITSUI ET AL.

636

sHF3

x

Control

0

treme illness, a model not typically found in animals. When systemic circulation is assisted by a nonpulsatile blood pump, t h e total flow and mean AoP increase according the amount of assistance, but, at the same time, pulse pressure decreases. We thus propose the pulsatility indicator to present the pulsatile characteristics quantitatively. With this parameter, the degree of the pulsatility can be defined clearly and will be helpful in forecasting the physiologic effect of pulsatile or nonpulsatile circulatory assistance in vivo.

REFERENCES I . Pae WE, Miller CA, Matthew Y, Pierce WS. Ventricular assist devices for post cardiotomy cardiogenic shock-a combined registry experience. J Thorac Curdiovusc Surr: I992 ;104:54 1-53, 2. Mizoguchi K , Damm J , Benkowsky R , Aher G , Bacak J. Svjkovsky P, Glueck J , Takatani S, Nos6 Y, Noon GP, DeBakey ME. Development of an axiai flow ventricular assist device: in vitro and in vivo evaluation. Artif Orguns 1995; 19:653-9. 3 . Schima H, Schmallergger H, Huber L, Birgmann I, Reindl C, Schmid C. Roschal K , Weiselthaler G , Trubel W, Losert U, Wolner E. An implantable seal-less centrifugal pump with integrated double-disk motor. A r f i f Orguns 1995;19: 63943. 4. Qian KX, Wang SS, Chu SH. In vivo evaluation of a pulsatile impeller total heart. A m Soc Artiflnrern Orguns J 1'994; 40:2 I 3-5. S. Montani JP, Adair TH, Summer RL, Coleman TG, Guyton AC. A simulation support system for solving large physiological models on microcomputers. In? J Biomed Con7pu/ 1989;24:41-54. 6 . Ferrari G , De Lazzari C , Mimmo R , Tosti G, Ambrosi t). A modular numerical model of the cardiovascular system for studying and training in the field of cardiovascular physiopathology. J Eiorned Eng 1992;14:91-107. I . Schima H , Honigschnabel J , Trubel W, Thoma H. Computer simulation of the circulatory system during support with a rotary blood pump. Trans A m Soc Artiflntern Organs 1990; 36:M252-M254. 8. Fukunaga S , Hamanaka Y, Sueda Y, Orihashi K , Matuura Y , Simulation of artificial heart hemodynamics (in Japanese). Technical M e e t i n g on L i n e a r D r i v e s of IEEJ 1992;LD-92-32:1-1 0. 9. Hayashi S. Study on circulatory assist by combined use of left ventricular assist device with intra aortic balloon pumping for heart failure (in Japanese). Hiroshimu M e d J 1991; 39: 15-28. 10. Maruyama T. Experimental studies of biventricular bypass using roller pump (in Japanese). Hiroshima M e d J 1993;41: 143-65. I I . Reul H . Cardiovascular simulation models. Life Support Systems 1984;2:77-98. 12. Voytic S , Babbs CF, Badylak SF. Simple electrical model of the circulation to explore design parameters for a skeletal muscle ventricle. J Heurt Trunspiant 1990;9:160-74. 13. Buckberg GD, Fixer DE, Archie JP, Hoffman JIE. Experimental subendocardial ischemia in dogs with normal coronary arteries. Circ R e s 1972;30:67-81. 14. Philips PA, Marty AT, Miyamoto AM. A clinical method for detecting subendocardial ischemia after cardiopulmonary bypass. J Thorac Cardiovasc Surg 1975;69:3&9.

I 300

-

0

750 [ms]

--

systole

diastole

FIG. 3. The aortic pressure curves of the numerical simulation are shown. Control, normal heart; sHFO, severe heart failure; sHF2, severe heart failure with assistance of 2.4 Umin; sHF3, severe heart failure with assistance of 4.8 Umin.

in Fig. 3. The AoP curve shifted downward with the increase of Cl,s (control 3 severe heart failure level 0 [sHFO]), and shifted upward with the increase in assistance flow (sHF +. sHF2), especially in diastole. When the assistance flow was total (sHF3 in Fig. 3 ) , the AoP curve became almost flat and the P1 became very small. DISCUSSION

Numerical simulation models of human cardiovascular systems have been proposed, and the influence of various circulatory assist systems have been studied. From the results of our simulation, hemodynamic parameters showed almost physiologic changes in conditions. Only the changes in CVP in heart failure are dissimilar to those seen clinically. It may be due to the simulation of heart failure, with decreasing the compliance change of only the left ventricle. There is no denying that, in the strictest sense, the data from the simulation are not the same as clinical or experimental data; however, the parameters can be changed independently in the simulation, and analysis can be performed under conditions of ex-

A r t f O r g u m . Vol. 20, No. 6,1996

I ,

Numerical Simulation of Nonpulsatile Left Ventricular Bypass.

A computer simulation was carried out to investigate the influence of nonpulsatile left ventricular assistance on hemodynamics. A simulation circuit w...
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