Performance criteria for optimization of cardiac assistance

Bulletin of Mathematical Biology, Vol. 41, pp. 217-227 Pergamon Press Ltd. 1979. Printed in Great Britain © Society for Mathematical Biology

0007-4985/79/0301-0217 $02.00/0

P E R F O R M A N C E CRITERIA F O R O P T I M I Z A T I O N OF

CARDIAC ASSISTANCE

liB. G. MIN

Department of Electrical Engineering, Rutgers University, Piscataway, NJ 08854, U.S.A.

A performance criterion and weighting factors for the optimal cardiac assistance are investigated by applying Tellegen's network theorem and tolerance analysis on animal experimental data for left ventricular (LV) bypass on the failing heart. Two major factors with respect to cardiac assistance (total power delivered to the peripheral circulatory system, and changes in temporal pattern of ventricular contraction) are represented by two performance criteria, J1 and J 2 , where J1 relates to the sum of LV and pump power, and J2 relates to the "'peakedness" factor of LV power. Tile total performance index ( J ) i s determined as the weighted sum of J1 and J2; J=wlJ1 +w2J2. The weighting factors, w~ and w2, are computed as inverses of the tolerance in the performance contours with respect to improvement of stroke work per minute from pre- to post-bypass condition.

1. Introduction. One basic question in optimizing the effectiveness of cardiac assistance by using mechanical devices is how to combine appropriately the following two major objectives of assistance for the failing heart (Kolff et al., 1969; Moulopoulos et al., 1973): (1) the increase of blood supplied to the systemic circulation, and (2) the improvement of the heart itself caused by changes of the temporal pattern of ventricular contraction. Pursuit of the first objective without considering its effect on the second factor will usually increase the rate of energy consumption of the ventricle. This could be detrimental, especially in the presence of acute myocardial infarction or compromised coronary circulation. On the other hand, marked diminution of the ventricular workload may lead to unacceptably low cardiac output or blood pressure, resulting in deterioration after the termination of cardiac assistance. While the importance of these two objectives can be intuitively understood, the establishment of quantitative criteria for the appropriate combination in assisted-circulation presents a complex problem (M0ulopoulos 217

218

B.G. MIN

et al., 1973; Kolff et al., 1969). In previous studies of optimal cardiac assistance ( M i n e t al., 1972; Anderson and Clark, 1975; Clark et al., 1973) questions relating to the determination of performance criteria and weighting factors have been largely avoided, and the necessary information was assumed to exist. That is, computerized automation of the assist device based upon monitoring certain variables considered to be of importance was accomplished by mathematical computation wherein changes of these variables led to adjustments of the assist device. In these cases regulation of the device was accomplished according to parameters that might not be necessarily pertinent. In this paper we are describing the objectives of cardiac assistance using ventricular power and the Tellegen network theorem (Penfield et al., 1970) and determining the weighting factors using a tolerance analysis from the performance contour diagram (Butler, 1971a,b; Karafin, 1971). Based upon this theoretical analysis and animal experimental studies of left ventricular bypass, a method of determining the performance index and weighting factors is presented.

2. Analysis.

(A) Single-stage dynamic optimal problem. Optimization of cardiac assistance can be considered as a single-stage optimal problem of a dynamic system ( M i n e t al., 1972; Bryson and Ho, 1969), as shown in

Iu(o) x(o) ]

fo

X(I)

Figure 1. A single-stagedynamic system for optimization of cardiac assistance Figure 1. A system is initially in a known state described by X(0), an ndimensional state vector. An m-dimensional control vector, U(0), determines a transition to a state described by X(1) through the

X (1)= f ° ( X (O), U(O))

(1)

In the present example, X(0) and X(1) describe hemodynamic states before and after cardiac assistance, respectively, and U(0) represents the pressure and flow of the assist device.

OPTIMIZATION

OF CARDIAC ASSISTANCE

219

Using this approach, the objective of the present study is to develop a scalar performance index (PI), J, which compares the "goodness" of the state response of X(1) with respect to X(0), when J is a function of X(0) and U (0).

J=L°(X(O), U(0))

(2)

A desirable property of J is for the minimization of J to result in the greatest improvement in stroke work per minute (the product of mean aortic pressure and mean aortic flow) in X(1) as compared to the initial stroke work in X(0). This can be accomplished by appropriately combining the two objectives of cardiac assistance described in the introduction. Thus, the performance index can be formulated by the weighted sum of two objectives, J = W l J 1 +w2J2,

(3)

whe.re J1 is the functional representation of the first objective, i.e., the improvement of hemodynamic power supplied to the peripheral system during assistance, and will be derived in the next section J2 is the functional representation of the second objective, i.e., the improvement of the temporal pattern of ventricular contraction, and will be derived in the next section and w~ and 1412 are the weighting factors of J1 and respect to the total performance index, J.

J2, respectively, with

(B) Application of Tellegen's theorem to assisted-circulation. The above two objectives of cardiac assistance can be quantitatively expressed by applying Tellegen's network theorem to assisted-circulation. The peripheral cardiovascular system is represented as a network model, q, where 1/ contains a total N number of lumped elements, linear or nonlinear, passive or active, time-varying or time-invariant. The left ventricle and the assist device are represented as two energy generating sources with periodic waveforms connected to the rest of the network at the port 1-1' and 2-2' in Figure 2. In this figure, t/1 corresponds to the state before assistance, and ~2 corresponds to the state during assistance. It is assumed that two networks, ql and t/z, have the same topology (same node and branch configurations). However, i/1 and q2 may have different elements or

220

B.G.

MIN

element values, and different initial conditions (Penfield et al., 1970), reflecting the variations caused by physiological compensatory mechanisms and changes due to the interaction between the cardiovascular system and the assist device (Welkowitz et al., 1973).

(B) During-Assistance

(A) Pre=Assistance

%=o [i

-I"

Plvl (~

%

-t2,

Figure 2. A network representation of the assisted circulation .

Using the Tellegen's theorem, the pressure and flow relationships in qx and q2 reduce to the following four separate theorems of conservation of instantaneous power and quasi-power (Penfield et al., 1970): N

Pt~a(t)Ql~,(t)= ~ P~l(t)Q~,(t)

(4)

a=l N

Pl~2(t)Ql~,(t)= ~ P~2(t)Q~l(t)

(5)

N

Pl~l (t)Ql.2(t)+Pvl (t)Qp2(t)= ~ P.I (t)Q~2(t)

(6)

a=l N

Pzv2(t)Ql~2(t)+Pp2(t)Qp2(t)= ~ P.2(t)Q~2(t)

(7)

o~=1

where

Ply1 and Q~vl are the left ventrical pressure and aortic flow in Figure 2-A for the pre-assistance state Pzv2 and Qlv2 a r e the left ventricular pressure and aortic flow in Figure 2-B during assisted-circulation Ppl is the pump pressure on ql and Qpl (pump flow in pre-assistance) equals zero

OPTIMIZATION OF CARDIAC ASSISTANCE

221

Pp2 and Qpz are the pump pressure and flow during assistance P~ and Q,1 are the pressure and flow for branch e in rh satisfying Kirchhoff's law P,2 and Q~2 are thepressure and flow for branch e in ~2 satisfying Kirchhoff's law. We can observe from these equations that the pump power, Pp2(t)Qpz(t), can contribute to changes in two major factors of assisted circulation by its effect on the first and third term of (7). For example, in (4) and (7), the sum of the instantaneous power delivered to each branch of the peripheral system can be expressed by the right-hand side of (4) for the preassistance case, and (7) for assisted circulation. Thus t h e parameter, P~, representing the first objective of assistance, is the sum of the magnitudes of the ventricular and pump power in (7) compared to the ventricular power in (4). While the first objective relates to magnitude, the second parameter, P2, describing the second objective of assisted circulation, can be represented in (7) by the temporal change in the instantaneous LV power during assistance i[Plvz(t)Qtv2(t)] produced by pump power [Ppz(t)Qpz(t)] as compared with [Pl, l(t)Qz~l(t)] of (4) for the pre-assistance state. This change in the temporal relationship in LV power is measured by the "peakedness" factor, K, as defined in (8). In a previous study (Min et al., 1978), this factor, determined by the location of the peak and the sharpness of the LV power waveform, was Shown to change due to left ventricular bypass and was significant in determining the efficacy of the assist device.

K = T.

AT

(8)

where Tp is the time from the beginning of ejection to the peak of the LV power AT is the time difference of two points in the Ptvl(t)Qzvl(t) [or Pzv2(t)Qtv2(t)] waveform'which have a magnitude of half the peak power. (C) Weighting factor determined from tolerance analysis. The weighting factors, wl and w2, in (3) are measured as the large-change sensitivity (Butler, 1971a,b; Karafin, 1971) or the inverse of the tolerances (Kelley, 11972) of P1 and P2. As an example, for an index with small tolerance, a deviation only slightly from the optimal value leads to performance which

222

B.G.

MIN

is unacceptable; i.e., the term is highly sensitive to the total PI and thus requires a higher weighting as compared to a less sensitive (or highly tolerant) index with respect to the total PI. The tolerances of each term in the PI are determined using th~ performance contour of an n-dimensional parameter deviation space, i.e., a space where each axis is the deviation of one parameter from its nominal value (Karafin, 1971). The region of acceptability, RA, is determined in the performance contour such that the parameter values represented by points inside this region are acceptable with respect to the given performance criteria and those outside the region are unacceptable. In the same space, the tolerance vector, z k, is associated with the largest parallelepiped within RA (Butler, 1971a, b; Karafin, 1971).

3. Experimental Study. To illustrate the above theoretical analysis of the performance criteria and weighting factors for optimal cardiac assistance , t h e technique is applied to measured data of pulsatile left ventricular bypass pumping (LVBP) of the failing heart of six mongrel dogs. A description of the animal preparation and the operation .of the LVBP is presented in another paper (Minet al., 1978). A partial LVBP was pertormed from the left atrium to the descending aorta using a pneumatically driven mechanical pump with various pump pressures and flows. The pump ejection was synchronized to ventricular diastole. After pumping the normal heart for 20 min, a diagonal branch of the left anterior descending coronary artery was ligated. Approximately 30min after ligation, LVBP on the failing heart was initiated. Two or three different sets of pumping conditions were carried out on each animal. One set of pumping conditions consisted of 30 min of uninterrupted pumping followed by a 20-min rest interval after termination of the pumping. Left ventricular and pump pressures and flows were measured before, during, and after bypass at 5-min intervals. 4. Results and Discussion. Table I summarizes stroke work per min (SW) for the pre- and post-bypass conditions, the percent changes in SW from pre- to post-bypass, total power (P1), and the "peakedness" factor (P2) both before and during bypass. The deviations of P1 and P2 from initial values (pre-bypass values), defined as vl and v2, respectively, are also shown in this table, Figure 3 shows the performance contour with axes v~ and v2 with respect to the percent change of SW, as shown by the number inside the parentheses. One example of a region of possibility (Butler, 1971a,b; Karafin, 1971), Rp, is chosen as the parallelepiped region, ABCD, inside the connected region of RA, where the assisted circulation with parameter

OPTIMIZATION OF CARDIAC ASSISTANCE

I

.< [...,

<

I

223

I

I

I

I

o

o'3

0

,-.,~ o o

I

:t3 ~.,

I

O 0

I

,~"h ("q P--

I

I

',"-~ 0'~ ~'h

-,,,~ 0,1 ~'~

I

I

I

0

I

,-"~

I

,-~ •~ O q O

-"~0

",'~ 0

0

C~,l Oq Oq

'.'~ " , ' - - ~ 0

.~

0

0

~

0

~

~

I~.~1

Oq

0~0

~ C ~

".,~ ¢ ~ I~'-

"

~

~

R ~ ~-"

o

,~..-~

o

0 ~¢~ I.~ "~l ....

~ ~.~.~

d.

~

,,,-~ I ' N

~ ' ~ "~" ~

~

0

~.~

...~

~

0

'a% ii ~, ~

-.~,.,~,.P,

.

r~

e,~ + I

~ II

224

B . G . MIN

.2.0

x (- 85)

1,4

x(-38)

,I.2 .I.0

~ (26

X(--16)

0.4

4 ~ ~(-38)

x(-251

x(-8) •QS-Q4 - 0 3 - 0 2 -0.1

0.1 o.'2 o'.3 o.z, o.5 06

x(-68)

(~7 o~8

v~

•O2x(--54 ) -0.4 x ( - 7 5 )

•i

PosiNve

-0.6

SW change x;NegativeSW change

-0.8

X (--53)

-ID

Figure 3. The performance contour in the left ventricular bypass. The percent changes of stroke work per minute (SW) are shown by the numbers inside the parentheses

values represented by points inside this region produced increased SW and those outside of this region produced decreased SW. This performance contour exhibits the following characteristics: (a) The two parameters (P1 and P2) are uncorrelated with respect to the percent change of SW, since the region of acceptability for one parameter does not depend strongly on the value of the second parameter, i.e., R A is parallel to the axes and does not include the 45 ° line in the performance contour. This is a typical form for two uncorrelated parameters (Butler, 1971b). (b) The region of possibility has a center at 0' with vl--1.0 and v2 = -0.1. This corresponds to 1 0 0 ~ increase in the total power and 1 0 ~ decrease in the "peakedness" factor. Tolerance are given as _ 8 0 ~ for P1 (distance from 0' to AB or CD) and _+20~ for P2 (distance from 0' to BC or AD). Thus P2 is less tolerant (more sensitive) than P1 by a factor of 4:1 in terms of stroke work improvement. (c) From Figure 3 we can also note that if the nominal operating points were centered at 0' in RA, the assisted-circulation could tolerate larger parameter variations from nominal until it becomes

OPTIMIZATION

O F CARI~IAC A S S I S T A N C E

225

unacceptable. Thus, one function of mechanical assistance may be considered to desensitize (Karafin, 1971) in a large change sense of the initial state X(0) (the failing condition) by removing the operating point from the initial 0 point to 0', the center of RA. Based upon this observation of the performance contour, a total performance criterion for optimal cardiac assistance with respect to improvement of stroke work can be expressed by the following "squarederror" term: J = w 1 J 1 ~- w 2 J 2 1

2

1

(vl -Vlo) + ~ (v2-V2o) 2

(9)

where 7"1 and T2 are tolerances for J1 and J2, respectively. Vlo and V2o are the nominal values of vt and v2 (the deviations of P1 and P2) respectively. Thus, minimization of J in (9) corresponds to operation of the bypass at the nominal (or optimal) point, v=V~o and v2=v2o. As the operating point moves away from this point, J increases with the relative weighting of wl and w2 to J~ and Jz, respectively. Substituting the data of Figure 3 for the positive change of SW, that is T1 =0.8, T2 =0.2, V~o=1.0, and V:o = -0.1, J is given as follows: J=0~(vl-l.0)2+

(v2+0.1) 2.

(10)

Using (10), the values of wlJ1, w2J2, and J are computed for each pumping condition in Table I. Figure 4 shows the relationship between the measured values of the percent change in SW from pre- to post-bypass condition and the computed total performance index, J. It shows that decrease of J during assistance is exponentially related to improvement in SW (percent change in SW = -102.841og10 J - 6 . 4 9 with a correlation coefficient of -0.876). Figure 5 shows the relationship of J vs v~ for several values of v2. It can be seen in this figure that when v2 = -0.1, 1 0 ~ decrease in K factor during assistance from pre-bypass, J values are extremely close to the optimal (J -~0) throughout the range of v2=0.8 to 1.2 (80~o to 120~o increase in the total power). From Figure 4, this value of J corresponds to approximately 100~ increase in SW. On the other hand, when v2 equals 0.4, the minimum value of J is 1.3, which corresponds to a 20~o decrease in SW. For other ranges of vl, assisted-circulation might have a poorer effect in G

226

B.G.

MIN

5.0

I

i

I

l

l

l

[

l

l 0

4.0

O

5.0 0 2.0 0

0

0

0

1.0

0

o

0

0 o

i

,

i

I

I

%0-,;o-~o-~-5o-,,o-3o-2o-,o

I

I

Percentage in S

Decreose

W

~o' ~'o 2o

o ,;

~'o~

i0

° I

70 do ~o ,oo

in SW

change

rmprovement"

in SW

F~gure 4. Relation between the measured values of the percent change in stroke work from pre- to post-bypass and the computed total performance index (J)

6.762

/"

i'/

5.410

Z I

4.057

f

........ ,'77 2.70~

J(v , v2=0.3) .........-/:

i%'-o.,.

I .35

7;:-J\ %.-..

J(v, ,va = 0.1) "

,.

.,

.

I

d t V

~V 2

-

~'~,/

U

~ . l

C'~/ / ,'~4" t ~'7

.

" %,'JJ(v ..... . . / d ( v i ,v 2 = 0.3)/ "" J " " ."""7-,, ...... , #.. ........ " ./1.1..~

7"'

-'~0

=-02)

,J(v,,~,~=o.o)

..... "~-

I v~

....

~

0.6

-

I

/

-~"/]" 'J(v,,v2 =-0 I)

-~ . . . . " - - : ~ S " I "-~z-L-"~ " I i.2

1.8

2.4

30

v,

Figure 5.

Relation of J vs vl for different values of Vz

post-bypass. The analytical result which indicates the significance of the temporal pattern of the ventricular contraction agrees with the previous experimental results in terms of the relationship between the absolute value of the K factor and the percent change of SW (Min et al., 1978).

OPTIMIZATION OF CARDIAC ASSISTANCE

227

I n conclusion, the simplicity a n d g e n e r a l i t y of T e t l e g e n ' s t h e o r e m m a k e s it possible to f o r m u l a t e a p e r f o r m a n c e criterion for a s s i s t e d - c i r c u l a t i o n o n an a n a l y t i c a l basis w i t h o u t the m a n y a s s u m p t i o n s of p r e v i o u s studies (Min et al., 1972; A n d e r s o n a n d C l a r k , 1975; C l a r k et al., 1973). T h e s e include the t i m e - i n v a r i a n c e of the m o d e l p a r a m e t e r s , linearity of the source, a n d the l i m i t a t i o n to s t e a d y state analysis. I n addition, the t o l e r a n c e design analysis p r o v i d e s a m e t h o d of d e t e r m i n i n g the w e i g h t i n g factors for e a c h p a r a m e t e r of the p e r f o r m a n c e index. This i n v e s t i g a t i o n was s u p p o r t e d b y C h a r l e s a n d J o h a n n a Busch M e m o r i a l F u n d a n d the R u t g e r s R e s e a r c h Council.

LITERATURE Anderson, C. M., and J. W. Clark, Jr. 1975. "Analog Simulation of Left Ventricular Bypass Mode Control." IEEE Trans. Biomed. Engng, BME-22, 384~392. Byson, A. E., Jr. and Y. C. Ho. 1969. Applied Optimal Control. Waltham, Mass: Blaisdell. Butler, E. M. 1971a. "Large Cha,nge Sensitivities for Statistical Design." Bell System Tech. J., 50, 1209-1224. , E. M. 1971b. "Realistic Design Using Large-Change Sensitivities and Performance Contours." IEEE Trans., CT-18, 58-66. Clark, J. W., G. R. Kane and H. M. Bourland. 1973. "On the Feasibility of Closed-loop Control of Intra-aortic Balloon Pumping." IEEE Trans. Biomed. Engng, BME-20, 404~ 412. Karafin, B. J. 1971. "The Optimum Assignment of Component Tolerances for Electrical Networks." Bell System Tech. J., 50, 1225 1242. Kelly, H. J. 1962. "Methods of Gradients." In Optimization Techniques (Leitmann, G., Ed.) Ch. 6. N.Y.: Academic Press. Kolff, W. J., S. D. Moulopoulos, C. S. Kwan-Gett and A. Kralios. 1969. "Mechanical Assistance to the Circulation: The Principle and the Methods." Prog. Cardiovas. Diseases, 12, 243-270. Min, B. G., M. Z. Abbassi, W. Welkowitz, J. B. Kostis, S. Eich and J. W. Mackenzie. 1978. "Significance of Left Ventricular Power Waveform on Changes in Stroke Work During Pulsatile Left Ventricular Bypass.". To be published in J. SurgicalRes. , B. G., W. Welkowitz, S. Fich, D. Jaron and A. Kantrowitz. 1972 "Dynamic Optimization on In-series Cardiac Assistance by Means of Intraaortic Balloon Pumping." Bull. Math. Biol., 37, 19-35. Moulopoulos, S. D., L. P. Anthopoulos, S. F. Stamatelopoulos and D . G . Bout~ts. 1973. "Optimal Changes in Stroke Work During Left Ventricular Bypass." J. AppL Physiol., 34, 12-17. Penfield, P., Jr., R. Spence and S. Duinker. 1970. Tellegen's Theorem and Electrical Networks. Cambridge: The M.I.T. Press. Welkowitz, W., B. Min and S. Fich. 1973. "The Interaction of Mechanical Left Ventricular Assistance and the Physiological Control System." In Regulation and Control in Physiological Systems (Iberal, A. S. and Guyton, A. C., Eds). Pp. 332-324. Pittsburgh: Instrument Soc. of Amer.

RECEIVED 5-17-77