Combining transmural left ventricular mechanics and energetics to predict oxygen demand

Annals of Biomedical Engineering, Vol. 16, pp. 495-513, 1988 Printed in the USA. All rights reserved.

0090-6964/88 $3.00 + .00 Copyright 9 1988 Pergamon Press plc

COMBINING TRANSMURAL LEFT VENTRICULAR MECHANICS AND ENERGETICS TO PREDICT OXYGEN DEMAND Shemy Carasso, Rafael Beyar, Alec G. Rooke,* Samuel Sideman Departments of Chemical and Biomedical Engineering The Julius Silver Institute of Biomedical Engineering Technion-Israel Institute of Technology, Haifa 3200, Israel *Department of Anesthesiology, University of Washington Seattle, Washington

(Received 8/4/87; Revised 1/29/88) This study relates to our earlier study which predicts the transmural distribution as well as the global left ventricular (L V) function and oxygen demand, based on the L V structure, geometry and sarcomere function. Here, we test the predicted global oxygen demand against experimental data in anesthetized, open chest dogs under changing working conditions. The experimental oxygen demand was calculated f r o m the arterio-venous difference in oxygen content times the measured coronary flow. L V load was manipulated by a combination o f a pressurized chamber connected to the femoral artery, phenylephrine infusion and an adjustable arteriovenous shunt. The heart was paced in two preset heart rates. The study demonstrates that the global predictions, based on the local distributed oxygen demand model are comparable to those obtained by other methods o f global metabolic predictions. However, unlike other global methods, the distributed model gives spatial information and predicts an endo/epi ratio o f oxygen demand ranging between 1.05 to 1.14, depending on the loading conditions, which is comparable to available experimental data. For the experimental conditions studied here (stroke volume, heart rate, aortic pressure), the theoretical analysis shows that only the end diastolic volume is significantly correlated to the endo/epi ratio o f the transmural oxygen demand. Keywords--Left ventricle, Mechanics, Energetics, Oxygen demand, Transmural distribution. INTRODUCTION T h e r e l a t i o n s h i p b e t w e e n t h e l e f t v e n t r i c u l a r ( L V ) m e c h a n i c a l a c t i v i t y a n d its o x y gen demand has been extensively studied and various methods have been suggested

Acknowledgment--This study was supported by a personal grant from Mr. Lou Pomerantz of Des Moines, IA, and sponsored by the MEP group, Women's Division, American Technion Society, NY, U.S.A. Particular thanks are due to Professor E.O. Feigl, Department of Physiology, University of Washington, Seattle, for his continued advice and support of this study. This study represents partial fulfillment of the requirements for the MD degree of S. Carasso in the Faculty of Medicine, the Technion-Israel Institute of Technology, Haifa, Israel. Address correspondence to Professor S. Sideman, Director, Heart System Research Center, The Julius Silver Institute of Biomedical Engineering, Technion liT, Haifa 32000, Israel. 495

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to predict the global LV oxygen demand from appropriate hemodynamic measurements (1,9,10,11,16,18,19,23,24,25). However, none of the techniques for global oxygen demand distinguish regional differences in oxygen consumption, and only a few attempt to address transmural distribution of oxygen demand in the LV wall (15,26). These attempts, however, lack the analytical capacity to interrelate the distributed myocardial properties and the integrated global function and oxygen demand. A recent comprehensive model of the LV mechanics (5) relates the sarcomeres' function to the global ventricular performance and elucidates the transmural distribution of the mechanical and energetical characteristics (5,6,7). By analogy to Suga's (20) pressure-volume area (PVA) relation to the global oxygen demand, the local stress-length area (SLA) relates the local muscle fiber function and the local oxygen demand (7), and reasonably predicts the experimentally established (12,13,14,25) transmural gradient in oxygen demand. However, the predictive accuracy of the SLA procedure in determining the global ventricular oxygen demand has not been tested against experimental data. Consequently, this study tests the ability to use the integrated spatial SLA procedure to predict the global LV oxygen demand by comparison to measurements made in anesthetized dogs (17) under variable loading conditions. In addition to evaluating the global LV oxygen demand, the distributed model is utilized to predict the endo/epi ratio of oxygen demand under different operational conditions. MATERIALS AND METHODS The Experimental Protocol

The data used to test the theoretical model of the LV oxygen consumption were taken from Rooke and Feigl's experimental study (17) in four closed chest dogs. Under sterile pretest surgery, electromagnetic flow probes were placed over the aorta, and complete arterioventricular block was generated by cauterization of the bundle of His. The experiments were performed two weeks later under morphine and chloralose anesthesia. Ventricular and ascending aortic pressures were measured with a double catheter tip pressure transducer. Coronary venous blood was constantly sampled from the coronary sinus at a low withdrawal rate to avoid contamination by right atrial blood. Coronary flow was measured in the left circumflex artery with a cannula-tipped ultrasonic Doppler-shift flow transducer. Heart rate was controlled by a bipolar pacing catheter in the right ventricle. The stroke volume was primarily determined by adjustable arteriovenous shunts. Systolic blood pressure was experimentally controlled with phenylephrine infusion and a pressure control reservoir connected to the femoral artery. The pressure control reservoir consisted of a collapsible plastic bag inside a box. The pressure in the box was set to a desired level. Blood moved from the animal into the bag when arterial pressure exceeded box pressure and out of it when the arterial pressure was lower. By manipulating the shunts, the phenylephrine infusion rate and the pressure in the control reservoir, the stroke volume and systolic blood pressure were carefully adjusted to achieve the desired hemodynamic parameter values. Myocardial oxygen consumption was calculated as the product of the circumflex coronary flow (in units of ml O2/min per 100 gm tissue) and the arteriovenous oxygen content difference. A series of eight measurements at different loading and heart rate conditions was performed in each dog.

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The Theoretical Mechanical Model The basic mechanical model of Beyar and Sideman (5) involves the following assumptions which are schematically shown in Fig. 1 and summarized in Table 1. A. The LV is assumed to be a nested shell spheroid wherein the ratio between the semimajor and the semiminor axes at the endocardium is held constant. The LV shell is constructed of fibers with angles that vary linearly from +60 ~ to - 6 0 ~ to the transverse plane across the wall thickness. B. The reference resting state is defined as the state at which no transmural pressure exists across the LV wall in the fully passive state. It is assumed that at this value the sarcomere lengths (SL) are constant across the wall and equal 1.9/~. The semim a j o r and semiminor axes, wall thickness, and volumes at this state are defined as ao, bo, ho, and Vo, respectively. C. The fibers exhibit the classical sarcomere physiology: the force-length relationship is exponential for the passive state and linear for the fully active state from 1.65 < SL < 2.2 ~. The active force has a plateau between 2.2 and 2.4 tz and declines thereafter. D. The muscle activation function, ACT, varies from the passive (ACT = 0) to the fully active (ACT = 1) states. As a first approximation, a half sinusoidal activation function is assumed. The duration of the muscle activation time, TC, is assumed to equal the ventricular contraction time since the electrical propagation time is relatively small. E. The force-velocity relationship is a linearly decreasing stress vs. strain rate function. F. The electrical activation propagates radially from the endocardium to the epicardium at a velocity of 0.3 m / s . G. The LV base twists counterclockwise relative to the apex during contraction. The magnitude o f the twist is a linear function o f the semiminor axis and is roughly 20-24 ~ for the normal ejection fractions.

TABLE 1. A Summary of the major assumptions used in the construction of the model of LV function (13,15). Mechanical Model

Oxygen Demand Model

Geometry: spheroidal nested shell fanlike fibrous structure. Reference state: unpressurized passive state

Oxygen demand is proportional to the local sarcomere stress-length area (SLA)

Fiber Characteristics: typical active and passive sarcomere stress length relationship Mechanics: forces are transmitted along the fibers; pressure gradient across each layer is calculated by Laplace law; the LV base twists 24 ~ relative to apex Electrical Propagation: radial (endo- to epicardial) Arterial Model: simple windkessel

SLA is defined by the " a c t i v e " area of the stress-length loop drawn by the sarcomere (B) and the "passive" area left of the active loop (Fig. 1)

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-" .--_-Z.'--. rJ)

A Strain rate e

hO

F

,o t o ~

C

G

SLo SLPo 2.2 2.4 (1.65) (1.5) SARCOMERELENGTH(p.)

i ] ~ r~ o

Q

WINDKESSEL ~ aes

H

FIGURE 1. Schematics of the assumptions used in the LV model. A: Ellipsoidal geometry with fan like fiber structure. B: The reference (unstressed) state, characterized by cavity volume. V o, semiminor bo and semimajor axis, ao, C: Sarcomere passive and active stress length relationship. D: Mechanical half sinusoidal activation function. E: Sarcomere stress-strain rate relationship, F: Radial propagation of the electrical excitation, G: Apex to base twist of the LV over its long axis. H: Windkessel arterial model.

H. A simple Windkessel afterload model with a peripheral resistance and an arterial capacitance is assumed for the arterial load. A summary of the mathematical equations used in this model is given in the Appendix.

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The Oxygen Demand Model Suga's global model (20,21,22) relates the pressure volume area (PVA) (Fig. 2) to MVOz, the average global oxygen demand (ml 02 per beat per gm tissue), by the following linear relationship: MVO 2 =

(1)

A . PVA + B

where A is an energy conversion constant, and B represents the passive energy for cellular processes that are not associated with the mechanical work. By analogy to Suga's model, it is assumed that the local oxygen demand VO2(Y), per beat, is linearly related to the normalized sarcomere stress-length-area (SLA,) by the following equation (7): (2)

VOz(y) = K1 . S L A . ( y ) + K2

where K1 and K2 are constants analogous to A and B in Eq. 1. The global MVOz is then calculated by integrating Eq. 2 across the LV wall:

MV02

-----

L

y=h

(3)

VO2(y ) 9S A ( y ) "dy =0

where SA(y) is the shell surface area for the layer at depth y from the endocardium, h is the wall thickness, and SLA, (y) is the sarcomere stress-length-area, normalized to the sarcomere length at zero active force. This normalization results in units of stress (mmHg) which represent the total mechanical energy per cm 3 of myocardium per beat. K1 and K2 are now determined by comparing M V 0 2 of Eq. 1 with that predicted by Eq. 3. The values are listed in Table 2.

40-

PRESSURE-VOLUME

STRESS-LENGTH ~,1150 . . . . . . . . . . . . . . . . . . . . . . . . . . .

GLOBAL

Q I0

0

0

40 8O Volume (ml)

120

i

I

I

165

L9

2.2

2'.4

SL (bL)

FIGURE 2. The stress-length area (SLA) applied to describe local oxygen demand compared with the pressure-volume area (PVA) method suggested by Suga (21). Note that in each case the oxygen demand relates to the mechanical active work performed (loop B), plus the passive energy expended without performing additional mechanical work (triangle A).

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The SLA area which correlates with the oxygen demand per beat (Fig. 2) is composed of the stress-length loop area (area B) and the area between the loop and the line representing the maximum sarcomere stress-length relationship (area A). The total area (A + B) is the maximum external work that a sarcomere can perform under the ideal loading conditions. However, under normal conditions the energy represented in area A is not transformed into mechanical energy and is mainly lost as heat. The Simulation Procedure

A forward simulation of the LV function requires that the initial condition and the system parameters be defined. The system parameters include the dimensions of the LV (at reference state), the LV mass, the properties of the sarcomeres (i.e., maximal stress at optimal length), the passive exponential stress-length relationship and the force-velocity relationship. Arterial system parameters include the peripheral resistance and the arterial capacitance. Temporal parameters, like heart rate (HR) and the contraction time (defined as the time from the beginning of the muscle activation to the end of the relaxation) are also included in the model. The initial end diastolic (ED) conditions are defined by the end diastolic LV volume and the diastolic arterial pressure. The complete cardiac cycle is simulated based on the input of the ventricular and the arterial system parameters, and the initial conditions. The details of the oxygen consumption calculation procedure is given elsewhere (7). Briefly stated, the stress in the fibers is first calculated stepwise, based on the initial conditions and the activation functions. The LV cavity pressure is calculated based on the geometry. Once the LV cavity pressure exceeds the aortic pressure, the aortic valve is opened, the ventricle is allowed to eject, and the LV volume changes. The instantaneous velocity of shortening and flow is calculated from the force velocity relationship of the sarcomeres. Ejection flow reversal is used as a criterion for aortic valve closure. The values of fiber stress and length as a function of time for each myocardial layer were later used to calculate the local SLA. Using Eqs. 2 and 3, the local and global values of the oxygen consumption are finally obtained. A d j u s t m e n t o f M o d e l Parameters to Experimental Data

The parameters used in the mechanical model are listed in Table 3. The preset parameters are based on direct experimental measurements and are not altered in any way. The "adjusting" parameters are the free parameters that are manipulated in

TABLE 2. Values of oxygen demand parameters. Method

Parameters = 1.8 x 1 0 - 5 ( m t O~.mmHg - ~ - b e a t - t )

B

= 3 , 1 0 - 2 ( m l 0 2 , 1 0 0 g-1 . b e a t - l )

Suga-PVA

A

Beyar & Sideman-SLA

K 1 = 1,7 x 1 0 - 7 ( m l 0 2 . m m H g - l . b e a t - 1 )

K 2 = 3 . 1 0 - 4 ( m l 0 2 . g -1 .beat -1 )

Rooke & FeigI-PWl

A 1 = 4 , 0 8 • 10 - 6 ml 0 2 . m m H g -1 x g 1 .beat 1

A 2 = 3 . 2 5 x 10 - 6 (ml 0 2 . g -1 .rnmHg -1 • beat -1 .m1-1 .kg)

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TABLE 3. Model parameters. Preset parameters 1. V m = LV muscle volume, ml, calculated f r o m measured LV w e i g h t (density 1.0 g . c m 2. h o = wall thickness at rest, cm, calculated (0.66 bo) bo = ellipsoidal s e m i m i n o r axis at rest, cm, calculated f r o m Vm ao = ellipsoidal semimajor axis at rest, cm, defined as = 2 b o 3. EDAP = end diastolic aortic pressure, m m H g , measured RES = peripheral resistance, calculated from measured values ( 6 0 . M B P - H R -~ .SV -1 ) HR = heart rate, bpm, measured

3)

Adjusting parameters 1. EDSA = end diastolic LV semiminor axis (preload), cm, calculated 2. CAP = capacitance of aortic root, calculated 3. TC = time of LV mechanical c o n t r a c t i o n , calculated A d j u s t e d parameters 1. S V = stroke volume, ml, calculated 2. SBP = m a x i m u m systolic aortic pressure, m m H g , calculated 3. ET = LV ejection time, sec, calculated Compared parameters 1. M V O 2 o x y g e n c o n s u m p t i o n , ml/min/O.1 kg tissue 2. SV = stroke volume, ml 3. SBP = m a x i m u m systolic aortic pressure, m m H g 4. ET = LV ejection time, s =

order to fit the "adjusted" parameters to the experimental data. Note that the calculations of the oxygen consumption are not involved in the adjusting procedure. The calculations are performed as follows: A. Given the LV mass, the reference semimajor axis ao, the semiminor LV axis bo and the wall thickness ho are calculated utilizing the following equation for the LV muscle volume: V~ = 4/3 7r[(bo + ho)2(2bo + ho) - 2b 3] 9

(4)

As noted in Table 2, ao = 2bo and ho = 0.66 bo (7). The resistance of the vascular system is calculated f r o m the experimental data as the mean aortic blood pressure divided by the cardiac output. The ED aortic pressure and heart rate are taken at their experimental values. B. Preload is defined as the ED semiminor axis (EDSA), because once the heart is loaded, the length of the semiminor and semimajor axes determine the sarcomere length at the beginning of contraction. Preload (EDSA), arterial capacitance (CAP), and the time of contraction (TC) are then adjusted iteratively until the calculated values for the systolic arterial pressures, stroke volume (SV), and the ejection time (ET) are within 2% of the measured experimental values. The procedure was carried out for each of the four hearts, for eight combinations o f two levels of stroke volume, two levels of systolic pressure and two heart rates. Contractility (defined by the m a x i m u m stress that the fiber can develop at optimal sarcomere length) is assumed to be constant in this set of experiments. C. The global oxygen consumption rates were calculated for each loading condition based on Suga's P V A procedure (20), as well as by integrating Beyar and Side-

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man's SLA procedure (7) for each layer over the LV mass. In addition, Rooke and Feigl's (17) experimental "Pressure Work Index" (PWI) was calculated according to the empirical formula

MV02

AI(SBP x HR) +

=

A 2

H R x SV (0.8 SBP + 0.2 DBP) + 1.43 BW

(5)

where A1 and A2 are constants given in Table 3, SBP = maximum systolic pressure, DBP diastolic aortic pressure, BW = body weight (kg) and M V 0 2 = LV oxygen consumption (in units of ml O2/min/0.1 kg tissue). The calculated values were compared to the measured ones by means o f a linear regression analysis (derived from a Texas Instruments Inc. program which was run on the Dragon 64 microcomputer), first for 8 points for each dog and then for 32 points obtained from the four dogs. D. E n d o / E p i ratio of predicted oxygen demand was calculated by the model for the various experimental loading conditions. RESULTS Global Oxygen D e m a n d

Eight loading conditions of each of the four dogs studied were analyzed. Figure 3 demonstrates the calculated performance loops for conditions a and c o f Table 4

0

ED vol: 70ml ...... FiB vol: 57ml

"-..~ "!'~

nn

PRESSURE

i

I~ ~'~

0

VOLUME

0

.......~

'~ 1 0 Volume (ml)

20

150 I

,oo

!!ii 60

r

9

8O

Time (sec)

c 3oo;

2001_ >

20

MI

O0 "OL2

o~

04 0.6 Time (sec)

08

too

~0

0

02 04 0.6 Time (sec)

FIGURE 3. Theoretical performance loops for two loading conditions (a,c) of the heart of dog number 2.

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TABLE 4. Preset loading parameters for dog No. 2. Loading Condition

EDAP (mmHg)

Resistance (mmHg/ml min -1 )

Heart Rate (bpm)

a b c d e f g h

102 144 142 117 94 86 74 88

2.39 4.65 6.98 3.39 4.60 2.26 2.42 2.93

118 118 78 78 78 118 78 118

EDAP = End Diastolic Aortic Pressure. Vm = 0.13 kg = left ventricular weight Resting Semiminor Axis, bo: 1.75 cm; for all loading conditions. Resting Semimajor Axis, ao: 3.50 cm; for all loading conditions. Resting Wall Thickness, ho: 1.16 cm; for all loading conditions.

in which systolic pressures were similar but stroke volume and heart rate were different. Tables 4-7 present an example of the computation procedure for one of the dogs. The preset parameters summarized in Table 4 include the wall thickness and the semiminor axis which correspond to an LV mass of 130.0 g. The ED arterial pressure, EDAP, the peripheral resistance, RES, and the heart rate for the eight loading conditions studies are also shown in Table 4. The adjusting parameters for the eight loading conditions are shown in Table 5. The ED semiminor axis, EDSA, the arterial capacitance, CAP, and the contraction time, TC, were each adjusted to yield

TABLE 5. Adjusting parameters for dog No. 2. TC (sec)

Loading Conditions

EDSA (cm)

CAP (ml-mmHg -1 )

(computer generated)

TC (sec) (experimental)

a b c d e f g h

2.06 1.9 1.89 2.03 1.78 1.85 1.94 1.79

0.26 0.30 0.32 0.35 0.4 0.38 0.41 0.4

0.34 0.35 0.39 0.38 0.41 0.37 0.42 0.36

.265 .280 .288 .277 -.276 .306 .279

Average EDSA = 1.91 _+ 0.O95

Average CAP = 0.353 ~ 0.O51

EDSA = end-diastolic semiminor axis, cm. CAP = arterial capacitance. TC = myocardial contraction time, sec.

Average TC = 0 . 3 7 6 _+ O.026 (Average = 118 = 0.36 +_ 0 . 0 1 6 TC at HR] (Average = 78 = 0 . 4 0 _+ .016 TC at HR]

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TABLE 6. Adjusted parameters for dog No. 2. Comparison of experimental Rooke and Feigl's data (19) and calculated values. SEP (mmHg)

ET (sec)

Leading Condition

Experimental

Model

a b c d e f g h

183 180 179 178 121 125 127 120

184.6 182.2 181.2 180.6 126.9 125.7 128.8 120.5

Experimental 0,173 0.149 0.151 0.172 0.167 0,176 0.210 0,164

SV (ml) Model

Experimental

Model

32.5 17.5 17.3 32.6 17.7 23.1 30.6 18.0

32.4 17.4 17.0 30.4 17.0 23.1 30.8 18.7

0.18 0.15 0.15 0.18 0.17 0.18 0.21 0.16

SBP = systolic aortic pressure, ET -- LV ejection time. SV = stroke volume.

the best fit in the adjusted parameters shown in Table 6. Note that the values for peak systolic pressure (SBP), ejection time (ET), and stroke volume as derived by the adjusting procedure did match the measured parameters closely. The oxygen demand values calculated by the PVA procedure, the SLA procedure, and Rooke and Feigl's pressure work index are given in Table 7, as are the experimentally measured values. The results of a linear regression analysis of the predicted vs. the measured oxygen demand for dog no. 2 and the four dogs together are depicted in Fig. 4. The results for each of the four dogs are summarized in Table 8. Table 9 has the results for the combined 32 data points. Clearly, both the PVA and the SLA values are comparable to those obtained by both experimental observations and those predicted by the pressure-work index. Comparison of Tables 8 and 9 (and particularly the intercepts) seem to indicate the inherent variability between the individual animals. Also, note that the correlation for the combined data utilizing the three calculation methods is excellent, although somewhat smaller than for the individual dogs. Prediction o f Endo/Epi Ratio o f Oxygen Demand The model was next used to predict the transmural oxygen demand for the conditions tested experimentally. The endo/epi ratio was thus calculated for (a) differ-

TABLE 7. Calculated and measured oxygen consumption for the eight loading conditions of dog No. 2 (in units of ml 02/min. 0.1 kg tissue). Method

a

b

c

d

e

f

g

h

PVA SLA PWl Experimental

17.17 16.49 19.36 18.49

13.39 12.70 15.26 14.00

8.43 8.20 10.44 10.42

10. 49 10.41 12.81 11.57

5.96 5.62 7.74 7.52

10.4 9.5 12.01 11. 83

7.51 7.19 9.57 9.58

9.26 8.47 10.8 10. 42

PWI = Pressure Work Index,

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Dog #2. MVO2 by SLA

A

12

38 c

E

102

4

Intercept =-2.52 C~176 : 099334

/ /

Four Dogs. MVO2by SLA

8!1 [ 4L

~/

~ ~

/

4

B

~176176 Meon Slope: 089 Intercept : - [ 08 CorCoef : 094614

8 12 16 20 24 MeosuredMVO (ml/min.lOOgr)

FIGURE 4. Comparison of the measured MV02 under eight loading conditions and culated by the SLA procedure for a single dog (no. 2) and four dogs, Note that the lation is good.

the values calaverage corre-

ent stroke volumes, (b) different heart rates, and (c) different aortic pressures. Paired t-test was used to compare the data. As seen in Table 10, the calculated endo/epi ratio of oxygen demand seems to increase with an increase in stroke volume and systolic pressure but is independent of the heart rate as an isolated parameter. A linear regression between each of the variables makes up the experimental parameters (i.e., EDV, ESV, HR, aortic pressure), and the endo/epi demand ratio shows that only the EDV can be significantly correlated (r = 0.85, p < 0.001) to the endo/epi ratio (Fig. 5). The r values for all the other parameters were 0.57, 0.47 and 0.008 for the stroke volume, ESV and HR, respectively. DISCUSSION The purpose of this study was to determine the predictive value of the "distributed" SLA procedure in describing the global oxygen demand of the heart, by comparing the model based predicted value of the global 02 demand for matched hemodynamic conditions to the experimental data in dogs. The calculated results are

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TABLE 8. Oxygen consumption correlation for the four experimental dogs. Standard Deviation

Procedure

Variable

Dog 1

Dog 2

Dog 3

Dog 4

PVA

Mean slope Intercept Correlation coefficient

0.96 -2.79 0.983

1.05 -2.15 0,987

1.00 -0.61 0.984

1.04 -2.72 0,975

1 . 0 1 2 5 3.5 x 10 2 -2.067 8 . 7 7 • 10 -1 0.982 4 . 4 • 10 3

SLA

Mean slope 0.93 Intercept -2.80 Correlation coefficient 0.988

1.02 -2.32 0.993

0.95 -0.68 0.982

0.98 -2.48 0.977

0.97 -2.07 0.985

3 . 3 9 x 10 -2 8.21 x 10 1 6.31 x 10 -3

PWl

Mean slope Intercept Correlation coefficient

1.08 -0.66 0.994

0.99 1.47 0.978

1.04 -0.64 0.979

1.025 -0.195 0.934

3.77 • 10 -2 9 . 6 9 • 10 -1 6 , 5 5 x 10 -3

0.99 -0.95 0.986

Average

TABLE 9. Regression analysis of oxygen consumption for the combined data points obtained from four dogs,

Mean slope Intercept Correlation coefficient

PVA

SLA

PWI

0.92 -0.95 0.937

0.89 - 1.08 0.946

0.94 0.85 0,946

TABLE 10. Endocardial to epicardial Experimental Parameters

VO 2

ratio.

Endo/Epi Ratio

Stroke volume 3 0 . 0 +_ 1.8 18.6 _+ 1.2

1.142• 1,065•

Heart rate 1 1 8 , 2 5 +_ 0 . 4 5 78,75 + 0.45

1. 090 +_ 0 . 0 6 1. 106 +_ 0 . 0 6

Systolic pressure 1 8 1 , 6 3 +_ 3 . 5 8 1 2 7 , 4 4 • 5.61

1.139 +_ 0 . 0 3 9 1. 057 +_ 0 . 0 4 2

indeed in excellent agreement with the globally measured values of oxygen demand, and are comparable to other methods (17,20) of determining global O2 demand. The mechanical model used here requires many assumptions (Table 1) and encompasses many variables, some of which can only be approximated for lack of direct experimental measurements. For instance, in the absence of absolute volume or wall thickness measurements, one has to assume a relationship between the LV mass, the reference volume, and the ED volume. Obviously, accurate volume measurements would help in the parameter adjustments. Sarcomere properties such as the force length and force velocity relationship were obtained from available literature data,

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1.20 1.18

000O O j

0 0

I

I

1.16 0

1.t 4 1.12 e

0 r

o

/

o

1.10 1D8 ID6 1.04

l~C~ ~

ID2

45

0 55

t._.

I 65

I 75

85

EDV (rnl)

FIGURE5.

The correlation between the predicted endo-epi ratio for 02 demand and the end diastolic volume.

since it is impossible to assess these data in vivo in the individual heart. These uncertainties, however, did not interfere with the resulting predictions of global O2 consumption. The oxygen consumption constants needed for the SLA technique were derived from Suga's PVA data. The constants in Suga's equation (Eq. 1) were determined previously in dogs that were subjected to very different experimental conditions than the dogs used here. The volume for the pressure-volume area (PVA) of the present data was calculated by utilizing EDSA, as explained below. Despite the experimental differences and the assumptions inherent in the SLA model, both PVA and SLA predictions of global LV oxygen requirements proved to be quite accurate. The measurement of 02 consumption in each of the loading conditions utilized electromagnetic coronary flow, and the difference of 02 saturations between arterial and coronary sinus blood. This method is obviously subjected to possible errors of the flow measurements, inadequate venous mixing of blood, as well as alternate venous drainage pathways. Nevertheless, this method seems to be valid within a reasonable practical accuracy for measuring the 02 demand. This study is confined to load and rate alternations without stipulating any changes in the contractility. No attempt is made here to answer whether the distributed model can predict oxygen demand reasonably well with changes in contractility. Suga et al. (22) raised the possibility that changes in the constants K~ and K2 are induced by contractility changes which may suggest a way to account for changes in contractility by correcting these constants. As schematically shown in Fig. 6, enhanced contractility with either epinephrine or calcium is associated with an upward shift of the VO2-PVA line, indicating that the increase in energy utilization for excitation contraction coupling (oxygen wasting effect) is associated with the enhancement of con-

508

S. Carasso et al.

No Epinephrine

--j

oJ

C> >

Epinephrine

/ ExcilotionContraction

ExcilalionContraction

Coupling

Coupling

Basal Metabolism

Basal Metobolisrn

PVA-'l"01o1 Mechanical

Energy (J/Beat)

FIGURE 6. The effect of epinephrine on the VOz-PVA relationship, Note that the excitation contraction coupling cost is increased with epinephrine while the basal metabolism as well as the VO2-PVA slope are kept constant,

tractile state. The oxygen wasting effect may thus be practically expressed as an increase in the value of the free constants B (Eq. 1) or K2 (Eq. 2). However, it is beyond the scope of this work to address changes in the contractile state. Although it was shown here that the model can accurately predict the global LV oxygen demand, its major advantage over the other models is in its local, distributed parameters of mechanics and energetics. Only supporting evidence is presently available to validate the model's prediction that the endocardial 02 demand is larger than epicardial demand. This prediction is supported by previous experimental observations that show that the subendocardial layers consume more oxygen than the subepicardial layers of the LV (13,14,25) and, consequently, that the endocardial layers are more susceptible to hypoxic injury. The applicability of the SLA procedure for predicting the transmural gradients for oxygen demand is further encouraged by the success of its application to the analysis of the transmural temperature profiles (2,3) and the epicardial temperature maps (4) of the LV in open-heart operations. Since local energy consumption is directly related to local heat production, this evidence is obviously only circumstantial, and additional data by direct measurement of the transmural oxygen demand distribution is highly desirable. The predictions of the endo/epi ratio of oxygen demand under the experimentally tested conditions are quite instructive. As seen in Table 10, an increase in stroke volume and blood pressure (experimental data) increases the (predicted) transmural gradient, whereas an increase in heart rate does not affect the predicted transmural

Ventricular Mechanics and Energetics

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m e t a b o l i c g ra d i e n t. T h e increase in the s t r o k e v o l u m e an d the a o r t i c pressure is reflected by an increase o f the E D v o l u m e . T h e latter was previously f o u n d to be the p r i m a r y d e t e r m i n a n t o f the t r a n s m u r a l m e t a b o l i c gradient (7); These theoretical predictions o f the e f f e c t o f l o a d o n the t r a n s m u r a l m e t a b o l i c g r a d i e n t h a v e n o t been e x p e r i m e n t a l l y verified, a n d d a t a are still n e e d e d to test a n d v a l i d a t e these m o d e l predictions. In s u m m a r y , the S L A m o d e l o f v e n t r i c u l a r o x y g e n d e m a n d yields excellent predictions o f global o x y g e n d e m a n d in dogs u n d e r stable c o n t r a c t i l e state. T h e g l o b a l predictive p o w e r o f t h a t m o d e l is e q u i v a l e n t , b u t n o t s u p e r i o r to, p r e v i o u s m o d e l s such as Suga's or R o o k e a n d Fejgl's. H o w e v e r , it has t h e a d d i t i o n a l f e a t u r e o f predicting the t r a n s m u r a l gradients in 0 2 d e m a n d , which are in a g r e e m e n t with physi o l o g i cal m e a s u r e m e n t s .

REFERENCES 1. Bailer, D.; Bretschneider, J.H.; Hellige, G. Validity of myocardial oxygen consumption parameters. Clin. Cardiol. 2:317-327; 1979. 2. Barta, E.; Sideman, S.; Beyar, R. Temperature distribution within the left ventricular wall of the heart. Int. J. Heat Mass Transfer 28:603-673; 1985. 3. Barta, E.; Sideman, S.; Beyar, R. Spatial and temporal temperature distribution in the healthy and locally diseased wall of the heart, lm. J. Heat Mass Transfer 29:86-94; 1986. 4. Barta, E.; Sideman, S.; Adachi, H.; Beyar, R. Heat transfer and temperature profiles during ischemia and infarction in the left ventricular wall. In: Sideman, S.; Beyar, R., eds. Activation, Metabolism and Perfusion of the Heart. Dordrecht/Boston: Martinus Nijhoff; 1987: pp. 683-688. 5. Beyar, R.; Sideman, S. A computer study of the left ventricular performance based on the fiber structure, sarcomere dynamics and transmural electrical propagation velocity. Circ. Res. 55:358-375; 1984. 6. Beyar, R.; Sideman, S. The interrelationship between the left ventricular contraction, transmural blood perfusion and spatial energy balance: A new model of the cardiac system. In: Sideman, S.; Beyar, R., eds. Simulation and Imaging of the Cardiac System. Dordrecht/Boston: Martinus Nijhoff; 1985:331-357. 7. Beyar, R.; Sideman, S. Left ventricular mechanics related to the local distribution of oxygen demand throughout the wall. Circ. Res. 58:664-677; 1986. 8. Beyar, R.; Sideman, S. Relating left ventricular dimensions to the maximum elastance by fiber mechanics. Am. J. Physiol. 251(Regul. Interp. & Comp. Physiol. 20):R627-R635; 1986. 9. Bing, R.J.; Hammond, M.M.; Handlesman, J.C.; Powers, S.M.; Spencer, F.C.; Eckenhoff, J.E.; Goodale, W.T.; Hafkenshell, J.H.; Kety, S.S. Measurement of coronary blood flow, oxygen consumption and efficiency of the left ventricle in man. Am. Heart J. 38:1-24; 1979. 10. Braunwald, E. Control of myocardial oxygen consumption: Physiologic and chemical consideration. Am. J. Cardiol. 27:426-432; 1971. 11. Bretschneider, H.J.; Die haemodynamischen determinanten des myokardialen sauerstoffverbrauches. In: Dengler, H.G., ed. Die Therapeutischen Anwendung Beta-sympathikolykscher Stoffe. Germany: Stuttgart 1979:45. 12. Feigl, E.L. Coronary physiology. Physiol. Rev. 63:1-204; 1983. 13. Gamble, W.J.; Lafarge, C.G.; Dyler, D.C.; Weisal, J.; Monroe, R.G. Regional coronary venous oxygen saturation and myocardial oxygen tension following abrupt changes in ventricular pressure in the isolated dog heart. Circ. Res. 34:672-681; 1974. 14. Hoffman, J.I.E. Determinants and prediction of transmural myocardial perfusion. Circulation 13:929-940; 1978. 15. Panerai, R. A model of cardiac muscle mechanics and energetics. J. Biomechanics 13:929-940; 1980. 16. Parmley, W.W.; Tyberg, J.V. Determination of myocardial oxygen demand. Prog. Cardiol. 5:19-36; 1976. 17. Rooke, G.A.; Feigl, E. Work as a correlate of canine left ventricular oxygen consumption, and the problem of catecholamine oxygen wasting. Circ. Res. 50:273-286; 1982. 18. Sarnoff, S.J.; Braunwald, E.; Welch, G.H.; Case, R.B.; Stansby, W.M.; Marcos, R. Hemodynamic determinants of oxygen consumption of the heart with special reference to the tension time index. Am. J. Physiol. 192:148-156; 1958.

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19. Skelton, C.L.; Sonnenblick, E.H. Myocardial energetics. In: Mirsky, I.; Ghista, I).N.; Sandler, H., eds. Cardiac Mechanics. New York: J. Wiley & Sons; 1974:112-140. 20. Suga, H. Total mechanicalenergy of a ventriclemodel and cardiac oxygenconsumption. Am. J. Physiol. 236:H498-H505; 1979. 21. Suga, H.; Hisano, R.; Hirata, S.; Hayashi, T.; Yamada, O.; Ninomiya, I. Heart rate-independent energetics and systolic pressure-volume area in dog heart. Am. J. Physiol. 224:H206-H214; 1983. 22. Suga, H.; Hisano, R.; Goto, Y.; Yamada, O.; Igarashi, Y. Effect of positive inotropic agents on the relation between oxygen consumption and systolic pressure volume area in canine left ventricle. Circ. Res. 53:306-318; 1983. 23. Weber, K.T. Seminars on myocardial oxygen utilization physiological and chemical correlates. Am. J. Cardiol. 44:719-721; 1979. 24. Weber, K.T., Janicki, J.S. The metabolic demand and oxygen supply of the heart: Physiological and chemical consideration. Am. J. Cardiol. 44:722-729; 1979. 25. Weiss, H.R.; Newbauer, J.A.; Lipp, J.A.; Sinha, A.K. Quantitative determination of regional oxygen consumption in the dog heart. Circ. Res. 42:394-401; 1978. 26. Wong, A.Y.K. Some proposals in cardiac muscle mechanics and energetics. Bull. Math. Biol. 35:357-399; 1973. NOMENCLATURE

A,B A~,A2 ao, bo

= c o n s t a n t s , Eq. 1 = c o n s t a n t s , Eq. 5 = s e m i m a j o r a n d s e m i m i n o r axis o f the LV in the unstressed reference states, cm BW = b o d y weight, kg DBP = diastolic aortic pressure, m m H g ho = LV wall thickness in the relaxed state, c m HR = heart rate, b p m (beats per m i n u t e ) KI,Kz = c o n s t a n t s related to local oxygen d e m a n d , Eq. 2 MBP = m e a n aortic b l o o d pressure, m m H g MV02 = LV oxygen d e m a n d , ml O 2 / b e a t / 0 . 1 kg LV PVA = pressure v o l u m e area, n o r m a l i z e d to 0.1 kg tissue, Eq. 1 PWI = pressure w o r k index SA(y) = shell surface area at y, cm 2 SBP = peak systolic aortic pressure, m m H G SL = sarcomere length, # SLo = sarcomere length in unstressed state ( = 1.9 tz) S L A ( y ) = stress-(sarcomere) length area (Fig. 2) SLAn ( y ) = n o r m a l i z e d S L A ( y ) (divided b y SLo) SV = stroke v o l u m e , ml VOz(y) = local oxygen d e m a n d i n ml O 2 / b e a t / g LV o f the m y o c a r d i a l shell at e n d o c a r d i a l depth y Vo = unstressed (reference) v o l u m e o f fully relaxed LV, ml Vm = LV muscle v o l u m e , ml y = distance f o r m the e n d o c a r d i u m , cm Oo = m a x i m u m fiber isometric stress at o p t i m u m sarcomere length, m m H G Of,max ~--- m a x i m u m fiber "isometric" stress at a n y value o f SL, m m H G

APPENDIX

The Mathematical Equations o f the L V Model Figure A1 is a schematic presentation of the geometries o f the assumed thick shell LV m o d e l in the reference, unstressed, a n d general states. It is a s s u m e d that:

Ventricular Mechanics and Energetics

511 a/b

=

(At)

k

throughout the contraction. The LV muscle volume is given by: 47r Vm = -'~ I(bo + ho)E(kbo + ho) - kb31 9

(A2)

A layer of thickness, dg, at a distance of g from the endocardium in the reference state, transforms after contraction into dy and y according to: SA(g) dy = dg - SA(y)

(A3)

Y = fo where SA(g) and SA(y) are the surface areas of the spheroidal shell at the corresponding distances. The strain rate of the different layers is obtained by differentiating Eq. A2 (after replacing bo and ho by b and y), assuming incompressibility of the cardiac muscle: 4kb2+kby+_2b2_b2y__ b ] ~y = --eb skb2 + 2ypk + b 2 + 4by + 3yZJ ~y -

ldy ydt

and

~b -

ldb bdt

(A5)

REFERENC U E NSTRESSE SD TATE

GENERAL STATE

.y

A

~'Yy B

FIGURE A1. The geometry of the thick shell LV model in the "reference" and general states. (Reproduced with permission of the American Heart Association.)

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S. Carasso et al.

The circumferential strain rate 4~f is calculated by: 9

eb

ecf-- b + y

b + y

.

(A6)

The fiber orientation a in the unstressed configuration is assumed to be a linear function of g: o~ - -

7r

27rg

3

3ho

(A7)

The circumferential and the meridoinal stresses Ooo and o , , are related to the pressure gradient by:

dPw(y,t)dy - [ ~176176 r2 3 '

(A8)

where r~ and r 2 are the cavity curvature radii and Pw(Y, t) is the tissue intramural pressure. Modifying and integrating Eq. A8 yields: y

Pw(y,t) =

[ b of + y [c~176

sin 2 a ]j d y '

(A9)

where W is the stress in the fiber direction and kl is given by: k l --

a+y b+y

(AI0)

Pw(Y, t) is used in the model of the coronary circulation as the tissue pressure. The normalized isometric stress Of (t) (defined as the stress developed by contracting a constant number o f fibers) is related to the fiber physiological properties and is given by

Of(t) =

(Of,max)Sin[

t -- r ( y )

T

~r + Ofp; O < t - r(y) < T 0 > t - r ( y ) and t - r ( y ) > T . ( A l l )

tO:p;

The electrical activation front moves radially from the endocardium to the epicardium at a velocity c, causing a time delay r(y). for the corresponding muscle layer. The maximum normalized isometric stress 0 f , m a x is given by Eq. A10 in the text. The passive stress afp is defined by:

~

D ( e D(x-1) - 1);

o:p =

(-e(1

-

x) ;

k > 1 x < 1.

(AI2)

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513

E, B, and D are empirical constants, and ~ is given by:

X-

SL

(A13)

SLPo

where SL is the sarcomere length and SLP o is the sarcomere length at zero passive stress. The stress strain rate relationship is now introduced as:

~cf --

~ef, max

ao

((If -- ao) 9

(A14)

The above equations governing LV function are solved in combination with a Windkessel arterial model which is formalized as: 1

t

P~o(t) = e-t/R'cIPo - ~ f ~ et/RCQ(t)dt]

(A15)

where P~o(t) and Q(t) are the aortic inlet pressure and flow, R and C are the resistance and capacitance o f the arterial system and Po is the end diastolic aortic pressure.