Time and Frequency Domain Representations of the Left Ventricle : Theoretical and Experimental Results

OF THE THESIS Time and Frequency Domain Representations of the Left Ventricle: Theoretical and Experimental Results by Dennis James Arena Thesis Director: Dr. Dov Jaron A frequency domain and a time domain model of the left ventricle are described in this work. These representations provide insight into the function of the healthy left ventricle and show how ventricular function may be altered by heart disease. In developing the frequency domain representation of the left ventricle, the flow and pressure waveforms generated by the left ventricle were described as superpositions of sinusoidal oscillations at different frequencies. Flow and pressure waveforms were obtained experimentally at two different left ventricular afterloads . . The two different afterloads were obtained utilizing an intraaortic balloon. The left ventricle was modeled by an equivalent source pressure and source impedance analogous to a Thevenin•s equivalent representation. The model parameters (source pressure and source impedance) appear to be sensitive to cardiovascular changes such as myocardial infarction and increased left ventricular end-diastolic pressure. The source pressure expressed in the time domain may be a useful myocardial contractility index. For aortic input impedance much greater than source impedance, a change in left ventricular afterload would result in small change in aortic pressure. In this case the left ventricle would be functioning as a pressure source. By comparing the source resistance and aortic input resistance, the left ventricle appears to be a D.C. pressure source. For analysis in the time domain, the left ventricle was represented by truncated, confocal ellipsoids approximated by a series of cylindrical shells. The properties of the left ventricle were distributed over the cylindrical sections. The timing and sequence of contraction of the cylindrical shells were prescribed to simulate the mechanical action of the left ventricle. Plow and pressure waveforms produced by the model were similar to those obtained experimentally. Results of the simulation indicate that the pressure distribution in the ventricular chamber may be a useful index for determining the status of the left ventricle.

Electrical activity in t he heart begins (under n ormal conditions) in the sinoatrial node located in the wall of the right atrium.  (9). When the pressure in the LV exceeds that in the aorta, the aortic valve opens and the LV ej ects blood into the aorta.
During the ejection phase of the cardiac cycle, the ven tricle shortens slightly (9) , but the ventricular chamber vo lume is reduced primarily by a reduction circumferentially rather than longitudinally (10) .
Near the end of the ejection phase, the ventricular myocardium ceases contraction and the pressure in the LV       Solution of the two equations provides expressions for the mqdel parameters: One non-linearity of the cardiovascular system is the aortic valve. The valve was not considered part of the Ex perimental runs utilizing this procedure were termed "L OW-HIGH" runs ( Figure 3.3).
In the second procedure, the balloon was inflated f or several minutes until the system became stable and he modynamic parameters reached a steady state. Hemodynamic parameters were recorded for several heart beats providing data for one load condition. The balloon was then deflated precisely at the end of diastole. Hemodynamic data recorded d uring the first heart cycle with the balloon deflated were u sed for the second load condition. Experimental runs utilizing this procedure were termed "HIGH-LOW" runs.
Since compensatory mechanisms tend to change left ventricular function in response to changes in the systemic load, it was essential to consider hemodynamic data before such changes occurred. consequently, hemodynamic data during only the first heart cycle following the change in load were used to calculate the model parameters (17).
In addition, a few times during the experiment, ao rtic pressure and flow were recorded for a few heart cycles with the balloon deflated. This provided data for th e calculation of aortic input impedance.

Data Analysis
Data were digitized on line at a rate of 120 s amples per second and stored. Typical time domain data collected for a "LOW-HIGH" run are displayed in Figure 3.3.
Fourier analysis was performed on the data off line.
Frequency domain data were corrected with respect to instrument frequency response and for the distance between aortic flow and pressure measurement sites (21).
The "steady state" cycles of flow and pressure were averaged to provide one load condition. To assure that the "steady state" condition had been reached, and to remove the effect of any abnormal cycles from the "steady state" av erage, the following procedure was utilized.   t hat the model parameters for the "HIGH-LOW" runs are in g eneral higher than the respective parameters for the " LOW-HIGH" runs. The LVEDP of the "HIGH-LOW" runs are also higher than those of the "LOW-HIGH" runs.
In order to study how the model parameters vary over a wide range of LVEDP, the results of the "HIGH-LOW" Poor-correlation to a linear regression was fo und for the fundamental components of the model pa rameters. Aortic input impedance was compared to the source i mpedance. Typical results are displayed in Figure 3.14 for t he D.C. and seven harmonics.

E. Discussion
The representation of the left ventricle as a s ource pressure in series with a source impedance has been tested on isolated heart preparations and in-vivo. The p arameters appear to be responsive to cardiovascular changes such as MI and increased LVEDP. It has been suggested that so urce pressure expressed in the time domain be used as a myocardial contractility index (22

LVEDP as a Function of Mean Aortic Pressure
It has been reported that an increase in coronary perfusion pressure is accompanied by an increase in left ventricular diastolic pressure (23) . A similar observation i s noted with increased aortic end-diastolic pressure (15).
I n the present work, LVEDP was found to be an almost l inearly increasing function of mean aortic pressure. In t he "HIGH-LOW" runs, the aorta was partially occluded during t he "steady state" by the inflated balloon. This caused an e levated mean aortic pressure which in turn caused an e levated LVEDP. Therefore, the D.C. components of the model parameters, apparently dependent on LVEDP, are higher for " HIGH-LOW" runs than for "LOW-HIGH" runs.

Time Domain Source Pressure has been
This led The rate at which left ventricular pressure rises recognized as indicative of the inotropic state. _ to the use of peak dp/dt as a contractility index More recently, dp/dt during isovolumic contraction was shown to be a useful index of cardiac contractility (  The time domain source pressure waveforms indicate t hat denervation caused dp/dt to decline and subsequent MI caused a further decline ( Figure 3.12). MI without denervation caused a decline in dp/dt (Figure 3.13). These results indicate the potential usefulness of dp/dt of the time domain source pressure as a contractility index.
It has been suggested that myocardial infarction interferes with the normal excitation-contraction linkage          .,;               Th e focal length (the distance from the center to either foc al point) is  A confocal ellipse has the same focal length,  where 2B = major axis, 2A = minor axis, and, (4)(5) Hence, for confocal ellipsoids (

Evaluation of Constants
In the present model the · ellipsoids are confocal only at end-diastole.
The end-diastolic dimensions were c hosen as shown in Table 4 (4-8) The value of RATIO used is given in Table 4.~. VOLUME is th e volume contained by the intact ellipsoids. The value of VOLUME was such that the truncated ellipsoids contained the de signated end-diastolic volume.
Th us, equations (4-8) and (4-9) were solved as two simultaneous equations in two unknowns, (a) and (b). The outer semiminor axis (A) was determined from the value of THICK (Table 4.1) and the inner semiminor axis (a): The outer semimajor axis was determined from equation (4-6) : b2 -a2 = s2 -A2 (4-6) The ellipsoids were truncated at the "basal plane" a s specified by the value of TRCTNC (Table 4 In addition to the end-diastolic geometric c onstraints, other specifications for the model were the t otal number of cylindrical segments, the end-diastolic p ressure, cardiac output, heart rate, duration of isovolumic co ntraction and duration of ejection.

1)
The ventricular chamber has a uniform pressure di stribution at end-diastole.

2)
Stress is circumferential and uniform throughout the wa ll of each cylindrical shell. 3) The strain in the wall of each cylindrical shell can be re presented by the strain at the midwall of the shell.

5)
Flow through each cylindrical shell is laminar and un idirectional.

6)
The cylindrical shells contract sequentially, beginning at the ventricular apex.

7)
The shells contract until a given stroke volume is produced, the radius of each shell changes sinusoidally and in proportion to the end-diastolic radius of the shell.

8)
If the radius of a cylindrical shell reaches its end-systolic radius before the other shells have completed co ntraction it remains at that radius until all the shells re ach their end-systolic radii.

9)
During contraction, the inertia of the myocardium is ne gligible compared to forces generating static pressure • .

c. Results
The seg ments.
All experimental results used for comparison with model res ults were obtained from dogs unless otherwise specified.
The constants for the end-diastolic geometry of the left ventricle were specified as shown in

Model Assumptions and Constraints
The end-diastolic constraints presented in Table   4. In developing the present model, it was assumed th at blood mass displaced during isovolumic contraction by co ntracting shells was distributed in the passive shells in such a way that the passive shells were equally strained.
Th is assumption is not supported by physiologic evidence and hence, may be a basic source of inaccuracy in the model.
The forces on either side of equation (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19) , however, are not equal, but differ by the force required to accelerate one half of the shell wall. That is, The acceleration of X(cm) was determined at each point in the simulated cardiac cycle: The force required to accelerate the myocardium in one half of the shell wall was determined at each point in the simulated cardiac cycle: