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Measurements, modeling, control and simulation— as applied to the human left ventricle for purposeful physiological monitoring
Measurements, Modeling, Control and SimulationAs Applied to the Human Left Ventricle for Purposeful Physiological Monitoring by DHANJOO
N. GHISTA,
and
SANDLER
HAROLD
DARYL
N. RASMUSSEN,
ROBERT
N. LINEBARGER
National Aeronautics and Space Administration Ames Research Center Mofet Field, California
ABSTRACT
Interdisciplinary
:
measurement, closed-chest
modeling,
engineering
nontraumatic
procedures)
chosen
the left ventricle
in supplying techniques
energy
displayed
Mathematical muscle
left ventricle circulatory predict
as reJected is presented
system
us to determine
out
in order
the
physiological
plays
are presented; in
and
cardiac
is
function.
with the help of left ventricular
rate. A control system for the of dynamic
the control
stress sfituation
a key role
t?be geometry
of the intact
the effects
performance;
the
we have
The importance
the modeling,
consumption
to incorporate
on the left ventricle’s
life.
abnormalities
is presented;
monitor
because it is the left ventricle
the performance
by, say, its oxygen
the effect of a certain
uentride’s
bring
of the left ventricle
data, enables
subject
to
geometry
in the
[involving
heart chambers,
and hence its performance
of the left ventricular
(on-line)
modeling
measurement
cardiac performance: system
interaction)
to physiologically
Of the four
to the body cells to enable them to sustain
for measurement
effectively
characteristics.
to characterize
blood into the circulatory
effort (with medical
of the intact human left ventricle
has been employed
heart and obtain its “state-of-health” that pumps
research
control and simulation
system
changes enables
(such as exercise)
in the us to
on the left
performance.
I. Introduction
In this paper we present a brief account of how interdisciplinary engineering research effort (with medical interaction) in the measurement modeling, control and simulation of the intact human left ventricle (involving closed-chest nontraumatic procedures) has been effectively employed to physiologically monitor the heart and obtain its “state-of-health” characteristics. The heart caters to the metabolic demands of the cells of the body by pumping blood through the circulatory system, which is schematically shown in Fig. 1. The heart is divided into four chambers (see Fig. Z), namely the left and right atriums, the left and right ventricles. Of the four chambers, we have chosen to concentrate on the left ventricle (as far as modeling goes) in order to characterize cardiac performance, because (as shown in Fig. l), it is the left ventricle (LV) that pumps the blood into the circulatory system :
545
D. N. Ghista, D. A’. Rasmussen, R. N. Linebarger and H. Sandler and hence its performance plays a key role in supplying energy to the body cells in order to maintain homeostasis. PULMONARY
CIRCULATION
ARTERIOLES AND CAPILLARIES m
ARTERIES/ \
/
~VEINS ONARY VEIN AORTA
!
VEINS/j
LARTERIES
ART%:LES CAPILLARIES Sk3TEMIC FIG.
1. A schematic RA RV LA LV
CIRCULATION
diagram
of the circulatory
syst,em.
= RIGHT ATRIUM = RIGHT VENTRICLE = LEFT ATRIUM = LEFT VENTRICLE ,xAORTA AORT VALV MITRAL VALVE
(a) THE FOUR CHAMBERS OF THE HEART
~
FIG. 2. Schematic
(b) COMPARISON OF THE SHAPES OF THE RIGHT AND LEFT VENTRICULAR CHAMBER
(Cl A SECTION SHOWING THE LEFT ATRIUM AND LEFT VENTRICLE
views of the left ventricular chamber chambers.
in relation
to the other heart
The left ventricular function during one cycle consists of four stages. These stages are explained with respect to the left ventricular chamber’s pressure variation, see Fig. 3(a). During the filling stage, blood enters the
546
Journal of The
Franklin Institute
Measurement,
,Wodeling, Control and Simulation of Human Left Ventricle
left ventricular chamber from the left atrium causing passive distension of the muscular left ventricular chamber. Now, the mitral valve closes [at point A on the pressure cycle in Fig. 3(a)] and the left ventricular muscle contracts, following its electrical stimulation. Since both valves of the
lSOVOLUMlC CONTRACTION
ISOVOLUMIC
RELAXATION ,.
EJECTION -FILLING STAGE
FILLINGItch --___ t
- PRESSURE
IN AORTA
PRESSURE IN THE LEFT VENTRICULAR CHAMBER
(a) THEFOUR STAGES OF THE LEFT VENTRICULARCYCLE
PRESSURE IN THE LEFT ATRIUM I
1st
SOUND FIG.
1
2nd
SOUND
3rd
SOUND
3. (a) The four stages of the left ventricular pressure cycle, (b) Correlation of the occurrence of the heart sound with the left ventricular pressure cycle.
chamber are closed, the contraction of the muscle against the fairly incompressible blood raises the chamber pressure significantly. This stage is referred to as the isovolumic phase, as indicated in Fig. 3(a). When the pressure in the left ventricular chamber exceeds that in the aorta (point B in the pressure cycle), the aortic valve opens and the blood is ejected into the aorta by the now-shortening contracted left ventricular muscle. When the pressure in the chamber falls below the aortic pressure (point C in the pressure cycle), the aortic valve closes. With both valves closed, the muscle now relaxes. This stage is referred to as isovolumic relaxation. The relaxation continues until the chamber pressure falls below the pressure in the left atrium (point D in the pressure cycle) when blood again enters the chamber. The cycle is then repeated. As we have seen, the left ventricular muscle does work in pumping blood into the circulatory system. It derives. energy by consuming oxygen from the coronary blood which perfuses it. The percentage of this energy input spent in doing useful work is a manifestation of the efficiency of the left ventricle, which is one of the performance parameters that we want to obtain. With this brief introduction, we will now present in three parts the three topics of measurement, modeling and simulation ; each topic will be geared
Vol. 292, No. 6, December 1971
547
D. N. Ghista, D. N. Rasmussen, R. N. Lineba.rger and H. Sadler towards the same end, that is, of determining the performance of the intact human LV. II.
Measurement
of Left
Ventricular
characteristics
Geometry
Importance of measurement. A continuous monitoring of the dynamic geometry of the LV gives us the volume of the LV at filling and its stroke volume. These two factors are important clinical indices of cardiac performance. A knowledge of the dynamic geometry of the LV is essential for its mathematical modeling, which, as we will see later, is employed to determine the mechanical properties of the left ventricular muscle. Size and shape give important clues to heart defects such as hypertrophy (enlarged heart) and congenital abnormalities. Changes in size and shape throughout a cycle reveal the participation of various segments in the filling and contraction processes. For instance, a segment of the wall may not be participating due to insufficient blood supply and instead causing a drag as in infarction.
FIG. 4. A
computer-driven cathode ray tube display of a three-dimensional model of a human left ventricle.
Techniques for measurement. The technique we have employed (and which is also most popular) for determining the dynamic geometry of the intact LV during a cardiac cycle is cineangiocardiography (l-3). Here a radio-opaque dye is injected directly into the LV by means of a catheter which is inserted into the LV chamber via the femoral artery. X-rays of the left ventricle are recorded in the anterioposterior and left lateral projections. This system produces two orthogonal outlines of the chamber for each frame reading. The outlines are traced on a, graphical input device for a computer. In the computer, the dimensions are corrected for optical magnification and distortion due to the recording unit and combined to form a three-dimensional model of the LV. The model (simulating the LV at an instant during a cardiac cycle) is shown in Fig. 4, as displayed on a computer graphics unit. The cross-section of the model is assumed to be elliptical, the major and
548
Journal of The Franklin Institute
Measurement,
Modeling, Control and Simulation of Human Left Ventricle
minor axis of each merideanal segment being obtained from the two X-ray projections. The possibly physiologically deleterious aspects of contrast media (such as toxicity, red blood cell aggregation and circulatory stasis) call for more research on the method of intracavitary ultrasound to determine the dimensions of the LV chamber (4). Herein, a catheter tip cylindrical ultrasonic transducer is introduced into the LV chamber. The cylindrical piezoelectric crystal emits ultrasonic energy perpendicular to its long axis through 360 arc. Sound waves which meet an acoustical surface perpendicularly are reflected back to the crystal. Consequently, when the crystal
FIG.
5. The
time
course of motion of a longitudinal line on the connecting the aortic valve and the apex.
surface
of the
LV
is in the position i (see Fig. 4) in the LV chamber, it will provide simultaneous measurements of only the shortest and the longest crystal-to-wall dimensions on a diameter sl of the ith latitudinal section of the chamber. The improvement of this technique, to (i) distinguish echoes from the outer surface of the heart (or its interface with the lungs, against which it rests), and (ii) derive simultaneous diametrical measurements at other latitudes, would make it useful. Purposeful display of the dynamic geometry. It is worth while to purposefully display the geometry so as to bring out the abnormalities in cardiac function. As the ventricle breathes (i.e. expands and contracts) during a cycle, its latitudes and longitudes (as shown in Fig. 4) describe “breathing modes”. Magnifications of their time-varying modes enable discovery of anomalous events in the wall, such as the nonparticipation of a segment of the wall, which can hence be diagnosed as isochemic. In Fig. 5, we see the time-course of motion of a longitudinal line connecting the aortic valve and the apex. Further, if we observe the simultaneous motions of a number of merideans, we could trace the characteristics of the propagation of the wave of excitation
Vol. 092, No. 6, December 19il
549
D. N. Ghista,, D. N. Rasmussen,
R. N. Lineburger
and H. Sandier
or depolarization such as variation in its speed over various segments, which again would help us detect “scarred” wall segments.
ZZZ. Modeling
We wish to mathematically model the LV in order to determine the performance of the intact LV muscle of a human subject such as (i) its pathological state from a knowledge of its modulus and (ii) its oxygen consumption rate. The LV structure undergoes passive strain (due to mechanical bloodpressure loading of its chamber) and active strains (due to electrical excitation, stiffening and consequential contraction of the LV muscle). Also, the muscle material property (i.e. its stiffness or modulus) changes continuously during one complete cycle. The problem definition for the mathematical modeling is governed by the nature of quantitative data available to us from monitoring the intact LV by closed-chest measurements. Hitherto, quasi-static models utilizing data consisting of LV chamber dimensions (obtained immediately before or after angiocardiography by a fluid-filled catheter and pressure transducer) have been employed. For quasi-static equilibrium states, such as that during an interval, the strains are small enough to justify use of linear small strain elasticity theory. Alternately, we are currently working on a dynamic model which will utilize the frequency spectrum of the heart sounds caused by the vibrations of the cardiochemic system (consisting of chamber walls, valves and the contained blood). Quasi-static modeling. It is natural to take the number of quasi-static time intervals equal to the number of frames per cycle at which the chamber pressure and dimension readings are taken. When we model the LV by employing elasticity theory, we determine the deformation of the LV structure to a certain pressure loading. For the relaxed muscle, we measure strain relative to the state at which the loading is at a minimum, at the start of ventricular filling. For the contracted muscle, we measure the strains relative to the fully shortened state, at the end of ejection; thereby, the active strains are filtered out. The quasi-static LV model that we have employed, see (5), is shown in Fig. 6. This elastic, homogeneous, isotropic model is derived by superposing live dilatation and hydrostatic stress systems. The parameters of the stress systems are obtained by satisfying the boundary conditions that the stresses on the inner and outer surface equal the chamber pressure and zero, respectively. The model is employed to determine t,he instantaneous modulus of the cardiac muscle (6) by equating the expression for a principal strain in a model element, which is a function of the modulus, to its instantaneous numerical value obtained from frame-by-frame dimensional data. The systolic modulus represents the effort of the LV in raising the pressure in the LV chamber above the aortic pressure and hence in ejecting the blood into the aorta.
550
Journal
of The Franklin
Institute
Measurement,
Modeling, Control and Simulation of Human Left Ventricle
We would like to indicate a further application of the model, namely, that of determining the left ventricular oxygen consumption rate. We have shown (7) that the sum UT, of the strain energy of the LV during a cardiac cycle and the energy spent in increasing the modulus of the LV muscle from the relaxed to contracted state during isovolumic contraction, represents the energy equivalent of the total oxygen consumption (QO,) of the intact heart.
I VENTRICLE
MODEL
FIG. 6. Comparison of the analytical model with the simulated left ventricle at an instant during the cardiac cycle.
T-l
?-I
PE
CE
PE
SE
CE ,
SE-
SERIES
PE-
PARALLEL
ELECTRICITY
CE-
CONTRACTILE
ELECTRICITY ELEMENT
SE i’
i!
FIG. 7. Alternative rheological models for the left ventricular muscle.
A noteworthy application of this model is to attempt to provide physiological monitoring. Accordingly, we propose the following criteria for evaluating LV performance: (1) if the value of Us is normal, then the LV is doing a normal amount of work, (2) in the event the subject has a valvular lesion, then a normal value of modulus indicates compensation to the disease, (3) if the value of UT is subnormal, then the LV muscle is being poorly perfused. For the model, just described, the muscle medium has been assumed to be uniform isotropic. We could represent the three-dimensional muscle medium (in dilatation and shear) by the rheological model shown in Fig. 7, which has been derived from experiments in an isolated muscle. Work is
Vol. 292, No. 6, December
1971
551
D. N. Ghista, D. N. Rasmussen, R. N. Linebarger and H. Sander currently in progress to determine the in vivo parameters of this threedimensional model; these parameters would then be related to in vice muscle pathology. Dynamic modeling. Here we employ the frequency content of the heart sounds obtained by means of a phonocardiograph. For convenience in modeling, we must only consider those sounds wherein the LV system is set into vibration. The first heart sound appears at the beginning of isovolumic interaction. Its second component (8) starts as the blood rushing back towards the mitral valve is decelerated by the closed mitral valve; the resulting oscillation of blood throws the LV into vibration. The frequency response analysis of the fluid-filled left ventricular thick-walled shell gives the response as a function of the modulus of the muscle. By equating this function to the numerical value of the frequency of the heart-sound spectrum (obtained by phonocardiography), the value of modulus is obtained. Since the muscle’s modulus reflects its pathological state, we can thus obtain an indirect evaluation of the “healt,h” state of the heart muscle.
IV. Control and Simulation
In the previous section we have modeled t,he LV’s mechanics by considering the LV as an isolated system. By measuring the LV’s pressure and dimensions, we have shown how we can characterize the current physiological state of the LV muscle by determining its modulus, work rate and oxygen consumption; each of these intrinsic quantities of the LV has limits which, if exceeded, would constitute heart failure. The pressure and dimensions, however, are continually regulated by physiological control mechanisms which enable the cardiovascular (CV) system to adapt to dynamic and, “stressed” physiological situations such as exercise, drugs, occasionally, hemorrhage, weightlessness and acceleration. So if we wish to predict the effect of a certain physiological stress situation on the heart performance (say, if we wish to check the extent to which the oxygen consumption rate would overreach on its reserve capacity, without exceeding the limit, under a certain physiological stress situation), then it is necessary to gain knowledge of the cardiovascular control mechanisms (through experimentation) and incorporate them in a CV control system model. Such a CV control system model, proposed by us, is shown in Fig. 8. Figure 8 shows how the neurochemical process provides the feedback for regulating the measurable left ventricular variables (chamber pressure and dimensions and the heart rate). The neurochemical feedback mechanism monitors the arterial pressure and oxygen content and regulates (i) the systemic resistance which resets the arterial pressure and in turn determines the ejection pressure in the LV chamber, (ii) the heart rate and (iii) the dimensions of the LV by controlling the contractibility of the LV muscle. The form of the equations governing the feedback loops 1, 2 and 3 (Fig. 6) relating the resistance, LV dimensions and heart rate to the arterial pressure
552
Jownal
of The Franklin Institute
Measurement,
Modeling, Control and Simulation of Human Left Ventricle
is determined by controlled animal experimentation. These equations are not presented here, for sake of brevity ; however, some idea as to then nature can be obtained from Refs. (9-11). The parameters of these equations have to be determined by making the control system model match (in real time) the actual CV system of the subject under examination. To carry out this matching, we clinically measure the subject’s left ventricular pressure MEASURABLE
QUANTITIES
-7 r----
(Affects
LV strain
by regulating
LV contractibility)
,
MATHEMATICAL MODEL OF THE LV ----
r1)
LV muscle’s
modulus
Neurochemical Process
Fig. 8. Proposed control system for assessing left ventricular performance.
and dimensions, heart rate and arterial pressure. Then by means of analog computer simulation of our control system model, we determine the values of the parameters of the governing equations (of the feedback loops) which give the subject’s measured clinical values of left ventricular dimensions and heart rate corresponding to the measured value of the arterial pressure. Use of the CV control system to determine the reserve capacity of a subject’s LV. In order to predict the extent to which the LV of a subject can withstand a certain physiological stress situation, we first match our control system model with the subject’s CV system, as discussed earlier. Then on an analog simulation of our control system model (Fig. S), we effect step increases in the variables such as heart rate and/or stroke volume (which are characteristic of the particular stress situation that we wish to simulate). On the analog computer, we obtain the corresponding LV dimensions and pressure. From these determinants, we then obtain (by means of our mathematical model of the LV) the modulus of the LV muscle and its oxygen consumption rate (as explained in the previous section and as indicated in Fig. 8). VVe thus measure the sensitivity of the modulus and the oxygen consumption rat,e to perturbations in heart rate and/or st’roke volume. This sensitivity analysis enables us to prescribe safe physiological stress limits (in the form of,
Vol.
292,
No.
6, December
1971
553
D. N. Ghista, D. N. Rasmussen, R. N. Lineburger and H. Sandier say, exercise) that will prevent heart failure: i.e. will prevent the subject’s cardiac and coronary capacities (represented by LV muscle modulus and oxygen consumption rate, respectively) from exceeding their reserve limits.
References (1) “Measurement of heart chamber volumes and dimensions”, Proc. Summer Workshop sponsored by the Council on Basic Science of the American Heart Assoc. in July 1967. Published by the American Heart Assoc., Inc. (2) H. Sandler and H. T. Dodge, “Use of single plane angiocardiograms for the calculation of left ventricular volume in man”, Am. Heart J., Vol. 75, p. 325, 1969. (3) H. T. Dodge, H. Sandier, D. H. Ballew and J. D. Lord, Jr., “The use of biplane angiocardiography for the measurement of left ventricular volume in man”, Am. Heart J., Vol. 60, p. 762, 1960. (4) R. A. Carleton, R. W. Sessions and J. S. Grattinger, “Diameter of heart measured by intracavitary ultrasound”, Med. Res. Engng, May-June 1969. (5) D. N. Ghista and H. Sandler, “Analytical model for the shape and forces in the left ventricle”, J. Biomechanics, Vol. 2, 1969. (6) D. N. Ghista and H. W. Vayo, “The time varying elastic properties of the left ventricular muscle”, Bull. Math. Biophys., Vol. 31, 1969. (7) D. N. Ghista and H. Sandier, “Indirect determination of the oxygen utilisation of human left ventricle”, J. Biomechawics, to be published. (8) R. F. Rushmer, “Cardiovascular Dynamics”, Philadelphia, Pa., W. B. Saunders, 1967. (9) F. S. Grodins, “Control Theory and Biological Systems”, New York, Columbia Univ. Press, 1963, (10) G. A. Bekey and L. W. Morrison, “Mathematical models of the cardiovascular system”, USCEE Rep. 220, U.S.C., Los Angeles, Calif. (11) H. A. Warner and A. Cox, “A mathematical model of neart rate control by sympathetic and vagus efferent information”, J. AppE. Physiol., Vol. 17, 1962.
554
Journal of The Franklin Institute
N. GHISTA,
and
SANDLER
HAROLD
DARYL
N. RASMUSSEN,
ROBERT
N. LINEBARGER
National Aeronautics and Space Administration Ames Research Center Mofet Field, California
ABSTRACT
Interdisciplinary
:
measurement, closed-chest
modeling,
engineering
nontraumatic
procedures)
chosen
the left ventricle
in supplying techniques
energy
displayed
Mathematical muscle
left ventricle circulatory predict
as reJected is presented
system
us to determine
out
in order
the
physiological
plays
are presented; in
and
cardiac
is
function.
with the help of left ventricular
rate. A control system for the of dynamic
the control
stress sfituation
a key role
t?be geometry
of the intact
the effects
performance;
the
we have
The importance
the modeling,
consumption
to incorporate
on the left ventricle’s
life.
abnormalities
is presented;
monitor
because it is the left ventricle
the performance
by, say, its oxygen
the effect of a certain
uentride’s
bring
of the left ventricle
data, enables
subject
to
geometry
in the
[involving
heart chambers,
and hence its performance
of the left ventricular
(on-line)
modeling
measurement
cardiac performance: system
interaction)
to physiologically
Of the four
to the body cells to enable them to sustain
for measurement
effectively
characteristics.
to characterize
blood into the circulatory
effort (with medical
of the intact human left ventricle
has been employed
heart and obtain its “state-of-health” that pumps
research
control and simulation
system
changes enables
(such as exercise)
in the us to
on the left
performance.
I. Introduction
In this paper we present a brief account of how interdisciplinary engineering research effort (with medical interaction) in the measurement modeling, control and simulation of the intact human left ventricle (involving closed-chest nontraumatic procedures) has been effectively employed to physiologically monitor the heart and obtain its “state-of-health” characteristics. The heart caters to the metabolic demands of the cells of the body by pumping blood through the circulatory system, which is schematically shown in Fig. 1. The heart is divided into four chambers (see Fig. Z), namely the left and right atriums, the left and right ventricles. Of the four chambers, we have chosen to concentrate on the left ventricle (as far as modeling goes) in order to characterize cardiac performance, because (as shown in Fig. l), it is the left ventricle (LV) that pumps the blood into the circulatory system :
545
D. N. Ghista, D. A’. Rasmussen, R. N. Linebarger and H. Sandler and hence its performance plays a key role in supplying energy to the body cells in order to maintain homeostasis. PULMONARY
CIRCULATION
ARTERIOLES AND CAPILLARIES m
ARTERIES/ \
/
~VEINS ONARY VEIN AORTA
!
VEINS/j
LARTERIES
ART%:LES CAPILLARIES Sk3TEMIC FIG.
1. A schematic RA RV LA LV
CIRCULATION
diagram
of the circulatory
syst,em.
= RIGHT ATRIUM = RIGHT VENTRICLE = LEFT ATRIUM = LEFT VENTRICLE ,xAORTA AORT VALV MITRAL VALVE
(a) THE FOUR CHAMBERS OF THE HEART
~
FIG. 2. Schematic
(b) COMPARISON OF THE SHAPES OF THE RIGHT AND LEFT VENTRICULAR CHAMBER
(Cl A SECTION SHOWING THE LEFT ATRIUM AND LEFT VENTRICLE
views of the left ventricular chamber chambers.
in relation
to the other heart
The left ventricular function during one cycle consists of four stages. These stages are explained with respect to the left ventricular chamber’s pressure variation, see Fig. 3(a). During the filling stage, blood enters the
546
Journal of The
Franklin Institute
Measurement,
,Wodeling, Control and Simulation of Human Left Ventricle
left ventricular chamber from the left atrium causing passive distension of the muscular left ventricular chamber. Now, the mitral valve closes [at point A on the pressure cycle in Fig. 3(a)] and the left ventricular muscle contracts, following its electrical stimulation. Since both valves of the
lSOVOLUMlC CONTRACTION
ISOVOLUMIC
RELAXATION ,.
EJECTION -FILLING STAGE
FILLINGItch --___ t
- PRESSURE
IN AORTA
PRESSURE IN THE LEFT VENTRICULAR CHAMBER
(a) THEFOUR STAGES OF THE LEFT VENTRICULARCYCLE
PRESSURE IN THE LEFT ATRIUM I
1st
SOUND FIG.
1
2nd
SOUND
3rd
SOUND
3. (a) The four stages of the left ventricular pressure cycle, (b) Correlation of the occurrence of the heart sound with the left ventricular pressure cycle.
chamber are closed, the contraction of the muscle against the fairly incompressible blood raises the chamber pressure significantly. This stage is referred to as the isovolumic phase, as indicated in Fig. 3(a). When the pressure in the left ventricular chamber exceeds that in the aorta (point B in the pressure cycle), the aortic valve opens and the blood is ejected into the aorta by the now-shortening contracted left ventricular muscle. When the pressure in the chamber falls below the aortic pressure (point C in the pressure cycle), the aortic valve closes. With both valves closed, the muscle now relaxes. This stage is referred to as isovolumic relaxation. The relaxation continues until the chamber pressure falls below the pressure in the left atrium (point D in the pressure cycle) when blood again enters the chamber. The cycle is then repeated. As we have seen, the left ventricular muscle does work in pumping blood into the circulatory system. It derives. energy by consuming oxygen from the coronary blood which perfuses it. The percentage of this energy input spent in doing useful work is a manifestation of the efficiency of the left ventricle, which is one of the performance parameters that we want to obtain. With this brief introduction, we will now present in three parts the three topics of measurement, modeling and simulation ; each topic will be geared
Vol. 292, No. 6, December 1971
547
D. N. Ghista, D. N. Rasmussen, R. N. Lineba.rger and H. Sadler towards the same end, that is, of determining the performance of the intact human LV. II.
Measurement
of Left
Ventricular
characteristics
Geometry
Importance of measurement. A continuous monitoring of the dynamic geometry of the LV gives us the volume of the LV at filling and its stroke volume. These two factors are important clinical indices of cardiac performance. A knowledge of the dynamic geometry of the LV is essential for its mathematical modeling, which, as we will see later, is employed to determine the mechanical properties of the left ventricular muscle. Size and shape give important clues to heart defects such as hypertrophy (enlarged heart) and congenital abnormalities. Changes in size and shape throughout a cycle reveal the participation of various segments in the filling and contraction processes. For instance, a segment of the wall may not be participating due to insufficient blood supply and instead causing a drag as in infarction.
FIG. 4. A
computer-driven cathode ray tube display of a three-dimensional model of a human left ventricle.
Techniques for measurement. The technique we have employed (and which is also most popular) for determining the dynamic geometry of the intact LV during a cardiac cycle is cineangiocardiography (l-3). Here a radio-opaque dye is injected directly into the LV by means of a catheter which is inserted into the LV chamber via the femoral artery. X-rays of the left ventricle are recorded in the anterioposterior and left lateral projections. This system produces two orthogonal outlines of the chamber for each frame reading. The outlines are traced on a, graphical input device for a computer. In the computer, the dimensions are corrected for optical magnification and distortion due to the recording unit and combined to form a three-dimensional model of the LV. The model (simulating the LV at an instant during a cardiac cycle) is shown in Fig. 4, as displayed on a computer graphics unit. The cross-section of the model is assumed to be elliptical, the major and
548
Journal of The Franklin Institute
Measurement,
Modeling, Control and Simulation of Human Left Ventricle
minor axis of each merideanal segment being obtained from the two X-ray projections. The possibly physiologically deleterious aspects of contrast media (such as toxicity, red blood cell aggregation and circulatory stasis) call for more research on the method of intracavitary ultrasound to determine the dimensions of the LV chamber (4). Herein, a catheter tip cylindrical ultrasonic transducer is introduced into the LV chamber. The cylindrical piezoelectric crystal emits ultrasonic energy perpendicular to its long axis through 360 arc. Sound waves which meet an acoustical surface perpendicularly are reflected back to the crystal. Consequently, when the crystal
FIG.
5. The
time
course of motion of a longitudinal line on the connecting the aortic valve and the apex.
surface
of the
LV
is in the position i (see Fig. 4) in the LV chamber, it will provide simultaneous measurements of only the shortest and the longest crystal-to-wall dimensions on a diameter sl of the ith latitudinal section of the chamber. The improvement of this technique, to (i) distinguish echoes from the outer surface of the heart (or its interface with the lungs, against which it rests), and (ii) derive simultaneous diametrical measurements at other latitudes, would make it useful. Purposeful display of the dynamic geometry. It is worth while to purposefully display the geometry so as to bring out the abnormalities in cardiac function. As the ventricle breathes (i.e. expands and contracts) during a cycle, its latitudes and longitudes (as shown in Fig. 4) describe “breathing modes”. Magnifications of their time-varying modes enable discovery of anomalous events in the wall, such as the nonparticipation of a segment of the wall, which can hence be diagnosed as isochemic. In Fig. 5, we see the time-course of motion of a longitudinal line connecting the aortic valve and the apex. Further, if we observe the simultaneous motions of a number of merideans, we could trace the characteristics of the propagation of the wave of excitation
Vol. 092, No. 6, December 19il
549
D. N. Ghista,, D. N. Rasmussen,
R. N. Lineburger
and H. Sandier
or depolarization such as variation in its speed over various segments, which again would help us detect “scarred” wall segments.
ZZZ. Modeling
We wish to mathematically model the LV in order to determine the performance of the intact LV muscle of a human subject such as (i) its pathological state from a knowledge of its modulus and (ii) its oxygen consumption rate. The LV structure undergoes passive strain (due to mechanical bloodpressure loading of its chamber) and active strains (due to electrical excitation, stiffening and consequential contraction of the LV muscle). Also, the muscle material property (i.e. its stiffness or modulus) changes continuously during one complete cycle. The problem definition for the mathematical modeling is governed by the nature of quantitative data available to us from monitoring the intact LV by closed-chest measurements. Hitherto, quasi-static models utilizing data consisting of LV chamber dimensions (obtained immediately before or after angiocardiography by a fluid-filled catheter and pressure transducer) have been employed. For quasi-static equilibrium states, such as that during an interval, the strains are small enough to justify use of linear small strain elasticity theory. Alternately, we are currently working on a dynamic model which will utilize the frequency spectrum of the heart sounds caused by the vibrations of the cardiochemic system (consisting of chamber walls, valves and the contained blood). Quasi-static modeling. It is natural to take the number of quasi-static time intervals equal to the number of frames per cycle at which the chamber pressure and dimension readings are taken. When we model the LV by employing elasticity theory, we determine the deformation of the LV structure to a certain pressure loading. For the relaxed muscle, we measure strain relative to the state at which the loading is at a minimum, at the start of ventricular filling. For the contracted muscle, we measure the strains relative to the fully shortened state, at the end of ejection; thereby, the active strains are filtered out. The quasi-static LV model that we have employed, see (5), is shown in Fig. 6. This elastic, homogeneous, isotropic model is derived by superposing live dilatation and hydrostatic stress systems. The parameters of the stress systems are obtained by satisfying the boundary conditions that the stresses on the inner and outer surface equal the chamber pressure and zero, respectively. The model is employed to determine t,he instantaneous modulus of the cardiac muscle (6) by equating the expression for a principal strain in a model element, which is a function of the modulus, to its instantaneous numerical value obtained from frame-by-frame dimensional data. The systolic modulus represents the effort of the LV in raising the pressure in the LV chamber above the aortic pressure and hence in ejecting the blood into the aorta.
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Measurement,
Modeling, Control and Simulation of Human Left Ventricle
We would like to indicate a further application of the model, namely, that of determining the left ventricular oxygen consumption rate. We have shown (7) that the sum UT, of the strain energy of the LV during a cardiac cycle and the energy spent in increasing the modulus of the LV muscle from the relaxed to contracted state during isovolumic contraction, represents the energy equivalent of the total oxygen consumption (QO,) of the intact heart.
I VENTRICLE
MODEL
FIG. 6. Comparison of the analytical model with the simulated left ventricle at an instant during the cardiac cycle.
T-l
?-I
PE
CE
PE
SE
CE ,
SE-
SERIES
PE-
PARALLEL
ELECTRICITY
CE-
CONTRACTILE
ELECTRICITY ELEMENT
SE i’
i!
FIG. 7. Alternative rheological models for the left ventricular muscle.
A noteworthy application of this model is to attempt to provide physiological monitoring. Accordingly, we propose the following criteria for evaluating LV performance: (1) if the value of Us is normal, then the LV is doing a normal amount of work, (2) in the event the subject has a valvular lesion, then a normal value of modulus indicates compensation to the disease, (3) if the value of UT is subnormal, then the LV muscle is being poorly perfused. For the model, just described, the muscle medium has been assumed to be uniform isotropic. We could represent the three-dimensional muscle medium (in dilatation and shear) by the rheological model shown in Fig. 7, which has been derived from experiments in an isolated muscle. Work is
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D. N. Ghista, D. N. Rasmussen, R. N. Linebarger and H. Sander currently in progress to determine the in vivo parameters of this threedimensional model; these parameters would then be related to in vice muscle pathology. Dynamic modeling. Here we employ the frequency content of the heart sounds obtained by means of a phonocardiograph. For convenience in modeling, we must only consider those sounds wherein the LV system is set into vibration. The first heart sound appears at the beginning of isovolumic interaction. Its second component (8) starts as the blood rushing back towards the mitral valve is decelerated by the closed mitral valve; the resulting oscillation of blood throws the LV into vibration. The frequency response analysis of the fluid-filled left ventricular thick-walled shell gives the response as a function of the modulus of the muscle. By equating this function to the numerical value of the frequency of the heart-sound spectrum (obtained by phonocardiography), the value of modulus is obtained. Since the muscle’s modulus reflects its pathological state, we can thus obtain an indirect evaluation of the “healt,h” state of the heart muscle.
IV. Control and Simulation
In the previous section we have modeled t,he LV’s mechanics by considering the LV as an isolated system. By measuring the LV’s pressure and dimensions, we have shown how we can characterize the current physiological state of the LV muscle by determining its modulus, work rate and oxygen consumption; each of these intrinsic quantities of the LV has limits which, if exceeded, would constitute heart failure. The pressure and dimensions, however, are continually regulated by physiological control mechanisms which enable the cardiovascular (CV) system to adapt to dynamic and, “stressed” physiological situations such as exercise, drugs, occasionally, hemorrhage, weightlessness and acceleration. So if we wish to predict the effect of a certain physiological stress situation on the heart performance (say, if we wish to check the extent to which the oxygen consumption rate would overreach on its reserve capacity, without exceeding the limit, under a certain physiological stress situation), then it is necessary to gain knowledge of the cardiovascular control mechanisms (through experimentation) and incorporate them in a CV control system model. Such a CV control system model, proposed by us, is shown in Fig. 8. Figure 8 shows how the neurochemical process provides the feedback for regulating the measurable left ventricular variables (chamber pressure and dimensions and the heart rate). The neurochemical feedback mechanism monitors the arterial pressure and oxygen content and regulates (i) the systemic resistance which resets the arterial pressure and in turn determines the ejection pressure in the LV chamber, (ii) the heart rate and (iii) the dimensions of the LV by controlling the contractibility of the LV muscle. The form of the equations governing the feedback loops 1, 2 and 3 (Fig. 6) relating the resistance, LV dimensions and heart rate to the arterial pressure
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Jownal
of The Franklin Institute
Measurement,
Modeling, Control and Simulation of Human Left Ventricle
is determined by controlled animal experimentation. These equations are not presented here, for sake of brevity ; however, some idea as to then nature can be obtained from Refs. (9-11). The parameters of these equations have to be determined by making the control system model match (in real time) the actual CV system of the subject under examination. To carry out this matching, we clinically measure the subject’s left ventricular pressure MEASURABLE
QUANTITIES
-7 r----
(Affects
LV strain
by regulating
LV contractibility)
,
MATHEMATICAL MODEL OF THE LV ----
r1)
LV muscle’s
modulus
Neurochemical Process
Fig. 8. Proposed control system for assessing left ventricular performance.
and dimensions, heart rate and arterial pressure. Then by means of analog computer simulation of our control system model, we determine the values of the parameters of the governing equations (of the feedback loops) which give the subject’s measured clinical values of left ventricular dimensions and heart rate corresponding to the measured value of the arterial pressure. Use of the CV control system to determine the reserve capacity of a subject’s LV. In order to predict the extent to which the LV of a subject can withstand a certain physiological stress situation, we first match our control system model with the subject’s CV system, as discussed earlier. Then on an analog simulation of our control system model (Fig. S), we effect step increases in the variables such as heart rate and/or stroke volume (which are characteristic of the particular stress situation that we wish to simulate). On the analog computer, we obtain the corresponding LV dimensions and pressure. From these determinants, we then obtain (by means of our mathematical model of the LV) the modulus of the LV muscle and its oxygen consumption rate (as explained in the previous section and as indicated in Fig. 8). VVe thus measure the sensitivity of the modulus and the oxygen consumption rat,e to perturbations in heart rate and/or st’roke volume. This sensitivity analysis enables us to prescribe safe physiological stress limits (in the form of,
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6, December
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D. N. Ghista, D. N. Rasmussen, R. N. Lineburger and H. Sandier say, exercise) that will prevent heart failure: i.e. will prevent the subject’s cardiac and coronary capacities (represented by LV muscle modulus and oxygen consumption rate, respectively) from exceeding their reserve limits.
References (1) “Measurement of heart chamber volumes and dimensions”, Proc. Summer Workshop sponsored by the Council on Basic Science of the American Heart Assoc. in July 1967. Published by the American Heart Assoc., Inc. (2) H. Sandler and H. T. Dodge, “Use of single plane angiocardiograms for the calculation of left ventricular volume in man”, Am. Heart J., Vol. 75, p. 325, 1969. (3) H. T. Dodge, H. Sandier, D. H. Ballew and J. D. Lord, Jr., “The use of biplane angiocardiography for the measurement of left ventricular volume in man”, Am. Heart J., Vol. 60, p. 762, 1960. (4) R. A. Carleton, R. W. Sessions and J. S. Grattinger, “Diameter of heart measured by intracavitary ultrasound”, Med. Res. Engng, May-June 1969. (5) D. N. Ghista and H. Sandler, “Analytical model for the shape and forces in the left ventricle”, J. Biomechanics, Vol. 2, 1969. (6) D. N. Ghista and H. W. Vayo, “The time varying elastic properties of the left ventricular muscle”, Bull. Math. Biophys., Vol. 31, 1969. (7) D. N. Ghista and H. Sandier, “Indirect determination of the oxygen utilisation of human left ventricle”, J. Biomechawics, to be published. (8) R. F. Rushmer, “Cardiovascular Dynamics”, Philadelphia, Pa., W. B. Saunders, 1967. (9) F. S. Grodins, “Control Theory and Biological Systems”, New York, Columbia Univ. Press, 1963, (10) G. A. Bekey and L. W. Morrison, “Mathematical models of the cardiovascular system”, USCEE Rep. 220, U.S.C., Los Angeles, Calif. (11) H. A. Warner and A. Cox, “A mathematical model of neart rate control by sympathetic and vagus efferent information”, J. AppE. Physiol., Vol. 17, 1962.
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Journal of The Franklin Institute