Assessment of left ventricular (LV) function

with an emphasis on contractility has been a challenge

in cardiac mechanics during the recent decades. The LV

function is usually described by the LV pressurevolume

(P-V) diagram. The standard P-V diagrams are

easy to interpret but difficult to obtain and require

invasive instrumentation for measuring the

corresponding volume and pressure data. In the present

study, we introduce a technique that can estimate the

viscoelastic properties of the LV based on harmonic

behavior of the ventricular chamber and it can be

applied non-invasively as well. The estimation technique

is based on modeling the actual long axis displacement

of the mitral annulus plane toward the cardiac base as a

linear damped oscillator with time-varying coefficients.

The time-varying parameters of the model were

estimated by a standard Recursive Linear Least

Squares (RLLS) technique. LV stiffness at end-systole

and end diastole was in the range of 61.86-136.00

dyne/g.cm and 1.25-21.02 dyne/g.cm, respectively. The

only input used in this model was the long axis

displacement of the annulus plane, which can also be

obtained non-invasively using tissue Doppler or MR

imaging

Gharib, Morteza

Gorman, Joseph H., III

Gorman, Robert C.

Kheradvar, Arash

Milano, Michele

English

Arash Kheradvar

Michele Milano

Robert C. Gorman

Joseph H. Gorman

Morteza Gharib

Crossref

Abstract— Assessment of left ventricular (LV) function 
with an emphasis on contractility has been a challenge 
in cardiac mechanics during the recent decades. The LV 
function is usually described by the LV pressure-
volume (P-V) diagram. The standard P-V diagrams are 
easy to interpret but difficult to obtain and require 
invasive instrumentation for measuring the 
corresponding volume and pressure data. In the present 
study, we introduce a technique that can estimate the 
viscoelastic properties of the LV based on harmonic 
behavior of the ventricular chamber and it can be 
applied non-invasively as well. The estimation technique 
is based on modeling the actual long axis displacement 
of the mitral annulus plane toward the cardiac base as a 
linear damped oscillator with time-varying coefficients. 
The time-varying parameters of the model were 
estimated by a standard Recursive Linear Least 
Squares (RLLS) technique. LV stiffness at end-systole 
and end diastole was in the range of 61.86-136.00 
dyne/g.cm and 1.25-21.02 dyne/g.cm, respectively. The 
only input used in this model was the long axis 
displacement of the annulus plane, which can also be 
obtained non-invasively using tissue Doppler or MR 
imaging. 
I. INTRODUCTION
SSESSMENT of left ventricular function with an 
emphasis on contractility has been a major challenge 
Manuscript received December 15, 2005. This work was supported in 
part by NIH grants HL-63954, HL-71137, HL-73021 and HL-76560. 
Arash Kheradvar is affiliated with California Institute of Technology, 
Pasadena, CA 91125 USA (phone: 1-626-395-3151; fax: 1-626-577-5258; 
email: arashkh@caltech.edu). 
Michele Milano is affiliated with California Institute of Technology, 
Pasadena, CA 91125 USA. 
Robert C. Gorman is affiliated with Harrison Department of Surgical 
Research, University of Pennsylvania School of Medicine, Philadelphia, 
PA. 
Joseph H. Gorman, III is affiliated with Harrison Department of 
Surgical Research, University of Pennsylvania School of Medicine, 
Philadelphia, PA. 
Morteza Gharib is affiliated with the California Institute of 
Technology, Pasadena, CA 91125 USA. 
in cardiac mechanics during the recent decades. To date, 
extensive work has been done to develop models for 
describing left ventricular (LV) dynamics. However, an 
applicable model that can differentiate between different 
pathophysiological states based on mechanical properties 
of the heart is still needed. With regards to describing 
ventricular function, Suga and Sagawa [1] introduced a 
diagram for instantaneous ventricular pressure-volume (P-
V) relationship. Based on their diagram, they described the 
ratio of instantaneous pressure to instantaneous volume 
( ))(/()( dVtVtP ) as the time varying stiffness for 
ventricular chamber. The standard P-V diagrams are easy 
to interpret and give a rough estimate of the mechanical 
work done by the LV. However, the pressure and the 
corresponding volume of the LV need to be measured 
invasively using sophisticated techniques, such as 
intravascular micromanometers and conductance catheters, 
which restrict the clinical applications of this model. 
Furthermore, this model ignores the viscoelasticity of the 
heart by not considering the viscous damping. It has also 
been shown that although dvdp  provides a useful portrait 
of simultaneous LV pressure and volume events, it does not 
represent actual LV physical properties.  
In the present study, we introduce a noninvasive 
technique to assess the LV contractile behavior during the 
entire cardiac cycle. The estimation technique is based on 
modeling the long axis displacement of the mitral annulus 
plane toward the cardiac base as a linear damped oscillator 
with time-varying coefficients. We derive longitudinal 
rather than global indices for stiffness and damping of the 
LV. Elastic deformations resulting from the changes in the 
mechanical properties of myocardium are represented as a 
spring model with variable coefficients. The viscous 
components of the model include a time-varying viscous 
damper, representing relaxation and the frictional energy 
loss. 
II. METHOD 
A. Mathematical Model 
Longitudinal displacement of the mitral annulus plane 
toward the apex during a cardiac cycle can be considered 
Estimation of elastic and viscous properties of 
the left ventricle based on annulus plane 
harmonic behavior 
Arash Kheradvar, Michele Milano, Robert C. Gorman, Joseph H. Gorman, III, Morteza Gharib 
A
Proceedings of the 28th IEEE
EMBS Annual International Conference
New York City, USA, Aug 30-Sept 3, 2006
ThA09.2
1-4244-0033-3/06/$20.00 ©2006 IEEE. 616
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analogous to motion of a damped harmonic oscillator with 
time-varying coefficients. The time dependency of the 
coefficients is an advantage of this model, which allows the 
model to describe systole, diastole and the transitional 
isovolumic phases of the cardiac cycle. This description of 
model is also consistent with the fact that the LV acts as 
two distinct pumps in a cardiac cycle; acting as a suction 
pump during diastole and as a positive-displacement pump 
during systole in which the pressure of the chamber 
depends on the wall displacement and the blood volume. 
The abbreviations and the acronyms used in the paper are 
summarized in table 1.  
Due to the natural time-dependency of the LV mass 
(blood and tissue), identification of LV mass, separated 
from the rest of the heart, as a function of time was 
impractical. Therefore, the equation of motion for a linear 
harmonic oscillator with time varying coefficients was 
divided by the instantaneous mass of the system and 
rewritten as: 
0)()( ytKythy  (1) 
where (y) is the zero-mean displacement of the system and 
(x) is the longitudinal base to apex displacement: 
)( xxy (2) 
The bar indicates mean value, the dot denotes 
differentiation with respect to time and “h” and “K” are 
the damping and elastic (stiffness) coefficients per unit 
mass, respectively. Equation (1) can also be reorganized to 
a constant-coefficient harmonic oscillator with time varying 
forcing term, as follows: 
ytKKythhyKyhy ))(())((  (3) 
In (3), “ h ” and “ K ” are the averaged values of damping 
(h) and stiffness (K) coefficients during a cardiac cycle, 
respectively.  
TABLE I
ABBREVIATIONS AND ACRONYMS
x longitudinal base to apex displacement 
y zero-mean displacement 
K mean of stiffness coefficient 
EDK end diastolic stiffness coefficient 
ESK end systolic stiffness coefficient 
h mean of damping coefficient 
K mean of the mean of stiffness coefficient 
h
mean of the mean of damping 
coefficient 
i sample index 
t time 
The function on the right-hand side of (3) is the intrinsic 
forcing function, which can represent the contractile 
elements of LV [2-4]. The intrinsic forcing function is 
described as: 
ytKKythhtf ))(())(()(  (4) 
Considering that we measured the longitudinal 
displacement “x(t)”, the parameters in this model, as well as 
the forcing function, would be estimated using a standard 
identification technique [5-6] which will be described after 
the data acquisition procedure. 
B. Animal Preparation 
In order to assess the model behavior, an animal study 
with limited cases was performed. Animal data were 
collected at the Harrison department of surgical research, 
University of Pennsylvania School of Medicine. To 
measure the left ventricular axial displacement and 
intraventricular pressure, ten healthy sheep between 35 and 
45 kg underwent left thoracotomy after induction of 
anesthesia. Sonomicrometry transducers (1.0 mm; 
Sonometrics Corp., London, Ontario, Canada) were 
implanted at the apex and base of the left ventricle to 
characterize ventricular motion. 
C.  Equation of motion and parameter estimations 
The relative displacements obtained from 
sonomicrometry transducers were substituted in equation 
(1). The problem of tracking the parameters was tackled by 
re-sorting to a class of recursive linear least squares 
algorithms [5]. To estimate the model parameters, equation 
(1) was re-written as: 
TKyyhy  (5) 
where TKh, , and Tyy, . The coefficients 
were estimated by a standard Recursive Linear Least 
Squares (RLLS) technique [5] coupled with an estimator 
for continuous time derivatives described in [6]. 
III. RESULTS 
The forced form of the oscillator (3) provides 
information about the global (averaged) state of elasticity 
and damping of the system, in addition to the instantaneous 
intrinsic longitudinal force generated over a cardiac cycle. 
Values of mean stiffness and damping coefficients for each 
case used in the forced form of the equation (3) are 
provided in table 2. To determine whether the coefficients 
were from the same distribution and possessed the same 
mean, we applied Student’s t-test for each sample of 
coefficients.  
617
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TABLE II 
MAGNITUDE OF COEFFICIENTS
K (dyne/g.cm) h (dyne.s/g.cm) ESK (dyne/g.cm) EDK (dyne/g.cm) 
magnitude P-value magnitude P-value magnitude magnitude 
Sheep1 49.11 0.16 0.05 0.90 85.81  2 .00 
Sheep2 51.32 0.50 0.02 0.90 83.83  5.84  
Sheep3 62.39 0.39 0.09 0.68 61.86  1.25  
Sheep4 47.16 0.12 0.01 0.64 89.64  16.62  
Sheep5 44.40 0.05 0.04 0.79 68.74  4.80  
Sheep6 56.86 0.83 0.04 0.79 86.41  5.43  
Sheep7 63.62 0.15 0.07 0.70 66.86  3.42  
Sheep8 83.40 0.05 0.12 0.68 134.50  7.07  
Sheep9 83.48 0.05 0.12 0.68 136.00  7.07  
Sheep10 47.84 0.06 0.02 0.85 86.79  21.02  
For each coefficient of stiffness and damping, we 
calculated the mean of the means of all 10 cases, “ K ”
and ” h ”, respectively. Then the null hypothesis was 
tested as each sample of coefficients had the same mean as 
“ K ” and ” h ” (table 2). The estimated “ K ” and ” h ”
for 10 healthy cases were 58.63±12.8 dyne/g.cm and zero 
dyne.s/g.cm, respectively. The p-value for each sample is 
shown in table 2. 
The forcing term of the equation (3) was also computed 
based on (4), using the estimated parameters, zero-mean 
longitudinal displacement (y) and zero-mean longitudinal 
velocity ( y ). Evolution of the forcing term within a 
cardiac cycle is shown in figure 1. 
Fig. 1. Time evolution of forcing function for all the 10 cases.
618
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I. CONCLUSION
We developed a technique that employs a dynamic model 
for longitudinal displacement of the annulus plane toward 
the apex. Using this technique enables us to estimate 
longitudinal elastic and damping coefficients for the left 
ventricle based only on the mitral annulus displacement. 
Although the estimated values of coefficients were 
considered one-dimensional parameters, their time-varying 
trends were consistent with the previous studies that 
considered dP/dV as an index of stiffness [1,8]. The only 
input used in this model is the real-time long axis 
displacement of the annulus plane, which can also be 
obtained non-invasively using tissue Doppler. The fact that 
the technique can be used as a non-invasive way to 
evaluate the LV function is the key advantage of this model 
with respect to other existing techniques. In all the cases, 
the stiffness of the model has a minimum at the end-
diastole, increasing during systole. It reaches a maximal 
peak at the end-systole. This observation was consistent 
with the P-V diagram of Suga and Sagawa [1] and the other 
models that described LV stiffness based on P-V 
relationship [7-11]. 
Another interesting observation resulting from the forced 
form of the harmonic oscillator was the similarity between 
the magnitude of K and h in all the 10 cases (p>0.05). 
However, in only two cases (sheep 4 and sheep 10) the 
magnitude of EDK  was greater than the rest of the cases. 
This can be a result of inaccuracy in defining the onset of 
end-diastole due to ECG irregularities during surgery. 
Based on the present model, the mean damping coefficient 
can be considered zero in healthy hearts, denoting that the 
viscous damping is minimal in the normal LV [12]. Further 
study is in progress to observe the variations of coefficients 
in cases where the physical state of the LV has been 
changed due to the acquired or congenital heart diseases.  
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Estimation of elastic and viscous properties of the left ventricle based on annulus plane harmonic behavior

Assessment of left ventricular elastic and viscous components based on ventricular harmonic behavior; Cardiovascular Engineering,

Instantaneous pressure-volume relationships and their ratio in the excised, supported canine left ventricle”

Late systolic mechanical properties of the left ventricle: deviation from elastance- resistance behavior”,

Least Squares Parameter Estimation of Continuous-Time ARX models from Discrete-Time Data”,

Length dependence of tension generation in rat skinned cardiac muscle: role of titin in the Frank-Starling mechanism of the heart. Circulation.

Noninvasive Single-Beat Determination of Left Ventricular EndSystolic Elastance in Humans”,

Relationship between early diastolic intraventricular pressure gradients, an index of elastic recoil, and improvements in systolic and diastolic function”, Circulation.

Short time-scale LV systolic dynamics: evidence for a common mechanism in both LV chamber and heart-muscle mechanics”, Circ Res 1991;68:

System identification – Theory for the user, Prentice-Hall, Englewood Cliffs,

Systolic mechanical properties of the left ventricle. Effects of volume and contractile state”, Circ Res

The Giant Protein Titin: A Major Player

Titin-based modulation of calcium sensitivity of active tension in mouse skinned cardiac myocytes”, Circ Res.

https://authors.library.caltech.edu/22521/1/Kheradvar2006p88782006_28Th_Annual_International_Conference_Of_The_Ieee_Engineering_In_Medicine_And_Biology_Society_Vols_1-15.pdf

Estimation of elastic and viscous properties of the left ventricle based on annulus plane harmonic behavior

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