Department of Mathematics and Computer Science, Royal Military
College of Canada, Kingston, Ontario, Canada K7K 7B4
A
mathematical approach that can be used to calculate the passive stress
in the ventricular wall is presented. The active fiber stress
(force/unit area) generated by the muscular fibers in the ventricular
wall is expressed by means of body force (force/unit volume of the
myocardium). It is shown that the total intramyocardial passive stress
induced in the passive medium of the myocardium can be expressed as the
sum of a passive stress induced by the left ventricular pressure and a
passive stress induced by the active fiber stress. Applications to
experimental data published in the literature are given. New results
are presented that show the relation among those two components of the
intramyocardial passive stress. New relations between the
intramyocardial passive stress, the slope (elastance) of the
pressure-volume relation, and the residual volume are also derived. The
results obtained give a better understanding of some aspects of the
mechanics of cardiac contraction and can provide a more detailed
interpretation of clinical conditions.
stress-strain relations; end-systolic pressure-volume
relation, maximum left ventricular elastance; cardiac mechanics; residual volume; Law of Laplace
 |
INTRODUCTION |
THE RELATION
BETWEEN WALL STRESS in the myocardium and the pressure in the
left ventricle has generally been designated as the Law of Laplace
(4, 22, 25, 31). A quasi-static approximation of left
ventricular contraction is used in general to study the Law of Laplace
because the inertia forces of the myocardium amount to ~1% of the
static forces and, as such, can be neglected (45). Theoretical studies were usually based on relations derived from the
theory of elasticity for passive media; the values of the intramyocardial passive stress calculated in this way were often reported as higher than the measured values of the stress (17, 26, 28, 30). Moreover, direct measurement is an invasive process
that can perturb the quantity to be measured by damaging the muscular
fibers (12, 14, 19, 24, 32, 33, 48), and, depending on the
technique of measurement used, one can easily have measurement errors
of up to 50% (16). It was suggested that some transducers
may record, along with the intramyocardial passive stress, a
part of the active fiber stress (16, 33). The
micropipette-transducer technique with a sensor 1,000 times smaller in
volume than those usually used may offer more reliable results
(27).
Instead of "intramyocardial pressure," the term "intramyocardial
passive stress" or "total passive stress" is used in this study
to designate the stress induced in the passive medium of the myocardium
by the combined action of the active fiber stress, the left ventricular
pressure (P), and the external pressure (Po) on the
pericardium. The intramyocardial passive stress (force/unit area) is a
tensorial quantity with six components (3 normal and 3 shear stresses).
Under the assumption of symmetric contraction of the left ventricle
used in this study, the intramyocardial passive stress reduces to a
vectorial quantity with magnitude 
tot, and in
cylindrical coordinates it has three normal components: radial
r, circumferential c, and
longitudinal L. A pressure on the contrary is a scalar
quantity, and the use of pressure to describe the stresses existing at
any time in the ventricular wall can be misleading (2);
this point is further discussed later on in the DISCUSSION
(see Eq. 13). The need to distinguish between the active and
passive medium in modeling the intramyocardial stress has been pointed
out by some researchers (1, 5, 13, 33, 34, 37-41). In
this study, the concept of body force (force/unit volume of the
myocardium) that was successfully used to model the radial active force
of the myocardium (37-41) is applied to calculate the
active and passive stresses induced by the muscular fibers in the
ventricular wall during the systolic phase. As will be seen, this
approach has clear advantages; it will lead to new relations among the
passive and active components of the intramyocardial stress, the left
ventricular elastance (E), and the residual volume
(Vd) (see Fig. 2).
Glossary
| a |
inner radius of the myocardium
|
| b |
outer radius of the myocardium
|
| D |
radial active force/unit volume of the myocardium
|
 |
radial fiber stress (radial active force/unit area) generated on
the inner surface of the myocardium
|
| E |
left ventricular elastance
|
h = b  a |
thickness of the myocardium
|
| P |
left ventricular pressure
|
| Po |
external pressure on the pericardium
|
| V |
left ventricular volume
|
| Vd |
residual volume and intercept of the of the end-systolic
pressure-volume relation with the volume axis
|
| Ved |
end-diastolic left ventricular volume (when the change in
left ventricular volume over time = 0)
|
| Ves |
end-systolic left-ventricular volume (when the change in left
ventricular volume over time = 0)
|
 ij |
ij = 1 for i = j; ij = 0 if i is
different from j; see DISCUSSION (Eq. 13)
|
| (c)p |
circumferential stress induced in the passive medium of the myocardium
by P and Po;
|
| (c)d |
circumferential stress induced in the passive medium of the myocardium
by  h
|
| c=(c)p+
(c)d |
circumferential stress induced in the passive medium of the myocardium
as a result of the combined action of P, Po, and
h, shortly called intramyocardial circumferential
stress
|
| r, L |
similar definitions as above for the radial stress
r and the longitudinal stress
L
|
 c, (c)p,
(c)d |
average values, same definition as before (cylindrical model)
|
| cs, (cs)p,
(cs)d |
average values, same definition as before (spherical model)
|
The subscript "p" is used to indicate those values
induced by pressure, and the subscript "d" is used to indicate
those values induced by the radial fiber stress. The overbar
"
" indicates average values.
| |
MATHEMATICAL MODEL |
A mathematical study of the contraction of the myocardium was
previously carried out based on the theory of large elastic deformation
(38, 39, 41). To simplify the mathematical formulation, the theory of linear elasticity is used in this study; it has been used
in many studies (10, 15-17, 27-30, 35) related
to the calculation of the stress in the myocardium, and it is also used
by many physiologists for clinical studies because of its relative
simplicity. Moreover, the possibility to use body force to model the
fiber stress in the linear theory of elasticity does not seem to have
been previously investigated. Simplification can sometimes be very
useful; it can lead to new insight and better orientation of the research.
In this study, the effect of the residual stress is neglected. The
myocardium is represented as a thick-walled cylinder with transverse isotropy contracting symmetrically, and inertia forces and
viscous forces are neglected in a quasi-static approximation of a
steady-state contraction. Anisotropy is modeled with the longitudinal
axis as the preferred axis of material symmetry rather than the fiber
axis, which varies by about 120° from epicardium to endocardium
(47). A helical muscular fiber in the myocardium is
projected as a circle on the cross section of the cylindrical wall. As
a consequence of the assumed symmetry of the problem, a resultant
radial active force (D; force/unit volume of the myocardium) will be
generated by the muscular fibers (see Fig.
1B). The radial active fiber
stress at a radial distance (r) in the cylindrical wall is
given by
D
dr, and the radial active force/unit area generated on the
inner surface of the myocardium is given by
D
dr = h, where is an
average value of D calculated by the mean value theorem;
h = b a, thickness of the
myocardium, a is the inner radius, and b is the
outer radius (note that h does not depend on
r).
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Fig. 1.
A: equilibrium of a half-cylinder subject to
pressure (P) and external pressure on the pericardium (Po).
The average circumferential stress is given by
(c)p = (P Po)
a/h; for P > Po, a tension
stress is developed in the circumferential direction. B:
equilibrium of a half-cylinder subject to a radial force/unit area
h acting on the inner surface of the half-cylinder.
The average circumferential stress is given by
(c)d = (h)a/h; when
h is directed inward, a compression stress in
developed in the circumferential direction.
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Average Stress
As a tutorial introduction, consider the half-cylinders shown in
Fig. 1. The equilibrium equation for a half-cylinder is obtained by
applying the equality
In Fig. 1A, it is assumed that only the external
forces P and Po are acting on the cylindrical wall, and the
equilibrium equation gives
|
(1a)
|
where L is the length of the myocardium.
Eq. 1a can be written in the form
|
(1b)
|
Similarly, in Fig. 1B, it is asumed that only
the radial active force/unit area h is acting on the
inner surface of the myocardium, and the equilibrium equation gives
|
(2)
|
The negative sign in Eq. 2 means that the
average circumferential passive stress
(c)d is compressive. In the case of
the myocardium, where both external forces P and Po and
radial active fiber stress are present, the combination of Eqs.
1b and 2 give the Law of Laplace for a cylindrical
geometry in the following form
|
(3)
|
where c = (c)p + (c)d is the average circumferencial
passive stress induced in the passive medium of the cylindrical wall,
and c will be normally negative (compression)
because h > P Po during the
systolic phase. Equation 3 clearly shows the difficulty that
arises when h is neglected and Eq. 1b is applied to an active medium like the myocardium. By applying only Eq. 1b to the myocardium, what we are actually calculating
is the equivalent of c (c)d.
Stress Components for a Cylindrical Model
A more exact derivation of the stress-strain relations derived
by using the theory of linear elasticity is given in many textbooks (36, 43). To avoid overloading the text with mathematical derivations, the necessary equations will be directly stated, and it
will be then shown how they can be applied to experimental data. Let us
first consider the passive stress (tot)p
induced in the passive medium of the myocardium by the external
pressures P and Po. For a symmetrical contraction of a
cylinder (36, 43), we have three components
<IT>−</IT>P<SUB>o</SUB>](/content/vol279/issue5/fulltext/H2519/img011.gif)
|
(4a)
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<IT>−</IT>P<SUB>o</SUB>](/content/vol279/issue5/fulltext/H2519/img012.gif)
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(4b)
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![(&sfgr;<SUB>L</SUB>)<SUB>p</SUB><IT>=&egr;</IT><SUB>Lp</SUB><IT>Y</IT><SUB>L</SUB><IT>+</IT>(<IT>&ugr;</IT><SUB>t</SUB><IT>Y</IT><SUB>L</SUB><IT>/Y</IT><SUB>t</SUB>)[(<IT>&sfgr;<SUB>r</SUB></IT>)<SUB>p</SUB><IT>+</IT>(<IT>&sfgr;<SUB>c</SUB></IT>)<SUB>p</SUB>]](/content/vol279/issue5/fulltext/H2519/img013.gif)
|
(4c)
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with
|
(4d)
|
where YL is Young's modulus of
elasticity in the longitudinal direction, Yt is
Young's modulus in the transversal direction, 
LP is
longitudinal strain, and 
t is Poisson's coefficient.
The boundary conditions are
|
(4e)
|
|
(4f)
|
Similarly, the passive stress
(tot)d induced in the passive medium of the
myocardium by the active fiber
stress [ D dr = h] has the following
components
](/content/vol279/issue5/fulltext/H2519/img017.gif)
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(5a)
|
](/content/vol279/issue5/fulltext/H2519/img018.gif)
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(5b)
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![(&sfgr;<SUB>L</SUB>)<SUB>d</SUB><IT>=&egr;</IT><SUB>Ld</SUB><IT>Y</IT><SUB>L</SUB><IT>+</IT>(<IT>&ugr;</IT><SUB>t</SUB><IT>Y</IT><SUB>L</SUB><IT>/Y</IT><SUB>t</SUB>)[(<IT>&sfgr;<SUB>r</SUB></IT>)<SUB>d</SUB><IT>+</IT>(<IT>&sfgr;<SUB>c</SUB></IT>)<SUB>d</SUB>]](/content/vol279/issue5/fulltext/H2519/img019.gif)
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(5c)
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with
|
(5d)
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The boundary conditions are
|
(5e)
|
|
(5f)
|
Combining Eqs. 4 and 5 together
gives the components of the intramyocardial passive stress due to the
combined action of P, Po, and h. One gets
|
(6a)
|
|
(6b)
|
|
(6c)
|
|
(6d)
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or written in full
<IT>−</IT>P<SUB>o</SUB>](/content/vol279/issue5/fulltext/H2519/img027.gif)
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(7a)
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<IT>−</IT>P<SUB>o</SUB>](/content/vol279/issue5/fulltext/H2519/img028.gif)
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(7b)
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|
(7c)
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with
|
(7d)
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The boundary conditions are
|
(7e)
|
|
(7f)
|
tot is the total intramyocardial passive stress,
which is supposed to be what is measured by a microtransducer inserted in the myocardium if the microtransducer is properly calibrated. Twist
and shear are neglected in the previous equations. As discussed in the
introduction and as will be seen in what follows, many of the problems
encountered in the calculation of the passive stress induced in the
passive medium of the myocardium come from using Eq. 4 instead of using Eq. 7. In the calculations that follow, it
is assumed that the external pressure Po 
0.
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EXPERIMENTAL APPLICATIONS |
Intramyocardial Stress Calculation
The first application is based on the experimental data
published by Mihailescu and Abel (27). The experiment in
Ref. 27 was carried out on excised hearts from adult cats, and the
stress in the myocardium was measured with glass micropipette
transducers and a controlled displacement of those transducers inside
the myocardium. Experimentally measured values of the intramyocardial passive stress were taken from Fig. 4, A and B,
of Ref. 27, together with the abscissas (x), and are
reproduced in Table 1. The values of the
intramyocardial passive stress were measured at three different
perfusion pressures (PP) (50, 75, and 100 mmHg PP) for a working heart
(Table 1; A) and a nonworking heart (Table 1; B).
Also, data for measured intramyocardial passive stress and calculated
radial passive stress in Table 2 are
taken from Fig. 6 of Ref. 27 for only one perfusion pressure (100 mmHg PP); the other two cases are similar. The ratios of measured passive stress to calculated radial passive stress were calculated in Table 2
for different radial distances (r = b xh) along the thickness of the myocardium. Note the
following: 1) In Table 1, the ratios of measured values
(A) to measured values (B) are practically constant within small fluctuations. 2) In Table 2, the
ratios of measured values (A) to calculated radial passive
stress are not constant. 3) In both Figs. 4 and 6 of Ref.
27, the authors compare their measurements with the calculated radial
passive stress (the equivalent of Eq. 4a), with
(ra)p = P at r = a (Eq. 4e) when in fact what
is meant is the application of Eq. 7, a and
e. The measured stress seems to be related to the total intramyocardial passive stress tot and not the
radial passive stress induced by P, as will be discussed in the
following.
Possible explanation of the previous results follows.
Measurement with micropipette.
In the measurement of the intramyocardial passive stress reported in
Ref. 27, the micropipette (if properly calibrated) seems to be sensing
the total intramyocardial passive stress tot in the
tissue, which under the assumption of symmetric contraction (twist and
shear neglected) is given by Eq. 7d. For the purpose of
calculation, the simplifying assumptions are made that the longitudinal
strain L0,
YLYt, and
t = 0.5. This is equivalent to assuming a
contraction of a two-dimensional incompressible cylinder. Under
these assumptions, Eq. 7d takes the form
![&sfgr;<SUB>tot</SUB><IT>=</IT>(P<IT>−<A><AC>D</AC><AC>&cjs1171;</AC></A>h</IT>)[<IT>a<SUP>2</SUP>/</IT>(<IT>b<SUP>2</SUP>−a<SUP>2</SUP></IT>)]<RAD><RCD><IT>3+2b<SUP>4</SUP>/r<SUP>4</SUP></IT></RCD></RAD>](/content/vol279/issue5/fulltext/H2519/img033.gif)
|
(8a)
|
This total stress can also be split into its component
(tot)p due to P, and
(tot)d due to
h, as follows
![(&sfgr;<SUB>tot</SUB>)<SUB>p</SUB><IT>=</IT>P[<IT>a<SUP>2</SUP>/</IT>(<IT>b<SUP>2</SUP>−a<SUP>2</SUP></IT>)]<RAD><RCD><IT>3+2b<SUP>4</SUP>/r<SUP>4</SUP></IT></RCD></RAD>](/content/vol279/issue5/fulltext/H2519/img034.gif)
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(8b)
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![(&sfgr;<SUB>tot</SUB>)<SUB>d</SUB><IT>=</IT>(−<IT><A><AC>D</AC><AC>&cjs1171;</AC></A>h</IT>)[<IT>a<SUP>2</SUP>/</IT>(<IT>b<SUP>2</SUP>−a<SUP>2</SUP></IT>)]<RAD><RCD>3<IT>+2b<SUP>4</SUP>/r<SUP>4</SUP></IT></RCD></RAD>](/content/vol279/issue5/fulltext/H2519/img035.gif)
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(8c)
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which is what would be obtained by using Eqs. 4d and 5d.
For the same x or radial distance r = b xh, the factor
[a2/(b2 a2)]
is constant, and, according to Eq. 8a, the ratios of
measured values (A) to measured values (B) is
|
(9)
|
and are nearly constant, as shown in Table 1, and do not depend on
the position of measurement r. Note also that
tot is negative (compression) during contraction and,
although only its absolute value (experimentally measured) is shown in
Table 1 to avoid repetition of the negative sign, the algebraic value should be used in the computation.
Calculation of the active fiber stress h.
The current trend is to consider that the maximum activation of the
cardiac muscle corresponds to the end-systolic pressure-volume relation
(ESPVR) in Fig. 2, which is the instant
when the elastance E reaches its maximum value
(Emax) near end systole and P Pmax/1.2, Pmax is the maximum left ventricular
pressure. But the peaks Emax and
Pmax do not occur simultaneously;
Emax is reached after Pmax is
reached during the systolic phase. The factor of 1.2 gives the
estimated value of P near end systole at which h is
maximum when the elastance Emax is reached and
is chosen based on previous studies (37-41).
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Fig. 2.
Simplified drawing of the pressure-volume relation in the
left ventricle. The line d3Vd is the
end-systolic pressure-volume relation with maximum slope (or elastance)
Emax. The left ventricular pressure
(Pm) is assumed constant during ejection for
simplicity. The loop
Vedd2d1Vm
represents the pressure-volume loop in a normal ejecting cycle.
(h)m is the maximal value of
the radial fiber stress (radial active force/unit area) generated on
the inner surface of the myocardium near end systole. For simplicity,
the subscript "m," denoting when
Emax is reached, is dropped from
h and P in the text, and
Vm Ves, the
end-systolic volume. Note that Vd is not necessarily
constant during the cardiac cycle.
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On the basis of these observations, Eq. 8a was used to
calculate h P and then h
for P = Pmax/1.2 with the same experimentally measured
values of tot given in Table 1 (A); the
results are shown in Table 3 only for the
case of 100 mmHg PP; and calculation for the other two cases is similar
(the subscript "m" for h and P shown
in Fig. 2 when Emax is reached and is dropped in
the notation for simplicity). Equation 8b is also used to
calculate (tot)p, and Eq. 8c is
used to calculate (tot)d; the results are also shown in Table 3. Calculation was carried out with
a = 0.74 cm, b = 1.54 cm, and
h = 0.8 cm, with the values taken from Ref. 27.
Although not indicated in Ref. 27, it was assumed that these values
correspond to the end-systolic dimensions of the left ventricle.
With the exception of the first result with r = 1.44 cm
(which is slightly low), all other calculations yield nearly constant value for h, which is as expected because
h = D
dr and does not depend on r, and its calculation
from Eq. 8a should give the same value for a given
combination of r and tot. This result shows
the consistency of the mathematical formalism used. h
is normally greater than P for a contracting myocardium. Previous
studies (37-41) indicate that, near end systole,
h/P 3 under normal physiological contraction
(maximum efficiency), h/P 2 for a mildly
depressed state of the heart (which corresponds to points
d1 and d5 coinciding in
Fig. 2), and h/P < 2 for a severely depressed state of the heart. A value of
h/P 2 in Table 3 is as expected because
experiments on excised hearts give results below the expected normal
physiological values. From Eqs. 4, 5, and
8, we have h/P (tot)d/(tot)p (c)d/(c)p (r)d/(r)p ,
and this ratio is nearly constant, as is evident from Table 3 and Figs.
3, 5, and 6, which gives an indication for the consistency of the
mathematical computations. This ratio can offer an interesting index to
study the contractility of the cardiac muscle.
ra = h P at the
surface of the endocardium (see Eq. 7e), and
h P is of the same order of magnitude as P
when h/P 2, which can create the wrong
impression that ra = P at the surface of
the endocardium (see Eq. 1 of Ref. 27 and Eq. 4,
a and e, of this study).
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Fig. 3.
A: plot of the intramyocardial passive stress
induced by the active fiber stress h
[(tot)d] (Eq. 8c) against the
intramyocardial passive stress induced by the left ventricular pressure
P
[(tot)p]
(Eq. 8b) for 3 groups of data (A in Table 1). +,
50 mmHg; x, 75 mmHg; and *, 100 mmHg perfusion pressure (PP).
B: same as in A, but the stress values are
divided by their respective PP.
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Calculation of the passive stress components.
As mentioned previously, the experimentally measured stress was
identified with tot in Eq. 8a through
Eq. 8b and 8c, were respectively used to
calculate (tot)p and
(tot)d. Equation 7,
a and b, were then respectively used to calculate
r and c, and Eqs. 4 and 5 were respectively used to calculate
(r)p,
(c)p, (r)d, and
(c)d. The results are shown in Tables 3 and
4 for the case of 100 mmHg PP. The
graphical representation of those results are shown in Figs.
3-7
for the three cases of perfusion pressure given in Table 1. In Fig.
3A, (tot)d is plotted against
(tot)p, and in Fig. 3B, the
values of (tot)d and
(tot)p are divided by their
respective perfusion pressure. The values of
(tot)p and (tot)d
are plotted in Fig. 4 against the radial distance r. In Fig.
5, the relation between the circumferential passive stresses
(c)p and (c)d is
shown. Note that the relation between these two stresses is very
similar to that reported for (tot)p and
(tot)d. From Tables 3 and 4, it is seen that
the calculated values of c are nearly 90% the measured
values tot (twist, shear, and longitudinal shortening
neglected). In Figs. 6 and 7, similar graphical relations between the
components of r are shown. All these results give
further evidence for the consistency of the mathematical formalism
used.
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Fig. 4.
Variation with the radial distance (r) in the
ventricular wall of the intramyocardial passive stress induced by the
active fiber stress h
[(tot)d] (Eq. 8c) and of the
intramyocardial passive stress induced by the left ventricular pressure
P [(tot)p] (Eq. 8b). Results
correspond to 3 groups of data (A in Table 1). +, 50 mmHg;
x, 75 mmHg; and *, 100 mmHg PP. The endocardium is at r = a = 0.74 cm, and the epicardium is at
r = b = 1.54 cm.
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Fig. 5.
Plot of the circumferential passive stress induced by the
active fiber stress h
[(c)d] (Eq. 5b) against the
circumferential passive stress induced by the left ventricular pressure
P [(c)p]
(Eq. 4b) for 3 groups of data (A in Table 1). +,
50 mmHg; x, 75 mmHg; and *, 100 mmHg PP. Other relations between
(c)d and (c)p are
similar to those between (tot)d and
(tot)p.
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Fig. 6.
A: plot of the radial passive stress induced
by the active fiber stress h
[(r)d] (Eq. 5a) against the
radial passive stress induced by the left ventricular pressure P
[(r)p]
(Eq. 4a) for 3 groups of data (A in Table 1). +,
50 mmHg; x, 75 mmHg; and *, 100 mmHg PP. B: same as in
A, but the stress values are divided by their respective
PP.
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Fig. 7.
Variation with the radial distance r in the
ventricular wall of the radial passive stress induced by the active
fiber stress h
[(r)d] (Eq. 5a) and
of the radial passive stress induced by the left ventricular
pressure P [(r)p] (Eq. 4a). Results correspond to 3 groups of data
(A in Table 1). +, 50 mmHg; x, 75 mmHg; and *, 100 mmHg PP.
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Relation Between Intramyocardial Passive Stress and
Elastance
In this second application, it is the relation between the
intramyocardial passive stress and the left ventricular elastance E that is the focus of our attention. The possibility of
such a relation was qualitatively discussed by Westerhof
(48); a mathematical formulation is given in what follows.
The equation of the pressure-volume relation in the left ventricle can
be written in the form (37-41)
|
(10a)
|
where Ved is the end-diastolic left ventricular
volume (when DV/Dt = 0) and V is the left
ventricular volume (
a2L).
Equation 10a can be split into two equations
|
(10b)
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|
(10c)
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Vd is the intercept of the pressure-volume relation
with the volume axis in Fig. 2. One can easily relate Eq.
10, for instance, to Eqs. 4a, 5a, and
7a for the radial stress, or to Eqs. 4b, 5b,
and 7b for the circumferential pressure, or to Eq.
8 for the total passive stress. One can also relate Eq.
10 to the average circumferential passive stress given by
Eqs. 1b, 2, and 3; one gets
|
(11a)
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(11b)
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(11c)
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The average circumferential passive stress c
is compressive (negative) during contraction. Note how the splitting of
Ved V into Ved Vd and V Vd corresponds to the splitting of the stress into
()p and ()d components. Equations for the
average circumferential passive stress for a spherical model can be
obtained in the same way. Equation 1 of Burns et al.
(3) and Eq. 10 show that one has for the
average circumferential passive stress of a spherical model
|
(12a)
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(12b)
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(12c)
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In Eq. 12, it will be assumed that Po 0 as usual, and the subscript "cs" is used for the circumferential
(spherical) model to avoid confusion between Eqs. 11 and 12. Note that (b2 a2)/a2 = (b a)(b + a)/a2 2h/a by taking b + a 2a; it is well known that there is a difference by a
factor of 2 between the spherical and the cylindrical model
(22). Equation 12 has been applied to the
results of Table 1 of Burns et al. (3) to calculate
h, Emax, and Vd in
Table 5; only four cases are used for the
purpose of illustration (the subscript "m" has also been dropped
from h in Table 5, for simplicity in the
notation). The experiments corresponding to Table 1 of Burns et al.
(3) were conducted on excised hearts of mongrel dogs to
study the effect of wall stress and left ventricular pressure on the
extent of shortening and stroke work in several contractions
originating from the same end-diastolic volume. Mean left ventricular
wall force was measured by a transmural auxotonic strain gauge, with
pins inserted to measure strain in the circumferential direction. The
stress is calculated by dividing the force measured by the estimated
cross-sectional area of the venticular wall. So the measurement
technique in this case is different from that used in Ref. 27 and seems
to measure directly the force in the circumferential direction.
In Table 5, experimental results with the same initial stretch
Ved were assumed to generate the same radial active fiber stress h (Frank-Starling mechanism), and it is also
supposed that the effect of afterload on h can be
neglected. A factor of 1.36 was used to change the stress unit from
grams per centimeters squared to millimeters of mercury. We have used
the value P = Pmax/1.2 in Eqs. 10a and 12, as explained before, and the values of
(cs)p
in Eq. 12a taken from Burns et al. (3) are also
divided by a factor of 1.2 because calculations in Burns et al.
(3) were carried out with Pmax instead of P = Pmax/1.2. It also supposed that when
Emax is reached in Fig. 2, one can take
Vm Ves, where
Vm is the left ventricular volume corresponding to Emax. These assumptions have already been
used (37-41) and do not change in any way the basic
results and conclusions of this study. It is interesting to note in
Table 5 that in cases of small stroke volume (near isovolumic
contraction), both (cs)d and
(cs)p increase, but the average
intramyocardial passive stress cs = (cs)p + (cs)d is small. Note that in a perfect isovolumic contraction with P = h,
cs = 0 under the hypothetical assumption that the
passive medium of the myocardium does not undergo any change of shape
(actually a change of shape does occur in an isovolumic contraction).
| |
DISCUSSION |
Several aspects of this work deserve some discussion to indicate
the limitations of this study, points of controversy, and possible
orientation for future research.
Linear Model
Several studies (17, 18, 46, 49) in the past have
indicated that the linear elastic model cannot correctly describe the
contraction of the myocardium. However, those studies were limited to
linear models in which the myocardium was treated as a passive medium.
Also, the present study assumes a symmetric contraction of the
cylindrical model that neglects the effect of twist and shear. Recent
studies on this aspect of the problem (21, 47) also
indicate that the approximation of a symmetric contraction will affect
the accuracy of the results.
Stress Measurement with Micropipette
The intramyocardial passive stress can be written as a
six-component tensor
|
(13)
|
where Ph is a hydrostatic pressure and
sij is the deviatoric stress; similar
decomposition can also be used for the active fiber stress (5,
41, 42). The indexes (i and j) equal
r, c, and L and indicate, in a cyclic way, the components of
the stress in cylindrical coordinates. What does the micropipette transducer in the experience of Mihailescu and Abel (27)
actually sense? These authors calculate the radial passive stress
(r)p and compare it with what
they claim is the measurement of intramyocardial pressure, which can be
understood as the equivalent of the hydrostatic pressure in Eq. 13. A source of confusion is that intramyocardial pressure has
been extensively used in the literature where in fact intramyocardial
passive stress is meant (2). It is suggested in this study
that the micropipette transducer is actually sensing the total force
acting on it, and, when properly calibrated, it measures the total
force/unit area, which is the total passive stress (Eqs. 7d and 8a). Stein et al. (44), using a
microtransducer, have reported that their measurements do not depend on
the axial rotation of the needle; however, they reported that it was
important to maintain the position of the microtransducer relative to
the myocardium unchanged whence inserted. Nematzadeh et al.
(32) also mention the necessity of consistent orientation
of the sensing surface of the microtransducer within the myocardium.
There is an apparent dependence of the measurement on orientation.
According to Brandi and McGregor (2), in view of the
complex intramyocardial structure, it is unjustified to speak of
"pressure" in the tissue (which is a scalar quantity similar to
Ph in Eq. 13). On the contrary, it can be
confidently predicted that the intramyocardial passive stress is a
tensorial quantity with six components, which, under the assumption of
symmetric contraction used in this study, reduces to a vectorial
quantity with three components: r,
c, and L.
Simplified Contraction Model
An important observation reported by Waldman et al.
(47) is that the principal shortening direction and fiber
direction were almost parallel in the outer wall, but perpendicular in
the inner wall where shortening was greatest near the circumferential direction and accompanied by substantial wall thickening (wall thickening can account for 25 to 50% of stroke volume; Refs. 8, 9, and
11). This explains the result of this study: that in the inner layers
of the myocardium the circumferential passive stress c
is ~90% the value of the total passive stress tot
which seems to be near or in the direction of maximum shortening.
Because twist, shear, and longitudinal shortening are neglected in the two-dimensional calculation of this study, this may explain why the
values of h calculated in the subepicardium at
r = 1.44 cm are lower than the other values reported in
Table 3. Another reason may be the experimental difficulty to measure
the total passive stress near the surface of the epicardium.
In conclusion, this study shows that the stress induced in the passive
medium of the myocardium (called intramyocardial passive stress in this
study) can be expressed as the resultant effect 1) of a
stress induced by the active fiber stress
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ABSTRACT
INTRODUCTION
