- Email: [email protected]
Effects of inertia and viscoelasticity in late rapid filling of the left ventricle
J. hmrchonics Vol. 14, No. 6. pp 443-445, Printed in Great Britain.
0021-9290%lJ06C443-03 re I9111 Pergamon
1981.
TECHNICAL
502.00/O Press Ltd.
NOTE
EFFECTS OF INERTIA AND VISCOELASTICITY IN LATE RAPID FILLING OF THE LEFT VENTRICLE* Abstract-A
theoretical solution for circumferential stress in the left ventricle is determined when wall mass is taken into account. Cavity pressure starts at lowest diastolic value in late rapid filli& and follows a parabolic arc in the pressure-time plane. The myocardium is composed of a viscoelastic material whose shear relaxation function is specified by an elastic solution for material stiffness in diastole. The dynamic stress oscillates about the quasi-static stress with two frequencies. Inertia produces the high frequency, whereas the shear relaxation function yields the low frequency. For a representative example fashioned from published canine data, the maximum absolute value of the difference between dynamic and auasi-static stresses is not significant. As far as stress is concerned, in this case constitutive considerations are not important. INTRODUCTION
Inertial forces arising within the left ventricle during systole were estimated by Tallarida et al. (1970) and found to be insignificant. Their conclusion provided justification for neglecting inertia in subsequent studies of other researchers, e.g. Rankin et al. (1977); Pouleur et al. (1979); and Hess et al. (1979). Mitchell et al. (1960), Horwitz and Bishop (1972), and Pouleur et al. (1979), however, suggested that inertia could account for some of the observed deviation of left intraventricular pressure from its passive value at a given volume (Moskowitz, 1981). The central question is whether one must replace an equilibrium equation by its corresponding equation of motion, i.e. at time t, whether to include the term &i where 5 is mass density, u is the radial displacement of the left ventricular free wall, and = d2/at2. This study is concerned with the inversion of the Laplace transformed dynamic circumferential stress solution, published by Moskowitz (1980), for a particular pressure waveform and material stiffness variation. Cavity pressure starts at the lowest diastolic value in late rapid filling, and follows a parabolic arc in the pressure-time plane. The myocardium is composed of a viscoelastic material whose shear relaxation function is specified by an elastic solution for material stiffness in diastole. TRANSFORMED
VISCOELASTIC STRESS
principle between transformed elastic and viscoelastic solutions is available: each elastic modulus is replaced by the product of its transform and Laplace complex variable s. The Laplace transformed viscoelastic stress field was obtained by Moskowitz (1980). The component stresses are produced by left intraventricular pressure in the absence of ventricular filling throughout diastole. Inertia was included in the analysis. It is associated with varying pressure and not with wall distention to accommodate filling. Neglecting the presence of a pericardium and right ventricle, the transformed circumferential component of dynamic stress #‘evaluated at the endocardial surface is t?(s) = during
1 + X: + G1 (s) 2k 1 - x: - c, (s) .Y3
(1)
late rapid filling, where al(s)
= ($5 pi x: In x,) s/C(s) = I-, s/C(s).
A bar denotes the Laplace transform of the time dependent variable over which it is placed. The mass density is given by 5, p0 denotes the outer myocardial radius (epicardium), and p1 the inner radius (endocardium), both fixed at the initial state, x1 = pl/po, 2k > 0 is the constant acceleration of left intraventricular pressure P(t) which obeys the parabolic law P(t) = kt’ + P(0)
(2)
at any instant I starting from t = 0 at minimum diastolic pressure P(0). The Laplace transformed shear relaxation function is represented by c(s). It is found from
CIRCUMFERENTIAL
The isothermal linear viscoelastic stress-strain relation expressed as a bounded linear functional mapping strains into stress fulfills a memory hypothesis which asserts that the current value of stress depends on the history of strain as well as the present (elastic) strain. Under restrictions that strain is a continuous function, and the material properties, described by relaxation functions, are functions of bounded variation, the Riesz representation theorem states that the bounded linear functional is a Stieltjes convolution integral with respect to the material properties. Further assumptions on smoothness and material isotropy bring the stress-strain relation to a form for which a Laplace correspondence
(Moskowitz
(1980)) to be V,a 2k G(s) = ~ ,-x:T+S
G(O)
where V, is the initial volume of the cavity, and a is a ventricular stiffness constant. After substitution of equation (4) in (1) and algebraic simplification, (1) becomes R(s) 2k CD(s) zz - -? S(s) s
* Received 1 January 1980; in revised form 15 November 1980. 443
(4)
with R(s) = s4 + a&
+ a.,
(5)
444
Technical Note
and
CONCLUSIONS The dynamic stress uD oscillates about the quasi-static stress u* with frequencies o1 and w*, wI z w2. Inertia produces the high frequency ol, whereas the shear relaxation function G yields the low frequency wl. We can filter-out the low frequency by allowing a4 + 0, b4 + 0 as V, + 0, in equation (5), so that R(s)/S(s) + (s’ + a#? + b2). The high frequency is then approximated by w1 = f ,/b,. In this case of constant material property, the dynamic stress performs a steady-state harmonic motion about the quasi-static stress, and their absolute difference is directly proportional to acceleration.
S(s) = s4 + b2s2 + b, polynomials of degree 4. The coefficients are the constants 1+x: a2 = r G(0) < 0
1
1 + x? V,a 2k < o 1-x: a4 =rl bz=
-~
1-x: r
G(0) > 0
1
EXAMPLE
INVERSION bD(s) is a rational algebraic function of complex variable s such that the degree of the polynomial in the denominator s3 S(s) is higher than that of the polynomial in the numerator R(s). so(t) = 2-l [aD( then equals the sum of the residues of CD(s)exp (st) at all the singular points of r?‘(s), the roots of s3 S(s) = 0 denoted by {s,} = 0, 0, 0, 01,B>Y,6. In order to ascertain the nature of the zeros of S(s) we write S(c) = 1’ + b2c + b4 with the aid of the transformation c = sz. Then, 21 = -b, + ,/(b: - 4b4) and s = &-d/r. Now, b2 > 2,/b4; hence, the two roots of S(c) = 0 are real and negative, and the four roots of S(s) = 0 must therefore be pure imaginary. Let these roots be distinguished by frequencies w1 # 02, a = -p=iw,andy= -6=io,.TheO-rootofs’S(s)=O has multiplicity m, = 3 ; the remaining roots are simple: tnz = tn3 = rnq = ms = 1. There are n = 5 distinct roots. Then, - (2k)-’ uD(t) = i 3 H,, tm,-q exp (s,t) r=, *=l
(6)
where the Heaviside factors are 1 H,q =
(q- 1) ! (m,--q)! ds’-’
The calculations are straightforward and therefore are omitted. We find that the difference between the dynamic and quasi-static circumferential stresses at time t is 4p(t) cos (WI t) - cos (w1 t) aD(t) - aQ(t) = 1_x2
1
(WI@ - kJA2
(7)
where p(t) = kt2 represents the change of left intraventricular pressure from its minimum diastolic value P(O), and 1+x: uQ(t) = __ 1_x2 P(f) I is the stress without inertia. Equation (7) is the theoretical solution for circumferential stress in the left ventricle when wall mass is taken into account. Suppose max uD(t) - u*(t)
I>0
I
= d’(t*)
I
- uQ(t*)
The maximum absolute difference in stress occurs at time t = t* which satisfies o1 sin (w,t*) - o2 sin &t*)
= 0.
(9)
It is of interest to use published experimental and theoretical results to estimate how much the dynamic stress actually departs from the quasi-static value. Let us take: the cavity volume V, = 23.1 cm3, myocardial radii p,, = 2.58 cm and p1 = 1.33 cm, all from the study of Rankin et al. (1977), so that x1 = 0.515. The mass density { = 1.05/981 g. s2. crne4 was used by Rackley (1976). In the experimental investigation of Pouleur et al. (1979) we find a pressure-time curve with halfacceleration k = 2484 g. cm-* . SC*.For the initial value of myocardial stiffness we use the theoretical result of Moskowitz (1980) G(0) = V,a[P(O) + (b/a)]/(l - xf) = 53.9g crneL at P(0) = 5.20 g cm-‘ (Pouleur et al. (1979)) and a = 0.068 cmm3, b = 1.359 g .crn-s, ventricular stiffness constants experimentally determined by Templeton et al. (1975). This completes the set of representative input data. The constant coefficients of S(s) therefore are b2 = 63,093 s-l and b, = 12,440,790 sm4. Clearly b2 > 2Jb4 as presupposed. From equations (5), (7), (8), and (9) we evaluate the frequencies oi = 251 s-l and o2 = 14.1 s-l, and the absolute value of the difference between dynamic and quasistatic stresses Ior’ - u*I = 0.427 g . cm-‘, which happens at time t* = 12 ms. Near the initial time t = 0, the dynamic stress is less than the corresponding stress without inertia. DISCUSSION If the eNect of inertia on stress associated with pressure is neglected, stress is the same, regardless of constitutive considerations, because as 4 -) 0 or rl + 0, i.e. allowing the mass density to be vanishingly small, uD + u* (viscoelastic or not). Inertia associated with pressure acceleration cannot be the primary cause of the deviation of observed from passive states. Equation (1) was obtained from the principles of linear viscoelasticity. At all times in the interval from lowest diastolic pressure to the initiation of slow filling the least upper bound of the magnitude of the displacement gradient at any point within the left ventricular free wall is assumed to be small compared with unity. Further, the displacements remain small relative to the inner radius. NO restriction is placed on stress. The maximum strain observed by Pouleur et d. (1979) in late rapid filling was about 8&,. Only part of this amount is due to pressure, the remaining portion arises from filling (Moskowitz, 1981). Linear strain-displacement is, therefore, a valid approximation. Moreover, no distinction need be made between ventricular configurations at the start and during the deformation process when determining stresses, strains, and displacements. Division ofApplied Mathematics, Graduate School ofApplied Science and Technology, Bergmann Building. Givat Ram, The Hebrew University. Jerusalem 91904. Israel
SAMUEL
E.
MOSKOWITZ
Technical REFERENCES Hess, 0. M., Grimm, J. and Krayenbuehl, H. P. (1979) Diastolic simple elastic and viscoelastic properties of the left ventricle in man. Circularion 59, 1178. Horwitz, L. D. and Bishop. V. S. (1972) Left ventricular pressure-dimension relationships in the conscious dog. Cardiowsc. Res. 6, 163. Mitchell. J. H., Linden, R. J. and Sarnoff, S. J. (1960) Influence of cardiac sympathetic and vagal nerve stimulation on the relation between left ventricular diastolic pressure and myocardial segment length. Circulation Res. 8, 1100 jvoskowitz, S. E. (1980) On the mechantcs of left ventricular diastole. J. Biomeck. 13, 301. Moskowitz. S. E. (1981) Deviation of observed from passive states m dlastole. (submrtted for publication) Pouleur, H., Karliner, J. S., LeWinter. M. M. and Covell, J. W. (1979) Diastolic viscous properties of the intact canine left ventricle. Circulation Res. 45, 410. Rackley, C. E. (1976) Quantitative evaluation of left ventricular function by radrographic techniques. Circulation 54, 862. Rankin, J. S., Arentzen, C. E., McHale, P. A., Ling, D. and Anderson, R. W. (1977) Viscoelastic properties of the diastolic left ventricle in the conscious dog. Circulation Res. 41, 31. Tallarida. R. J., Rusy. B. F. and Loughnane, M. H. (1970)Left ventricular wall acceleration and the law of Laplace. C‘ardrotasc. Res. 4, 2 17. Templeton, G. H., Wildenthal. K., Wilierson. J. T. and Mitchell. J. H. (1975) Influence of acute myocardial depression on left ventricular stiffness and itselastic viscous components. .I. clin. Inresr. 56, 278.
NOMENCLATURE
6
root root root root s2
445
Note mass density, g sz crne4 radial coordinate, cm circumferential stress, g cm -Z frequency, s - ’ function function ventricular stiffness constant, cm-’ constant coefficienls ventricular stiffness constant, g cm-’ constant coefficients
J-’ pressure constant half-acceleration, g cm -’ s_’ multiplicity of r-root intraventricular pressure change, g .crn-’ Laplace complex variable time, s radial displacement, cm nondimensional radial coordinate shear modulus, g cmm2, material stiffness, shear relaxation function Heaviside factor: r = 1.2.. . ,I and q = I. 2.. , VI, left intraventricular pressure, g cm-* polynomial polynomial volume, cm’
Subscripts c cavity Superscripts dynamic quasi-static : Markings -9-1 inverse Laplace transform Laplace transformed time dependent (s,) sequence of roots
*
SZ/dtZ maximum
variable
0021-9290%lJ06C443-03 re I9111 Pergamon
1981.
TECHNICAL
502.00/O Press Ltd.
NOTE
EFFECTS OF INERTIA AND VISCOELASTICITY IN LATE RAPID FILLING OF THE LEFT VENTRICLE* Abstract-A
theoretical solution for circumferential stress in the left ventricle is determined when wall mass is taken into account. Cavity pressure starts at lowest diastolic value in late rapid filli& and follows a parabolic arc in the pressure-time plane. The myocardium is composed of a viscoelastic material whose shear relaxation function is specified by an elastic solution for material stiffness in diastole. The dynamic stress oscillates about the quasi-static stress with two frequencies. Inertia produces the high frequency, whereas the shear relaxation function yields the low frequency. For a representative example fashioned from published canine data, the maximum absolute value of the difference between dynamic and auasi-static stresses is not significant. As far as stress is concerned, in this case constitutive considerations are not important. INTRODUCTION
Inertial forces arising within the left ventricle during systole were estimated by Tallarida et al. (1970) and found to be insignificant. Their conclusion provided justification for neglecting inertia in subsequent studies of other researchers, e.g. Rankin et al. (1977); Pouleur et al. (1979); and Hess et al. (1979). Mitchell et al. (1960), Horwitz and Bishop (1972), and Pouleur et al. (1979), however, suggested that inertia could account for some of the observed deviation of left intraventricular pressure from its passive value at a given volume (Moskowitz, 1981). The central question is whether one must replace an equilibrium equation by its corresponding equation of motion, i.e. at time t, whether to include the term &i where 5 is mass density, u is the radial displacement of the left ventricular free wall, and = d2/at2. This study is concerned with the inversion of the Laplace transformed dynamic circumferential stress solution, published by Moskowitz (1980), for a particular pressure waveform and material stiffness variation. Cavity pressure starts at the lowest diastolic value in late rapid filling, and follows a parabolic arc in the pressure-time plane. The myocardium is composed of a viscoelastic material whose shear relaxation function is specified by an elastic solution for material stiffness in diastole. TRANSFORMED
VISCOELASTIC STRESS
principle between transformed elastic and viscoelastic solutions is available: each elastic modulus is replaced by the product of its transform and Laplace complex variable s. The Laplace transformed viscoelastic stress field was obtained by Moskowitz (1980). The component stresses are produced by left intraventricular pressure in the absence of ventricular filling throughout diastole. Inertia was included in the analysis. It is associated with varying pressure and not with wall distention to accommodate filling. Neglecting the presence of a pericardium and right ventricle, the transformed circumferential component of dynamic stress #‘evaluated at the endocardial surface is t?(s) = during
1 + X: + G1 (s) 2k 1 - x: - c, (s) .Y3
(1)
late rapid filling, where al(s)
= ($5 pi x: In x,) s/C(s) = I-, s/C(s).
A bar denotes the Laplace transform of the time dependent variable over which it is placed. The mass density is given by 5, p0 denotes the outer myocardial radius (epicardium), and p1 the inner radius (endocardium), both fixed at the initial state, x1 = pl/po, 2k > 0 is the constant acceleration of left intraventricular pressure P(t) which obeys the parabolic law P(t) = kt’ + P(0)
(2)
at any instant I starting from t = 0 at minimum diastolic pressure P(0). The Laplace transformed shear relaxation function is represented by c(s). It is found from
CIRCUMFERENTIAL
The isothermal linear viscoelastic stress-strain relation expressed as a bounded linear functional mapping strains into stress fulfills a memory hypothesis which asserts that the current value of stress depends on the history of strain as well as the present (elastic) strain. Under restrictions that strain is a continuous function, and the material properties, described by relaxation functions, are functions of bounded variation, the Riesz representation theorem states that the bounded linear functional is a Stieltjes convolution integral with respect to the material properties. Further assumptions on smoothness and material isotropy bring the stress-strain relation to a form for which a Laplace correspondence
(Moskowitz
(1980)) to be V,a 2k G(s) = ~ ,-x:T+S
G(O)
where V, is the initial volume of the cavity, and a is a ventricular stiffness constant. After substitution of equation (4) in (1) and algebraic simplification, (1) becomes R(s) 2k CD(s) zz - -? S(s) s
* Received 1 January 1980; in revised form 15 November 1980. 443
(4)
with R(s) = s4 + a&
+ a.,
(5)
444
Technical Note
and
CONCLUSIONS The dynamic stress uD oscillates about the quasi-static stress u* with frequencies o1 and w*, wI z w2. Inertia produces the high frequency ol, whereas the shear relaxation function G yields the low frequency wl. We can filter-out the low frequency by allowing a4 + 0, b4 + 0 as V, + 0, in equation (5), so that R(s)/S(s) + (s’ + a#? + b2). The high frequency is then approximated by w1 = f ,/b,. In this case of constant material property, the dynamic stress performs a steady-state harmonic motion about the quasi-static stress, and their absolute difference is directly proportional to acceleration.
S(s) = s4 + b2s2 + b, polynomials of degree 4. The coefficients are the constants 1+x: a2 = r G(0) < 0
1
1 + x? V,a 2k < o 1-x: a4 =rl bz=
-~
1-x: r
G(0) > 0
1
EXAMPLE
INVERSION bD(s) is a rational algebraic function of complex variable s such that the degree of the polynomial in the denominator s3 S(s) is higher than that of the polynomial in the numerator R(s). so(t) = 2-l [aD( then equals the sum of the residues of CD(s)exp (st) at all the singular points of r?‘(s), the roots of s3 S(s) = 0 denoted by {s,} = 0, 0, 0, 01,B>Y,6. In order to ascertain the nature of the zeros of S(s) we write S(c) = 1’ + b2c + b4 with the aid of the transformation c = sz. Then, 21 = -b, + ,/(b: - 4b4) and s = &-d/r. Now, b2 > 2,/b4; hence, the two roots of S(c) = 0 are real and negative, and the four roots of S(s) = 0 must therefore be pure imaginary. Let these roots be distinguished by frequencies w1 # 02, a = -p=iw,andy= -6=io,.TheO-rootofs’S(s)=O has multiplicity m, = 3 ; the remaining roots are simple: tnz = tn3 = rnq = ms = 1. There are n = 5 distinct roots. Then, - (2k)-’ uD(t) = i 3 H,, tm,-q exp (s,t) r=, *=l
(6)
where the Heaviside factors are 1 H,q =
(q- 1) ! (m,--q)! ds’-’
The calculations are straightforward and therefore are omitted. We find that the difference between the dynamic and quasi-static circumferential stresses at time t is 4p(t) cos (WI t) - cos (w1 t) aD(t) - aQ(t) = 1_x2
1
(WI@ - kJA2
(7)
where p(t) = kt2 represents the change of left intraventricular pressure from its minimum diastolic value P(O), and 1+x: uQ(t) = __ 1_x2 P(f) I is the stress without inertia. Equation (7) is the theoretical solution for circumferential stress in the left ventricle when wall mass is taken into account. Suppose max uD(t) - u*(t)
I>0
I
= d’(t*)
I
- uQ(t*)
The maximum absolute difference in stress occurs at time t = t* which satisfies o1 sin (w,t*) - o2 sin &t*)
= 0.
(9)
It is of interest to use published experimental and theoretical results to estimate how much the dynamic stress actually departs from the quasi-static value. Let us take: the cavity volume V, = 23.1 cm3, myocardial radii p,, = 2.58 cm and p1 = 1.33 cm, all from the study of Rankin et al. (1977), so that x1 = 0.515. The mass density { = 1.05/981 g. s2. crne4 was used by Rackley (1976). In the experimental investigation of Pouleur et al. (1979) we find a pressure-time curve with halfacceleration k = 2484 g. cm-* . SC*.For the initial value of myocardial stiffness we use the theoretical result of Moskowitz (1980) G(0) = V,a[P(O) + (b/a)]/(l - xf) = 53.9g crneL at P(0) = 5.20 g cm-‘ (Pouleur et al. (1979)) and a = 0.068 cmm3, b = 1.359 g .crn-s, ventricular stiffness constants experimentally determined by Templeton et al. (1975). This completes the set of representative input data. The constant coefficients of S(s) therefore are b2 = 63,093 s-l and b, = 12,440,790 sm4. Clearly b2 > 2Jb4 as presupposed. From equations (5), (7), (8), and (9) we evaluate the frequencies oi = 251 s-l and o2 = 14.1 s-l, and the absolute value of the difference between dynamic and quasistatic stresses Ior’ - u*I = 0.427 g . cm-‘, which happens at time t* = 12 ms. Near the initial time t = 0, the dynamic stress is less than the corresponding stress without inertia. DISCUSSION If the eNect of inertia on stress associated with pressure is neglected, stress is the same, regardless of constitutive considerations, because as 4 -) 0 or rl + 0, i.e. allowing the mass density to be vanishingly small, uD + u* (viscoelastic or not). Inertia associated with pressure acceleration cannot be the primary cause of the deviation of observed from passive states. Equation (1) was obtained from the principles of linear viscoelasticity. At all times in the interval from lowest diastolic pressure to the initiation of slow filling the least upper bound of the magnitude of the displacement gradient at any point within the left ventricular free wall is assumed to be small compared with unity. Further, the displacements remain small relative to the inner radius. NO restriction is placed on stress. The maximum strain observed by Pouleur et d. (1979) in late rapid filling was about 8&,. Only part of this amount is due to pressure, the remaining portion arises from filling (Moskowitz, 1981). Linear strain-displacement is, therefore, a valid approximation. Moreover, no distinction need be made between ventricular configurations at the start and during the deformation process when determining stresses, strains, and displacements. Division ofApplied Mathematics, Graduate School ofApplied Science and Technology, Bergmann Building. Givat Ram, The Hebrew University. Jerusalem 91904. Israel
SAMUEL
E.
MOSKOWITZ
Technical REFERENCES Hess, 0. M., Grimm, J. and Krayenbuehl, H. P. (1979) Diastolic simple elastic and viscoelastic properties of the left ventricle in man. Circularion 59, 1178. Horwitz, L. D. and Bishop. V. S. (1972) Left ventricular pressure-dimension relationships in the conscious dog. Cardiowsc. Res. 6, 163. Mitchell. J. H., Linden, R. J. and Sarnoff, S. J. (1960) Influence of cardiac sympathetic and vagal nerve stimulation on the relation between left ventricular diastolic pressure and myocardial segment length. Circulation Res. 8, 1100 jvoskowitz, S. E. (1980) On the mechantcs of left ventricular diastole. J. Biomeck. 13, 301. Moskowitz. S. E. (1981) Deviation of observed from passive states m dlastole. (submrtted for publication) Pouleur, H., Karliner, J. S., LeWinter. M. M. and Covell, J. W. (1979) Diastolic viscous properties of the intact canine left ventricle. Circulation Res. 45, 410. Rackley, C. E. (1976) Quantitative evaluation of left ventricular function by radrographic techniques. Circulation 54, 862. Rankin, J. S., Arentzen, C. E., McHale, P. A., Ling, D. and Anderson, R. W. (1977) Viscoelastic properties of the diastolic left ventricle in the conscious dog. Circulation Res. 41, 31. Tallarida. R. J., Rusy. B. F. and Loughnane, M. H. (1970)Left ventricular wall acceleration and the law of Laplace. C‘ardrotasc. Res. 4, 2 17. Templeton, G. H., Wildenthal. K., Wilierson. J. T. and Mitchell. J. H. (1975) Influence of acute myocardial depression on left ventricular stiffness and itselastic viscous components. .I. clin. Inresr. 56, 278.
NOMENCLATURE
6
root root root root s2
445
Note mass density, g sz crne4 radial coordinate, cm circumferential stress, g cm -Z frequency, s - ’ function function ventricular stiffness constant, cm-’ constant coefficienls ventricular stiffness constant, g cm-’ constant coefficients
J-’ pressure constant half-acceleration, g cm -’ s_’ multiplicity of r-root intraventricular pressure change, g .crn-’ Laplace complex variable time, s radial displacement, cm nondimensional radial coordinate shear modulus, g cmm2, material stiffness, shear relaxation function Heaviside factor: r = 1.2.. . ,I and q = I. 2.. , VI, left intraventricular pressure, g cm-* polynomial polynomial volume, cm’
Subscripts c cavity Superscripts dynamic quasi-static : Markings -9-1 inverse Laplace transform Laplace transformed time dependent (s,) sequence of roots
*
SZ/dtZ maximum
variable