INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

ONE RESPONSE OF THE LEFT VENTRICLE to prolonged arterial hypertension is the development of concentric hypertrophy (2, 13). This adaptation lowers the stress experienced by the ventricular wall and normalizes the elevated stress resulting from increased pressure. Laplace's law of the heart relates wall stress to pressure in terms of the ratio of wall thickness to lumen radius. Lower ratios (relatively thin walls) result in greater internal stress. Because of this pattern of stress increases, there is a lower limit to the thickness of the myocardial wall at a particular operating pressure (2, 21).

Investigators have found that normal wall thickness-to-radius ratios are nearly constant over a group of mammalian species that are widely disparate in size but relatively close in blood pressure (5, 19). Ratios that are much lower than this normal value are indicative of pathological eccentric hypertrophy, which results in elevated wall stress at normal pressures. It is not clear, however, what is constraining the ratio in the opposite direction.

A small amount of nonobstructive hypertrophy would lower wall stress, and this is believed to be one benefit of exercise (9). It is, therefore, not obvious why concentric hypertrophy often regresses after developing in response to transient conditions such as pregnancy (22) or repaired valve stenosis (4). The stress reduction is apparently not enough of an advantage for the mild hypertrophy to be maintained. It is unclear what disadvantages may result from concentric hypertrophy that would counterbalance the advantage of stress reduction.

Recently suggested counteracting factors have included problems in perfusion observed in the endocardial layers of concentrically hypertrophied ventricles (15, 24), nonischemic decrease in myocardial function (12), and the increased oxygen demand due to greater deformation occurring at the endocardium (1). Whereas the significant cellular and biochemical issues underlying cardiac function have been extensively investigated, comparatively less attention has been directed toward analysis of the purely physics-based factors that may play a role (3, 11, 18, 21, 29). Because deformation or internal rearrangement of the myocardium, whether achieved by slippage between layers or by cellular distortion, occurs in a viscous medium, the effects of viscosity are a possible factor in determining chamber geometry and behavior. This study is an examination of the magnitude and patterns of predicted viscous effects in a thick-walled chamber such as the left ventricle. These effects are 1) stresses that resist contraction (viscous drag) and 2) extra energy consumed to overcome these stresses (viscous energy losses).

Glossary

	Proportionality constant between lumen radius r and intermediate radius
A	Chemical affinities
	Proportion of total wall volume inside intermediate radius
C	Circumference at intermediate radius
ij	Kroneker delta
	Viscosity coefficient
h	Wall thickness
	Intermediate radius within the myocardium
P	Hydrostatic pressure
r	Radius of the lumen (endocardial radius)
R	Radius at epicardium
[S]	Entropy production
T	Temperature
vc	Circumferential components of velocity
vi	Components of velocity
v	Radial components of velocity
Vw	Wall (myocardial) volume
w	Reaction rates
Wj	Heat flow
ij	Components of the stress tensor
x	Distance along intermediate radius

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

Geometric model. The model selected for analysis was a homogeneous, thick-walled spherical shell. This model was chosen to elucidate the patterns of viscous effects that are due only to bulk geometry and the fundamental laws of hydrodynamics. Whereas the actual left ventricle of the heart has a much greater complexity than that of the model, the ventricle must still, in the mean, behave within the boundaries established by the fundamental physics of this model. More specifically, whereas patterns of fibrous structure, layering, and particular cytoskeletal structure within the wall may change the absolute level of viscous effects, the overall pattern of stress and energy losses across the thickness of the wall will be determined by the constraints of geometry and hydrodynamics.

Stresses within a thick-walled shell. There will be components of stress (force per unit area) due to viscosity stemming from any gradient in velocities of contraction between adjacent elements in a mass (shearing). These stresses are expressed counter to the direction of motion of the mass element. The stress components are proportional to the first spatial derivatives of the velocities taken in directions normal to the velocities themselves (10)

(1)

Here P is hydrostatic pressure and _ij is the Kroneker delta, which equals unity when i = j; it is zero otherwise. Note that the double subscript, as used here, does not indicate summation but signifies the row and column of an overall stress tensor. Eq. 1 can be split into diagonal (_ij = 1) and off-diagonal (_ij = 0) terms

(2)

Note that the sign convention used here requires that , the viscosity coefficient, be positive so that the viscous stress component will be counter to the direction of motion. Most importantly, note also that Eq. 2, right, as well as Eq. 2, left, second term, represents stress components that occur independently of external loads or internal restoring forces. They occur with all contractions, even those generating no transmural pressure.

Viscous energy losses. In any system, some portion of the total energy flux is lost as heat (dissipated) in entropy production. The equation for total net energy losses, [S], in a nonequilibrium system such as myocardium is (10, 14)

(3)

In this equation, W_j is heat flow, T is temperature, _ij is the components of the stress tensor, v_i is the components of velocity, A is the affinities of the chemical reactions, and w is the reaction rates. The first term represents entropy produced due to temperature gradients and heat flows. The second term is the contribution due to chemical affinity, a function of changes in chemical potential, and chemical flow. The third term represents the entropy increase due to velocity gradients and stress. This last term is the viscous energy loss addressed in the present study.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

General framework of analysis. The first part of RESULTS addresses the patterns of motion that are dictated by the geometry of contraction of a thick shell with constant wall volume. The analysis in the section below shows that the constancy of wall volume, combined with shell geometry, requires that inner and outer regions must behave in fundamentally differing ways for contraction to take place. Compared with the outer regions, the inner regions of the wall are subject to much greater rearrangements in both the radial and circumferential directions during contraction. Accordingly, the spatial rates of change (gradients) in displacement and velocity are also much greater toward the inner surface of the shell.

Patterns of stress and dissipation across the wall. Viscous stresses in a deforming mass are determined by the gradients in motion of adjacent mass elements. To determine what these gradients are in the present model, one can define an intermediate radius , within the myocardium, that represents the position of a particular physical fiber throughout contraction of a thick-walled spherical shell (see Fig. 1). This fiber will be positioned at a changing radius within which is enclosed a constant proportion  of total wall volume, V_w. The radius of the lumen itself, r, is related to , the intermediate radius, by a variable factor, , that depends on r itself and the proportion .

View larger version (18K): [in this window] [in a new window]   Fig. 1.   Cross-sectional diagram of the spherical shell model used for analysis. R and r, radii of the outer and inner surfaces, respectively; , intermediate radius that encloses a constant proportion of the total volume of the shell.

(4)

(5)

(6)

(7)

(8)

View larger version (20K): [in this window] [in a new window]   Fig. 2.   A: plots of the relative magnitudes of radial (rad) stress due to viscous drag at different radii and at different proportional wall thicknesses typical of end diastole (ED), end systole (ES), and end systole in a hypertrophic (hypert) ventricle. The endocardium is defined as being at a radius of 1, and the magnitude at the epicardium is assumed to remain constant. B: plots of circumferential (cir) stress due to viscous drag.

View larger version (19K): [in this window] [in a new window]   Fig. 3.   A: plots of the relative magnitudes of dissipation of energy resulting from radial stress due to viscous drag at different radii and at different proportional wall thicknesses typical of end diastole, end systole, and end systole in a hypertrophic ventricle. The endocardium is defined as being at a radius of 1, and the magnitude at the epicardium is assumed to remain constant. B: plots of dissipation of energy resulting from circumferential stress due to viscous drag.

Are energy losses large enough to be significant? Whereas the stress and entropy equations have terms that behave qualitatively in ways that would explain observed limits to myocardial thickening, it is not established whether these quantities are of sufficient magnitude to be significant in cardiac function and pathology. To determine a possible answer to this question, the present study investigated possible sizes of energy losses rather than the more difficult estimation of local stress values.

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View larger version (12K): [in this window] [in a new window]   Fig. 4.   Lumped model used by Chiu et al. (6, 7) for modeling viscosity in myocardium. CE, contractile element; PE, parallel elastic element; SE, series elastic element; V, viscous elements.

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	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX REFERENCES

The preceding analysis demonstrates that the interaction of myocardial geometry and viscosity is a qualitatively and quantitatively significant factor in constraining the maximum relative thickness of the left ventricular myocardium. This interaction is thus strongly implicated in the limitations observed in the extent and persistence of hypertrophy.

The constraint imposed by viscosity on ventricular geometry is the opposite of the minimum thickness constraint imposed by Laplace's law. The overall picture is one of a balance between load increases at higher relative distensions (eccentric hypertrophy) due to the Laplace law and load increases at lower relative distensions (concentric hypertrophy) due to viscosity. This pattern of variation results in a minimum total load at some particular ratio of wall thickness to lumen radius. Such a minimum indicates a mechanism for the observed small variation in end-systolic relative wall thickness over large ranges of ventricular mass, both within and between mammalian species (5, 19). Systolic thickening beyond a certain point will result in a cost, in total stress and energy dissipation, that will exceed the benefit to be gained in cardiac output or mechanical advantage against pressure. Thus there would be no net advantage to be gained by maintenance of concentric hypertrophy after load is normalized, such as after pregnancy or after valve repair.

The nonisotropic structure of the actual myocardium could change the quantitative levels of stress and energy loss but not the overall epicardial-endocardial pattern set by physics and demonstrated in RESULTS. It is certainly reasonable to assume that some aspects of the microstructure of the myocardium have reduction of viscosity as their function. Due to the extreme steepness of the increase in viscous effects with thickening, it is clear that only relatively small, overall increments in thickening could be achieved without large increases in stress and energy loss.

The extent of similarity of mammalian relative wall thicknesses at systole may be further determined, not only by the existence of a minimum load as indicated in RESULTS, but also by the steepness of the increase in viscous stress and energy loss with further contraction. The eighth-power dependence contained in Eq. 7 presents a sharply defined barrier to further contraction, which will occur within a tight range of the relative extent of contraction. Due to the steepness of the function, the variations in velocity, cytoskeletal mechanics, and viscosity that undoubtedly occur between species would have minimal effects on the particular proportional extent of contraction that marks the limits of what is physiologically attainable. It would be interesting to investigate whether significant variations in overall geometry, such as the elongated geometry in the ferret heart, are accompanied by significant variations in the thickness-to-radius ratio.

The widely observed dysfunction of the endocardial layers during tachycardia in hypertrophy (12, 15, 24) is completely consistent with the predictions of the currently presented analysis. Indeed, the marked endocardial-epicardial difference in energy dissipation would suggest that dysfunction may occur without any other underlying pathologies, such as impeded blood flow (15, 24) or Ca²⁺-ATPase (12) levels. Much more oxygen must be consumed at the endocardium than at the epicardium to overcome local viscous resistance to contraction. The selective ischemia (15, 24) and the greater oxygen demand (1) observed at the endocardium agree with this physics-based pattern of energy demand.

The predictions of the present study agree with the observations of Young et al. (29). Their results, obtained using magnetic resonance tagging, suggest that greater mechanical work is expended in wall shearing (as opposed to lumen reduction) in hypertrophic than in normal hearts. Additionally, these studies show that, as approximated by the present circumferentially homogenous model, radial strain is maximal and total strain along the fibers is minimal (i.e., little change in angular position is observed).

The increase in the endocardial shear with contraction (as shown graphically in Figs. 2 and 3) may indeed result in the attainment of a point of maximum load beyond which sarcomeres cannot proceed. As successively greater thicknesses of the endocardial myocardium reach this point of limiting viscous load from shear, more and more of the lumen pressure load must be born by the remaining outer layers of muscle. Indeed, this is the pattern observed by Aoyagi et al. (1) with increasing dobutamine administration. After a certain point, only the velocity of the contraction increases and not the extent of the contraction, a result the investigators attribute to the load imposed by increased distortion necessary at greater extents of contraction. As demonstrated here, such increased distortion would result in increased viscous resistance to contraction and increased energy lost to overcoming viscosity.

The results of LeGrice et al. (17) with a tachycardia model are also in agreement with the present results. Whereas their work does not address concentric hypertrophy, it does demonstrate the mismatch between oxygen supply and demand that occurs at the endocardium and not the epicardium. Considering the epicardium-to-endocardium gradient in predicted energy dissipation even at normal systole, as shown in Fig. 3, it is reasonable to expect that the extra stress of tachycardia would affect the endocardium much sooner than the epicardium. The greater overall velocities of contraction would increase the energy dissipation in all layers, but the endocardial layers would be more likely to exceed physiological limits because these layers are already functioning at a greater level of viscous loss.

The present results apparently contradict those of Izzi et al. (16), who were not able to find differences in efficiency between normal and hypertrophic hearts. If more energy were wasted overcoming viscosity in hypertrophic hearts, it would be expected that these hearts would display lower energetic efficiency. However, the study contained relatively large standard deviations in the results, such that an existing difference in efficiencies may not have been detected. Calculation of the statistical power (8) of the t-test that was used to compare regression slopes gives a result of 0.10 (1-tailed test,  = 0.05). This means that there was only a 10% probability that a real difference of the size observed would indeed have produced a statistically significant result in a given experiment. In fact, a real difference as large as 40% between the efficiencies of the two groups of hearts would still have had a 20% probability of producing a statistically nonsignificant t-value in a given experiment. Additionally, the hearts were studied under isovolumic conditions; therefore, not all viscous properties would have been observable. Considering these factors, a real and substantial difference might have existed between the normal and hypertrophic hearts in this study in agreement with the present analysis.

Limitations of the present study. The geometry of the left ventricle is much more closely approximated by an ellipsoidal shell than a spherical shell. The choice was made to present the analysis in terms of a sphere due to the much simpler mathematics involved. The qualitative pattern of the results also applies to ellipsoids and even cylinders. If the analysis is carried out in terms of a cylinder, the inverse eighth- and sixth-power terms are replaced by inverse sixth- and fourth-power terms, respectively. An ellipsoidal ventricular geometry may be represented as something between these two models, implying that the dependence of viscous losses, while possibly not inverse eighth power, are at least on the order of inverse seventh power. Such a dependence would not change any of the implications of the present work.