Peterson D.R., Bronzino J.D. (Eds.) Biomechanics: Principles and Applications

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7-10 Biomechanics

300

200

100

Velocity (m/sec)

0.1 0.2 0.4 0.6 1 2 4 6 10

Mass (g)

20 40 60 100 200 400 1000

58 ft-lbs

Fracture

et al.

Lewis

Threshold liver

Skin V50

VC=1 m/sec

Far-field

masses

FIGURE 7.6 Tolerance levelsfor bluntloading as afunctionof impact massand velocity.The plotincludesinformation

from automotive impact situations and from high-speed military projectile impacts. The Lobdell model is effective over

the entire range of impact conditions. (Modiﬁed from Quatros J.H., Proceedings of the 14th International Symposium

on Ballistics, Quebec, Canada, September 26–29, 1993. With permission.)

TABLE 7.1 Human Tolerance for Chest and Abdomen Impact

Chest Abdomen

Criteria Frontal Lateral Frontal Lateral Criteria

Acceleration Acceleration

3 msec limit 60 g

TTI 85–90 g

ASA 30 g

AIS 4+ 45 g 39 g AIS 4+

Force Force

Sternum 3.3 kN

Chest +shoulder 8.8 kN 10.2 kN

AIS 3+ 2.9 kN 3.1 kN AIS 3+

AIS 4+ 5.5 kN 3.8 kN 6.7 kN AIS 4+

Pressure Pressure

187 kPa 166 kPa AIS 3+

216 kPa AIS 4+

Compression Compression

Rib fracture 20%

Stable ribcage 32% 38% AIS 3+

Flail chest 40% 38% 48% 44% AIS 4+

Viscous Viscous

AIS 3+ 1.0 m/sec AIS 3+

AIS 4+ 1.3 m/sec 1.47 m/sec 1.4 m/sec 1.98 m/sec AIS 4+

Source: (Adapted from Cavanaugh J.M., The Biomechanics of Thoracic Trauma, In

Accidental Injury: Biomechanics and Prevention, Nahum A.M. and Melvin J.W., (Eds.),

pp. 362–391, Springer-Verlag, New York, 1993 and Rouhana S.W., Biomechanics of Ab-

dominal Trauma, In Accidental Injury: Biomechanics and Prevention, Nahum A.M. and

Melvin J.W., (Eds.), pp. 391–428, Springer-Verlag, New York, 1993.)

Biomechanics of Chest and Abdomen Impact 7-11

0.8

0.6

0.4

0.2

0 0.5 1

(VC)

max

(m/sec)

1.5 2 2.5

Risk of severe injury (AIS 4+)

=1.08 m/sec

n=37 cadavers

FIGURE 7.7 Typical Logist injury probability function relating the risk of serious injury to the viscous response

of the chest. (From Viano D.C., Bull. NY Acad. Med., 2nd Series, 64: 376–421, 1988. With permission.)

there is injury. An additional factor is biomechanical response scaling for individuals of different size and

weight. The commonly accepted procedure involves equal stress and velocity, which enabled Mertz et al.

[1989] to predict injury tolerances and biomechanical responses for different size adult dummies.

Injury risk assessment is frequently used. It evaluates the probability of injury as a continuous function

of a biomechanical response. A Logist function relates injury probability p to a biomechanical response

x by p(x) = [1 + exp(α − βx)]

−1

where α and β are parameters derived from statistical analysis

of biomechanical data. This function provides a sigmoidal relationship with three distinct regions in

Figure 7.7. For low biomechanical response levels, there is a low probability of injury. Similarly, for very

high levels, the risk asymptotes to 100%. The transition region between the two extremes involves risk,

which is proportional to the biomechanical response. A sigmoidal function is typical of human tolerance

because it represents the distribution in weak through strong subjects in a population exposed to impact.

Table 7.2 summarizes available parameters for chest and abdominal injury risk assessment.

TABLE 7.2 Injury Probability Functions for Blunt Impact

Body Region ED

25%

αβX

Frontal Impact

Chest (AIS 4+)

VC 1.0 m/sec 11.42 11.56 25.6 0.000 0.68

C 34% 10.49 0.277 15.9 0.000 0.52

Lateral Impact

Chest (AIS 4+)

VC 1.5 m/sec 10.02 6.08 13.7 0.000 0.77

C 38% 31.22 0.79 13.5 0.000 0.76

Abdomen (AIS 4+)

VC 2.0 m/sec 8.64 3.81 6.1 0.013 0.60

C 47% 16.29 0.35 4.6 0.032 0.48

Pelvis (pubic ramus facture)

C 27% 84.02 3.07 11.5 0.001 0.91

Source: Modiﬁed from Viano et al. J. Biomech., 22: 403–417, 1989.

7-12 Biomechanics

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Kroell, C.K., Schneider, D.C., and Nahum, A.M., Impact Tolerance and Response to the Human Thorax II,

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Lau, I.V. and Viano, D.C., Inﬂuence of Impact Velocity on the Severity of Nonpenetrating Hepatic Injury,

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Lau, I.V. and Viano, D.C., The Viscous Criterion-Bases and Application of an Injury Severity Index for

Soft Tissue, Proceedings of the 30th Stapp Car Crash Conference, pp. 123–142, SAE Paper No. 861882,

Society of Automotive Engineers, Warrendale, PA, 1986.

Lau, I.V., Horsch, J.D. et al., Biomechanics of Liver Injury by Steering Wheel Loading, J. Trauma, 27:

225–237, 1987.

Lau, I.V. and Viano, D.C., How and When Blunt Injury Occurs: Implications to Frontal and Side Impact

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Lobdell, T.E., Kroell, C.K., Schneider, D.C., Hering, W.E., and Nahum, A.M., Impact Response of the

Human Thorax, In Human Impact Response Measurement and Simulation, King W.F. and Mertz H.J.

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Warrendale, PA, 1971.

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7-14 Biomechanics

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Cardiac Biomechanics

Andrew D. McCulloch

University of California-San Diego

8.1 Introduction ..........................................

8-1

8.2 Cardiac Geometry and Structure ......................8-1

Ventricular Geometry

•

Myoﬁber Architecture

•

Extracellular

Matrix Organization

8.3 Cardiac Pump Function ...............................

8-8

Ventricular Hemodynamics

•

Ventricular Pressure–Volume

Relations and Energetics

8.4 Myocardial Material Properties ........................8-12

Muscle Contractile Properties

•

Resting Myocardial Properties

8.5 Regional Ventricular Mechanics: Stress and Strain ......8-18

Acknowledgments ..........................................8-20

References ..................................................

8-20

8.1 Introduction

The primary function of the heart, to pump blood through the circulatory system, is fundamentally me-

chanical. In this chapter, cardiac function is discussed in the context of the mechanics of the ventricular

walls from the perspective of the determinants of myocardial stresses and strains (Table 8.1). Many physi-

ological, pathophysiological, and clinical factors are directly or indirectly affected by myocardial stress and

strain (Table 8.2). Of course, the factors in Table 8.1 and Table 8.2 are closely interrelated — most of the

factors affected by myocardial stress and strain in turn affect the stress and strain in the ventricular wall. For

example, changes in wall stress due to altered hemodynamic load may cause ventricular remodeling, which

in turn alters geometry, structure, and material properties. This chapter is organized around the governing

determinants in Table 8.1, but mention is made where appropriate to some of the factors in Table 8.2.

8.2 Cardiac Geometry and Structure

The mammalian heart consists of four pumping chambers, the left and right atria and ventricles commu-

nicating through the atrioventricular (mitral and tricuspid) valves, which are structurally connected by

chordae tendineae to papillary muscles that extend from the anterior and posterior aspects of the right

and left ventricular lumens. The muscular cardiac wall is perfused via the coronary vessels that originate

at the left and right coronary ostia located in the sinuses of Valsalva immediately distal to the aortic valve

leaﬂets. Surrounding the whole heart is the collagenous parietal pericardium that fuses with the diaphragm

and great vessels. These are the anatomical structures that are most commonly studied in the ﬁeld of

8-1

8-2 Biomechanics

TABLE 8.1 Basic Determinants of Myocardial Stress and Strain

Geometry and Structure

Three-dimensional shape Wall thickness

Curvature

Stress-free and unloaded reference conﬁgurations

Tissue structure Muscle ﬁber architecture

Connective tissue organization

Pericardium, epicardium, and endocardium

Coronary vascular anatomy

Boundary/Initial Conditions

Pressure Filling pressure (preload)

Arterial pressure (afterload)

Direct and indirect ventricular interactions

Thoracic and pericardial pressure

Constraints Effects of inspiration and expiration

Constraints due to the pericardium and its attachments

Valves and ﬁbrous valve annuli, chordae tendineae

Great vessels, lungs

Material Properties

Resting or passive Nonlinear ﬁnite elasticity

Quasilinear viscoelasticity

Anisotropy

Biphasic poroelasticity

Active dynamic Activation sequence

Myoﬁber isometric and isotonic contractile dynamics

Sarcomere length and length history

Cellular calcium kinetics and metabolic energy supply

TABLE 8.2 Factors Affected by Myocardial Stress and Strain

Direct factors Regional muscle work

Myocardial oxygen demand and energetics

Coronary blood ﬂow

Electrophysiological responses Action potential duration (QT interval)

Repolarization (T wave morphology)

Excitability

Risk of arrhythmia

Development and morphogenesis Growth rate

Cardiac looping and septation

Valve formation

Vulnerability to injury Ischemia

Arrhythmia

Cell dropout

Aneurysm rupture

Remodeling, repair, and adaptation Eccentric and concentric hypertrophy

Fibrosis

Scar formation

Progression of disease Transition from hypertrophy to failure

Ventricular dilation

Infarct expansion

Response to reperfusion

Aneurysm formation

Cardiac Biomechanics 8-3

TABLE 8.3 Representative Left Ventricular Minor-Axis Dimensions

Inner Outer Wall

Radius Radius Thickness:

Species Comments (mm) (mm) Inner Radius

Dog (21 kg) Unloaded diastole (0 mmHg) 16 26 0.62

Normal diastole (2–12 mmHg) 19 28 0.47

Dilated diastole (24–40 mmHg) 22 30 0.36

Normal systole (1–9 mmHg EDP) 14 26 0.86

Long axis, apex-equator (normal diastole) 42 47 0.12

Young rats Unloaded diastole (0 mmHg) 1.4 3.5 1.50

Mature rats Unloaded diastole (0 mmHg) 3.2 5.8 0.81

Human Normal 24 32 0.34

Compensated pressure overload 27 42 0.56

Compensated volume overload 32 42 0.33

Dog data from Ross et al., Circ. Res., 21, 409–421, 1967 [129], and Streeter and Hanna, Circ.

Res., 33, 639–655, 1973 [2]. Human data from Grossman, Am. J. Med., 69, 576–583, 1980 [130],

Grossman, et al., J. Clin. Invest., 56, 56–64, 1975 [131]. Rat data are from unpublished observations

in the author’s laboratory.

cardiac mechanics. Particular emphasis in this chapter is given to the ventricular walls, which are the most

important for the pumping function of the heart. Most studies of cardiac mechanics have focused on the

left ventricle, but many of the important conclusions apply equally to the right ventricle.

8.2.1 Ventricular Geometry

From the perspective of engineering mechanics, the ventricles are three-dimensional thick-walled pressure

vesselswith substantial variations in wall thicknessand principal curvatures both regionally and temporally

through the cardiac cycle. The ventricular walls in the normal heart are thickest at the equator and base

of the left ventricle and thinnest at the left ventricular apex and right ventricular free wall. There are also

variations in the principal dimensions of the left ventricle with species, age, phase of the cardiac cycle, and

disease (Table 8.3). But, in general, the ratio of wall thickness to radius is too high to be treated accurately

by all but the most sophisticated thick-wall shell theories [1].

Ventricular geometry has been studied in most quantitative detail in the dog heart [2,3]. Geometric

models have been very useful in the analysis, especially the use of confocal and nonconfocal ellipses

of revolution to describe the epicardial and endocardial surfaces of the left and right ventricular walls

(Figure 8.1). The canine left ventricle is reasonably modeled by a thick ellipsoid of revolution truncated at

the base. The crescentic right ventricle wraps about 180

◦

degrees around the heart wall circumferentially

and extends longitudinally about two-thirds of the distance from the base to the apex. Using a truncated

ellipsoidal model, left ventricular geometry in the dog can be deﬁned by the major and minor radii of two

surfaces, the left ventricular endocardium, and a surface deﬁning the free wall epicardium and the septal

endocardium of the right ventricle. Streeter and Hanna [2] described the position of the basal plane using

a truncation factor f

deﬁned as the ratio between the longitudinal distances from equator-to-base and

equator-to-apex. Hence, the overall longitudinal distance from base to apex is (1 + f

) times the major

radius of the ellipse. Since variations in f

between diastole and systole are relatively small (0.45 to 0.51),

they suggested a constant value of 0.5.

The focal length d of an ellipsoid is deﬁned from the major and minor radii (a and b)byd

= a

− b

andvaries onlyslightly inthe dog from endocardiumtoepicardiumbetween end-diastole (37.3 to 37.9mm)

and end-systole (37.7 to 37.1 mm) [2]. Hence, within the accuracy that the boundaries of the left ventricular

wall can be treated as ellipsoids of revolution, the assumption that the ellipsoids are confocal appears to

be a good one. This has motivated the choice of prolate spheroidal (elliptic-hyperbolic-polar) coordinates

(λ, μ, θ ) as a system for economically representing ventricular geometries obtained postmortem or by

8-4 Biomechanics

␪

␮

␭

FIGURE 8.1 Truncated ellipsoid representation of ventricular geometry, showing major left ventricular radius (a),

minor radius (b), focal length (d), and prolate spheroidal coordinates (λ, μ, θ).

noninvasive tomography [3,4]. The Cartesian coordinates of a point are given in terms of its prolate

spheroidal coordinates by

= d cosh λ cos μ

= d sinh λ sin μ cos θ (8.1)

= d sinh λ sin μ sin θ

Here, the focal length d deﬁnes a family of coordinate systems that vary from spherical polar when d = 0

to cylindrical polar in the limit when d →∞. A surface of constant transmural coordinate λ (Figure 8.1)

is an ellipse of revolution with major radius a = d cosh λ and minor radius b = d sinh λ. In an ellipsoidal

model with a truncation factor of 0.5, the longitudinal coordinate μ varies from zero at the apex to 120

◦

the base. Integrating the Jacobian in prolate spheroidal coordinates gives the volume of the wall or cavity:



2π



((sinh

λ +sin

μ) sinh λ sin μ)dλ dμ dθ

2πd



(1 −cos μ

) cosh

λ −



1 −cos



cosh λ



(8.2)

The scaling between heart mass M

and body mass M within or between species is commonly described

by the allometric formula,

= kM

(8.3)

Using combined measurements from a variety of mammalian species with M expressed in kilograms, the

coefﬁcient k is 5.8 g and the power α is close to unity (0.98) [5]. Within individual species, the ratio of

heart weight to body weight is somewhat lower in mature rabbits and rats (about 2 g/kg) than in humans

(5 g/kg) and higher in horses and dogs (8 g/kg) [6]. The rate α of heart growth with body weight decreases