Biomechanics Principles and Applications - Donald R. Peterson & Joseph D. Bronzino

7-10	Biomechanics

300

Velocity (m/sec)

200	Threshold		58		masses
			ft
			-		Far-field
			lbs
			liver
100	Lewis	Fracture
80
60	Skin		et	al.
		V50
40

VC=1 m/sec

20

10

0.1

0.2

0.4 0.6

1

2

4

6

10

20

40 60 100

200

400

1000

Mass (g)

FIGURE 7.6 Tolerance levels for blunt loading as a function of impact mass and velocity. The plot includes information 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+
AIS 4+			45 g				39 g
Force									Force
Sternum	3.3 kN
Chest + shoulder	8.8 kN		10.2 kN						AIS 3+
AIS 3+					2.9 kN		3.1 kN
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%								AIS 3+
Stable ribcage	32%				38%
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 Abdominal 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

Risk of severe injury (AIS 4+)

1

0.8

0.6							ED50=1.08 m/sec
0.4
							n=37 cadavers
0.2
0
						1.5		2		2.5
0		0.5		1

(VC)max (m/sec)

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 − β x)]−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	ED25%	α	β	X 2	p	R
		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

References

Bir, C. and Viano, D.C., Biomechanics of Commotio Cordis. J. Trauma, 47(3): 468–473, 1999.

Bir, C., Viano, D.C., and King, A.I., Human Response of the Thorax to Blunt Ballistic Impacts. J. Biomech., 37(1): 73–79, 2004.

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.

Cavanaugh, J.M. et al., Injury and Response of the Thorax in Side Impact Cadaveric Tests, Proceedings of the 37th Stapp Car Crash Conference, pp. 199–222, SAE Paper No. 933127, Society of Automotive Engineers, Warrendale, PA, 1993.

Eiband, A.M., Human Tolerance to Rapidly Applied Acceleration. A Survey of the Literature. National Aeronautics and Space Administration, Washington DC, NASA Memo No. 5-19-59E, 1959.

Foster, J.K., Kortge, J.O., and Wolanin, M.J., Hybrid III-A Biomechanically-Based Crash Test Dummy, Stapp Car Crash Conference, pp. 975–1014, SAE Paper No. 770938, Society of Automotive Engineers, Warrendale, PA, 1977.

Gadd, C.W. and Patrick, L.M., Systems Versus Laboratory Impact Tests for Estimating Injury Hazards, SAE Paper No. 680053, Society of Automotive Engineers, Warrendale, PA, 1968.

Jonsson, A., Clemedson, C.J. et al., Dynamic Factors Influencing the Production of Lung Injury in Rabbits Subjected to Blunt Chest Wall Impact, Aviation, Space Environ. Med., 50: 325–337, 1979.

King, A.I., Regional Tolerance to Impact Acceleration, In SP-622, SAE 850852, Society of Automotive Engineers, Warrendale, PA, 1985.

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

Proceedings of the 15th Stapp Car Crash Conference, pp. 84–134, SAE Paper No. 710851, Society of Automotive Engineers, Warrendale, PA, 1971.

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

Proceedings of the 18th Stapp Car Crash Conference, pp. 383–457, SAE Paper No. 741187, Society of Automotive Engineers, Warrendale, PA, 1974.

Lau, I.V. and Viano, D.C., Influence of Impact Velocity on the Severity of Nonpenetrating Hepatic Injury, J. Trauma, 21(2): 115–123, 1981.

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 Protection. Proceedings of the 32nd Stapp Car Crash Conference, pp. 81–100, SAE Paper No. 881714, Society of Automotive Engineers, Warrendale, PA, 1988.

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. (Eds.), Plenum Press, New York, pp. 201–245, 1973.

Melvin, J.W., King, A.I., and Alem, N.M., AATD System Technical Characteristics, Design Concepts, and Trauma Assessment Criteria, AATD task E-F Final Report, DOT-HS-807-224, US Department of Transportation, National Highway Traffic Safety Administration, Washington, DC, 1988.

Melvin, J.W. and Weber, K. (Eds.), Review of Biomechanical Response and Injury in the Automotive Environment, AATD Task B Final Report, DOT-HS-807-224, US Department of Transportation, National Highway Traffic Safety Administration, Washington, DC, 1988.

Mertz, H.J. and Gadd, C.W., Thoracic Tolerance to Whole-Body Deceleration, Proceedings of the 15th Stapp Car Crash Conference, pp. 135–157, SAE Paper No. 710852, Society of Automotive Engineers, Warrendale, PA, 1971.

Biomechanics of Chest and Abdomen Impact

7-13

Mertz, H.J., Irwin, A. et al., Size, Weight and Biomechanical Impact Response Requirements for Adult Size Small Female and Large Male Dummies, SAE Paper No. 890756, Society of Automotive Engineers, Warrendale, PA, 1989.

Mertz, H.J., Anthropomorphic Test Devices, In Accidental Injury: Biomechanics and Prevention, Nahum A.M. and Melvin J.W. (Eds.), pp. 66–84, Springer-Verlag, New York, 1993.

Morgan, R.M., Marcus, J.H., and Eppinger, R.H., Side Impact — The Biofidelity of NHTSA’s Proposed ATD and Efficacy of TTI, Proceedings of the 30th Stapp Car Crash Conference, pp. 27–40, SAE Paper No. 861877, Society of Automotive Engineers, Warrendale, PA, 1986.

Patrick, L.M., Kroell, C.K., and Mertz, H.J., Forces on the Human Body in Simulated Crashes, Proceedings of the 9th Stapp Car Crash Conference, SAE, pp. 237–260, Society of Automotive Engineers, Warrendale, PA, 1965.

Patrick, L.M., Mertz, H.J., and Kroell, C.K., Cadaver Knee, Chest, and Head Impact Loads, Proceedings of the 11th Stapp Car Crash Conference, pp. 168–182, SAE Paper No. 670913, Society of Automotive Engineers, Warrendale, PA, 1967.

Quatros, J.H., Terminal Ballistics of Non-lethal Projectiles, Proceedings of the 14th International Symposium on Ballistics, Quebec, Canada, September 26–29, 1993.

Rouhana, S.W. et al., Assessing Submarining and Abdominal Injury Risk in the Hybrid III Family of Dummies, Proceedings of the 33rd Stapp Car Crash Conference, pp. 257–279, SAE Paper No. 892440, Society of Automotive Engineers, Warrendale, PA, 1989.

Rouhana, S.W., Biomechanics of Abdominal Trauma, In Accidental Injury: Biomechanics and Prevention, Nahum A.M. and Melvin J.W. (Eds.), pp. 391–428, Springer-Verlag, New York, 1993.

Scherer, R.D., Kirkish, S.L., McCleary, J.P., Rouhana, S.W. et al., SIDS-IIs Beta\u+− prototype dummy biomechanical responses. SAE 983151, Proceedings of the 42nd Stapp Car Crash Conference, Society of Automotive Engineers, Warrendale, PA, 1998.

Schneider, L.W., Haffner, M.P. et al., Development of an Advanced ATD Thorax for Improved Injury Assessment in Frontal Crash Environments, Proceedings of the 36th Stapp Car Crash Conference, pp. 129–156, SAE Paper No. 922520, Society of Automotive Engineers, Warrendale, PA, 1992.

Shah, C.S., Yang, K.H., Hardy, W.N., Wang, H.K., and King, A.I., Development of a Computer Model to Predict Aortic Rupture due to Impact Loading. SAE 2001-22-0007, Society of Automotive Engineers, Warrendale, PA, Stapp Car Crash J., 45: 161–182, 2001.

Society of Automotive Engineers, Human Tolerance to Impact Conditions as Related to Motor Vehicle Design, SAE J885, Society of Automotive Engineers, Warrendale, PA, 1986.

Stapp, J.P., Voluntary Human Tolerance Levels, In Impact Injury and Crash Protection, Gurdjian, E.S., Lange, W.A., Patrick, L.M., and Thomas, L.M. (Eds.), pp. 308–349, Charles C Thomas, Springfield, IL, 1970.

Stein, P.D., Sabbah, H.N. et al., Response of the Heart to Nonpenetrating Cardiac Trauma. J. Trauma, 22(5): 364–373, 1982.

Sturdivan, L.M., Viano, D.C., and Champion, H., Analysis of Injury Criteria to Assess Chest and Abdominal Injury Risks in Blunt and Ballistic Impacts. J. Trauma, 56: 651–663, 2004.

Viano, D.C., King, A.I. et al., Injury Biomechanics Research: An Essential Element in the Prevention of Trauma, J. Biomech., 22: 403–417, 1989.

Viano, D.C. and Lau, I.V., A Viscous Tolerance Criterion for Soft Tissue Injury Assessment, J. Biomech., 21: 387–399, 1988.

Viano, D.C., Cause and Control of Automotive Trauma, Bull. NY Acad. Med., 2nd Series, 64: 376–421, 1988.

Viano, D.C., Biomechanical Responses and Injuries in Blunt Lateral Impact, Proceedings of the 33rd Stapp Car Crash Conference, pp. 113–142, SAE Paper No. 892432, Society of Automotive Engineers, Warrendale, PA, 1989.

7-14	Biomechanics

Viano, D.C., Evaluation of the Benefit of Energy-Absorbing Materials for Side Impact Protection,

Proceedings of the 31st Stapp Car Crash Conference, pp. 185–224, SAE Paper No. 872213, Society of Automotive Engineers, Warrendale, PA, 1987.

Viano, D.C., Andrzejak, D.V., Polley, T.Z., and King, A.I., Mechanism of Fatal Chest Injury by Baseball Impact: Development of an Experimental Model, Clin. J. Sport Med., 2: 166–171, 1992.

Viano, D.C. and Andrzejak, D.V., Biomechanics of Abdominal Injury by Armrest Loading. J. Trauma, 34(1): 105–115, 1993.

8

Cardiac Biomechanics

	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
Andrew D. McCulloch	Acknowledgments . . . . . . . . . . . . .		. . . . . . . . . . . . . . . . . . . . . . . . . . . . .	8-20
University of California-San Diego	References . . . . . . . . . . . . . . . . . . . . .		. . . . . . . . . . . . . . . . . . . . . . . . . . . . .	8-20

8.1 Introduction

The primary function of the heart, to pump blood through the circulatory system, is fundamentally mechanical. 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 physiological, 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 communicating 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 Dimensionsa
			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

aDog 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 vessels with substantial variations in wall thickness and 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 fb 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 + fb) times the major radius of the ellipse. Since variations in fb 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) by d2 = a2 − b2, and varies only slightly in the dog from endocardium to epicardium between end-diastole (37.3 to 37.9 mm) 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

X1

b X2

d

a

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

x1	= d cosh λ cos μ	(8.1)
x2	= d sinh λ sin μ cos θ
x3	= 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◦ at the base. Integrating the Jacobian in prolate spheroidal coordinates gives the volume of the wall or cavity:

		2π	μ2 λ2
d3	0	0	((sinh2 λ + sin2 μ) sinh λ sin μ)dλ dμ dθ
			λ1
=	2π d3		(1 − cos μ2) cosh3 λ −	1 − cos3 μ2 cosh λ	λ2	(8.2)
		3			λ1

The scaling between heart mass MH and body mass M within or between species is commonly described by the allometric formula,

MH = k Mα

(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