Continuum Biomechanics of Soft Tissue: Successes and Challenges J. D. Humphrey Department of Biomedical Engineering and M.E DeBakey Institute Texas A&M University, College Station TX U.S.A.

Biomechanics - A Long History of Past Successes Leonardo ( ) Galileo ( ) W. Harvey ( ) R. Descartes ( ) G. Borelli ( ) L. Euler ( ) … and many others

Biomechanics comes of age ~ 1965 Renaissance in continuum mechanics ( ) Structural biology (1951, 1954) Digital computer (late 1950s - mid 1960s) Finite element methods (1956) Space race ( ) Note: L. Pauling (1995) attributes the birth of modern biology to the methods of theoretical physics / mathematics

Constitutive Relations - The Key to Success “we see that the greatest need lies in the direction of collecting data in multiaxial loading conditions and formulating a theory for the general rheological behavior of living tissues when stresses and strains vary with time in an arbitrary manner.” Y.C. Fung (1973)

An Example - Vascular Mechanics

Early Stress Analyses (~1979)

Importance of Residual Stress (~1986)

Importance of Smooth Muscle (~1999)

From Complexity Comes Simplicity Nonlinear Material Properties and Large Strain Anisotropy (circumferential muscle, axial collagen) Residual Stresses Smooth Muscle Activation Heterogeneity (functionally graded) Question – what optimization rules govern the development and adaptation of vessels?

Future Challenges & Promises “There are however innumerable other local motions which on account of the minuteness of the moving particles cannot be detected, such as the motions of the particles in hot bodies, fermenting bodies, in putrescent bodies, growing bodies, in the organs of sensation and so forth. If any one shall have the good fortune to discover all these, I might almost say that he will have laid bare the whole nature of bodies so far as the mechanical causes of things are concerned.” Sir I. Newton ( )

ECM Synthesis Cytokines MMP’s TIMP’s Cell Traction Receptor Binding Constitutive Relations Cell Cycle Cross-linking Growth Factors Integrins Mechanical Loads Balance Relations Cell Migration Modeling in Mechanobiology

Developmental Biomechanics “Here and elsewhere we shall not obtain the best insights into things until we actually see them growing from the beginning” Aristotle ( B.C.) “without the aid of mechanicians, and others skilled in simulation and modelling, developmental biology will remain a prisoner of our inadequate and conflicting physical intuitions and methaphors.” A.K Harris (1994) See: LA Taber (1995) Appl Mech Rev 48:

Growth & Remodeling Growth - an increase in mass that is achieved locally via an increase in the number (hyperplasia, migration) or size (hypertrophy) of cells and via a synthesis of extracellular matrix that exceeds removal. Remodeling - a change in structure that is achieved by reorganizing existing constituents (cross-links) or by producing new constituents having a different organization.

Regulators of Vascular G&R

Conceptually - G&R

Constrained mixture stress-response

Stretches for a constrained mixture

Kinetics of production and removal

Solid-Fluid Coupling - Aneurysms

Linearization – Stability in the Small

Elastodynamics of Aneurysms

Phase Plane – Time Plot

Denaturation of Collagen

Thermomechanical Testing of Collagenous Membranes

Heat-induced Changes in Mechanical Properties

In Summary, the Need… “The success of reductionist and molecular approaches in modern medical science has led to an explosion of information, but progress in integrating information has lagged… Mathematical models provide a rational approach for integrating this ocean of data, as well as providing deep insight into biological processes.” 1998 BECON Report

The Promise... Molecular & Cellular Biomechanics Developmental Biomechanics Growth & Remodeling Injury & Rehab Functional Tissue Engineering Muscle Mechanics Solid-Fluid Interactions Biothermomechanics

The Bioengineer’s Bookshelf Alberts et al. (2002) Molecular Biology of the Cell. Garland Publishers. Fawcett DW (1986) A Textbook of Histology. W.B. Saunders. Medical Dictionary (e.g., Dorland’s) and a good “system-specific” text on physiology, e.g., Milnor (1990) Cardiovascular Physiology, Oxford.

Acknowledgments Funding by the NIH, NSF, Texas-ATP, and Whitaker Foundation References Humphrey JD (2002) Cardiovascular Solid Mechanics: Cells, Tissues, and Organs, Springer-Verlag, NY Humphrey JD (2003) Continuum biomechanics of soft biological tissues. Proc R Soc Lond A 459: 3-46 Humphrey JD (2003) Continuum thermomechanics and the clinical treatment of disease and injury. Appl Mech Rev 56: Biomechanics and Modeling in Mechanobiology