The kinematic aspects of biological growth models are reviewed by paying attention to the handful of crucial ideas on which modern treatments rest. Both surface and volumetric growth are considered. A critical appraisal is presented of the geometric and physical features of the models. Links are made to the mathematical treatment of growth and evolving interface phenomena in other physical problems. Computational issues are pointed out wherever appropriate.
Issue Section:
Review Articles
1.
Thompson
, D. A.
, 1961, On Growth and Form
, Cambridge University Press
, Cambridge, England
.2.
DiCarlo
, A.
, and Quiligotti
, S.
, 2002, “Growth and Balance
,” Mech. Res. Commun.
0093-6413, 29
, pp. 449
–456
.3.
Garikipati
, K.
, Arruda
, E. M.
, Grosh
, K.
, Narayanan
, H.
, and Calve
, S.
, 2004, “A Continuum Treatment of Growth in Biological Tissue: Mass Transport Coupled With Mechanics
,” J. Mech. Phys. Solids
0022-5096, 52
(7
), pp. 1595
–1625
.4.
Loret
, B.
, and Simoes
, F. M. F.
, 2005, “A Framework for Deformation, Generalized Diffusion, Mass Transfer and Growth in Multi-Species Multi-Phase Biological Tissues
,” Eur. J. Mech. A/Solids
0997-7538, 24
, pp. 757
–781
.5.
Ambrosi
, D.
, and Mollica
, F.
, 2004, “The Role of Stress in the Growth of a Multicell Spheroid
,” J. Math. Biol.
0303-6812, 48
, pp. 477
–499
.6.
Ambrosi
, D.
, and Guana
, F.
, 2007, “Stress-Modulated Growth
,” Math. Mech. Solids
1081-2865, 12
, pp. 319
–342
.7.
Ateshian
, G. A.
, 2007, “On the Theory of Reactive Mixtures to Model Biological Growth
,” Biomech. Model. Mechanobiol.
, 6
, pp. 423
–445
. 1617-79598.
Taber
, L. A.
, 1995, “Biomechanics of Growth, Remodelling and Morphogenesis
,” Appl. Mech. Rev.
0003-6900, 48
, pp. 487
–545
.9.
Skalak
, R.
, Dasgupta
, G.
, Moss
, M.
, Otten
, E.
, Dullemeijer
, P.
, and Vilmann
, H.
, 1982, “Analytical Description of Growth
,” J. Theor. Biol.
0022-5193, 94
, pp. 555
–577
.10.
Skalak
, R.
, Farrow
, D. A.
, and Hoger
, A.
, 1997, “Kinematics of Surface Growth
,” J. Math. Biol.
, 35
, pp. 869
–907
. 0303-681211.
Eringen
, A. C.
, and Ingram
, J. D.
, 1965, “A Continuum Theory of Chemically Reacting Media—I
,” Int. J. Eng. Sci.
0020-7225, 3
, pp. 197
–212
.12.
Cahn
, J. W.
, and Hilliard
, J. E.
, 1958, “Free Energy of a Nonuniform System-I: Interfacial Free Energy
,” J. Chem. Phys.
0021-9606, 28
(2
), pp. 258
–267
.13.
Sethian
, J. A.
, 1996, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science
, Cambridge University Press
, Cambridge, England
.14.
Osher
, S.
, and Fedkiw
, R.
, 2003, Level Set Methods and Dynamic Implicit Surfaces
, Springer
, New York
.15.
Rao
, V. S.
, Hughes
, T. J. R.
, and Garikipati
, K.
, 2000, “On Modelling Thermal Oxidation of Silicon II: Numerical Aspects
,” Int. J. Numer. Methods Eng.
0029-5981, 47
(1/3
), pp. 359
–378
.16.
Garikipati
, K.
, and Rao
, V. S.
, 2001, “Recent Advances in Models for Thermal Oxidation of Silicon
,” J. Comput. Phys.
, 174
, 138
–170
. 0022-509617.
Mourad
, H. M.
, Dolbow
, J.
, and Garikipati
, K.
, 2005, “An Assumed-Gradient Finite Element Method for the Level Set Equation
,” Int. J. Numer. Methods Eng.
0029-5981, 64
, pp. 1009
–1032
.18.
Mourad
, H. M.
, and Garikipati
, K.
, 2006, “Advances in the Numerical Treatment of Grain Boundary Migration: Coupling With Mass Transport and Mechanics
,” Comput. Methods Appl. Mech. Eng.
, 196
, pp. 595
–607
. 0045-782519.
Hsu
, F. -H.
, 1968, “The Influence of Mechanical Loads on the Form of a Growing Elastic Body
,” J. Biomech.
0021-9290, 1
, pp. 303
–311
.20.
Rodriguez
, E. K.
, Hoger
, A.
, and McCulloch
, A. D.
, 1994, “Stress-Dependent Finite Growth in Soft Elastic Tissues
,” J. Biomech.
0021-9290, 27
, pp. 455
–467
.21.
Lee
, E. H.
, 1969, “Elastic-Plastic Deformation at Finite Strains
,” ASME J. Appl. Mech.
, 36
, pp. 1
–6
. 0021-893622.
Humphrey
, J. D.
, 1999, “Remodelling of Collagenous Tissue at Fixed Lengths
,” ASME J. Biomech. Eng.
0148-0731, 121
, pp. 591
–597
.23.
Taber
, L. A.
, and Humphrey
, J. D.
, 2001, “Stress-Modulated Growth, Residual Stress and Vascular Heterogeneity
,” ASME J. Biomech. Eng.
0148-0731, 123
, pp. 528
–535
.24.
Epstein
, M.
, and Maugin
, G. A.
, 2000, “Thermomechanics of Volumetric Growth in Uniform Bodies
,” Int. J. Plast.
0749-6419, 16
, pp. 951
–978
.25.
Chen
, Y. -C.
, and Hoger
, A.
, 2000, “Constitutive Functions of Elastic Materials in Finite Growth and Deformation
,” J. Elast.
0374-3535, 59
, pp. 175
–193
.26.
Kuhl
, E.
, and Steinmann
, P.
, 2003, “Theory and Numerics of Geometrically-Nonlinear Open System Mechanics
,” Int. J. Numer. Methods Eng.
0029-5981, 58
, 1593
–1615
.27.
Ambrosi
, D.
, and Mollica
, F.
, 2002, “On the Mechanics of a Growing Tumor
,” Int. J. Eng. Sci.
0020-7225, 40
, pp. 1297
–1316
.28.
Klisch
, S. M.
, Shen
, S. S.
, and Suh
, R. L.
, 2003, “A Growth Mixture Theory for Cartilage With Application to Growth-Related Experiments on Cartilage Explants
,” ASME J. Biomech. Eng.
0148-0731, 125
, pp. 169
–179
.29.
Skalak
, R.
, Zargaryan
, S.
, Jain
, R. K.
, Netti
, P. A.
, and Hoger
, A.
, 1996, “The Genesis of Residual Stress by Growth
,” J. Math. Biol.
0303-6812, 34
, pp. 889
–914
.30.
Love
, A. E. H.
, 1927, A Treatise on the Mathematical Theory of Elasticity
, Cambridge University Press
, Cambridge, England
.31.
Gurtin
, M. E.
, 1972, “The Linear Theory of Elasticity
,” Mechanics of Solids
, Vol. 2
, C.
Truesdell
, ed., Springer-Verlag
, Berlin
, pp. 1
–295
.32.
Hirth
, J. P.
, and Lothe
, J.
, 1992, Theory of Dislocations
, Krieger
, Malabar, FL
.33.
Blume
, J. A.
, 1989, “Compatibility Conditions for a Left Cauchy–Green Strain Field
,” J. Elast.
0374-3535, 21
, pp. 271
–308
.34.
Kondo
, K.
, 1955, “Geometry of Deformations and Stresses
,” Unifying Study of the Basic Problems in Engineering Sciences by Means of Geometry
, I. K.
Kondo
, ed., Gakujustu Bunken Fukyu-Kai
, pp. 1
–17
.35.
Bilby
, B. A.
, Bullough
, R.
, and Smith
, E.
, 1955, “Continuous Distributions of Dislocations—A New Application of the Methods of Non-Riemannian Geometry
,” Proc. R. Soc. London, Ser. A
1364-5021, 231
, pp. 263
–273
.36.
Kröner
, E.
, and Anthony
, K. H.
, 1975, “Dislocations and Disclinations in Material Structures—Basic Topological Concepts
,” Annu. Rev. Mater. Sci.
, 5
, pp. 43
–72
. 0084-660037.
Choung
, C. J.
, and Fung
, Y. C.
, 1986, “Residual Stress in Arteries
,” Frontiers in Biomechanics
, G. W.
Schmid-Schoenbein
, S. L.
Woo
, and B. W.
Zweifach
, eds., Springer
, New York
, pp. 117
–129
.38.
Liu
, S. Q.
, and Fung
, Y. C.
, 1988, “Zero-Stress States of Arteries
,” ASME J. Biomech. Eng.
0148-0731, 110
, pp. 82
–84
.39.
Xie
, J. P.
, Liu
, S. Q.
, Yang
, R. F.
, and Fung
, Y. C.
, 1991, “The Zero-Stress State of Rat Veins and Vena Cava
,” ASME J. Biomech. Eng.
0148-0731, 113
, pp. 36
–41
.40.
Omens
, J. H.
, and Fung
, Y. C.
, 1990, “Residual Strain in Rat Left Ventricle
,” Circ. Res.
, 66
, pp. 37
–45
. 0009-733041.
Han
, H. C.
, and Fung
, Y. C.
, 1991, “Residual Strains in Porcine and Canine Trachea
,” J. Biomech.
0021-9290, 24
, pp. 307
–315
.42.
Humphrey
, J. D.
, and Rajagopal
, K. R.
, 2002, “A Constrained Mixture Model for Growth and Remodeling of Soft Tissues
,” Math. Models Meth. Appl. Sci.
0218-2025, 12
(3
), pp. 407
–430
.43.
Truesdell
, C.
, and Noll
, W.
, 1965, “The Non-linear Field Theories
,” Handbuch der Physik, band III
, Springer
, Berlin
.44.
Narayanan
, H.
, Arruda
, E. M.
, Grosh
, K.
, and Garikipati
, K.
, 2008, “The Micromechanics of Fluid-Solid Interactions During Growth in Porous Soft Biological Tissue
,” Biomech. Model. Mechanobiol.
, in press45.
Gleason
, R. L.
, and Humphrey
, J. D.
, 2005, “A 2D Constrained Mixture Model for Arterial Adaptations to Large Changes in Flow, Pressure and Axial Stretch
,” Mathematical Medicine and Biology-A Journal of the IMA
, 22
(4
), pp. 347
–369
.46.
Lanir
, Y.
, 1978, “Structure-Strength Relations in Mammalian Tendon
,” Biophys. J.
, 24
, pp. 541
–554
. 0006-349547.
Fung
, Y. C.
, 1993, Biomechanics: Mechanical Properties of Living Tissues
, Springer-Verlag
, Berlin
.48.
Provenzano
, P.
, Lakes
, R.
, Keenan
, T.
, and Vanderby
, R.
, 2001, “Nonlinear Ligament Viscoelasticity
,” Ann. Biomed. Eng.
0090-6964, 29
, pp. 908
–914
.49.
DiSilvestro
, M. R.
, Zhu
, Q. L.
, Wong
, M.
, Jurvelin
, J. S.
, and Suh
, J. K. F.
, 2001, “Biphasic Poroviscoelastic Simulation of the Unconfined Compression of Articular Cartilage: I-Simultaneous Prediction of Reaction Force and Lateral Displacement
,” ASME J. Biomech. Eng.
0148-0731, 123
(2
), pp. 191
–197
.50.
Bischoff
, J. E.
, Arruda
, E. M.
, and Grosh
, K.
, 2002, “Orthotropic Elasticity in Terms of an Arbitrary Molecular Chain Model
,” ASME J. Appl. Mech.
0021-8936, 69
, pp. 198
–201
.51.
Holzapfel
, G. A.
, Glasser
, T. C.
, and Ogden
, R. W.
, 2004, “Comparison of a Multi-Layer Structural Model for Arterial Walls With a Fung-Type Model, and Issues of Material Stability
,” ASME J. Biomech. Eng.
0148-0731, 126
, pp. 264
–274
.52.
Cacho
, F.
, Elbischger
, P. J.
, Rodrigues
, J. F.
, Doblare
, M.
, and Holzapfel
, G. A.
, 2007, “A Constitutive Model for Fibrous Tissues Considering Collagen Fiber Crimp
,” Int. J. Non-Linear Mech.
0020-7462, 42
, pp. 391
–402
.53.
Garikipati
, K.
, Goektepe
, S.
, and Miehe
, C.
, 2008, “Elastica-Based Strain Energy Functions for Soft Biological Tissue
,” J. Mech. Phys. Solids
, 56
, pp. 1693
–1713
. 0022-509654.
Holzapfel
, G. A.
, and Ogden
, R. W.
, 2006, Mechanics of Biological Tissue
, Springer-Verlag
, Berlin
.55.
Guyton
, A.
, and Hall
, J.
, 1996, Textbook of Medical Physiology
, W.B. Saunders
, Philadelphia, PA
.56.
Alberts
, B.
, Bray
, D.
, Lewis
, J.
, Raff
, M.
, Roberts
, K.
, and Watson
, J. D.
, 2002, Molecular Biology of the Cell
, 4th ed., Garland
, New York
.57.
Taber
, L. A.
, and Eggers
, D. W.
, 1996, “Theoretical Study of Stress-Modulated Growth in the Aorta
,” J. Theor. Biol.
0022-5193, 180
, pp. 343
–357
.58.
Taber
, L. A.
, 1998, “Biomechanical Growth Laws for Muscle Tissue
,” J. Theor. Biol.
0022-5193, 193
, pp. 201
–213
.59.
Kroon
, M.
, and Holzapfel
, G. A.
, 2007, “A Model for Saccular Cerebral Aneurysm Growth by Collagen Fibre Remodelling
,” J. Theor. Biol.
0022-5193, 247
, pp. 775
–787
.60.
Ramasubramanian
, A.
, and Taber
, L.
, 2008, “Computational Modelling of Morphogenesis Regulated by Mechanical Feedback
,” Biomech. Model. Mechanobiol.
1617-7959, 7
(2
), pp. 77
–91
.Copyright © 2009
by American Society of Mechanical Engineers
You do not currently have access to this content.