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.

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
. 1617-7959
8.
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
. 0303-6812
11.
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
. 0022-5096
17.
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
. 0045-7825
19.
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
. 0021-8936
22.
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
. 0084-6600
37.
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
. 0009-7330
41.
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 press
45.
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
. 0006-3495
47.
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
. 0022-5096
54.
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
.
You do not currently have access to this content.