The first objective of this review and re-evaluation is to present a brief history of efforts to mathematically model the growth of tissues. The second objective is to place this historical material in a current perspective where it may be of help in future research. The overall objective is to look backward in order to see ways forward. It is noted that two distinct methods of imaging or modeling the growth of an organism were inspired over 70 years ago by Thompson’s (1915, “XXVII Morphology and Mathematics,” Trans. - R. Soc. Edinbrgh, 50, pp. 857–895; 1942, On Growth and Form, Cambridge University Press, Cambridge, UK) method of coordinate transformations to study the growth and form of organisms. One is based on the solid mechanics concept of the deformation of an object, and the other is based on the fluid mechanics concept of the velocity field of a fluid. The solid mechanics model is called the distributed continuous growth (DCG) model by Skalak (1981, “Growth as a Finite Displacement Field,” Proceedings of the IUTAM Symposium on Finite Elasticity, D. E. Carlson and R. T. Shield, eds., Nijhoff, The Hague, pp. 348–355) and Skalak et al. (1982, “Analytical Description of Growth,” J. Theor. Biol., 94, pp. 555–577), and the fluid mechanics model is called the graphical growth velocity field representation (GVFR) by Cowin (2010, “Continuum Kinematical Modeling of Mass Increasing Biological Growth,” Int. J. Eng. Sci., 48, pp. 1137–1145). The GVFR is a minimum or simple model based only on the assumption that a velocity field may be used effectively to illustrate experimental results concerning the temporal evolution of the size and shape of the organism that reveals the centers of growth and growth gradients first described by Huxley (1924, “Constant Differential Growth-Ratios and Their Significance,” Nature (London), 114, pp. 895–896; 1972, Problems of Relative Growth, 2nd ed., L. MacVeagh, ed., Dover, New York). It is the method with an independent future that some earlier writers considered as an aspect of the DCG model. The development of the DCG hypothesis and the mixture theory models into models for the predicted growth of an organism is taking longer because these models are complicated and the development and refinement of the basic concepts are slower.

1.
Skalak
,
R.
, 1981, “
Growth as a Finite Displacement Field
,”
Proceedings of the IUTAM Symposium on Finite Elasticity
,
D. E.
Carlson
and
R. T.
Shield
, eds.,
Nijhoff
,
The Hague
, pp.
348
355
.
2.
Skalak
,
R.
,
DaGupta
,
G.
,
Moss
,
M.
,
Otten
,
E.
,
Dullemeijer
,
P.
, and
Vilmann
,
H.
, 1982, “
Analytical Description of Growth
,”
J. Theor. Biol.
0022-5193,
94
, pp.
555
577
.
3.
Medawar
,
P. B.
, 1941, “
The ‘Laws’ of Biological Growth
,”
Nature (London)
0028-0836,
148
, pp.
772
774
.
4.
Malthus
,
T.
, 1798,
An Essay on the Principle of Population
,
J. Johnson
,
London
.
5.
Gompertz
,
B.
, 1825, “
On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies
,”
Philos. Trans. R. Soc. London
0962-8428,
115
, pp.
513
583
.
6.
Verhulst
,
P. -F.
, 1838, “
Notice sur la loi que la population suit dans son accroissement
,”
Corr. Math. et Phys.
,
10
, pp.
113
121
.
7.
Verhulst
,
P. -F.
, 1845, “
Recherches mathématiques sur la loi d'accroissement de la population (Mathematical Researches Into the Law of Population Growth Increase)
,”
Nouveaux Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles
,
Belgian Royal Academy
,
Brussels
,
18
, pp.
1
42
.
8.
Verhulst
,
P. -F.
, 1847, “
Deuxième mémoire sur la loi d’accroissement de la population
,”
Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique
,
Belgian Royal Academy
,
Brussels
,
20
, pp.
1
32
.
9.
Winsor
,
C. P.
, 1932, “
The Gompertz Curve as a Growth Curve
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
18
, pp.
1
8
.
10.
Wright
,
S.
, 1926, “
Book Review of Pearl
,”
J. Am. Stat. Assoc.
0003-1291,
21
, pp.
493
497
.
11.
Pearl
,
R.
, 1925,
The Biology of Population Growth
,
Knopf
,
New York
.
12.
Pearl
,
R.
, and
Reed
,
L. J.
, 1920, “
On the Rate of Growth of the Population of the United States Since 1790 and Its Mathematical Representation
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
6
, pp.
275
288
.
13.
Davidson
,
F. A.
, 1928, “
Growth and Senescence in Purebred Jersey Cows
,”
Bull. Univ. Ill. Agric. Exp. Sta
,
302
(
21
), pp.
183
230
.
14.
Weymouth
,
F. W.
, and
Thompson
,
S. H.
, 1931, “
The Age and Growth of the Pacific Cockle (Cardium Corbis, Martyn)
,”
Bull. U.S. Bur. Fish.
,
46
, Bulletin No. 1101, pp.
633
641
.
15.
Weymouth
,
F. W.
,
McMillin
,
H. C.
, and
Rich
,
W. H.
, 1931, “
Latitude and Relative Growth in the Razor Clam, Siliqua patula
,”
J. Exp. Biol.
0022-0949,
8
, pp.
228
249
.
16.
Laird
,
A. K.
, 1964, “
Dynamics of Tumor Growth
,”
Br. J. Cancer
0007-0920,
18
, pp.
490
502
.
17.
Laird
,
A. K.
,
Tyler
,
S. A.
, and
Barton
,
A. D.
, 1965, “
Dynamics of Normal Growth
,”
Growth
0017-4793,
29
, pp.
233
248
.
18.
Ranferi Gutiérrez
,
M.
,
Reyes
,
M. A.
, and
Rosu
,
H. C.
, 2010, “
A Note on Verhulst’s Logistic Equation and Related Logistic Maps
,”
J. Phys. A: Math. Theor.
1751-8113,
43
, p.
205204
.
19.
Cowin
,
S. C.
, and
Hegedus
,
D. H.
, 1976, “
Bone Remodeling I: A Theory of Adaptive Elasticity
,”
J. Elast.
0374-3535,
6
, pp.
313
326
.
20.
Richards
,
F. J.
, 1959, “
A Flexible Growth Function for Empirical Use
,”
J. Exp. Bot.
0022-0957,
10
, pp.
290
301
.
21.
Zwanzig
,
R.
, 1973, “
Generalized Verhulst Laws for Population Growth
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
70
, pp.
3048
3051
.
22.
Minot
,
C. S.
, 1908,
The Problem of Age, Growth and Death
,
Putnam
,
New York
.
23.
Medawar
,
P. B.
, 1940, “
The Growth, Growth Energy, and Ageing of the Chicken’s Heart
,”
Proc. R. Soc. London, Ser. B
0962-8452,
129
, pp.
332
355
.
24.
Forys
,
U.
, and
Marciniak-Czochra
,
A.
, 2003, “
Logistic Equations in Tumor Growth Modelling
,”
Int. J. Appl. Math Comput. Sci.
0867-857X,
13
, pp.
317
325
.
25.
Minot
,
C. S.
, 1891, “
Senescence and Rejuvenation
,”
J. Physiol. (London)
0022-3751,
12
(
2
), pp.
97
153
.
26.
Medawar
,
P. B.
, 1952,
An Unsolved Problem of Biology
,
H.K. Lewis
,
London
.
27.
Stewart
,
S. A.
, and
German
,
R. Z.
, 1999, “
Sexual Dimorphism and Ontologenetic Allometry of Soft Tissues in Rattus norvegicus
,”
J. Morphol.
0362-2525,
242
, pp.
57
66
.
28.
Fiorello
,
C. V.
, and
German
,
R. Z.
, 1997, “
Heterochronies Within Species: Craniofacial Growth in Giant, Standard, and Dwarf Rabbits
,”
Evolution (Lawrence, Kans.)
0014-3820,
51
, pp.
250
261
.
29.
Thompson
,
D. W.
, 1942,
On Growth and Form
,
Cambridge University Press
,
Cambridge, UK
.
30.
Wang
,
S.
, and
Nagrath
,
D.
, 2011, “
Liver Tissue Engineering
,”
Biomaterials for Tissue Engineering Applications: A Review of the Past and Future Trends
,
J.
Burdick
and
R.
Mauck
, eds.,
Springer
,
New York
, in press.
31.
Levick
,
J. R.
, 1995,
An Introduction to Cardiovascular Physiology
, 2nd ed.,
Butterworth-Heinemann
,
Boston
.
32.
von Bertalanffy
,
L.
, 1950, “
The Theory of Open Systems in Physics and Biology
,”
Science
0036-8075,
111
, pp.
23
29
.
33.
Huxley
,
J. S.
, 1924, “
Constant Differential Growth-Ratios and Their Significance
,”
Nature (London)
0028-0836,
114
, pp.
895
896
.
34.
Huxley
,
J. S.
, 1972,
Problems of Relative Growth
, 2nd ed.,
L.
MacVeagh
, ed.,
Dover
,
New York
.
35.
Huxley
,
J. S.
, 1927, “
Further Work on Heterogonic Growth
,”
Biol. Zentralbl.
,
47
, pp.
151
163
.
36.
Huxley
,
J. S.
, 1931, “
Notes on Differential Growth
,”
Am. Nat.
0003-0147,
65
, pp.
289
315
.
37.
Huxley
,
J. S.
, and
Tessier
,
G.
, 1936, “
Terminology of Relative Growth
,”
Nature (London)
0028-0836,
137
, pp.
780
781
.
38.
Huxley
,
J. S.
,
Needham
,
J.
, and
Lerner
,
I. M.
, 1941, “
Terminology of Relative Growth-Rates
,”
Nature (London)
0028-0836,
148
, p.
225
.
39.
Kavanagh
,
A. J.
, and
Richards
,
O. W.
, 1942, “
Mathematical Analysis of the Relative Growth of Organisms
,”
Proc. Rochester Acad. Sci.
,
8
, pp.
150
174
.
40.
Waddington
,
C. H.
, 1950, “
The Biological Foundations of Measurements of Growth and Form
,”
Proc. R. Soc. London, Ser. B
0962-8452,
137
, pp.
509
515
.
41.
Gould
,
S. J.
, 1966, “
Allometry and Size in Ontogeny and Phylogeny
,”
Biol. Rev. Cambridge Philos. Soc.
0006-3231,
41
, pp.
587
638
.
42.
Blackstone
,
N. W.
, 1987, “
Allometry and Relative Growth: Pattern and Process in Evolutionary Studies
,”
Syst. Zool.
0039-7989,
36
, pp.
76
78
.
43.
Richards
,
O. W.
, and
Kavanagh
,
A. J.
, 1943, “
The Analysis of the Relative Growth Gradients and Changing Form of Growing Gradients Illustrated by the Tobacco Leaf
,”
Am. Nat.
0003-0147,
77
, p.
385
399
.
44.
Woodger
,
J. H.
, 1945, “
On Biological Transformations
,”
Essays in Growth and Form
,
W. E.
Le Gros Clark
and
P. B.
Medawar
, eds.,
Clarendon
,
Oxford
, pp.
92
120
.
45.
Richards
,
O. W.
, and
Riley
,
G. A.
, 1937, “
The Growth of Amphibian Larvae Illustrated by Transformed Coordinates
,”
J. Exp. Zool.
0022-104X,
77
, pp.
159
167
.
46.
Blount
,
R. F.
, 1935, “
Size Relationships as Influenced by Pituitary Rudiment Implantation and Extirpation in the Urodele Embryo
,”
J. Exp. Zool.
0022-104X,
70
, pp.
131
185
.
47.
Medawar
,
P. B.
, 1944, “
The Shape of the Human Being as a Function of Time
,”
Proc. R. Soc. London, Ser. B
0962-8452,
132
, pp.
133
141
.
48.
Silk
,
W. K.
, and
Erickson
,
R. O.
, 1979, “
Kinematics of Plant Growth
,”
Am. J. Bot.
0002-9122,
65
, pp.
481
501
.
49.
Truesdell
,
C. A.
, and
Toupin
,
R. A.
, 1960, “
The Classical Field Theories
,”
Handbuch der Physik
,
S.
Flugge
, ed.,
Springer-Verlag
,
Berlin
, Vol.
III/1
.
50.
Richards
,
O. W.
, and
Kavanagh
,
A. J.
, 1945, “
The Analysis of Growing Form
,”
Essays in Growth and Form
,
W. E.
Le Gros Clark
and
P. B.
Medawar
, eds.,
Clarendon
,
Oxford
, pp.
188
230
.
51.
Avery
,
G. S.
, 1933, “
Structure and Development of the Tobacco Leaf
,”
Am. J. Bot.
0002-9122,
20
, pp.
565
591
.
52.
Erickson
,
R. O.
, 1966, “
Relative Elemental Rates and Anisotropy of Growth in Area: A Computer Programme
,”
J. Exp. Bot.
0022-0957,
17
, pp.
390
403
.
53.
Erickson
,
R. O.
, 1976, “
Modeling of Plant Growth
,”
Annu. Rev. Plant Physiol.
0066-4294,
27
, pp.
407
434
.
54.
Cox
,
R. W.
, and
Peacock
,
M. A.
, 1977, “
The Fine Structure of Developing Elastic Cartilage
,”
J. Anat.
0021-8782,
123
, pp.
283
296
.
55.
Cox
,
R. W.
, and
Peacock
,
M. A.
, 1978, “
The Velocity Field of Growing Ear Cartilage
,”
J. Anat.
0021-8782,
126
, pp.
555
566
.
56.
Cox
,
R. W.
, and
Peacock
,
M. A.
, 1979, “
The Growth of Elastic Cartilage
,”
J. Anat.
0021-8782,
128
, pp.
207
213
.
58.
Thompson
,
D. W.
, 1915, “
XXVII Morphology and Mathematics
,”
Trans. - R. Soc. Edinbrgh
0080-4568,
50
, pp.
857
895
.
59.
Medawar
,
P. B.
, 1945, “
Size, Shape and Age
,”
Essays in Growth and Form
,
W. E.
Le Gros Clark
and
P. B.
Medawar
, eds.,
Clarendon
,
Oxford
, pp.
157
187
.
60.
Noll
,
W.
, 1959, “
The Foundations of Classical Mechanics in the Light of Recent Advances in Continuum Mechanics
,”
Proceedings of the International Symposium, University of California, Berkeley, 1958, Studies in Logic and the Foundations of Mathematics
,
North-Holland
,
Amsterdam
, pp.
266
281
.
61.
Cowin
,
S. C.
, 2010, “
Continuum Kinematical Modeling of Mass Increasing Biological Growth
,”
Int. J. Eng. Sci.
0020-7225,
48
, pp.
1137
1145
.
62.
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
.
63.
Tobler
,
W. R.
, 1963, “
D’Arcy Thompson and the Analysis of Growth and Form
,”
Papers of the Michigan Academy of Science, Arts, and Letters
,
48
, pp.
385
90
.
64.
Tobler
,
W. R.
, 1978, “
Comparison of Plane Forms
,”
Geogr. Anal.
0016-7363,
X.2
, pp.
154
172
.
65.
Tobler
,
W. R.
, 1994, “
Bidimensional Regression
,”
Geogr. Anal.
0016-7363,
26
, pp.
186
212
.
66.
Bookstein
,
F. L.
, 1978,
The Measurement of Biological Shape and Shape Change
,
Springer
,
New York
.
67.
Truesdell
,
C. A.
, 1957, “
Sulle basi della termomeccania
,”
Rend. Accad. Naz. Lincei
0392-7881,
22
, pp.
33
38
.
68.
Bowen
,
R. M.
, 1976, “
Theory of Mixtures
,”
Mixtures and EM Field Theories
(
Continuum Physics
),
Academic Press
,
New York
, Vol.
III
, pp.
1
127
.
69.
Rajagopal
,
K. R.
, and
Tao
,
L.
, 1995,
Mechanics of Mixtures
,
World Scientific
,
Singapore
.
70.
Bowen
,
R. M.
, 1967, “
Toward a Thermodynamics and Mechanics of Mixtures
,”
Arch. Ration. Mech. Anal.
0003-9527,
24
, pp.
370
403
.
71.
Atkin
,
R. J.
, and
Craine
,
R. E.
, 1976, “
Continuum Theories of Mixtures: Basic Theory and Historical Development
,”
Q. J. Mech. Appl. Math.
0033-5614,
29
, pp.
209
244
.
72.
Atkin
,
R. J.
, and
Craine
,
R. E.
, 1976, “
Continuum Theories of Mixtures: Applications
,”
J. Inst. Math. Appl.
0020-2932,
17
, pp.
153
207
.
73.
de Boer
,
R.
, 1996, “
Highlights in the Historical Development of the Porous Media Theory: Toward a Consistent Macroscopic Theory
,”
Appl. Mech. Rev.
0003-6900,
49
, pp.
201
262
.
74.
de Boer
,
R.
, 2000,
Theory of Porous Media: Highlights in the Historical Development and Current State
,
Springer-Verlag
,
Berlin
.
75.
Bowen
,
R. M.
, 1980, “
Incompressible Porous Media Models by Use of the Theory of Mixtures
,”
Int. J. Eng. Sci.
0020-7225,
18
, pp.
1129
1148
.
76.
Bowen
,
R. M.
, 1982, “
Compressible Porous Media Models by Use of the Theory of Mixtures
,”
Int. J. Eng. Sci.
0020-7225,
20
, pp.
697
735
.
77.
Coleman
,
B. D.
, and
Noll
,
W.
, 1963, “
The Thermodynamics of Elastic Materials With Heat Conduction
,”
Arch. Ration. Mech. Anal.
0003-9527,
13
, pp.
167
178
.
78.
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
, pp.
407
430
.
79.
Humphrey
,
J. D.
, and
Rajagopal
,
K. R.
, 2003, “
A Constrained Mixture Model for Arterial Adaptations to a Sustained Step Change in Blood Flow
,”
Biomech. Model. Mechanobiol.
1617-7959,
2
, pp.
109
126
.
80.
Ateshian
,
G. A.
, 2007, “
On the Theory of Reactive Mixtures for Modeling Biological Growth
,”
Biomech. Model. Mechanobiol.
1617-7959,
6
, pp.
423
445
.
81.
Fick
,
A.
, 1855, “
Über Diffusion
,”
Ann. Phys.
0003-3804,
94
, pp.
59
86
.
82.
Stefan
,
J.
, 1871, “
Über das Gleichgewicht und die Bewegung, insbesondere die Diffusion von Gasmengen, Sitzgsber
,”
Akad. Wiss. Wein
,
63
, pp.
63
124
.
83.
Cowin
,
S. C.
, and
Cardoso
,
L.
, 2011, “
An Alternative Approach to Mixture Theory Based Poroelasticity—A Larger RVE
,”
Mech. Mater.
0167-6636, submitted.
84.
Biot
,
M. A.
, 1941, “
General Theory of Three-Dimensional Consolidation
,”
J. Appl. Phys.
0021-8979,
12
, pp.
155
164
.
85.
Biot
,
M. A.
, 1956, “
Theory of Propagation of Elastic Waves in a Fluid Saturated Porous Solid. II. Higher Frequency Range
,”
J. Acoust. Soc. Am.
0001-4966,
28
, pp.
179
191
.
86.
Biot
,
M. A.
, 1962, “
Generalized Theory of Acoustic Propagation in Porous Dissipative Media
,”
J. Acoust. Soc. Am.
0001-4966,
34
, pp.
1254
1264
.
87.
Cowin
,
S. C.
, 1985, “
The Relationship Between the Elasticity Tensor and the Fabric Tensor
,”
Mech. Mater.
0167-6636,
4
, pp.
137
147
.
88.
Cowin
,
S. C.
, 2004, “
Anisotropic Poroelasticity: Fabric Tensor Formulation
,”
Mech. Mater.
0167-6636,
36
, pp.
665
677
.
89.
Cowin
,
S. C.
, and
Cardoso
,
L.
, 2011, “
Fabric Dependence of Poroelastic Wave Propagation in Anisotropic Porous Media
,”
Biomech. Model. Mechanobiol.
1617-7959,
10
, pp.
39
65
.
90.
Cardoso
,
L.
, and
Cowin
,
S. C.
, “
Fabric Dependence of Quasi-Waves in Anisotropic Porous Media
,”
J. Acoust. Soc. Am.
0001-4966, in press.
91.
Jolicoeur
,
P.
, and
Mosimann
,
J. E.
, 1960, “
Size and Shape Variation in the Painted Turtle. A Principal Component Analysis
,”
Growth
0017-4793,
24
, pp.
339
354
.
93.
Hooker
,
P. J.
, 1965, “
Benjamin Gompertz 5 March 1779–14 July 1865
,”
J. Inst. Actuar.
,
91
, pp.
203
212
.
94.
Minot
,
C. S.
, 1892,
Human Embryology
,
Macmillan
,
New York
.
95.
Morse
,
E. S.
, 1920, “
Biographical Memoir of Charles Sedgwick Minot 1852–1914, National Academy of Sciences
,”
Biogr. Mem.
,
IX
, pp.
261
285
.
96.
Thompson
,
D. W.
, 1966,
On Growth and Form
, abridged edition,
Cambridge University Press
,
Cambridge, UK
.
97.
Richards
,
O. W.
, 1955, “
D’Arcy W. Thompson’s Mathematical Transformation and the Analysis of Growth
,”
Ann. N.Y. Acad. Sci. U.S.A
,
63
, pp.
456
473
.
98.
Desmond
,
A.
, 1994,
Huxley: The Devil’s Disciple
,
Michael Joseph
,
London
.
99.
Mitchison
,
N. A.
, 1990, “
Peter Brian Medawar: 28 February 1915-2 October 1987
,”
Biogr. Mem. Fellows R. Soc.
0080-4606,
35
, pp.
283
301
.
You do not currently have access to this content.