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Research Papers

The Specific Growth Rates of Tissues: A Review and a Re-Evaluation

[+] Author and Article Information
Stephen C. Cowin1

Department of Biomedical Engineering, The City College of New York, 138th Street and Convent Avenue, New York, NY 10031

1

Also at 2166 Broadway, Apartment 12D, New York, NY 10024.

J Biomech Eng 133(4), 041001 (Feb 11, 2011) (20 pages) doi:10.1115/1.4003341 History: Received November 29, 2010; Revised December 10, 2010; Posted January 03, 2011; Published February 11, 2011; Online February 11, 2011

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 (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.

Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 6

Thompson’s famous coordinate transformation of a porcupine fish into a sunfish, from Thompson (29)

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Figure 7

The human form at six stages from the fifth month of fetal life to maturity rescaled to be the same overall length, from Medawar (47)

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Figure 8

Stage IV of growth of a tobacco leaf from Richards and Kavanagh (43)

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Figure 9

Stages I–III of growth of a tobacco leaf from Richards and Kavanagh (43)

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Figure 10

All stages of growth of a tobacco leaf rescaled to be the same overall length from Richards and Kavanagh (43)

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Figure 11

Drawings of the selected cocklebur leaf on three successive days are shown with the rectangular array of reference points marked on the leaf surface, from Erickson (52)

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Figure 12

The contours of the constant values of the growth, representing the values of the divergence of the velocity field for the cocklebur leaf of Fig. 1, from Erickson (52)

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Figure 13

The field lines of the gradient of the specific growth rate and level lines of the specific growth rate for the growing right ear of a 15 day old rabbit, from Cox and Peacock (56)

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Figure 14

The field lines of the gradient of the specific growth rate and level lines of the specific growth rate for the growing right ear of a 49 day old rabbit, from Cox and Peacock (56)

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Figure 15

The field lines of the gradient of the specific growth rate and level lines of the specific growth rate for the growing right ear of an 87 day old rabbit, from Cox and Peacock (56)

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Figure 16

The development of the wing of Drosophila; nine images at successive stages of the growth and development of the Drosophila wing; these stages are numbered 1–9 from the upper right-hand image to the lower left-hand image, from Waddington (40)

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Figure 17

Tobler’s (64) representation of tobacco leaf growth. From left to right: young and older leaves shown superimposed with grid coordinates, homologous points, and displacements. Distorted grid and transformed outline. Homogeneously deformed ellipses showing the magnitude of growth in all directions at each point and the enlargement of area. Axes of ellipses showing the directions of principal growth. The length of the small leaf, which is not to scale, is 85 mm. The large leaf measures 300 mm.

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Figure 18

Tobler’s (64) representation of tobacco leaf growth illustrated on the young leaf. The principal axes and the contours of equation distortion are shown.

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Figure 19

Comparison of the GVFR of the older tobacco leaf data (right) with the prediction of the DCG hypothesis (left)

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Figure 20

(a) A RVE of the Biot (39) type. (b) The Eulerian point in mixture theory.

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Figure 4

A Gompertz plot of the weight of female and male rat livers as a function of time, from Stewart and German (27)

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Figure 5

The portal vein delivers enriched blood to the liver from the gastrointestinal tract and the spleen

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Figure 2

The Gompertz and logistic sigmoidal curves, modified from Winsor (9)

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Figure 3

Semilog plots of the specific growth rate (P−1)Ṗ for both the Gompertz and logistic equations, from Laird (17)

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