A new cell-level finite element formulation is presented and used to investigate how epithelia and other planar collections of viscous cells might deform during events such as embryo morphogenesis and wound healing. Forces arising from cytoskeletal components, cytoplasm viscosity, and cell-cell adhesions are included. Individual cells are modeled using multiple finite elements, and cell rearrangements can occur. Simulations of cell-sheet stretching indicate that the initial stages of sheet stretching are characterized by changes in cell shape, while subsequent stages are governed by cell rearrangement. Inferences can be made from the simulations about the forces that act in real cell sheets when suitable experimental data are available. [S0148-0731(00)01404-7]

1.
Trinkaus, J. P., 1984, Cells into Organs: The Forces That Shape the Embryo, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ.
2.
Clausi
,
D. A.
, and
Brodland
,
G. W.
,
1993
, “
Mechanical Evaluation of Theories of Neurulation Using Computer Simulations
,”
Development
,
118
, pp.
1013
1023
.
3.
Moury
,
J. D.
, and
Schoenwolf
,
G. C.
,
1995
, “
Cooperative Model of Epithelial Shaping and Bending During Avian Neurulation: Autonomous Movements of the Neural Plate, Autonomous Movements of the Epidermis, and Interactions in the Neural Plate/Epidermis Transition Zone
,”
Develop. Dynam.
,
204
, pp.
323
337
.
4.
Kalnins
,
V. I.
,
Sandig
,
M.
,
Hergott
,
G. J.
, and
Nagai
,
H.
,
1995
, “
Microfilament Organization and Wound Repair in Retinal Pigment Epithelium
,”
Biochem. Cell Biol.
,
73
, pp.
709
722
.
5.
Honda
,
H.
,
Ogita
,
Y.
,
Higuchi
,
S.
, and
Kani
,
K.
,
1982
, “
Cell Movements in a Living Mammalian Tissue: Long-Term Observation of Individual Cells in Wounded Corneal Endothelia of Cats
,”
J. Morphol.
,
174
, pp.
25
39
.
6.
His, W., 1874, Unsere Ko¨rperform und das Physiologische Problem ihrer Enstehung, Briefe an einen Befreundeten Naturforscher, Leipzig, F. C. W. Vogel.
7.
Roux
,
W.
,
1888
, “
Beitra¨ge zur Entwickelungsmechanik Des Embryo. U¨ber die Ku¨nstliche Hervorbringung Halber Embryonen Durch Zersto¨rung einer der Beiden Ersten Furchungskugeln, Sowie U¨ber die Nachentwickelung (Post-Generation) der Fehlenden Ko¨rperha¨lfte
,”
Virchows Arch. Pathol. Anat. Physiol. Klin. Med.
,
114
, pp.
113
153
, 289–291.
8.
Lewis
,
W. H.
,
1947
, “
Mechanics of Invagination
,”
Anat. Rec.
,
97
, pp.
139
156
.
9.
Jacobson
,
A. G.
,
1978
, “
Some Forces That Shape the Nervous System
,”
Zoon—A J. Zoology
,
6
, pp.
13
21
.
10.
Lee
,
H.
, and
Nagele
,
R. G.
,
1988
, “
Intrinsic Forces Alone Are Sufficient to Cause Closure of the Neural Tube in the Chick
,”
Experientia
,
44
, pp.
60
61
.
11.
Schoenwolf
,
G. C.
, and
Smith
,
J. L.
,
1990
, “
Mechanics of Neurulation: Traditional Viewpoint and Recent Advances
,”
Development
,
109
, pp.
243
270
.
12.
Brodland
,
G. W.
, and
Clausi
,
D. A.
,
1995
, “
Cytoskeletal Mechanics of Neurulation: Insights Obtained From Computer Simulations
,”
Biochem. Cell Biol.
,
73
, Nos.
7 and 8
, pp.
545
553
.
13.
Brodland
,
G. W.
, and
Clausi
,
D. A.
,
1994
, “
Embryonic Tissue Morphogenesis Modeled by FEM
,”
ASME J. Biomech. Eng.
,
116
, pp.
146
155
.
14.
Davidson
,
L. A.
,
Koehl
,
M. A. R.
,
Keller
,
R.
, and
Oster
,
F.
,
1995
, “
How Do Sea Urchins Invaginate? Using Biomechanics to Distinguish Between Mechanisms of Primary Invagination
,”
Development
,
121
, pp.
2005
2018
.
15.
Brodland, G. W., 1997, “Computer Modelling of Morphogenesis: Recent Advances and Future Prospects,” (Keynote Address) the 16th Canadian Congress of Applied Mechanics (CANCAM), Universite´ Laval, Que´bec, Canada, June 1–5, 2, pp. 85–95.
16.
Keller
,
R. E.
,
1982
, “
Time-Lapse Cinemicrographic Analysis of Superficial Cell Behavior During and Prior to Gastrulation in Xenopus laevis
,”
J. Morphol.
,
157
, pp.
223
248
.
17.
Wilson
,
P. A.
,
Oster
,
G.
, and
Keller
,
R.
,
1989
, “
Cell Rearrangement and Segmentation in Xenopus: Direct Observation of Cultured Explants
,”
Development
,
105
, pp.
155
166
.
18.
Keller, R. E., Shih, J., and Wilson, P., 1991, “Cell Motility, Control and Function of Convergence and Extension During Gastrulation in Xenopus,” in: Gastrulation: Movements, Patterns and Molecules, R. E. Keller, H. Wallis, X. Clark, Jr., and F. Griffin, eds., Plenum Press, New York, pp. 101–119.
19.
Schoenwolf
,
G. C.
, and
Alvarez
,
I. S.
,
1989
, “
Roles of Neuroepithelial Cell Rearrangement and Division in Shaping of the Avian Neural Plate
,”
Development
,
106
, pp.
427
439
.
20.
Schoenwolf
,
G. C.
, and
Yuan
,
S.
,
1995
, “
Experimental Analyses of the Rearrangement of Ectodermal Cells During Gastrulation and Neurulation in Avian Embryos
,”
Cell Tissue Res.
,
280
, pp.
243
251
.
21.
Brodland, G. W., and Chen, H. H., 2000, “The Mechanics of Heterotypic Cell Aggregates: Insights From Computer Simulations,” ASME J. Biomech. Eng., accepted for publication.
22.
Burnside
,
M. B.
,
1973
, “
Microtubules and Microfilaments in Amphibian Neurulation
,”
Am. Zool.
,
13
, pp.
989
1006
.
23.
Gordon
,
R.
, and
Brodland
,
G. W.
,
1987
, “
The Cytoskeletal Mechanics of Brain Morphogenesis: Cell State Splitters and Primary Neural Induction
,”
Cell Biophys.
,
11
, pp.
177
237
.
24.
Gordon
,
S. R.
, and
Essner
,
E.
,
1987
, “
Investigations on Circumferential Microfilament Bundles in Rat Retinal Pigment Epithelium
,”
Eur. J. Cell Biol.
,
44
, pp.
97
104
.
25.
Nagele
,
R. G.
, and
Lee
,
H. Y.
,
1979
, “
Ultra-Structural Changes in Cells Associated With Inter-Kinetic Nuclear Migration in the Developing Chick Neuroepithelium
,”
J. Exp. Zool.
,
210
, pp.
89
106
.
26.
Schoenwolf
,
G. C.
,
Folsom
,
D.
, and
Moe
,
A.
,
1988
, “
A Reexamination of the Role of Microfilaments in Neurulation in the Chick Embryo
,”
Anat. Rec.
,
220
, pp.
87
102
.
27.
Solnica-Krezel
,
L.
, and
Driever
,
W.
,
1994
, “
Microtubule Arrays of the Zebra Fish Yolk Cell: Organization and Function During Epiboly
,”
Development
,
120
, pp.
2443
2455
.
28.
Shapiro
,
L.
,
Fannon
,
A. M.
,
Kwong
,
P. D.
,
Thompson
,
A.
,
Lehmann
,
M. S.
,
Grubel
,
G.
,
Legrand
,
J. F.
,
Als-Nielsen
,
J.
,
Colman
,
D. R.
, and
Hendrickson
,
W. A.
,
1995
, “
Structural Basis of Cell-Cell Adhesion by Cadherins
,”
Nature (London)
,
374
, pp.
327
337
.
29.
Gumbiner
,
B. M.
,
1996
, “
Cell Adhesion: the Molecular Basis of Tissue Architecture and Morphogenesis
,”
Cell
,
84
, pp.
345
357
.
30.
Brodland
,
G. W.
, and
Gordon
,
R.
,
1990
, “
Intermediate Filaments May Prevent Buckling of Compressively Loaded Microtubules
,”
ASME J. Biomech. Eng.
,
112
, pp.
319
321
.
31.
Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, M., and Watson, J., 1997, Molecular Biology of the Cell, 2nd ed., Garland Publishing, New York.
32.
Odell
,
G. M.
,
Oster
,
G.
,
Alberch
,
P.
, and
Burnside
,
B.
,
1981
, “
The Mechanical Basis of Morphogenesis. I. Epithelial Folding and Invagination
,”
Dev. Biol.
,
85
, pp.
446
462
.
33.
Hilfer
,
S. R.
, and
Hifler
,
E. S.
,
1983
, “
Computer Simulation of Organogenesis: An Approach to the Analysis of Shape Changes in Epithelial Organs
,”
Dev. Biol.
,
97
, pp.
444
453
.
34.
Honda
,
H.
, and
Yamanaka
,
H.
,
1986
, “
Transformation of a Polygonal Cellular Pattern During Sexual Maturation of the Avian Oviduct Epithelium: Computer Simulation
,”
J. Embryol. Exp. Morphol.
,
98
, pp.
1
19
.
35.
Jacobson
,
A. G.
,
Oster
,
G.
,
Odell
,
G. M.
, and
Cheng
,
L. Y.
,
1986
, “
Neurulation and the Cortical Tractoring Model for Epithelial Folding
,”
J. Embryol. Exp. Morphol.
,
96
, pp.
19
49
.
36.
Weliky
,
M.
, and
Oster
,
G.
,
1990
, “
The Mechanical Basis of Cell Rearrangement
,”
Development
,
109
, pp.
373
386
.
37.
Glazier
,
J. A.
, and
Graner
,
F.
,
1993
, “
Simulation of the Differential Adhesion Driven Rearrangement of Biological Cells
,”
Phys. Rev. E
,
47
, No.
3
, pp.
2122
2154
.
38.
Chen, H. H., and Brodland, G. W., 1997, “Finite Element Simulation of Differential Adhesion-Driven Cell Sorting and Spreading,” Proc. 16th Canadian Congress on Applied Mechanics CANCAM-97 Que´bec City, Canada, June 1–5, pp. 597–598.
39.
Foty
,
R. A.
,
Forgacs
,
G.
,
Pfleger
,
C. M.
, and
Steinberg
,
M. S.
,
1994
, “
Liquid Properties of Embryonic Tissue: Measurement of Interfacial Tensions
,”
Phys. Rev. Lett.
,
72
, No.
14
, pp.
2298
2301
.
40.
Steinberg
,
M. S.
,
1996
, “
Adhesion in Development: A Historical Overview
,”
Dev. Biol.
,
180
, pp.
377
388
.
41.
Ruoslahti
,
E.
, and
O¨brink
,
B.
,
1996
, “
Common Principles in Cell Adhesion
,”
Exp. Cell Res.
,
277
, pp.
1
11
.
42.
Davies, J. T., and Rideal, E. K., 1963, Interfacial Phenomena, 2nd ed., Academic Press, New York, pp. 1–52.
43.
Mow, V. C., Guilak, F., Tran-Son-Tay, R., and Hochmuth, R. M., eds., 1994, Cell Mechanics and Cellular Engineering, Springer-Verlag, New York.
44.
Galou
,
M.
,
Gao
,
J.
,
Humbert
,
J.
,
Mericskay
,
M.
,
Li
,
Z. L.
,
Paulin
,
D.
, and
Vicart
,
P.
,
1997
, “
The Importance of Intermediate Filaments in the Adaptation of Tissues to Mechanical Stress: Evidence From Gene Knockout Studies
,”
Biol. Cell
,
89
, pp.
85
97
.
45.
Zienkiewicz, O. C., and Taylor, R. L., 1989, The Finite Element Method, Vol. 2, 4th ed., McGraw-Hill, London.
46.
CRC, 1972, Standard Mathematical Tables, 20th ed., The Chemical Rubber Co., Cleveland, p. 369.
47.
Bowyer
,
A.
,
1981
, “
Computing Dirichlet Tessellations
,”
Comput. J. (UK)
,
24
, No.
2
, pp.
162
166
.
48.
Taber
,
L. A.
,
Lin
,
I.-E.
, and
Clark
,
E. B.
,
1995
, “
Mechanics of Cardiac Looping
,”
Develop. Dynam.
,
203
, pp.
42
50
.
49.
Dong
,
C.
, and
Skalak
,
R.
,
1992
, “
Leukocyte Deformability: Finite Element Modeling of Large Viscoelastic Deformation
,”
J. Theor. Biol.
,
158
, pp.
173
193
.
50.
Fung, Y. C., 1981, Biomechanics: Mechanical Properties of Living Tissues, Springer, New York.
51.
Malvern, L. E., 1969, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Inc., Englewood Cliffs, NJ.
52.
Bausch
,
A.
,
Moller
,
W.
, and
Sackmann
,
E.
,
1999
, “
Measurement of Local Viscoelasticity and Forces in Living Cells by Magnetic Tweezers
,”
Biophys. J.
,
76
, pp.
573
579
.
53.
Brodland
,
G. W.
,
Scott
,
M. J.
,
MacLean
,
A. F.
,
Globus
,
M.
,
Vethamany-Globus
,
S.
,
Gordon
,
R.
,
Veldhuis
,
J. H.
, and
Del Maestro
,
R.
,
1996
, “
Morphogenetic Movements During Axolotl Neural Tube Formation Tracked by Digital Imaging
,”
Roux’s Arch. Develop. Biol.
,
205
, pp.
311
318
.
54.
Bordzilovskaya, N., Dettlaff, T., Duhon, S., and Malacinski, G., 1989, “Developmental-Stage Series of Axolotl Embryos,” J. Armstrong and G. Malacinski, eds., Developmental Biology of the Axolotl, Oxford University Press, New York, pp. 201–219.
You do not currently have access to this content.