During morphogenesis, epithelia (cell sheets) undergo complex deformations as they stretch, bend, and twist to form the embryo. Often these changes in shape create multivalued surfaces that can be problematic for strain measurements. This paper presents a method for quantifying deformation of such surfaces. The method requires four-dimensional spatiotemporal coordinates of a finite number of surface markers, acquired using standard imaging techniques. From the coordinates of the markers, various deformation measures are computed as functions of time and space using straightforward matrix algebra. This method accommodates sparse randomly scattered marker arrays, with reasonable errors in marker locations. The accuracy of the method is examined for some sample problems with exact solutions. Then, the utility of the method is illustrated by using it to measure surface stretch ratios and shear in the looping heart and developing brain of the early chick embryo. In these examples, microspheres are tracked using optical coherence tomography. This technique provides a new tool that can be used in studies of the mechanics of morphogenesis.

1.
Davies
,
J. A.
, 2005,
Mechanisms of Morphogenesis: The Creation of Biological Form
,
Elsevier
,
San Diego
.
2.
Gilbert
,
S. F.
, 2003,
Developmental Biology
,
7th Ed.
,
Sinauer Associates
,
Sunderland, MA
.
3.
Zamir
,
E. A.
,
Czirok
,
A.
,
Rongish
,
B. J.
, and
Little
,
C. D.
, 2005, “
A Digital Image-Based Method for Computational Tissue Fate Mapping During Early Avian Morphogenesis
,”
Ann. Biomed. Eng.
0090-6964,
33
, pp.
854
865
.
4.
Zamir
,
E. A.
,
Czirok
,
A.
,
Cui
,
C.
,
Little
,
C. D.
, and
Rongish
,
B. J.
, 2006, “
Mesodermal Cell Displacements During Avian Gastrulation are Due to Both Individual Cell-Autonomous and Convective Tissue Movements
,”
Proc. Natl. Acad. Sci. U.S.A.
0027-8424,
103
, pp.
19806
19811
.
5.
Ramasubramanian
,
A.
,
Latacha
,
K. S.
,
Benjamin
,
J. M.
,
Voronov
,
D. A.
,
Ravi
,
A.
, and
Taber
,
L. A.
, 2006, “
Computational Model for Early Cardiac Looping
,”
Ann. Biomed. Eng.
0090-6964,
34
, pp.
1655
1669
.
6.
Van Essen
,
D. C.
,
Drury
,
H. A.
,
Dickson
,
J.
,
Harwell
,
J.
,
Hanlon
,
D.
, and
Anderson
,
C. H.
, 2001, “
An Integrated Software Suite for Surface-Based Analyses of Cerebral Cortex
,”
J. Am. Med. Inform Assoc.
1067-5027,
8
, pp.
443
459
.
7.
Novozhilov
,
V. V.
, 1970,
Thin Shell Theory
,
Wolters-Noordhoff
,
Groningen, The Netherlands
.
8.
Axelrad
,
E. L.
, 1987,
Theory of Flexible Shells
,
North-Holland
,
Amsterdam
.
9.
Taber
,
L. A.
, 2004,
Nonlinear Theory of Elasticity: Applications in Biomechanics
,
World Scientific
,
Singapore
.
10.
Hamburger
,
V.
, and
Hamilton
,
H. L.
, 1951, “
A Series of Normal Stages in the Development of the Chick Embryo
,”
J. Morphol.
0362-2525,
88
, pp.
49
92
.
11.
Manner
,
J.
, 2000, “
Cardiac Looping in the Chick Embryo: A Morphological Review With Special Reference to Terminological and Biomechanical Aspects of the Looping Process
,”
Anat. Rec.
0003-276X,
259
, pp.
248
262
.
12.
Filas
,
B. A.
,
Efimov
,
I. R.
, and
Taber
,
L. A.
, 2007, “
Optical Coherence Tomography as a Tool for Measuring Morphogenetic Deformation of the Looping Heart
,”
Anat. Rec.
0003-276X,
290
, pp.
1057
1068
.
13.
Voronov
,
D. A.
, and
Taber
,
L. A.
, 2002, “
Cardiac Looping in Experimental Conditions: Effects of Extraembryonic Forces
,”
Dev. Dyn.
1058-8388,
224
, pp.
413
421
.
14.
Manasek
,
F. J.
, and
Monroe
,
R. G.
, 1972, “
Early Cardiac Morphogenesis is Independent of Function
,”
Dev. Biol.
0012-1606,
27
, pp.
584
588
.
15.
Clark
,
E. B.
,
Hu
,
N.
,
Turner
,
D. R.
,
Litter
,
J. E.
, and
Hansen
,
J.
, 1991, “
Effect of Chronic Verapamil Treatment on Ventricular Function and Growth in Chick Embryos
,”
Am. J. Physiol.
0002-9513,
261
, pp.
H166
H171
.
16.
Nerurkar
,
N. L.
,
Ramasubramanian
,
A.
, and
Taber
,
L. A.
, 2006, “
Morphogenetic Adaptation of the Looping Embryonic Heart to Altered Mechanical Loads
,”
Dev. Dyn.
1058-8388,
235
, pp.
1822
1829
.
17.
Van Straaten
,
H. W.
,
Janssen
,
H. C.
,
Peeters
,
M. C.
,
Copp
,
A. J.
, and
Hekking
,
J. W.
, 1996, “
Neural Tube Closure in the Chick Embryo is Multiphasic
,”
Dev. Dyn.
1058-8388,
207
, pp.
309
318
.
18.
Desmond
,
M. E.
,
Levitan
,
M. L.
, and
Haas
,
A. R.
, 2005, “
Internal Luminal Pressure During Early Chick Embryonic Brain Growth: Descriptive and Empirical Observations
,”
Anat. Rec.
0003-276X,
285
, pp.
737
747
.
19.
Huang
,
D.
,
Swanson
,
E. A.
,
Lin
,
C. P.
,
Schuman
,
J. S.
,
Stinson
,
W. G.
,
Chang
,
W.
,
Hee
,
M. R.
,
Flotte
,
T.
,
Gregory
,
K.
, and
Puliafito
,
C. A.
, 1991, “
Optical Coherence Tomography
,”
Science
0036-8075,
254
, pp.
1178
1181
.
20.
Fujimoto
,
J. G.
, 2003, “
Optical Coherence Tomography for Ultrahigh Resolution in Vivo Imaging
,”
Nat. Biotechnol.
1087-0156,
21
, pp.
1361
1367
.
21.
Ramasubramanian
,
A.
, and
Taber
,
L. A.
, 2008, “
Computational Modeling of Morphogenesis Regulated by Mechanical Feedback
,”
Biomech. Model. Mechanobiol.
,
2
, pp.
77
91
.
22.
Welker
,
W.
, 1990, “
Why Does the Cerebral Cortex Fissure and Fold? A Review of the Determinants of Gyri and Sulci
,”
Cerebral Cortex
,
E. G.
Jones
and
A.
Peters
, eds.
Plenum
,
New York
, pp.
3
136
.
23.
Neal
,
J.
,
Takahashi
,
M.
,
Silva
,
M.
,
Tiao
,
G.
,
Walsh
,
C. A.
, and
Sheen
,
V. L.
, 2007, “
Insights into the Gyrification of Developing Ferret Brain by Magnetic Resonance Imaging
,”
J. Anat.
0021-8782,
210
, pp.
66
77
.
24.
Hashima
,
A. R.
,
Young
,
A. A.
,
McCulloch
,
A. D.
, and
Waldman
,
L. K.
, 1993, “
Nonhomogeneous Analysis of Epicardial Strain Distributions During Acute Myocardial Ischemia in the Dog
,”
J. Biomech.
0021-9290,
26
, pp.
19
35
.
25.
Alford
,
P. W.
, and
Taber
,
L. A.
, 2003, “
Regional Epicardial Strain in the Embryonic Chick Heart During the Early Looping Stages
,”
J. Biomech.
0021-9290,
36
, pp.
1135
1141
.
26.
Taber
,
L. A.
,
Sun
,
H.
,
Clark
,
E. B.
, and
Keller
,
B. B.
, 1994, “
Epicardial Strains in Embryonic Chick Ventricle at Stages 16 Through 24
,”
Circ. Res.
0009-7330,
75
, pp.
896
903
.
27.
Tobita
,
K.
, and
Keller
,
B. B.
, 2000, “
Maturation of End-Systolic Stress-Strain Relations in Chick Embryonic Myocardium
,”
Am. J. Physiol.
0002-9513,
279
, pp.
H216
H224
.
28.
Tobita
,
K.
, and
Keller
,
B. B.
, 2000, “
Right and Left Ventricular Wall Deformation Patterns in Normal and Left Heart Hypoplasia Chick Embryos
,”
Am. J. Physiol.
0002-9513,
279
, pp.
H959
H969
29.
Meier
,
G. D.
,
Bove
,
A. A.
,
Santamore
,
W. P.
, and
Lynch
,
P. R.
, 1980, “
Contractile Function in Canine Right Ventricle
,”
Am. J. Physiol.
0002-9513,
239
, pp.
H794
H804
.
30.
Villarreal
,
F. J.
,
Waldman
,
L. K.
, and
Lew
,
W. Y.
, 1988, “
Technique for Measuring Regional Two-Dimensional Finite Strains in Canine Left Ventricle
,”
Circ. Res.
0009-7330,
62
, pp.
711
721
.
31.
Omens
,
J. H.
,
Farr
,
D. D.
,
McCulloch
,
A. D.
, and
Waldman
,
L. K.
, 1996, “
Comparison of Two Techniques for Measuring Two-Dimensional Strain in Rat Left Ventricles
,”
Am. J. Physiol.
0002-9513,
271
, pp.
H1256
H1261
.
32.
Kindberg
,
K.
,
Karlsson
,
M.
,
Ingels
,
N. B.
, Jr.
, and
Criscione
,
J. C.
, 2007, “
Nonhomogeneous Strain From Sparse Marker Arrays for Analysis of Transmural Myocardial Mechanics
,”
ASME J. Biomech. Eng.
0148-0731,
129
, pp.
603
610
.
33.
Soufan
,
A. T.
,
van den
,
B. G.
,
Ruijter
,
J. M.
,
de Boer
,
P. A.
,
van den Hoff
,
M. J.
, and
Moorman
,
A. F.
, 2006, “
Regionalized Sequence of Myocardial Cell Growth and Proliferation Characterizes Early Chamber Formation
,”
Circ. Res.
0009-7330,
99
, pp.
545
552
.
34.
Soufan
,
A. T.
,
van den
,
B. G.
,
Moerland
,
P. D.
,
Massink
,
M. M.
,
van den Hoff
,
M. J.
,
Moorman
,
A. F.
, and
Ruijter
,
J. M.
, 2007, “
Three-Dimensional Measurement and Visualization of Morphogenesis Applied to Cardiac Embryology
,”
J. Microsc.
0022-2720,
225
, pp.
269
274
.
You do not currently have access to this content.