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Research Papers

A New Method for Measuring Deformation of Folding Surfaces During Morphogenesis

[+] Author and Article Information
Benjamen A. Filas1

Department of Biomedical Engineering, Washington University, St. Louis, MO 63130

Andrew K. Knutsen1

Department of Mechanical, Aerospace and Structural Engineering, Washington University, St. Louis, MO 63130

Philip V. Bayly

Department of Mechanical, Aerospace and Structural Engineering, Washington University, St. Louis, MO 63130

Larry A. Taber2

Department of Biomedical Engineering, Washington University, St. Louis, MO 63130lat@wustl.edu

In this section, Latin indices take the values 1,2,3 and Greek indices take the values 1,2. The usual summation convention on repeated indices is implied.

http://brainmap.wustl.edu/caret

For the brain, beads were placed on the inner surface, but the reconstructed images show only the outer surface clearly. Thus, the tracked beads were shifted dorsally in Fig. 3 to allow their relative positions on the brain surface to be seen. However, the actual bead centroids on the inner surface were used in the strain analysis.

1

These authors contributed equally to this work.

2

Corresponding author.

J Biomech Eng 130(6), 061010 (Oct 14, 2008) (9 pages) doi:10.1115/1.2979866 History: Received January 15, 2008; Revised April 11, 2008; Published October 14, 2008

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.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of surface geometry. S: reference surface; s: deformed surface; Xi, xi: local Cartesian coordinate systems; ei: local Cartesian base vectors; Gi: local covariant base vectors at a bead on S; gi: convected base vectors at same bead on s. Note that, in general, e1 and e2 are only approximately tangent to S, and the tangent base vectors (G1, G2 and g1, g2) are not orthogonal.

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Figure 2

Motion of beads attached to the surface of a looping chick heart (ventral view of OCT reconstructions). (A) Stage 11+; (B) stage 12. Outlined region contains beads that initially congregated along the right side of the heart (A) and subsequently rotated toward the outer curvature and the backside of the heart (B). The asterisks in (A) and (B) show a region of low bead density near the primitive atrium (PA) where strains were not calculated. (C) Surface representation and beads (open circles) used in deformation analysis generated from images of the stage 12 heart. V: ventricle; CT: conotruncus. Scale bar: 250μm.

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Figure 3

Motion of beads attached to the inner surface of an embryonic chick brain (ventral view of OCT reconstructions). (A) Stage 11. (B) Stage 12. (C) Surface representation and beads (open circles) used in deformation analysis generated from images of stage 12 brain. F: forebrain; OV: optic vesicle; M: midbrain; H: hindbrain. Scale bar: 250μm.

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Figure 4

Wrapping of a flat sheet into a partial cylinder. Surfaces are shown with respect to the global (X,Y,Z) coordinate system. ((A) and (B)) Estimated principal Eulerian strains using a dense, regular, marker array (1300 markers/unit area); ((C) and (D)) estimated principal Eulerian strains using a less dense marker array (200 randomly distributed markers; 40 markers/unit area; see marker locations in upper panels). Differences between the calculated and actual principal strains were on the order of 10−14 for the dense marker array and 10−2 for the sparse marker array (Table 1). The gray regions in the strain plots indicate regions where strain was not calculated due to insufficient marker density.

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Figure 5

Bending and torsion of a cylinder: comparison of exact and estimated Eulerian strains. Surfaces are shown with respect to the global (X,Y,Z) coordinate system. ((A) and (B)) Exact values of first and second principal strains. ((C) and (D)) Estimated values obtained with a dense regular array of markers (2904 markers/unit area; radius of curvature, ρ≈0.20; fitting radius, r=0.20). ((E) and (F)) Differences between the actual and estimated principal strains (see Table 1). (G) and (H) Surface representations of the undeformed and deformed cylinders.

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Figure 6

Bending and torsion of a cylinder: effect of fitting radius on Eulerian strains. ((A) and (B)) First (maximum) principal strain estimated using different fitting radii (r=0.05, r=0.25) with a dense, randomly scattered, set of markers (2904 marker per unit area); ((C) and (D)) differences between actual and estimated principal strains (see Table 1).

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Figure 7

Bending and torsion of a cylinder: effect of random error in marker coordinates on first (maximum) principal Eulerian strain, E1. Random perturbations were added to both the reference (X,Y,Z) and deformed (x,y,z) coordinates (maximum error magnitude was ±1% of the corresponding cylinder dimension). (A) Estimates with fitting radius r=0.05. (B) Estimates with fitting radius r=0.10. (C) Estimates with fitting radius r=0.25. ((D)–(F)) Respective differences between estimated and exact strain values. Strains are mapped onto the true (error-free) surface of the deformed cylinder. At r=0.05, the added random errors noticeably affect strain estimates. Increasing r to 0.10 smooths strain estimates while providing accurate surface fitting. Increasing the fitting radius too much (i.e., r=0.25>ρ∕2) visibly increases the fitting error (see Table 1).

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Figure 8

Circumferential stretch ratio (λ1) longitudinal stretch ratio (λ2), and shear (γ) mapped onto a stage 12 (fully c-looped) embryonic chick heart. Quantities were computed relative to the configuration at stage 11⋆ (approximately 5h earlier). The circumferential and longitudinal directions were defined locally as the directions of maximum and minimum curvatures, respectively. Orientations show the ventral, lateral, and dorsal surfaces of the heart. V: ventricle; PA: primitive atrium; CT: conotruncus.

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Figure 9

Stretch ratios in the directions of maximum curvature (λ1) and minimum curvature (λ2), and shear (γ) mapped onto a stage 12 embryonic chick brain. Deformation measures were calculated relative to a stage 11 reference state (≈6h incubation). Note that, because of the complex geometry, the principal axes of curvature are not uniquely related to anatomical axes. As indicated by arrows, the longitudinal and circumferential directions in the midbrain and hindbrain correspond to the directions of minimum and maximal curvatures, respectively. In the forebrain, the situation is reversed. Orientations show the ventral and dorsal surfaces of the brain. F: forebrain; OV: optic vesicle; M: midbrain; H: hindbrain.

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