A New Method for Measuring Deformation of Folding Surfaces during Morphogenesis

Abstract

During morphogenesis, epithelia (cell sheets) undergo complex deformations as they stretch, bend, and twist to form the embryo. Often these changes in shape create multi-valued 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 (OCT). This technique provides a new tool that can be used in studies of the mechanics of morphogenesis.

1. Introduction

In the embryo, many morphogenetic processes involve complex three-dimensional (3-D) deformations of cell sheets, or epithelia [1,2]. In studies of morphomechanics, it is useful to quantify these deformations, and researchers have measured strain distributions in epithelia during gastrulation [3,4] and early heart development [5]. These analyses, however, generally have been restricted to single-valued surfaces of relatively modest curvature. This limitation is especially problematic when an epithelium folds, a common occurrence during embryogenesis.

Here, we present a technique for measuring surface strain that can accommodate virtually any type of deformation during epithelial morphogenesis. Deformation gradients, strains, and other kinematic quantities are computed using straightforward matrix algebra. After the basic theory is presented, the accuracy of the method is examined for some sample problems with exact solutions. Then, to illustrate the application of the method to physical examples of morphogenesis, it is used to measure surface stretch ratios and shear in the looping heart and developing brain of the early chick embryo. Our method is applicable to a wide range of morphogenetic problems, as well as to functional studies, e.g., strain measurements in the beating heart.

2. Methods

2.1 Theoretical and Numerical Methods

2.1.1 Images, Surfaces, and Markers

Deformation is measured by following the displacements of a set of fiducial points (markers) attached to a surface. The surface and the marker locations are typically derived from image volumes acquired, for example, by magnetic resonance imaging (MRI), computed tomography (CT), or optical coherence tomography (OCT). Surfaces are created from segmented image volumes via previously-described algorithms (e.g., CARET [6]) or standard software (MATLAB, The Mathworks Inc., Natick, MA). The resulting surface created by these algorithms consists of a set of triangular faces, each defined by the (global) coordinates of its three vertices and by a normal unit vector.

To characterize deformation kinematics, accurate measurements of displacements of points on the surface are needed. In the absence of natural landmarks that can be tracked over time, markers, such as opaque or reflective beads, are attached to the physical surface, so that they move with the material. The location of each marker is tracked over the duration of the imaging study. Typically, the number of markers is much smaller than the number of vertices that represent the surface. Hence, the distance between vertices is usually small compared to the marker spacing.

2.1.2 Coordinate Systems

Surfaces are originally described with respect to a global Cartesian coordinate system. The base unit vectors of the global system are aligned with the axes of the 3-D image volume (typically a stack of 2-D images).

Local Cartesian coordinate systems are defined in order to analyze the local deformation of surfaces near specific points. Each bead, or other fiducial marker, is used in turn as the origin of such a local system. A local, approximately normal, unit vector, e₃, is taken to be the average of the normals of the five nearest faces on the surface. An orthogonal, approximately tangent unit vector, e₁, is defined by setting one component to zero and enforcing e₃·e₁ = 0 and | e₁ | = 1. Finally, another orthogonal, approximately tangent unit vector, e₂, is obtained directly using e₂ = e₃ × e₁ (Fig. 1).

Figure 1.

Schematic diagram of surface geometry. S: reference surface; s: deformed surface; X_i, x_i: local Cartesian coordinate systems; e_i: local Cartesian base vectors; G_i: local covariant base vectors at a bead on S; g_i: 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.

2.1.3 Analysis of Deformation

The analysis is based on the general nonlinear membrane theory of shells [7,8]. Consider the deformation of the reference surface S into the current surface s (Fig. 1). For each locally analyzed region, the positions of points on these surfaces are described relative to the local Cartesian axes defined above. Relative to these axes, a point with material coordinates Xⁱ on S moves to the spatial coordinates xⁱ on s.^* Tracking individual markers allows us to write the spatial coordinates for a finite number of points in terms of material coordinates, i.e., xⁱ = xⁱ (Xⁱ). Next, we assume that the local region of surface S to be analyzed can be described by the relationship X³ = X³ (X¹, X²), where X³ is a single-valued function of the local coordinates X¹ and X². Hence, we can write xⁱ = xⁱ (X¹, X², X³ (X¹, X²)) = xⁱ (X¹, X²).

The position vectors to a point on S and its deformed image on s are, respectively,

$$\begin{array}{c}\mathbf{R}=X^{\alpha }{\mathbf{e}}_{\alpha }+X^3\left(X^{\alpha }\right){\mathbf{e}}_3\\ \mathbf{r}=x^{\beta }\left(X^{\alpha }\right){\mathbf{e}}_{\alpha }+x^3\left(X^{\alpha }\right){\mathbf{e}}_3.\end{array}$$ (1)

Covariant base vectors in S and convected base vectors in s are given by the respective relations (Taber 2004)

$${\mathbf{G}}_{\alpha }=\frac{\partial \mathbf{R}}{\partial X^{\alpha }}=\mathbf{R}{,}_{\alpha },{\mathbf{g}}_{\alpha }=\frac{\partial \mathbf{r}}{\partial X^{\alpha }}=\mathbf{r}{,}_{\alpha }.$$ (2)

Note that these base vectors are tangent to the surfaces, but are generally not orthogonal or unit vectors (Fig. 1). Substituting Eqs. (1) into (2) yields

$${\mathbf{G}}_{\alpha }={\mathbf{e}}_{\alpha }+{\mathbf{X}}^3{,}_{\alpha }{\mathbf{e}}_3,{\mathbf{g}}_{\alpha }={\mathbf{x}}^{\beta }{,}_{\alpha }{\mathbf{e}}_{\beta }+{\mathbf{x}}^3{,}_{\alpha }{\mathbf{e}}_3.$$ (3)

In addition, unit vectors normal to S and s, respectively, are given by

$${\mathbf{G}}_3=\frac{{\mathbf{G}}_1\times {\mathbf{G}}_2}{|{\mathbf{G}}_1\times {\mathbf{G}}_2|},{\mathbf{g}}_3=\frac{{\mathbf{g}}_1\times {\mathbf{g}}_2}{|{\mathbf{g}}_1\times {\mathbf{g}}_2|}.$$ (4)

Finally, contravariant base vectors, Gⁱ and gⁱ, are defined by the relations

$${\mathbf{G}}_{\mathrm{i}}·{\mathbf{G}}^{\mathrm{j}}={\mathbf{g}}_{\mathrm{i}}·{\mathbf{g}}^{\mathrm{j}}={\mathrm{\delta }}_i^j$$ (5)

where ${\delta }_i^j$ is the Kronecker delta.

In the membrane theory of shells, base vectors typically are written in terms of coordinates that lie entirely within the surface. Here, however, the surface is considered explicitly embedded in 3-D space, with the base vectors written in terms of coordinates that are (approximately) tangent and normal to the surface at this point. As shown next, this approach makes the analysis relatively simple to program in MATLAB using matrix algebra. The only explicit expressions needed in this analysis are those for the covariant base vectors of Eqs. (3) and (4). (It is important to note that this method requires the surface normals G₃ and g₃ to be defined as unit vectors, to prevent erroneous transverse deformation from entering the calculations.)

Equations (3) and (4) provide base vectors in the component forms

$$\begin{array}{l}{\mathbf{G}}_i=G_{ij}{\mathbf{e}}^j,{\mathbf{g}}_i=g_{ij}{\mathbf{e}}^j\\ {\mathbf{G}}^i=G^{ij}{\mathbf{e}}_j,{\mathbf{g}}^i=g^{ij}{\mathbf{e}}_j.\end{array}$$ (6)

Writing all vectors and tensors in terms of components relative to the Cartesian unit basis eⁱ = e_i allows us to use matrix algebra from here on. With bracketed quantities denoting 3×3 matrices, the components of the base vectors are represented by

$$\begin{array}{l}\left[\mathrm{G}\right]=\lfloor {\mathrm{G}}_{\text{ij}}\rfloor ,\left[\mathrm{g}\right]=\lfloor {\mathrm{g}}_{\text{ij}}\rfloor \\ \left[{\mathrm{G}}^{*}\right]=\left[{\mathrm{G}}^{\text{ij}}\right],\left[{\mathrm{g}}^{*}\right]=\left[{\mathrm{g}}^{\text{ij}}\right].\end{array}$$ (7)

With the components of [G] and [g] known, Eq. (5) yields matrix equations for [G^*]and [g^*] in the form

$$\begin{array}{l}\left[\mathrm{G}\right]{\left[{\mathrm{G}}^{*}\right]}^{\mathrm{T}}=\left[\mathrm{I}\right]\to \left[{\mathrm{G}}^{*}\right]={\left[\mathrm{G}\right]}^{-\mathrm{T}}\\ \left[\mathrm{g}\right]{\left[{\mathrm{g}}^{*}\right]}^{\mathrm{T}}=\left[\mathrm{I}\right]\to \left[{\mathrm{g}}^{*}\right]={\left[\mathrm{g}\right]}^{-\mathrm{T}}\end{array}$$ (8)

where [I] is the identity matrix and T denotes transpose.

The deformation gradient tensor is given by [9]

$$\mathbf{F}={\mathbf{g}}_{\mathrm{j}}{\mathbf{G}}^{\mathrm{i}}\to \left[\mathrm{F}\right]={\left[\mathrm{g}\right]}^{\mathrm{T}}\left[{\mathrm{G}}^{*}\right]$$ (9)

With F now known, it is straightforward to compute any deformation measure of interest. For example, the components of the right and left Cauchy-Green deformation tensors, respectively, are given by

$$\left[C\right]={\left[F\right]}^T\left[F\right],\left[B\right]=\left[F\right]{\left[F\right]}^T$$ (10)

which provide the components of the Lagrangian and Eulerian strain tensors [9]

$$\left[E\right]=\frac{1}{2}\left(\left[C\right]-\left[I\right]\right),\left[e\right]=\frac{1}{2}\left(\left[I\right]-{\left[B\right]}^{-1}\right).$$ (11)

Here, we again emphasize that all matrix components are defined relative to the local Cartesian axes.

2.1.4 Stretch Ratios and Strains in Specific Directions

Because the local Cartesian system does not follow the surface at every point, the components of the above tensors generally have no direct physical interpretation. However, meaningful quantities can be extracted relatively easily. For example, if N₁ and N₂ are orthogonal unit vectors tangent to S, then physical Lagrangian strain components relative to these directions are given by

$${\mathrm{E}}_{\mathrm{\alpha }\mathrm{\beta }}={\left\{\mathrm{N}\right\}}_{\mathrm{\alpha }}^{\mathrm{T}}\left[\mathrm{E}\right]{\left\{\mathrm{N}\right\}}_{\beta }$$ (12)

where curly braces indicate vectors. Similarly, if n₁ and n₂ are orthogonal unit vectors tangent to s, then physical Eulerian strain components are given by

$${\mathrm{e}}_{\mathrm{\alpha }\mathrm{\beta }}={\left\{\mathrm{n}\right\}}_{\alpha }^T\left[\mathrm{e}\right]{\left\{\mathrm{n}\right\}}_{\beta }.$$ (13)

Alternatively, stretch ratios in the direction N on S or n on s can be computed from the respective relations [9]

$${{\mathrm{\Lambda }}^2}_{\left(N\right)}={\left\{\mathrm{N}\right\}}^{\mathrm{T}}\left[\mathrm{C}\right]\left\{\mathrm{N}\right\},{{\lambda }^{-2}}_{\left(n\right)}={\left\{\mathrm{n}\right\}}^{\mathrm{T}}{\left[\mathrm{B}\right]}^{-1}\left\{\mathrm{n}\right\}.$$ (14)

Measures of change in angle between undeformed line elements (in the directions of the unit vectors N_α on S) and their deformed images (in the directions n_α on s) are given by the shear Γ_{(Nα Nβ)} = γ_{(nα nβ)} = (n_α · n_β) − (N_α · N_β) [9]. With n_α = F · N_α/|F · N_α|, the shear is calculated from either of the relations

$${\mathrm{\Gamma }}_{\left(N_{\alpha }{N}_{\beta }\right)}={\left\{\mathrm{N}\right\}}_{\alpha }^T\left(\frac{\left[\mathrm{C}\right]}{{\mathrm{\Lambda }}_{\left({\mathrm{N}}_{\alpha }\right)}{\mathrm{\Lambda }}_{({\mathrm{N}}_{\beta })}}-\left[I\right]\right){\left\{\mathrm{N}\right\}}_{\beta },{\gamma }_{\left(n_{\alpha }{n}_{\beta }\right)}={\left\{\mathrm{n}\right\}}_{\alpha }^T\left(\left[I\right]-\frac{{\left[\mathrm{B}\right]}^{-1}}{{\lambda }_{\left({\mathrm{n}}_{\alpha }\right)}{\lambda }_{({\mathrm{n}}_{\beta })}}\right){\left\{\mathrm{n}\right\}}_{\beta }.$$ (15)

For the heart and brain problems, strains were computed relative to unit vectors along the local directions of maximum and minimum curvature of s, corresponding approximately to local circumferential and longitudinal directions in the heart or brain tube at any time during development. The components of the curvature tensor κ for s are provided by

$${\kappa }_{\alpha \beta }=\frac{{\partial }^2{x}^3}{\partial x^{\alpha }\partial x^{\beta }}.$$ (16)

With the 2×2 curvature matrix defined by [κ] = [κ_αβ], the eigenvalue problem

$$\left(\left[\kappa \right]-k\left[\mathrm{I}\right]\right)\left\{\mathrm{v}\right\}=0$$ (17)

yields the principal curvatures k_α and directions {v}_α. Relative to these orthogonal directions, Eqs. (13) and (14)₂ then provide the corresponding Eulerian strains and stretch ratios, respectively.

Principal stretch ratios Λ_i = λ_i are obtained by solving either of the eigenvalue problems

$$\left(\left[\mathrm{C}\right]-{\mathrm{\Lambda }}^2\left[\mathrm{I}\right]\right)\left\{\mathrm{M}\right\}=0,\left(\left[\mathrm{B}\right]-{\lambda }^2\left[\mathrm{I}\right]\right)\left\{\mathrm{m}\right\}=0$$ (18)

where {M}_i and {m}_i are eigenvectors in S and s, respectively. Because the normal vectors do not change length, these relations yield unity for one principal stretch ratio, with the corresponding eigenvector normal to the surface.

2.1.5 Piecewise Fitting of Local Surface Functions

The above strain analysis is valid for reference surfaces that can be described by a single-valued function X³(X¹, X²). During morphogenesis, however, surfaces often fold, leading to multiple values of X³ for each X¹, X² combination in a global coordinate system. Hence, we compute strains in a piecewise manner, where each region of the surface is single-valued relative to its own local coordinate system (see Coordinate Systems section above). Relative to this system, the analysis requires expressions for the functions X³(X¹, X²) and xⁱ(X¹, X²) for i = 1,2,3. It is important to note that the use of convected base vectors in the strain analysis means that only the reference surface for each region need be single-valued.

Local surfaces were fit to the finite set of points to give local approximations for X³(X¹, X²) and xⁱ(X¹, X²). Each marker in turn was assigned to be the origin of a local Cartesian coordinate system. The absolute distances of all of the other markers from the local origin were calculated, and surfaces were fitted using markers within a user-specified radius, r. All markers outside that distance (r) were ignored. The fitting functions were taken as second-order polynomial functions of the form

$$f\left(X^1,X^2\right)=a_0+a_1{X}^1+a_2{X}^1+a_3{X}^1{X}^2+a_4{\left(X^1\right)}^2+a_5{\left(X^2\right)}^2.$$ (19)

The coefficients a_i were determined in a least squares sense (using the mldivide function in MATLAB). Then, the derivatives of these functions at the local origin were calculated and inserted into Eqs. (3). This process was repeated for all markers.

2.1.5 Length Scales: Curvature, Wavelength, Marker Spacing, and Fitting Radius

The characteristic length scales of the surface and the marker distribution can be used to bound the size of the fitting region. Length scales of a curved surface include the minimum radius of curvature, and the wavelength of undulations. For a quadratic function (e.g., Eq. (19)) to provide accurate strain estimates using our method, the error in the derivatives of the fitted surface (compared to the true surface), should be small (less than 10%). A quadratic fit to a cylindrical surface can be shown by straightforward calculation to satisfy this criterion if

$$\mathrm{r}<0.43\mathrm{\rho }$$ (20)

where ρ is the radius of curvature of the cylinder. A quadratic fit to a sinusoidal function can be shown to satisfy the derivative error criterion if

$$\mathrm{r}<0.071L$$ (21)

where L is the wavelength of the sinusoid. Note that the wavelength of a sinusoid of amplitude a is related to its minimum radius of curvature by $L=2\pi \sqrt{\rho a}$, so that for curves with amplitudes of the order of the radius of curvature, the criteria of Eqs. (20) and (21) are numerically similar. These criteria lead to the following guideline: the radius of the fitting region should be less than half the local radius of curvature of the surface. Note that the local radius of curvature may be found from high-resolution, image generated surfaces.

On the other hand, the fitting region must include enough markers so that the six free parameters of Eq. (19) can be estimated accurately. For example, if at least 8 points are desired in almost all fits, an average ± std. deviation of 20±4 points per fitting region is appropriate. If the marker density is μ markers/unit area (an average marker separation of $\sqrt{1/\mu }$ length units), the radius of the fitting region should satisfy

$$r^2\pi \mu >20.$$ (22)

In the current work, a fixed radius for all fitting regions was selected based on these criteria. In principle, the fitting region could be varied for different parts of the surface. Note that these criteria are based on the surface geometry, under the assumption that variations in strain occur at similar length scales. Since variations in marker distribution, measurement error, and actual deformation could cause underlying assumptions to be violated, strain estimates were rejected if either of two fitting requirements were not met: (1) A minimum number of markers (N_min = 8) must be found within the specified fitting radius, r; and (2) the residual error of each fit must be less than a specified fraction (usually 0.3) of the variance of the data. Finally, although the fitting process reduces the effects of random variations, errors in displacement measurements should be considerably smaller than the displacements themselves.

2.1.6 Visualization of Deformation Measures

Polynomial fits and strain estimates are typically obtained from a relatively sparse set of markers scattered on a surface, which is imaged at high-resolution in 3-D (for example by MRI or OCT). In order to depict the strain on these densely-parameterized surfaces, the strain at a given vertex is estimated from the weighted sum of the strains at marker locations within the radius, r/2. The weighting factor is inversely proportional to the distance between the vertex and the marker. Vertices on the surface that are not close enough to any marker with a valid strain estimate will not display a strain value.

2.2 Experimental Methods

Surface strains were measured in the developing heart and brain of the chick embryo during stages 10–12 of Hamburger and Hamilton [10]. During this early phase of development, the relatively straight heart tube bends ventrally and twists rightward into a c-shaped tube (Fig. 2) [11]. In the ventral view, rotation due to torsion causes markers that are initially located near the right side of the heart tube to move behind the heart (Fig. 2A,B). At the same time, the brain forms its major subdivisions (forebrain, midbrain, and hindbrain, see Fig. 3).

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. Regions outlined in red denote region where a high number of beads initially congregated along the right side of heart (A) and subsequently rotated toward the outer curvature and the backside of the heart (B). Blue asterisks (A,B) show a region of low bead density near the primitive atrium (PA) where strains were not calculated. (C) Surface representation and beads (blue circles) used in deformation analysis generated from images of the stage 12 heart. V: ventricle; CT: conotruncus. Scale bar: 250 µm.

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 (blue circles) used in deformation analysis generated from images of stage 12 brain. F: forebrain; OV: optic vesicle; M: midbrain; H: hindbrain. Scale bar: 250 µm.

2.2.1 Embryo Preparation

Experimental methods are similar to those presented in Filas et al. [12]. Briefly, white Leghorn chicken eggs were incubated at 38°C for approximately 34 or 42 hours to stage 10 (heart) or stage 11 (brain). The embryo was extracted from the egg using a filter paper carrier method [13] and, to gain direct access to the heart, the membrane covering the ventral surface of the heart (the splanchnopleure) was removed using a fine glass needle. Embryos were cultured at 38° C in a heated Delta T culture dish (Bioptechs, Butler, PA) in Dulbecco’s Modified Eagle’s Medium (Sigma). To prevent artifacts caused by surface tension, the embryo was submerged in the media; thus, it was superfused with a 95% oxygen, 5% carbon dioxide mixture to prevent hypoxia and to ensure proper physiological pH [13].

In the heart experiments, morphogenetic deformation was isolated from deformation due to the heartbeat by using verapamil (25 µM) to stop sarcomeric contraction. Previous studies have shown that c-looping is normal in embryos cultured in verapamil [14,15].

It is important to note that cardiac bending and torsion normally occur simultaneously during c-looping. When the splanchnopleure is removed, however, torsion is delayed until approximately stage 11 but is completed normally by stage 12 [16]. This behavior is not crucial for the present purpose, but it should be kept in mind when interpreting results.

2.2.2 Strain Measurements

For the heart, carboxylated blue polystyrene microspheres with a mean diameter of 6 µm (Polysciences, Warrington, PA) were sprinkled onto the epicardium at stage 10 with a syringe. Time lapse microscopy studies confirmed that the beads remained firmly attached during development (data not shown). For the brain, beads were injected into the brain tube at stage 11 using a pulled glass micropipette. We found this stage to be ideal for injection because the cranial neuropore had sealed completely [17], thus preventing markers from escaping through this cavity. Additionally, internal pressure in the brain is relatively low at this time [18], allowing the hole made at the bead injection site to heal quickly. Because the brain is surrounded by other tissue, larger polystyrene microspheres were used in these injections (10 µm diameter, Bangs Labs, Fishers, IN) to enhance image contrast. For both the heart and brain, markers were tracked until stage 12 (48 total hours of a 21-day incubation period).

The relatively new imaging technique of OCT [19,20] was used to image the embryo noninvasively approximately once per hour at high resolution (<10 µm) and in 3-D (image stacks) during the ex-ovo incubation period (≈ 12 hours, stage 10 to 12, for the heart; ≈ 6 hours, stage 11–12, for the brain). The 3-D coordinates of the microspheres were determined at each time point using thresholding functions in commercially available software (Volocity, Improvision), and deformation measures were calculated relative to the reference configuration using the methods described above.

To map computed quantities onto the surface of the heart or brain, cross sections obtained by OCT were manually segmented using the CARET (Computerized Anatomical Reconstruction and Editing Tool Kit) software package [6] available at http://brainmap.wustl.edu/caret. Reconstructed volumes were subsequently transferred into MATLAB and aligned with bead centroids.^† Calculated quantities were assigned to each vertex of the reconstructed surface as described above.

3. Results

The accuracy of the method is first evaluated for two surfaces of relatively simple geometry undergoing specified deformations. Then, the method is applied to the problems of early heart and brain morphogenesis.

3.1 Cylindrical Bending of a Sheet

The undeformed surface is a flat sheet defined in a global Cartesian (X, Y, Z) coordinate system by 0 ≤ X ≤ 1, −2.5 ≤ Y ≤ 2.5, and Z = 0 The sheet is deformed to partially enclose a circular cylinder, as described by (Fig. 4)

$$x=X,y=\frac{3}{\pi }\text{cos}\left(\frac{\pi }{3}Y\right),z=\frac{3}{\pi }\left(1-\text{sin}\left(\frac{\pi }{3}Y\right)\right).$$ (23)

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,B) Estimated principal Eulerian strains using a dense, regular, marker array (1300 markers/unit area); (C,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 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). Grey regions in the strain plots indicate regions where strain was not calculated due to insufficient marker density.

All strains are zero, as the plane curls without stretch or shear (the ends of the deformed sheet do not meet). With a dense array of markers (1300 markers/unit area, average marker spacing $\sqrt{1/\mu }=0.028$ units, final radius of curvature ρ = 0.96 units, fitting radius r = 0.10 units), estimates of first and second principal strains are almost exactly zero when deforming the plane into a cylinder and when unfurling the cylinder back to the original flat surface (Fig. 4, Table 1). With a less dense random array of markers (40 markers/unit area, $\sqrt{1/\mu }=0.16$ units), the radius of the fitting region was increased (r = 0.4 units) to include similar numbers of markers in each fit. Errors increased slightly, but remained small (Fig. 4, Table 1).

Table 1.

Summary of errors in strain estimates (compared to exact values) for known deformations of surfaces. Effects of marker density (µ) and fitting radius (r) are shown. E̅₁, E̅₂: average absolute error in first and second principal strains; σ₁, σ₂: standard deviation of absolute errors in the first and second principal strains; ρ: minimum radius of curvature; N̅ : the average number of points used in each polynomial fit.

		Sheet to Cylinder (Fig. 4)		Cylinder Bending and Torsion (Fig. 6)		Cylinder Bending and Torsion with Marker Location Error (Fig. 7)
Parameters of surface and marker distribution	$1/\sqrt{\mu }$	0.03	0.16	0.02	0.02	0.02	0.02	0.02
	ρ	0.96	0.96	0.20	0.20	0.20	0.20	0.20
	r	0.10	0.40	0.05	0.25	0.05	0.10	0.25
	N̅	40.2	16.9	24.0	578	23.0	87.4	571
Results	E̅1	0.0000	0.0009	0.0007	0.0191	0.0140	0.0037	0.0188
	σ1	0.0000	0.0022	0.0005	0.0109	0.0753	0.0034	0.0109
	E̅2	0.0007	0.0125	0.0069	0.0268	0.0098	0.0023	0.0114
	σ2	0.0001	0.0059	0.0141	0.0339	0.0411	0.0021	0.0059

3.2 Bending and Torsion of a Cylinder

Next we consider a cylinder specified by R = 0.2, 0 ≤ Θ 2π, and −1 ≤ Z ≤ 1 (Fig. 5). After converting from polar to global Cartesian coordinates (X = R cos Θ, Y = R sin Θ), the cylinder was bent and sheared with the deformed surface coordinates given by

$$x\text{'}=X\text{cos}\left(\alpha Z\right)+Z\text{sin}\left(\alpha Z\right)y\text{'}=Yz\text{'}=-X\text{sin}\left(\alpha Z\right)+Z\text{cos}\left(\alpha Z\right)$$ (24)

where α is a constant that describes the degree of bending (in this case α = 0.4). The bent cylinder was subsequently twisted with fully deformed surface coordinates given by

$$x=x\text{'}\text{cos}\left(\beta z\text{'}\right)+y\text{'}\text{sin}\left(\beta z\text{'}\right)y=-x\text{'}\text{sin}\left(\beta z\text{'}\right)+y\text{'}\text{cos}\left(\beta z\text{'}\right)z=z\text{'}$$ (25)

where β is a constant that describes the degree of twisting (in this case β = 1).

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,B) Exact values of first and second principal strains. (C,D) Estimated values obtained with a dense, regular, array of markers (2904 markers/unit area; radius of curvature ρ ≈ 0.20; fitting radius, r = 0.05). (E,F) differences between actual and estimated principal strains (see Table 1). (G,H) Surface representations of the undeformed and deformed cylinder.

Results were first obtained with a dense set of regularly spaced markers (2904 markers/unit area, $\sqrt{1/\mu }=0.018$ units, ρ = 0.2 units; r = 0.05 units). The average absolute errors of the principal strain values were less than 0.01 with this particular fitting radius (Fig. 5).

Because strain estimates depend upon the appropriate size of the fitting region relative to key length scales of the data, this parameter (r) was adjusted (Fig. 6, Table 1). When the size of the fitting region was not large relative to the average distance between randomly scattered markers (r = 0.05), small patches arise where insufficient marker density prevents a fit from being performed (Fig. 6A). When the size of the fitting region was large relative to the radius of curvature of the cylinder (r = 0.25), errors arise due to insufficient fitting accuracy (Fig. 6D). the quadratic function is inadequate to capture the local surface curvature.

Figure 6.

Bending and torsion of a cylinder: effect of fitting radius on Eulerian strains. (A,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,D) differences between actual and estimated principal strains (see Table 1).

To illustrate the potential effects of measurement error, random variations in marker coordinate values were applied in both the undeformed and deformed configurations (maximum coordinate variation was 1% of the corresponding cylinder dimension). Strain estimates were again compared with the exact strain values for different sizes of the fitting region (Fig. 7). When the fitting radius is small, error in displacement measurements adds visible “noise” to strain estimates (Fig. 7, Table 1); increasing the radius of the fitting region (r) reduces the effect of random measurement errors. On the other hand, if the fitting region is large compared to the scale of actual spatial variations (e.g., r = 0.25 in this example), error is introduced (Fig. 7, Table 1).

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 smoothes 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).

3.3 Cardiac Looping

Illustrative data are shown from one heart with 45 beads tracked for about 5 hours from stage 11⁺ to 12, a period that captures most of the torsion (rotation) that occurs following removal of the splanchnopleure (Fig. 2). Note that many of the beads initially near the right side of the heart (Fig. 2A, red outline) moved behind the heart during rotation (Fig. 2B, red outline). Stretch ratios in the direction of maximum curvature (λ₁, circumferential) and minimum curvature (λ₂, longitudinal) and shear (γ) were computed and mapped onto the fully deformed (c-looped) stage 12 heart surface (Fig. 8). Stretch ratios were predominately positive, while shear was significant near the ends of the heart tube.

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 five hours earlier). The circumferential and longitudinal directions were defined locally as the directions of maximum and minimum curvature, respectively. Orientations show the ventral, lateral, and dorsal surfaces of the heart. V: ventricle; PA: primitive atrium; CT: conotruncus.

Bead positions were used to verify that the strains were registered correctly to the surface (Fig. 2C). For this example the minimum radius of curvature was estimated to be roughly 175 µm. A fitting radius of 65 µm was used, allowing acceptable fits at 44 of the 45 markers with an average (± standard deviation) of 25.2±7.1 markers per fit. The average residual error of fitting, as a fraction of the variance of the original data, was 0.079.

3.4 Brain Development

In this example, 56 beads were tracked from stage 11 to 12 (6 hour incubation, (Fig. 3), and stretch ratios were calculated in the directions of maximum (λ₁) and minimum (λ₂) curvature (Fig. 9). As indicated by arrows in Fig. 9, the direction of maximum curvature approximates the circumferential direction everywhere except the forebrain, where it approximates the longitudinal direction. For this brain, the minimum radius of curvature of the OCT-generated surface was approximately 75 µm (excluding the optic vesicles, see Fig. 3). A fitting radius of 35 µm provided acceptable fits at 49 of 56 markers with 16.0±5.2 markers per fit. The average normalized residual error of fitting was 0.130. Stretch ratios in the direction of maximum curvature are generally greater than one in all regions, indicating expansion of the wall (Fig. 9). In the midbrain and hindbrain, stretch ratios in the direction of minimum curvature (longitudinal direction) were slightly less than one. Shear becomes a significant component of the deformation in the forebrain.

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 (≈ 6 hr 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 curvature, 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.

4. Discussion

A new method has been presented for computing the deformation of folding surfaces with multiple-valued coordinates. Multi-valued surfaces, which arise in 3-D morphogenetic phenomena such as invagination [21] and brain folding [22,23] pose difficulties for methods that rely on a single global fitting function [24,12]. To address this problem, we analyze locally single-valued patches of the surface.

Our method extends previous work on strain measurements that are based on tracking the motions of tissue labels using non-invasive imaging technologies such as OCT, MRI, or light microscopy. Several studies have used triangles of markers to measure strains due to the heartbeat in the developing heart [25,26,27,28] and the mature heart [29,30,31]. The spacing between the markers must be closely controlled to limit the effects of measurement error and to avoid missing large strain fluctuations within the triangle [25].

Nonhomogeneous strain analyses using arrays of multiple (>3) markers have also been implemented, although not throughout multi-valued surfaces. In one of the earliest of these strain analyses, Hashima et al. (1993) fit polynomial (cubic Hermite) surfaces to 3-D marker coordinates in an end-diastolic reference state and subsequent deformed states during the cardiac cycle in a canine heart. This enabled longitudinal, circumferential, and shear strains to be calculated over the entire domain of their marker array. More recently, a polynomial least squares fitting approach has been used to calculate strains from combined marker arrays [32]. Particle image velocimetry has also proved useful in describing nonhomogeneous, morphogenetic strains in quail embryos, but so far this approach has only been applied in 2-D [3,4].

Recently, we have used similar techniques to measure strains in limited regions of the looping chick heart over a limited period of development [12]. The current method allows us to track deformations for longer periods of time over all regions of the heart containing surface labels. It is important to recognize that, in addition to deformation attributed directly to mechanical stress (e.g., bending), morphogenetic strains can be caused by cell proliferation, cell growth, and cytoskeletal contraction [33,34].

4.1 Numerical Examples

Deformation of a flat sheet into a cylinder with the same surface dimensions (or vice versa) involves large displacements and rotations, but no strain. The current method provides accurate strain estimates even with a relatively sparse set of markers (Fig. 4, Table 1). The utility of the current method for analyzing complicated 3-D surfaces is further supported by its application to the bending and torsion of a cylinder (Fig. 5–Fig. 7, Table 1). In both of these examples, with adequate marker distributions and accurate measurement of marker locations, strain estimates coincide closely with exact values.

4.2 Effects of Marker Density, Measurement Error, and Fitting Radius

Analysis of the bent and twisted cylinder also illustrates the effects of marker density and measurement error. As noted above, strain estimates correspond closely to exact solutions when a dense marker array is used. However, in practice, discrepancies can arise because of practical issues intrinsic to polynomial fitting. When the fitting radius is too small, too few markers may be included in the fitting region, and fitting parameters may not be obtained, or estimates may be sensitive to measurement error. When the fitting radius is too large, highly curved or spatially complex features will be excessively smoothed. The choice of fitting radius to balance these effects is often largely heuristic [24], but should reflect features of the data. Characteristic length scales of the surface (radius of curvature, wavelength) are reasonable first estimates for characteristic length scales of deformation fields. Like any method to characterize local deformation, this technique inherently relies on accurate measurement of surface displacement at a sufficiently dense set of marker locations.

4.3 Cardiac Looping

During looping, the wall of the heart tube consists of a thin outer layer of myocardium, a relatively thick middle layer of extracellular matrix called cardiac jelly, and a thin endocardium that lines the lumen [11]. From the venous to the arterial end, a series of regions form along the tube: the primitive atrium, ventricle, and conotruncus (Fig. 2C).

Estimates of longitudinal and circumferential stretch ratios and shear obtained by the current approach are consistent with measurements from past studies [12,5]. Generally, the heart tube inflates (due to expanding cardiac jelly) and lengthens from stage 11⁺ to 12, giving stretch ratios greater than unity in most regions. Circumferential stretch is highest near the poles of the ventricle, showing a higher rate of inflation in these regions (Fig. 8). The longitudinal stretch ratio increases toward the conotruncus, indicating a higher rate of lengthening in this region approaching stage 12. Shear is relatively small except near the ends of the heart tube, where most of the torsion occurs. (The ends of the heart are still attached to the body of the embryo at this time.) These results are consistent with past observations [12].

4.4 Brain Development

Because the early brain tube is embedded in surrounding tissue, beads could not be placed on its outer surface. Hence, we measured stains at the inner luminal surface, which encloses embryonic cerebrospinal fluid. To our knowledge, this marks the first measurement of strain in the developing brain in any type of embryo. Therefore, comparison with other quantitative data is not possible, but some observations are appropriate.

In the direction of maximum curvature in the midbrain and hindbrain (i.e., the circumferential direction), stretch ratios from stage 11–12 were positive and highest in the expanding ‘neck’ region leading into the forebrain. This region undergoes significant shape change during this 6-hour incubation period (Fig. 9). As fluid pressure builds inside the brain, the internal cavities of the midbrain and hindbrain undergo differential circumferential expansion to form the characteristic bulges [18], consistent with our stretch estimates. Stretch in the direction of minimum curvature is slightly less than one in the bulges and the forebrain. This reflects slight longitudinal shortening in the bulges and slight shortening in the circumferential direction in the forebrain. This is also consistent with past data showing relatively little growth in the brain in the longitudinal direction between these developmental stages [17]. Higher longitudinal stretch ratios (λ₁) on the dorsal side of the forebrain and ‘neck’ relative to the ventral side reflect the motion of beads towards each other, as the brain undergoes cranial flexure, i.e., it bends downward during this period (Fig. 9). Shear was significant in the forebrain (though not in other regions), reflecting non-uniform, possibly anisotropic, growth in this region (Fig. 9). These observations represent only an illustrative case study of deformation in the embryonic brain. Clearly, deformation should be characterized in more embryos over longer periods of incubation before firm conclusions can be drawn.

The advantage of the current technique for this particular data set is clear. In the reference configuration, beads are scattered around the lumen, which is a multi-valued surface. The complete strain distribution would have been impossible to estimate by the approach of Filas et al (2007).

4.5 Conclusions

A new method for calculating surface strains was developed and used to compute deformation measures in the looping embryonic heart and the developing embryonic brain. The technique was validated by application to large deformations of curved, multi-valued surfaces. This method accommodates sparse, randomly scattered marker arrays, with reasonable errors in the location of markers. Application of this method to data from time series of OCT images of developing chick embryos provides maps of longitudinal and circumferential strain that illuminate the mechanics of morphogenesis.