Fig. 2
Mathematical description of epithelial-sheet deformation. a Geometrical view of tissue deformation. When the scale of organs is much larger than that of a cell, tissue-level deformation can be described as a map of continuum. The deformation map of a surface has 3D and 2D representations (x = ϕ(X) and \({\boldsymbol{u}}=\tilde{\phi }({\boldsymbol{U}})\), respectively). H(X) and h(x) are functions that indicate how to transform a 3D coordinate into 2D at different time points (T1 and T2). With these functions, the relationship \(\phi ={h}^{-1}\circ \tilde{\phi }\circ H\) holds. In our method, using positional data from sparsely labeled cells, the 2D map modeled by lattice deformation is estimated; the deformation of the gray region from \(\tilde{\Omega }\) to \(\tilde{\phi }(\tilde{\Omega })\) shows an example. The regions \(\tilde{\Omega }\) and \(\tilde{\phi }(\tilde{\Omega })\) can be linked to their 3D representations (i.e., Ω and ϕ(Ω)) through the functions H −1(U) and h −1(u), respectively. After the map is determined, local tissue deformation can be calculated from the deformation gradient tensor \(\tilde{{\boldsymbol{F}}}\). In the figure, the local deformation around a point \(\tilde{p}\) in a 2D coordinate system (from \(\Delta {\tilde{\Omega }}_{p}\) to \(\tilde{\boldsymbol{F}}(\Delta {\tilde{\Omega }}_{p})\)) is shown. As a result of compressing a curved surface into a flat plane, the metric differs depending on the position in the 2D coordinate systems (\({\tilde{G}}_{\alpha \beta }({\boldsymbol{U}})\) and \({\tilde{g}}_{\alpha \beta }({\boldsymbol{u}})\)). The bottom figure shows examples of the positional dependence of metric calculated from the spherical harmonics used for modeling the 3D morphologies at time points T1 and T2 shown in the figure. Different ellipses with different colors show the metric anisotropy at their positions. b Using the metric \(\tilde{g}\), the neighborhood of each point in the 3D representation (Δx TΔx = const., shown as red circles) can be approximated by \(\Delta {{\boldsymbol{u}}}^{T}\tilde{g}\Delta {\boldsymbol{u}}=\) const. in 2D. c In the process of estimating deformation maps from landmark positions, we assumed that the observed positions include an additive noise obeying isotropic Gaussian distribution (ξ i ) on the tangent plane at each point in the 3D representation. In the 2D representation, the noise distribution becomes anisotropic (\({\tilde{{\boldsymbol{\xi }}}}_{i}\)) due to metric anisotropy. The variance-covariance matrix was modeled as \({\tilde{\Sigma }}_{(i)}\cong {\sigma }^{2}{\tilde{g}}^{-1}({{\boldsymbol{u}}}_{i})\) using the metric at the observed position in the 2D coordinate system of each landmark. σ represents noise magnitude. Including this noise anisotropy into the likelihood function improves the estimation performance (see Fig. 4)
