Mathematical modeling for developmental processes.

We review several mathematical models and concepts in developmental biology that have been established over the last decade. (1) Feedback vertex set: Ascidian embryos contain cells of seven types, and cell fate is controlled by ~100 interacting genes. The "feedback vertex set" of the directed graph of the gene regulatory network consists of a small number of genes. By experimentally manipulating them, we can differentiate cells into any cell type. (2) Tissue deformation: Describing morphological changes in tissues and relating them to gene expression and other cellular processes is key in understanding morphogenesis. Expansion and anisotropy of the tissue are described by a "deformation tensor" at each location. A study on chick limb bud formation revealed that both the volume growth rate and anisotropy in deformation differed significantly between locations and stages. (3) Mechanobiology: Forces operating on each cell may alter cell shape and gene expression, which may subsequently exert forces on their surroundings. Measurements of force, tissue shape, and gene expression help us understand autonomous tissue deformation. (4) Adaptive design of development: An optimal growth schedule in fluctuating environments explains the growth response to starvation in Drosophila larvae. Adaptive placement of morphogen sources makes development robust to noises.

Geometry Does Impact on the Plane Strain Directions of the Human Left Ventricle, Irrespective of Disease.

The directions of primary strain lines of local deformation in Epicardial and Endocardial layers have been the subject of debate in recent years. Different methods led to different conclusions and a complete assessment of strain direction patterns in large and variable (in terms of pathology) cohorts of healthy and diseased patients is still lacking. Here, we use local deformation tensors in order to evaluate the angle of strain lines with respect to the horizontal circumferential direction in both Epi- and Endo-layers. We evaluated this on a large group of 193 subjects including 82 healthy control and 111 patients belonging to a great variety of pathological conditions. We found that Epicardial strain lines obliquely directed while those of Endocardium are almost circumferential. This result occurs irrespective of pathological condition. We propose that the geometric vinculum characterizing Endocardium and Epicardium in terms of different lever arm length and orientation of muscular fibers during contraction inescapably requires Endocardial strain lines to be circumferentially oriented and this is corroborated by experimental results. Further investigations on transmural structure of myocytes could couple results presented here in order to furnish additional experimental explanations.

Orientation, pattern center refinement and deformation state extraction through global optimization algorithms.

Global optimization algorithms have been adopted to the simultaneously refinement of orientation and pattern center for electron backscatter diffraction patterns as well as deformation state extraction. The hyperparameter space and mutation schemes of differential evolution (DE) algorithm has been thoroughly investigated and showed to be a more efficient algorithm than the particle swarm optimization (PSO) algorithm. The optimal hyperparameters for DE generally depend on conditions such as the number of variables to be optimized and the size of bounded search space but reasonably close initial values for crossover probability is 0.9, mutation factor is 0.5, population size is ten times the number of variables, and number of iterations is at least 100. Validation on a set of simulated undeformed single crystal nickel patterns reveals a mean accuracy of ≈0.03° and ≈0.01% detector width across a large field of view. In addition, validation using noisy simulated deformed patterns with known deformation state and pattern center shows that the mean accuracy of shear strain and rotation components is ≈0.001 and for the normal strain ≈0.002.

Higher-order singular value decomposition-based lung parcellation for breathing motion management.

Positron emission tomography (PET) imaging of the lungs is confounded by respiratory motion-induced blurring artifacts that degrade quantitative accuracy. Gating and motion-compensated image reconstruction are frequently used to correct these motion artifacts in PET. In the absence of voxel-by-voxel deformation measures, surrogate signals from external markers are used to track internal motion and generate gated PET images. The objective of our work is to develop a group-level parcellation framework for the lungs to guide the placement of markers depending on the location of the internal target region. We present a data-driven framework based on higher-order singular value decomposition (HOSVD) of deformation tensors that enables identification of synchronous areas inside the torso and on the skin surface. Four-dimensional (4-D) magnetic resonance (MR) imaging based on a specialized radial pulse sequence with a one-dimensional slice-projection navigator was used for motion capture under free-breathing conditions. The deformation tensors were computed by nonrigidly registering the gated MR images. Group-level motion signatures obtained via HOSVD were used to cluster the voxels both inside the volume and on the surface. To characterize the parcellation result, we computed correlation measures across the different regions of interest (ROIs). To assess the robustness of the parcellation technique, leave-one-out cross-validation was performed over the subject cohort, and the dependence of the result on varying numbers of gates and singular value thresholds was examined. Overall, the parcellation results were largely consistent across these test cases with Jaccard indices reflecting high degrees of overlap. Finally, a PET simulation study was performed which showed that, depending on the location of the lesion, the selection of a synchronous ROI may lead to noticeable gains in the recovery coefficient. Accurate quantitative interpretation of PET images is important for lung cancer management. Therefore, a guided motion monitoring approach is of utmost importance in the context of pulmonary PET imaging.

On modelling large deformations of heterogeneous biological tissues using a mixed finite element formulation.

This study addresses the issue of modelling material heterogeneity of incompressible bodies. It is seen that when using a mixed (displacement-pressure) finite element formulation, the basis functions used for pressure field may not be able to capture the nonlinearity of material parameters, resulting in pseudo-residual stresses. This problem can be resolved by modifying the constitutive relation using Flory's decomposition of the deformation gradient. A two-parameter Mooney-Rivlin constitutive relation is used to demonstrate the methodology. It is shown that for incompressible materials, the modification does not alter the mechanical behaviour described by the original constitutive model. In fact, the modified constitutive equation shows a better predictability when compared against analytical solutions. Two strategies of describing the material variation (i.e. linear and step change) are explained, and their solutions are evaluated for an ideal two-material interfacing problem. When compared with the standard tied coupling approach, the step change method exhibited a much better agreement because of its ability to capture abrupt changes of the material properties. The modified equation in conjunction with integration point-based material heterogeneity is then used to simulate the deformations of heterogeneous biological structures to illustrate its applications.