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. 2016 Feb;8(2):230-242.
doi: 10.1039/c5ib00270b. Epub 2016 Jan 29.

Precisely Parameterized Experimental and Computational Models of Tissue Organization

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Free PMC article

Precisely Parameterized Experimental and Computational Models of Tissue Organization

Jared M Molitoris et al. Integr Biol (Camb). .
Free PMC article

Abstract

Patterns of cellular organization in diverse tissues frequently display a complex geometry and topology tightly related to the tissue function. Progressive disorganization of tissue morphology can lead to pathologic remodeling, necessitating the development of experimental and theoretical methods of analysis of the tolerance of normal tissue function to structural alterations. A systematic way to investigate the relationship of diverse cell organization to tissue function is to engineer two-dimensional cell monolayers replicating key aspects of the in vivo tissue architecture. However, it is still not clear how this can be accomplished on a tissue level scale in a parameterized fashion, allowing for a mathematically precise definition of the model tissue organization and properties down to a cellular scale with a parameter dependent gradual change in model tissue organization. Here, we describe and use a method of designing precisely parameterized, geometrically complex patterns that are then used to control cell alignment and communication of model tissues. We demonstrate direct application of this method to guiding the growth of cardiac cell cultures and developing mathematical models of cell function that correspond to the underlying experimental patterns. Several anisotropic patterned cultures spanning a broad range of multicellular organization, mimicking the cardiac tissue organization of different regions of the heart, were found to be similar to each other and to isotropic cell monolayers in terms of local cell-cell interactions, reflected in similar confluency, morphology and connexin-43 expression. However, in agreement with the model predictions, different anisotropic patterns of cell organization, paralleling in vivo alterations of cardiac tissue morphology, resulted in variable and novel functional responses with important implications for the initiation and maintenance of cardiac arrhythmias. We conclude that variations of tissue geometry and topology can dramatically affect cardiac tissue function even if the constituent cells are themselves similar, and that the proposed method can provide a general strategy to experimentally and computationally investigate when such variation can lead to impaired tissue function.

Figures

Fig. 1
Fig. 1
Appropriate selection of the initial vector field, the seed pattern and the propagation rules, determines the type of pattern generated. (A) The same circular geometry can give rise to diverse vector fields based on the initial conditions of the simulation, including the positioning of the sources, e.g., at the center (left panel), leading to radially expanding flowlines, or along the radius (right panel), leading to concentrically curved flowlines. (B) The role of the seed pattern is illustrated in the generation of the final pattern. In the left panel, a set of bands (shown in red) and gaps (region between the bands) on the circumference of a small initial circle, form the initial conditions that give rise to a starburst pattern, indicated by the grey bands. The band propagation is indicated by the blue arrows. In the right panel, propagation of the initial conditions defined by alternating bands and gaps along the radius of the circle results in a concentric pattern; (C) the resulting patterns can be used for further analysis of the role of cell patterning. e.g. They can be translated into microfabricated masks and ultimately cell patterns, or be used as inputs into computational models. See additional information in the ESI.†
Fig. 2
Fig. 2
Varying the seed pattern and the rules of seed pattern propagation can result in distinct patterns that are based on the same vector field, with further diversity in patterns introduced by generating different underlying vector fields. (A) Direction of simulated fluid flow in a circular region resulting from the initial condition specification, i.e., a source at the center of the circle and a sink at the circumference. (B) The vector field defined by this flow can be used to generate several different patterns. Branching is allowed for both bands and gaps in (C) and (D), but only allowed for the bands in (E) and (F). In (E), the gaps are additionally constrained to not grow beyond a certain maximum width, while in (F) the gaps can grow in an unconstrained fashion. Additionally, varying the relative widths of the bands and gaps in the initial conditions, and the fractional widths of newly generated bands or gaps following a branching, are also conditions that give rise to a diverse range of patterns in (C–F). Changing the boundary conditions during the generation of the vector field, e.g., the direction of the flow to be non-normal to the perimeter of the patterned region, can result in a spiral pattern (G). By defining more than one source, e.g., a line source at the bottom edge of the rectangle and a point source in the center of the rectangle in (H), or by defining a source and a sink as two singularities within the rectangle (I), it is possible to further increase the pattern diversity. As shown in (I), one can define both the emergence and the fusion of adjoining bands (or gaps) depending on whether the local flow lines diverge or converge. See additional information in the ESI.†
Fig. 3
Fig. 3
Schematic of interplay between modeling and experimental approaches. The vector field is the basis for the generation of a desired pattern of bands and gaps, and can be used to generate a patterned substratum of extracellular matrix, e.g., through microcontact printing. Cells are then patterned on this surface, and biologically relevant experiments, such as electrophysiological studies, are performed. Simultaneously, the vector field is also the basis for the generation of a computational model, which is used to simulate functional consequences of alteration in the model tissue organization, e.g., the electrical conduction. Mutual refinement of the model and experimental design results in a better understanding of the system.
Fig. 4
Fig. 4
Gradual morphing of patterns resulting from variation of the simulation region and the boundary conditions. Vector fields with overlaid boundary conditions, where the red and blue boundaries are the source and sink respectively (above) and their resulting patterns of bands and gaps (below) are displayed. A linear (striped) pattern is generated based on the two horizontal boundaries acting as a source and sink (A). Gradual rotation of the source and sink boundaries as well as the simulation region leads to a semi-circular pattern with concentric rings (B) and eventually a circular pattern with concentric rings (C). The concentric ring pattern can also be generated as the limiting case of a family of patterns, where the source is a small inner circle, the sink is a larger outer circle and the direction of flow is constrained to be at an angle to the sink boundary (tangential for concentric circles). Using these source and sink boundaries, and by progressive variation of the direction of the sink boundary flux from tangential towards normal, we can generate spiral patterns of different shapes (D and E). The black curved arrows next to the flow fields are meant to be a visual aid to represent the degree of curvature of the flow lines (their length is not correlated with actual curl values). When the boundary flux at the sink is normal to the boundary, it leads to a ‘starburst’ pattern (F). The asterisks on (A), (C) and (F) indicate that these vector flows can be used to generate the linear, concentric and starburst patterns, similar to those used in further experimental and computational studies. The starburst pattern in Fig. 2E represents the pattern used for experimental studies.
Fig. 5
Fig. 5
Characterization of fibronectin transfer and culture properties in isotropic and anisotropic NRVM monolayers. Fibronectin immunostaining (left column), merged cTnI and DAPI (middle column), and merged actin and Cx43 (right column) images of isotropic (A–C), linear (D–F), concentric (G–I), and starburst (J–L) patterned cultures reveal similar cellular morphology, myocyte composition, and levels of Cx43 expression. The linear, concentric, and starburst patterns were constructed by designing microcontact printing masks. Scale bar = 50 μm.
Fig. 6
Fig. 6
Analysis and quantification of cellular alignment with their underlying fibronectin patterns. A Sobel filter was used to assign an average orientation for 36 sub-regions for each pattern type based on the orientation of the actin fibrils within the cells or the fibronectin lines. These orientations are indicated by white lines on the patterned cell and fibronectin images. Within each sub-region the orientation from the cell image was then laid on top of the orientation from the fibronectin image into a merged image showing that the actin orientation within the cells qualitatively follows the direction of the underlying fibronectin patterns, which are based on the orientation of their underlying vector fields. To quantitatively prove this, the orientation of each cell sub-region was plotted against that of the fibronectin patterns (top graph), the difference between them was plotted (bottom graph), and a t-test was used to show that the averages of these differences in orientation were not statistically significant for any of the anisotropic patterns (n.s. = not significant).
Fig. 7
Fig. 7
Characterization of the conduction properties of isotropic and anisotropic NRVM cultures and optimization of the corresponding computational model. Representative isochronal maps of AP propagation (rightmost panels) in isotropic (A) and linear anisotropic (B) monolayers are matched up with the modeling predictions (middle panels) based on the model fiber orientation (leftmost panels), and model parameters are adjusted to match the experimental measurements. In the isochronal maps, the selected paths of wavefront propagation along which CV was measured are shown by black arrows, and their corresponding values are displayed. The color scales indicate the time values measured in milliseconds. The optimized model was then used for prediction of conduction properties in more complex cellular patterns (Fig. 8), based on alternate vector fields.
Fig. 8
Fig. 8
Contrasting modeling predictions and experimental analysis, based on the initial characterization in Fig. 7. (A) and (F) The geometry of the concentric and starburst fiber patterns based on the vector fields used to generate the corresponding patterns in Fig. 5; (B) and (G) simulations of action potential propagation following a point stimulus at the center of each monolayer pattern shown in (A) and (F) respectively; (C) and (H) simulations of action potential propagation following a point stimulus delivered off-center in each monolayer pattern shown in (A) and (F) respectively; (D) and (I) experimental validation of the predictions in (B) and (G) respectively, using the experimentally constructed patterns, as shown in Fig. 5; (E) and (J) experimental validation of the predictions in (C) and (H) respectively. In the experimental isochronal maps, the selected paths of wavefront propagation along which CV was calculated are shown by black arrows, and their corresponding values are displayed. The color scales indicate time values measured in seconds.
Fig. 9
Fig. 9
Bar-plot of average experimental conduction velocities for each pattern. All values were statistically different from the isotropic ICV. Similarities were noted between linear TCV and LCV and their respective concentric and starburst CVs at regions away from the center of the patterns. These similarities also applied to the anisotropy ratio of CV (AR) for all three anisotropic patterns as discussed in the text.

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