Linking discrete and continuous models of cell birth and migration.

Self-organization of individuals within large collectives occurs throughout biology. Mathematical models can help elucidate the individual-level mechanisms behind these dynamics, but analytical tractability often comes at the cost of biological intuition. Discrete models provide straightforward interpretations by tracking each individual yet can be computationally expensive. Alternatively, continuous models supply a large-scale perspective by representing the 'effective' dynamics of infinite agents, but their results are often difficult to translate into experimentally relevant insights. We address this challenge by quantitatively linking spatio-temporal dynamics of continuous models and individual-based data in settings with biologically realistic, time-varying cell numbers. Specifically, we introduce and fit scaling parameters in continuous models to account for discrepancies that can arise from low cell numbers and localized interactions. We illustrate our approach on an example motivated by zebrafish-skin pattern formation, in which we create a continuous framework describing the movement and proliferation of a single cell population by upscaling rules from a discrete model. Our resulting continuous models accurately depict ensemble average agent-based solutions when migration or proliferation act alone. Interestingly, the same parameters are not optimal when both processes act simultaneously, highlighting a rich difference in how combining migration and proliferation affects discrete and continuous dynamics.

Comparative analysis of continuum angiogenesis models.

Although discrete approaches are increasingly employed to model biological phenomena, it remains unclear how complex, population-level behaviours in such frameworks arise from the rules used to represent interactions between individuals. Discrete-to-continuum approaches, which are used to derive systems of coarse-grained equations describing the mean-field dynamics of a microscopic model, can provide insight into such emergent behaviour. Coarse-grained models often contain nonlinear terms that depend on the microscopic rules of the discrete framework, however, and such nonlinearities can make a model difficult to mathematically analyse. By contrast, models developed using phenomenological approaches are typically easier to investigate but have a more obscure connection to the underlying microscopic system. To our knowledge, there has been little work done to compare solutions of phenomenological and coarse-grained models. Here we address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological "snail-trail" model for angiogenesis to solutions of a nonlinear system of partial differential equations (PDEs) derived via a systematic coarse-graining procedure (Pillay et al. in Phys Rev E 95(1):012410, 2017. https://doi.org/10.1103/PhysRevE.95.012410 ). For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to identical PDEs within the domain interior. Numerical and analytical results confirm that pointwise differences between solutions to the two continuum models are small if these conditions hold, and demonstrate how perturbation methods can be used to determine when a phenomenological model provides a good approximation to a more detailed coarse-grained system for the same biological process.

Evaluating snail-trail frameworks for leader-follower behavior with agent-based modeling.

Branched networks constitute a ubiquitous structure in biology, arising in plants, lungs, and the circulatory system; however, the mechanisms behind their creation are not well understood. A commonly used model for network morphogenesis proposes that sprouts develop through interactions between leader (tip) cells and follower (stalk) cells. In this description, tip cells emerge from existing structures, travel up chemoattractant gradients, and form new networks by guiding the movement of stalk cells. Such dynamics have been mathematically represented by continuum "snail-trail" models in which the tip cell flux contributes to the stalk cell proliferation rate. Although snail-trail models constitute a classical depiction of leader-follower behavior, their accuracy has yet to be evaluated in a rigorous quantitative setting. Here, we extend the snail-trail modeling framework to two spatial dimensions by introducing a novel multiplicative factor to the stalk cell rate equation, which corrects for neglected network creation in directions other than that of the migrating front. Our derivation of this factor demonstrates that snail-trail models are valid descriptions of cell dynamics when chemotaxis dominates cell movement. We confirm that our snail-trail model accurately predicts the dynamics of tip and stalk cells in an existing agent-based model (ABM) for network formation [Pillay et al., Phys. Rev. E 95, 012410 (2017)10.1103/PhysRevE.95.012410]. We also derive conditions for which it is appropriate to use a reduced, one-dimensional snail-trail model to analyze ABM results. Our analysis identifies key metrics for cell migration that may be used to anticipate when simple snail-trail models will accurately describe experimentally observed cell dynamics in network formation.

Variation in the Medial Patellofemoral Ligament Origin in the Skeletally Immature Knee: An Anatomic Study.

BACKGROUND: The medial patellofemoral ligament (MPFL) is frequently reconstructed to treat recurrent patellar instability. The femoral origin of the MPFL is well described in adults but not in the skeletally immature knee. PURPOSE: To identify a radiographic landmark for the femoral MPFL attachment in the skeletally immature knee and study its relationship to the distal femoral physis. STUDY DESIGN: Descriptive laboratory study. METHODS: Thirty-six cadaveric specimens between 2 and 11 years old were dissected and examined (29 male and 7 female). Metallic markers were placed at the proximal and distal borders of the MPFL femoral origin footprint. Computed tomography scans with 0.625-mm slices in the axial, coronal, and sagittal planes were used to measure the maximum ossified height and ossified depth. The measurements were used to describe the position of the midpoint MPFL attachment with respect to the posterior-anterior and distal-proximal dimensions of the femoral condyle on the sagittal view and to describe the distance from the physis to the femoral origin of the MPFL. RESULTS: In 23 of 36 specimens, the femoral origin of the MPFL was distal to the physis. Thirteen of the 36 specimens had an MPFL origin at or proximal to the physis, with a more proximal MPFL origin consistently seen in older specimens. The distance of the MPFL origin to the physis ranged from 15.1 mm distal to the physis to 8.3 mm proximal to the physis. The mean midpoint of the MPFL femoral origin was located 3.0 ± 5.5 mm distal to the physis for all specimens. For specimens aged <7 years, the mean MPFL origin was 4.7 mm distal to the physis, and for specimens aged ≥7 years, the mean MPFL origin was 0.8 mm proximal to the femoral physis. The MPFL origin was more proximal and anterior for those aged ≥7 years and more distal and posterior for those aged <7 years. CONCLUSION: Surgical reconstruction of the MPFL is a common treatment to restore patellar stability. There appears to be significant variability in the origin of the MPFL in skeletally immature specimens. This study demonstrated that the MPFL origin was more proximal and anterior with respect to the physis in the older age group. The MPFL origin footprint may be customized for different age groups. CLINICAL RELEVANCE: This information shows anatomic variation of the MPFL origin with age, with older specimens having a footprint that was more proximal and anterior than younger specimens. Customization of the surgical technique might be considered based on patient age.