Soliton approximation in continuum models of leader-follower behavior

Complex biological processes involve collective behavior of entities (bacteria, cells, animals) over many length and time scales and can be described by discrete models that track individuals or by continuum models involving densities and fields. We consider hybrid stochastic agent-based models of branching morphogenesis and angiogenesis (new blood vessel creation from preexisting vasculature), which treat cells as individuals that are guided by underlying continuous chemical and/or mechanical fields. In these descriptions, leader (tip) cells emerge from existing branches and follower (stalk) cells build the new sprout in their wake. Vessel branching and fusion (anastomosis) occur as a result of tip and stalk cell dynamics. Coarse graining these hybrid models in appropriate limits produces continuum partial differential equations (PDEs) for endothelial cell densities that are more analytically tractable. While these models differ in nonlinearity, they produce similar equations at leading order when chemotaxis is dominant. We analyze this leading order system in a simple quasi-one-dimensional geometry and show that the numerical solution of the leading order PDE is well described by a soliton wave that evolves from vessel to source. This wave is an attractor for intermediate times until it arrives at the hypoxic region releasing the growth factor. The mathematical techniques used here thus identify common features of discrete and continuum approaches and provide insight into general biological mechanisms governing their collective dynamics.

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.