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A Hybrid Discrete-Continuum Mathematical Model of Pattern Prediction in the Developing Retinal Vasculature

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Abstract

Pathological angiogenesis has been extensively explored by the mathematical modelling community over the past few decades, specifically in the contexts of tumour-induced vascularisation and wound healing. However, there have been relatively few attempts to model angiogenesis associated with normal development, despite the availability of animal models with experimentally accessible and highly ordered vascular topologies: for example, growth and development of the vascular plexus layers in the murine retina. The current study aims to address this issue through the development of a hybrid discrete-continuum mathematical model of the developing retinal vasculature in neonatal mice that is closely coupled with an ongoing experimental programme. The model of the functional vasculature is informed by a range of morphological and molecular data obtained over a period of several days, from 6 days prior to birth to approximately 8 days after birth.

The spatio-temporal formation of the superficial retinal vascular plexus (RVP) in wild-type mice occurs in a well-defined sequence. Prior to birth, astrocytes migrate from the optic nerve over the surface of the inner retina in response to a chemotactic gradient of PDGF-A, formed at an earlier stage by migrating retinal ganglion cells (RGCs). Astrocytes express a variety of chemotactic and haptotactic proteins, including VEGF and fibronectin (respectively), which subsequently induce endothelial cell sprouting and modulate growth of the RVP. The developing RVP is not an inert structure; however, the vascular bed adapts and remodels in response to a wide variety of metabolic and biomolecular stimuli. The main focus of this investigation is to understand how these interacting cellular, molecular, and metabolic cues regulate RVP growth and formation.

In an earlier one-dimensional continuum model of astrocyte and endothelial migration, we showed that the measured frontal velocities of the two cell types could be accurately reproduced by means of a system of five coupled partial differential equations (Aubert et al. in Bull. Math. Biol. 73:2430–2451, 2011). However, this approach was unable to generate spatial information and structural detail for the entire retinal surface. Building upon this earlier work, a more realistic two-dimensional hybrid PDE-discrete model is derived here that tracks the migration of individual astrocytes and endothelial tip cells towards the outer retinal boundary. Blood perfusion is included throughout plexus development and the emergent retinal architectures adapt and remodel in response to various biological factors. The resulting in silico RVP structures are compared with whole-mounted retinal vasculatures at various stages of development, and the agreement is found to be excellent. Having successfully benchmarked the model against wild-type data, the effect of transgenic over-expression of various genes is predicted, based on the ocular-specific expression of VEGF-A during murine development. These results can be used to help inform future experimental investigations of signalling pathways in ocular conditions characterised by aberrant angiogenesis.

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Acknowledgements

The authors gratefully acknowledge financial support from the BBSRC: Grant # BB/F002254/1 and BB/F002807/1.

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Authors and Affiliations

  1. Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland, UK

    S. R. McDougall & M. G. Watson

  2. Biomedical Sciences Research Institute, University of Ulster, Coleraine, BT52 1SA, Northern Ireland, UK

    A. H. Devlin & C. A. Mitchell

  3. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK

    M. A. J. Chaplain

Authors
  1. S. R. McDougall
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  2. M. G. Watson
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  3. A. H. Devlin
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  4. C. A. Mitchell
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  5. M. A. J. Chaplain
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Corresponding author

Correspondence to M. G. Watson.

Appendix A

Appendix A

Table 1 Parameter values used in all base-case simulations
Full size table
Table 2 Parameter values used in Fig. 9 (all other relevant values unchanged)
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We require some further details to explain the process of EC and AC sprout branching. At the beginning of the simulation each initial sprout is set to have an age of zero, but this is supplemented by also assigning each sprout a random time point in its cell cycle. Upon each subsequent occurrence of branching, we assume that one new sprout maintains the direction of its parent sprout while the other sprout direction is chosen randomly. In the former, we set the age of the sprout, and its cell cycle position, to be zero. In the latter, we again set the sprout age to zero, but here we assign a random cell cycle position. We define an additional two parameters, namely a threshold age for branching (t branch=0.076 days) and a mitosis time describing the length of the cell cycle (t mitosis=0.709 days). In order for branching to occur, both the age of the parent sprout and its individual cell cycle time must exceed these critical values. In addition to this, the probabilities of AC and EC branching are related to the concentrations of PDGF and VEGF, respectively.

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McDougall, S.R., Watson, M.G., Devlin, A.H. et al. A Hybrid Discrete-Continuum Mathematical Model of Pattern Prediction in the Developing Retinal Vasculature. Bull Math Biol 74, 2272–2314 (2012). https://doi.org/10.1007/s11538-012-9754-9

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