MOLEGRO · A stochastic model of eye lens growth with implications for cortical cataract formation

A stochastic model of eye lens growth with implications for cortical cataract formation

The purpose of the ocular lens is to focus light sharply onto the retina. To perform this demanding task, the lens must be built precisely, remain transparent throughout life, and retain its flexibility. The lens is a geometrically simple structure (an oblate spheroid) composed of only two types of cells (epithelial cells on the surface of the anterior capsule, and elongated fiber cells within the body of the lens). There is no cell turnover in this system, and archived cells are deposited, layer upon layer, in the fiber cell compartment. One unusual and defining property of the lens is that it grows continuously throughout life. The growth process determines its size and shape (and hence, its refractive properties) and may also influence the development of lens pathologies (cataract). The complexity of biological structures has generally confounded a rigorous mathematical approach to understanding organogenesis. This is particularly true when we attempt to model the growth process over extended time frames (i.e., the entire lifespan). The simple geometric form of the lens provides an opportunity to circumvent these difficulties and gain important insights into the organ building process. The current Marie Curie IOF project established a close collaboration between two groups of researchers. On the biomedical side are members of Dr. Steven Bassnett’s laboratory at the School of Medicine at Washington University in St. Louis, USA. Dr. Bassnett is an internationally recognized leader in the field of lens cell biology. The lab is in possession of excellent experimental data sets consisting of large volumetric images, multiple time-point lineage traces, and region-specific mitotic index measurements. The mathematical expertise of this interdisciplinary collaboration is contributed by the group led by Dr. Hrvoje Sikic, from the Probability and Statistics division of the Department of Mathematics, Faculty of Science, University of Zagreb, Croatia. The Probability and Statistics division is internationally recognized for its contributions to the potential theory of stochastic processes and harmonic analysis. The goal of this long term collaboration is to develop the first ever mathematical model of lens growth. The project was completed on August 31, 2017. After an initial period of consultation, which predates the official start of the project, the core of basic ideas was formed and presented at several international conferences (Kailua-Kona 2012, Hawaii; ARVO 2012, Ft. Lauderdale, FL; Berlin 2012, Germany). The basic idea underlying the growth model (that the process is governed by random divisions of epithelial cells and the geometry of the lens such that the number of cells which are “pushed” into the fiber compartment is determined by the increment in the volume growth) was presented in a 20-minute invited lecture at the 2014 meeting of the Association for Research in Vision and Ophthalmology (ARVO) meeting in Orlando, FL. The realization that this “penny pusher” model is accompanied by the zonal structure of the epithelium (consisting of four zones in the case of the mouse lens) and the change in the footprint area of individual epithelial cells, which is crucial for the dynamics among the cells within a single cell-cycle period, has been established and published in an article in the leading eye research journal (IOVS, 2015). The article attracted attention of several scientific newspapers (among others it was reported in Science Daily, ARVO Media Release, Cataract News Today). Having published biological aspects of lens growth, we turned our attention to the development of a fully detailed mathematical model. We emphasize that the model is not of a statistic-numerical nature (i.e. it is not a “fitting data into a curve” type model). Rather, it flows from a carefully developed axiomatic system, which starts from basic biological principles, and builds subsequent properties based on precise logic and exact mathematical computations. The model was developed on data collected from the mouse lens and then tested against growth measurements from a relatively narrow time period (from 4-12 weeks of age). This allowed us to keep constant some of the key model parameters. The results were published in one of the two leading biomath modeling journals (J Theor Biol, 2015). This is an important accomplishment for the project, since it represents the first ever mathematical model of the growth process of the eye lens. Since then we have collected a complete growth data set that spans the entire life of a mouse (from the prenatal period to the end of life). These data were used to developed a full lifespan model, with various specifics that are observed in the early life (shortened cell-cycle, explosive growth, build-up of the zonal structure, control of proliferation rate and footprint area in various zones, possible pathologies that result from the early growth issues), the middle life (slowing down of the growth process, overshoot in the number of cells, intricacies of the “two pedal” governing system, which combines proliferation rate with footprint area control, flow of cells in the epithelium), and the late life (compaction of fiber cells, stabilization of the “two pedal” system, reverse flow of cells and connections with possible cataract effects). We have published two major papers (Roy Soc Open Sci 2017, Progress in Ret Eye 2017) on this topic, which in addition to providing insights into the lens growth process also introduces concepts that are likely to be important in the growth of any organ system (for example, precision issues, coefficient of variation, isolation of unwanted cells). We have also begun to model the spatial movement of individual epithelial cells and their progeny. Some of the results were presented through posters and lectures at several recent conferences (Hawaii 2015, ARVO 2015, ARVO 2016; at the last conference Professor Bassnett received an international prize for his contributions to the eye research, which quotes the “penny pusher” model as one of the fundamental contributions to the field). Professor Sikic has also developed a collaboration on some related mathematical topics (MRA structure, low frequencies vs. high frequencies); a lengthy paper (over 150 pages of new material) has been completed. During the third year of the project Professor Sikic presented our results throughout Croatia and Europe. In addition to the research lectures, he also lectured a year-long course on biomath modeling to doctoral students at the University of Zagreb, and an interdisciplinary faculty committee was formed at the University of Zagreb to develop a new masters degree program in biomedical mathematics. We also organized “Workshop on Quantitative Modeling in Biomedicine”, June 5-7, 2017, at the Faculty of Science, University of Zagreb, which also marked the conclusion of the project. The workshop was reported on in Croatian media.

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