Mathematical and computational models of the retina in health, development and disease

Optimization-Based Pairwise Interaction Point Process (O-PIPP): A Precise and Universal Retinal Mosaic Modeling Approach

Purpose

A retinal mosaic, the spatial organization of a population of homotypic neurons, is thought to sample a specific visual feature into the feedforward visual pathway. The purpose of this study was to propose a universal modeling approach for precisely generating retinal mosaics and overcoming the limitations of previous models, especially in modeling abnormal mosaic patterns under disease conditions.

Methods

Here, we developed the optimization-based pairwise interaction point process (O-PIPP). It incorporates optimization techniques into previous simulation approaches, enabling directional control of the simulation process according to the user-designed optimization target. For the convenience of the community, we implemented the O-PIPP approach into a Python package and a website application.

Results

We showed that the O-PIPP can generate more precise neural spatial patterns of healthy and diseased mosaics compared to previous phenomenological approaches. Notably, through modeling the retinal neural circuitry with O-PIPP-simulated retinitis pigmentosa cone mosaics, we elucidated how the cone mosaic rearrangement impacted the information processing of ganglion cells.

Conclusions

The O-PIPP provides a precise and universal tool to simulate realistic mosaics, which could help to investigate the function of retinal mosaics in vision.

Spatially-dependent model for rods and cones in the retina

We develop a mathematical model for photoreceptors in the retina. We focus on rod and cone outer segment dynamics and interactions with a nutrient source associated with the retinal pigment epithelium cells. Rod and cone densities (number per unit area of retinal surface) are known to have significant spatial dependence in the retina with cones located primarily near the fovea and the rods located primarily away from the fovea. Our model accounts for this spatial dependence of the rod and cone photoreceptor density as well as for the possibility of nutrient diffusion. We present equilibrium and dynamic solutions, discuss their relation to existing models, and estimate model parameters through comparisons with available experimental measurements of both spatial and temporal photoreceptor characteristics. Our model compares well with existing data on spatially-dependent regrowth of photoreceptor outer segments in the macular region of Rhesus Monkeys. Our predictions are also consistent with existing data on the spatial dependence of photoreceptor outer segment length near the fovea in healthy human subjects. We focus primarily on the healthy eye but our model could be the basis for future efforts designed to explore various retinal pathologies, eye-related injuries, and treatments of these conditions.

Homeostasis at different backgrounds: The roles of overlayed feedback structures in vertebrate photoadaptation

We have studied the resetting behavior of eight basic integral controller motifs with respect to different but constant backgrounds. We found that the controllers split symmetrically into two classes: one class, based on derepression of the compensatory flux, leads to more rapid resetting kinetics as backgrounds increase. The other class, which directly activates the compensatory flux, shows a slowing down in the resetting at increased backgrounds. We found a striking analogy between the resetting kinetics of vertebrate photoreceptors and controllers based on derepression, i.e. vertebrate rod or cone cells show decreased sensitivities and accelerated response kinetics as background illuminations increase. The central molecular model of vertebrate photoadaptation consists of an overlay of three negative feedback loops with cytosolic calcium ([Formula: see text]), cyclic guanosine monophosphate (cGMP) and cyclic nucleotide-gated (CNG) channels as components. While in one of the feedback loops the extrusion of [Formula: see text] by potassium-dependent sodium-calcium exchangers (NCKX) can lead to integral control with cGMP as the controlled variable, the expected robust perfect adaptation of cGMP is lost, because of the two other feedback loops. They avoid that [Formula: see text] levels become too high and toxic. Looking at psychophysical laws, we found that in all of the above mentioned basic controllers Weber's law is followed when a "just noticeable difference" (threshold) of 1% of the controlled variable's set-point was considered. Applying comparable threshold pulses or steps to the photoadaptation model we find, in agreement with experimental results, that Weber's law is followed for relatively high backgrounds, while Stephens' power law gives a better description when backgrounds are low. Limitations of our photoadaption model, in particular with respect to potassium/sodium homeostasis, are discussed. Finally, we discuss possible implication of background perturbations in biological controllers when compensatory fluxes are based on activation.

A mathematical model of GLUT1 modulation in rods and RPE and its differential impact in cell metabolism

We present a mathematical model of key glucose metabolic pathways in two cells of the human retina: the rods and the retinal pigmented epithelium (RPE). Computational simulations of glucose transporter 1 (GLUT1) inhibition in the model accurately reproduce experimental data from conditional knockout mice and reveal that modification of GLUT1 expression levels of both cells differentially impacts their metabolism. We hypothesize that, under glucose scarcity, the RPE's energy producing pathways are altered in order to preserve its functionality, impacting the photoreceptors' outer segment renewal. On the other hand, when glucose is limited in the rods, aerobic glycolysis is preserved, which maintains the lactate contribution to the RPE.

Self-Organization of the Retina during Eye Development, Retinal Regeneration In Vivo, and in Retinal 3D Organoids In Vitro

Self-organization is a process that ensures histogenesis of the eye retina. This highly intricate phenomenon is not sufficiently studied due to its biological complexity and genetic heterogeneity. The review aims to summarize the existing central theories and ideas for a better understanding of retinal self-organization, as well as to address various practical problems of retinal biomedicine. The phenomenon of self-organization is discussed in the spatiotemporal context and illustrated by key findings during vertebrate retina development in vivo and retinal regeneration in amphibians in situ. Described also are histotypic 3D structures obtained from the disaggregated retinal progenitor cells of birds and retinal 3D organoids derived from the mouse and human pluripotent stem cells. The review highlights integral parts of retinal development in these conditions. On the cellular level, these include competence, differentiation, proliferation, apoptosis, cooperative movements, and migration. On the physical level, the focus is on the mechanical properties of cell- and cell layer-derived forces and on the molecular level on factors responsible for gene regulation, such as transcription factors, signaling molecules, and epigenetic changes. Finally, the self-organization phenomenon is discussed as a basis for the production of retinal organoids, a promising model for a wide range of basic scientific and medical applications.

Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis

Retinitis pigmentosa (RP) is the most common inherited retinal dystrophy with a prevalence of about 1 in 4,000, affecting approximately 1.5 million people worldwide. Patients with RP experience progressive visual field loss as the retina degenerates, destroying light-sensitive photoreceptor cells (rods and cones), with rods affected earlier and more severely than cones. Spatio-temporal patterns of retinal degeneration in human RP have been well characterised; however, the mechanism(s) giving rise to these patterns have not been conclusively determined. One such mechanism, which has received a wealth of experimental support, is described by the trophic factor hypothesis. This hypothesis suggests that rods produce a trophic factor necessary for cone survival; the loss of rods depletes this factor, leading to cone degeneration. In this article, we formulate a partial differential equation mathematical model of RP in one spatial dimension, spanning the region between the retinal centre (fovea) and the retinal edge (ora serrata). Using this model we derive and solve an inverse problem, revealing for the first time experimentally testable conditions under which the trophic factor mechanism will qualitatively recapitulate the spatio-temporal patterns of retinal regeneration observed in human RP.

Mathematical Models of Retinitis Pigmentosa: The Trophic Factor Hypothesis

Retinitis pigmentosa (RP) is the term used to denote a group of inherited retinal-degenerative conditions that cause progressive sight loss. Individuals with this condition lose their light-sensitive photoreceptor cells, known as rods and cones, over a period of years to decades; degeneration starting in the retinal periphery, and spreading peripherally and centrally over time. RP is a rod-cone dystrophy, meaning that rod health and function are affected earlier and more severely than that of cones. Rods degenerate due to an underlying mutation, whereas the reasons for cone degeneration are unknown. A number of mechanisms have been proposed to explain secondary cone loss and the spatio-temporal patterns of retinal degeneration in RP. One of the most promising is the trophic factor hypothesis, which suggests that rods produce a factor necessary for cone survival, such that, when rods degenerate, cone degeneration follows. In this paper we formulate and analyse mathematical models of human RP under the trophic factor hypothesis. These models are constructed as systems of reaction-diffusion partial differential equations in one spatial dimension, and are solved and analysed using a combination of numerical and analytical methods. We predict the conditions under which cones will degenerate following the loss of a patch of rods from the retina, the critical trophic factor treatment rate required to prevent cone degeneration following rod loss and the spatio-temporal patterns of cone loss that would result if the trophic factor mechanism alone were responsible for retinal degeneration.

The role of RdCVFL in a mathematical model of photoreceptor interactions

Recent experimental and mathematical work has shown the interdependence of the rod and cone photoreceptors with the retinal pigment epithelium in maintaining sight. Accelerated intake of glucose into the cones via the theoredoxin-like rod-derived cone viability factor (RdCVF) is needed as aerobic glycolysis is the primary source of energy production. Reactive oxidative species (ROS) result from the rod and cone metabolism and recent experimental work has shown that the long form of RdCVF (RdCVFL) helps mitigate the negative effects of ROS. In this work we investigate the role of RdCVFL in maintaining the health of the photoreceptors. The results of our mathematical model show the necessity of RdCVFL and also demonstrate additional stable modes that are present in this system. The sensitivity analysis shows the importance of glucose uptake, nutrient levels, and ROS mitigation in maintaining rod and cone health in light-damaged mouse models. Together, these suggests areas on which to focus treatment in order to prolong the photoreceptors, especially in situations where ROS is a contributing factor to their death such as retinitis pigmentosa.

Anomalous Angiogenesis in Retina

Age-related macular degeneration (AMD) may cause severe loss of vision or blindness, particularly in elderly people. Exudative AMD is characterized by the angiogenesis of blood vessels growing from underneath the macula, crossing the blood–retina barrier (which comprises Bruch’s membrane (BM) and the retinal pigmentation epithelium (RPE)), leaking blood and fluid into the retina and knocking off photoreceptors. Here, we simulate a computational model of angiogenesis from the choroid blood vessels via a cellular Potts model, as well as BM, RPE cells, drusen deposits and photoreceptors. Our results indicate that improving AMD may require fixing the impaired lateral adhesion between RPE cells and with BM, as well as diminishing Vessel Endothelial Growth Factor (VEGF) and Jagged proteins that affect the Notch signaling pathway. Our numerical simulations suggest that anti-VEGF and anti-Jagged therapies could temporarily halt exudative AMD while addressing impaired cellular adhesion, which could be more effective over a longer time-span.

Physiological constraints of visual pathway lead to more efficient coding of information in retina

Nowadays, numerous studies have investigated the modeling of efficient neural encoding processes in the retina of the eye to encode the sensory data. Retina, as the innermost coat of the eye, is the first and the most important area of the visual neural system of mammalians, which is responsible for neural processes. Retina encodes the information of light intensity into a sequence of spikes, and sends them to retinal ganglion cells (RGCs) for further processing. An appropriate retinal encoding model should be adapted to the real retina as much as possible by considering the physiological constraints of the visual pathway to transfer most of the information of the input signal to the brain without too much redundancy of the channel. In this paper, inspired from the existing linear models of retinal encoding process, which have employed input noise and the spatial locality of the RGCs receptive fields (RFs) in the calculation of the encoding matrix, two extra physiological constraints, adapted from the real retina are taken into account so as to achieve a more realistic model for themammalian retina. These new constraints that are the correlation between RGCs and the spatial locality of the photoreceptors' projective fields (PFs), are modeled in a mathematical form and analyzed in detail. To quantify fidelity of the proposed encoding matrix and prove its superiority over existing models, various parameters of the models are calculated and presented in this paper: mean square error between the original and reconstructed image (MSE), the redundancy of the channel, the amount of information transferred through the channel, and the amount of wasted capacity for carrying input noise, to name a few. The results of these calculations show that the proposed model transfers input information with less redundancy of the channel. In other words, it reduces a portion of channel capacity which is wasted for carrying the input noise in comparison to the existing models. Also, due to considering extra physiological constraints in the proposed model, it is acceptable to have a slightly higher amount of MSE value in order to become similar to the real retina.

Cholesterol Regulation in Age-Related Macular Degeneration: A Framework for Mathematical Modelling of Drusen Biogenesis

In age-related macular degeneration (AMD), there is, in common with many other age-related diseases, the need to distinguish between changes in the ageing eye that lead to disease and those changes that are considered part of a healthy, ageing eye. Various studies investigating the multitude of mechanisms involved in the aetiology of AMD exist within the field of ophthalmology and related medical fields, yet many aspects of it remain poorly understood and only a limited number of therapies are available. A recent study relates drusen's topographically cellular characteristics to the neural retina's metabolic needs and associated cholesterol involvement within the retina. In particular, there is a need to fully understand the maintenance of cholesterol homeostasis in the retina to prevent normal ageing processes from being perturbed towards maculopathy. Here, we present an extensive review of the clinical and physiological features of the ageing retina, as well as mechanisms implicated in pathology, synthesised from a vast body of the published literature. We use this novel synthesis to construct a comprehensive process schematic, encompassing all key species and physiological processes such as nutrients, waste and lipoprotein management. We are therefore able to express these processes in a mathematical language via a comprehensive modelling framework, comprising a set of twenty-three equations spanning three distinct biological compartments. This very general modelling framework may now be adapted to more focused studies on individual mechanisms, processes or components underlying of the many facets of AMD. As an example of such a focused application, we conclude this article with a one-compartment, four-species model of the retinal pigment epithelium, which considers the parametric conditions under which either cholesterol homeostasis or unregulated accumulation of cholesterol may obtain in the ageing eye.

Mathematical models for the effect of anti-vascular endothelial growth factor on visual acuity

The standard of care treatment for neovascular age-related macular degeneration, delivered as ocular injection, is based on anti-vascular endothelial growth factor (anti-VEGF). The course of treatment may need to be modified quickly for certain patients based on their response. Models that track both the concentration and the response to an anti-VEGF treatment are presented. The specific focus is to assess the existence of analytical solutions for the different types of models. Both an ODE-based model and a map-based model illustrate the dependence of the solution on various biological parameters and allow the measurement of patient-specific parameters from experimental data. A PDE-based model incorporates diffusive effects. The results are consistent with observed values, and could provide a framework for practitioners to understand the effect of the therapy on the progression of the disease in both responsive and non-responsive patients.

Hif1a and Hif2a can be safely inactivated in cone photoreceptors

Impaired tissue oxygenation results in hypoxia and leads to the activation of hypoxia-inducible transcription factors (HIF). A chronic, HIF-triggered molecular response to hypoxia may be an important factor in the etiology of age-related macular degeneration (AMD) and is likely activated before any clinical manifestation of the disease. Thus, HIF1 and HIF2 recently emerged as potential therapeutic targets for AMD. To address and evaluate potential consequences of anti-HIF therapies for retinal physiology and function, we generated mouse lines that have Hif1a, or both Hif1a and Hif2a ablated specifically in cone photoreceptors. The knockdown of Hifs in cones did not cause detectable pathological alterations such as loss of cone photoreceptors, retinal degeneration or abnormalities of the retinal vasculature, had no impact on retinal function and resulted in a similar tolerance to hypoxic exposure. Our data indicate that HIF transcription factors are dispensable for maintaining normal cone function and survival in retinas of adult mice. This study provides the groundwork necessary to establish safety profiles for strategies aiming at antagonizing HIF1A and HIF2A function in cone photoreceptors for the treatment of retinal degenerative diseases that involve a hypoxic component such as AMD.

Computational modeling of retinal hypoxia and photoreceptor degeneration in patients with age-related macular degeneration

Although drusen have long been acknowledged as a primary hallmark of dry age-related macular degeneration (AMD) their role in the disease remains unclear. We hypothesize that drusen accumulation increases the barrier to metabolite transport ultimately resulting in photoreceptor cell death. To investigate this hypothesis, a computational model was developed to evaluate steady-state oxygen distribution in the retina. Optical coherence tomography images from fifteen AMD patients and six control subjects were segmented and translated into 3D in silico representations of retinal morphology. A finite element model was then used to determine the steady-state oxygen distribution throughout the retina for both generic and patient-specific retinal morphology. Oxygen levels were compared to the change in retinal thickness at a later time point to observe possible correlations. The generic finite element model of oxygen concentration in the retina agreed closely with both experimental measurements from literature and clinical observations, including the minimal pathological drusen size identified by AREDS (64 μm). Modeling oxygen distribution in the outer retina of AMD patients showed a substantially stronger correlation between hypoxia and future retinal thinning (Pearson correlation coefficient, r = 0.2162) than between drusen height and retinal thinning (r = 0.0303) indicating the potential value of this physiology-based approach. This study presents proof-of-concept for the potential utility of finite element modeling in evaluating retinal health and also suggests a potential link between transport and AMD pathogenesis. This strategy may prove useful as a prognostic tool for predicting the clinical risk of AMD progression.

RDH12 Mutations Cause a Severe Retinal Degeneration With Relatively Spared Rod Function

Purpose

To describe the retinal phenotype of pediatric patients with mutations in the retinol dehydrogenase 12 (RDH12) gene.

Methods

Twenty-one patients from 14 families (ages 2-17 years) with RDH12-associated inherited retinal degeneration (RDH12-IRD) underwent a complete ophthalmic exam and imaging with spectral domain optical coherence tomography (SD-OCT) and near infrared and short-wavelength fundus autofluorescence. Visual field extent was measured with Goldmann kinetic perimetry, visual thresholds with dark-adapted static perimetry or with dark-adapted chromatic full-field stimulus testing (FST) and transient pupillometry.

Results

Visual acuity ranged from 20/40 to light perception. There was parafoveal depigmentation or atrophic maculopathies accompanied by midperipheral intraretinal pigment migration. SD-OCT revealed foveal thinning in all patients and detectable but thinned outer nuclear layer (ONL) at greater eccentricities from the fovea. Photoreceptor outer segment (POS) signals were only detectable in small pockets within the central retina. Measurable kinetic visual fields were limited to small (<5-10°) central islands of vision. Electroretinograms were reported as undetectable or severely reduced in amplitude. FST sensitivities to a 467 nm stimulus were rod-mediated and reduced on average by ∼2.5 log units. A thinned central ONL colocalized with severely reduced to nondetectable cone-mediated sensitivities. Pupillometry confirmed the psychophysically measured abnormalities.

Conclusions

RDH12-IRD causes an early-onset, retina-wide disease with particularly severe central retinal abnormalities associated with relatively less severe rod photoreceptor dysfunction, a pattern consistent with an early-onset cone-rod dystrophy. Severely abnormal POS but detectable ONL in the pericentral and peripapillary retina suggest these regions may become targets for gene therapy.

Modeling and simulation of phototransduction cascade in vertebrate rod photoreceptors

Background

The activation of phototransduction cascade in rod photoreceptors has been well studied in literature, but there is a lack of a mature kinetic model structure covering both the activation and inactivation processes.

Methods

In this work, a kinetic model structure is developed to describe the major activation and inactivation processes in vertebrate rod photoreceptors with the electroretinogram (ERG) as output. Simulation was performed to validate developed model structure.

Results

The developed model structure could fit experimental data with small error.

Conclusions

The result indicated that the developed model structure could show the inactivation process of phototransduction cascades in the rod photoreceptors.

Wogonin induces retinal neuron-like differentiation of bone marrow stem cells by inhibiting Notch-1 signaling

Age-related macular degeneration and retinitis pigmentosa are major causes of irreversible vision loss in the elderly and, despite sustained efforts, current treatments are largely ineffective. Wogonin is a bioactive plant flavonoid possessing a range of beneficial properties, including neuroprotective effects. We investigated the ability of wogonin to promote retinal neuron-like differentiation of bone marrow stem cells (BMSCs) and assessed the involvement of Notch-1 signaling in this process. Cultured mouse BMSCs were left untreated or exposed to neurotrophic factors in the presence or absence of wogonin, and western blotting, RT-PCR and immunofluorescence were used to identify changes in molecular markers of stemness and neuroretinal differentiation. Proteins in the Notch-1 signaling pathway, a main negative regulator of neurogenesis, were also examined by western blotting. We found that expression of stem cell markers was reduced, while markers of mature retinal neurons, bipolar cells and photoreceptors were increased in wogonin-treated BMSCs. Wogonin also dose-dependently decreased expression of Notch-1 signaling proteins. Moreover, blockade of Notch-1 both mimicked and enhanced the effect of wogonin to facilitate BMSC differentiation into retinal neuron-like cells. Wogonin thus appears to promote retinal neuron-like differentiation of BMSCs by antagonizing the inhibitory actions of Notch-1 signaling on neurogenesis and may be useful in the treatment of retinal degenerative diseases.

The lens growth process

The factors that regulate the size of organs to ensure that they fit within an organism are not well understood. A simple organ, the ocular lens serves as a useful model with which to tackle this problem. In many systems, considerable variance in the organ growth process is tolerable. This is almost certainly not the case in the lens, which in addition to fitting comfortably within the eyeball, must also be of the correct size and shape to focus light sharply onto the retina. Furthermore, the lens does not perform its optical function in isolation. Its growth, which continues throughout life, must therefore be coordinated with that of other tissues in the optical train. Here, we review the lens growth process in detail, from pioneering clinical investigations in the late nineteenth century to insights gleaned more recently in the course of cell and molecular studies. During embryonic development, the lens forms from an invagination of surface ectoderm. Consequently, the progenitor cell population is located at its surface and differentiated cells are confined to the interior. The interactions that regulate cell fate thus occur within the obligate ellipsoidal geometry of the lens. In this context, mathematical models are particularly appropriate tools with which to examine the growth process. In addition to identifying key growth determinants, such models constitute a framework for integrating cell biological and optical data, helping clarify the relationship between gene expression in the lens and image quality at the retinal plane.

Enabling Anyone to Translate Clinically Relevant Ideas to Therapies

How do we inspire new ideas that could lead to potential treatments for rare or neglected diseases, and allow for serendipity that could help to catalyze them? How many potentially good ideas are lost because they are never tested? What if those ideas could have lead to new therapeutic approaches and major healthcare advances? If a clinician or anyone for that matter, has a new idea they want to test to develop a molecule or therapeutic that they could translate to the clinic, how would they do it without a laboratory or funding? These are not idle theoretical questions but addressing them could have potentially huge economic implications for nations. If we fail to capture the diversity of ideas and test them we may also lose out on the next blockbuster treatments. Many of those involved in the process of ideation may be discouraged and simply not know where to go. We try to address these questions and describe how there are options to raising funding, how even small scale investments can foster preclinical or clinical translation, and how there are several approaches to outsourcing the experiments, whether to collaborators or commercial enterprises. While these are not new or far from complete solutions, they are first steps that can be taken by virtually anyone while we work on other solutions to build a more concrete structure for the "idea-hypothesis testing-proof of concept-translation-breakthrough pathway".