Mathematical Modeling Lends Insight into the Progression of Retinitis Pigmentosa

By Lina Sorg

Retinitis pigmentosa (RP) is a group of rare, genetic heterogeneous retinal diseases that cause photoreceptor cells in the retina to degenerate and die, resulting in vision loss. It is the leading source of inherited retinal degeneration-associated blindness. The progression of RP occurs in two major phases. First, patients experience night blindness and tunnel vision due to the degeneration and death of rod photoreceptor cells. They then undergo a decline in central vision that ultimately leads to blindness and corresponds to the degeneration and death of cone photoreceptor cells.

Photoreceptors are located in the outer nuclear layer of the retina and play a key role in the vision process (see Figure 1). When light enters the eye through the cornea, photopigments in the outer segment of the photoreceptor cells capture the light photons. The photons are then converted to chemical signals that are sent to the brain via the optic nerve. To prevent the toxic effects of cumulative photooxidative products throughout this sight process, photoreceptors undergo a daily shedding and renewal of the outer segments. When photoreceptor degeneration occurs, the renewal process is interrupted, the outer segments shrink, and vision is compromised.

During a minisymposia presentation at the 2021 SIAM Annual Meeting, which is taking place virtually this week, Danielle Brager of the National Institute of Standards and Technology used mathematical models to investigate photoreceptor degeneration and retinitis pigmentosa. Specifically, she employs a series of ordinary differential equations (ODEs) to investigate the mechanisms that cause photoreceptor cell death in RP.

Brager opened her talk with an overview of three different hypotheses that explain the secondary wave of cone photoreceptor cell death in existing models.

1. The toxic substance hypothesis maintains that when rod photoreceptor cells die, they release a toxic substance that causes the death of neighboring cone photoceptor cells, thereby releasing more toxins that trigger even more cell death. Yet studies have shown that cone death often occurs many months after rod degeneration, which challenges this theory. 
2. The oxygen toxicity hypothesis indicates that more oxygen is available in the outer retina when rod photoreceptor cells die. The oxygen then reaches a dangerous level of hyperoxia, which causes additional photoreceptor cells to die and release even more oxygen. 
3. The trophic (growth) factor hypothesis claims that rods emit a trophic/growth factor that is necessary for cone survival. When rod photoreceptor cells die, this trophic factor is no longer available to sustain cone photoreceptor cells.

Brager focuses on the trophic factor hypothesis and incorporates a growth factor in her model called the rod-derived cone viability factor (RdCVF). In 2015, researchers discovered that the RdCVF protein promotes cone survival by stimulating aerobic glycolysis, which prompts renewal of the cone outer segment. This new understanding of RdCVF inspired the construction of Brager’s model.

Her model makes several assumptions. First, Brager assumes that mutations are present and therefore splits the rod population into two phenotype categories: normal and mutated. She also assumes that the photoreceptors share a common nutrient source—the trophic pool—that is mediated by the retinal pigment epithelium (RPE). The RPE supplies the photoreceptors with nutrients to regenerate the outer segments. In addition, Brager equates outer segment renewal and shedding with birth and death. “When a photoreceptor cell dies, there is no spontaneous birth,” she said. “In other words, there is no new photoreceptor to take its place.” Finally, she assumes that rod-cone interactions are moderated by RdCVF secretion.

Next, Brager introduced the model’s four state variables:

• \(R_n\): Sum of the proportion of the full length of each normal rod outer segment
• \(R_m\): Sum of the proportion of the full length of each mutated rod outer segment
• \(C\): Sum of the proportion of the full length of each cone outer segment
• \(T\): Trophic pool/nutrients that are supplied via RPE.

Brager’s ODEs model nutrient production logistically and show that nutrients are withdrawn from the trophic pool and used to generate new outer segments. This process is complicated by the fact that RdCVF also provides a contribution for outer segment renewal that comes from both normal and mutated rods, meaning that the model must account for glucose uptake and aerobic glycolysis in the presence of RdCVF.

Brager utilizes ordinary least squares and retinal degeneration 1 mouse cone transcript data—from a mouse with RP—to estimate the model parameters. Her model has nine equilibrium points, seven of which are physiologically relevant (see Figure 2). Each point corresponds to a stage in the progression of retinal degeneration. “Using these equilibrium points, we determined the conditions that are necessary for stability,” Brager said. “We interpret those conditions as criteria needed for a person with RP to stay within a particular stage in the disease.”

Brager’s model reveals five pathways to blindness. Two of these pathways correspond to RP and three correspond to cone-rod dystrophy: a group of diseases wherein the cones are affected by genetic mutation that triggers cone death followed by rod death. To examine the uncertainty in her model, Brager identifies parameter sets where each equilibrium point is stable and performs sensitivity analysis using those parameter sets. 

Brager is currently working to develop a spatial model on which she intends to perform optimal control to explore other avenues that might prevent or halt the progression of RP. In the meantime, she is pleased with the significance of her ODE model. “We have developed a mathematical model that accounts for the new understanding of the mode of action of RdCVF,” Brager said. “This model allows us to open the possibility of using in silico experiments to test treatment options in the absence of rods. It incorporates crucial biology that leads to additional parameters that can cause and thus be used to prevent blindness.”

Lina Sorg is the managing editor of SIAM News.