Bulletin of Mathematical Biology

, Volume 76, Issue 2, pp 292–313 | Cite as

Optimal Control in the Treatment of Retinitis Pigmentosa

  • E. T. Camacho
  • L. A. Melara
  • M. C. Villalobos
  • S. Wirkus
Original Article

Abstract

Numerous therapies have been implemented in an effort to minimize the debilitating effects of the degenerative eye disease Retinitis Pigmentosa (RP), yet none have provided satisfactory long-term solution. To date there is no treatment that can halt the degeneration of photoreceptors. The recent discovery of the RdCVF protein has provided researchers with a potential therapy that could slow the secondary wave of cone death. In this work, we build on an existing mathematical model of photoreceptor interactions in the presence of RP and incorporate various treatment regiments via RdCVF. Our results show that an optimal control exists for the administration of RdCVF. In addition, our numerical solutions show the experimentally observed rescue effect that the RdCVF has on the cones.

Keywords

Optimal control Rod-derived viability factor 

References

  1. Banks, H. T., Davidian, M., Samuels, J. R., Jr., & Sutton, K. L. (2009). An inverse problem statistical methodology summary. In Mathematical and statistical estimation approaches in epidemiology (pp. 249–302). New York: Springer. CrossRefGoogle Scholar
  2. Besharse, J., & Bok, D. (2011). The retina and its disorders. San Diego: Academic Press. Google Scholar
  3. Bok, D. (1985). Retinal photoreceptor-pigment epithelium interactions: Friedenwald lecture. Investig. Ophthalmol. Vis. Sci., 26(12), 1659–1694. Google Scholar
  4. Bovolenta, P., & Cisneros, E. (2009). Retinitis pigmentosa: cone photoreceptors starving to death. Nat. Neurosci., 12, 5–7. CrossRefGoogle Scholar
  5. Bruce Szamier, R., Berson, E. L., Klein, R., & Meyers, S. (1979). Sex-linked retinitis pigmentosa: ultrastructure of photoreceptors and pigment epithelium. Investig. Ophthalmol. Vis. Sci., 18, 145–160. Google Scholar
  6. Camacho, E. T., & Wirkus, S. (2013). Tracing the progression of retinitis pigmentosa via photoreceptor interactions. J. Theor. Biol., 317C, 105–118. CrossRefGoogle Scholar
  7. Camacho, E. T., Colón Vélez, M. A., Hernández, D. J., Bernier, U. R., van Laarhoven, J., & Wirkus, S. (2010). A mathematical model for photoreceptor interactions. J. Theor. Biol., 21, 638–646. CrossRefMathSciNetGoogle Scholar
  8. Colón Vélez, M. A., Hernández, D. J., Bernier, U. R., van Laarhoven, J., & Camacho, E. T. (2003). A mathematical model of photoreceptor interactions. Department of Biological Statistics and Computational Biology Technical Report BU-1640-M, Cornell University, 2003, 25–69. Google Scholar
  9. Daiger, S. P., Sullivan, L. S., & Bowne, S. J. (2013). RetNet: retinal information network. University of Texas Health Center, https://sph.uth.edu/retnet/. Accessed 21 June 2013.
  10. DePillis, L. G., Fister, K. R., Gu, W., Head, T., Maples, K., Murugan, A., Neal, T., & Yoshida, K. (2007). Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls. Math. Biosci., 209, 292–315. CrossRefMathSciNetMATHGoogle Scholar
  11. DePillis, L. G., Fister, K. R., Gu, W., Head, T., Maples, K., Neal, T., Murugan, A., & Kozai, K. (2008). Optimal control of mixed immunotherapy and chemotherapy of tumors. J. Biol. Syst., 16(1), 51–80. CrossRefMATHGoogle Scholar
  12. Fleming, W. H., & Rishel, R. W. (1975). Deterministic and stochastic optimal control. New York: Springer. CrossRefMATHGoogle Scholar
  13. Frasson, M., Picaud, S., Léveillard, T., Simonutti, M., Mohand-Saïd, S., Dreyfus, H., Hicks, D., & Sahel, J. (1999). Glial cell line-derived neurotrophic factor induces histologic and functional protection of rod photoreceptors in the rd/rd mouse. Investig. Ophthalmol. Vis. Sci., 40, 2724–2734. Google Scholar
  14. Guérin, C. J., Lewis, G. P., Fisher, S. K., & Anderson, D. H. (1993). Recovery of photoreceptor outer segment length and analysis of membrane assembly rates in regenerating primate photoreceptor outer segments. Investig. Ophthalmol. Vis. Sci., 34, 175–183. Google Scholar
  15. Hamel, C. (2006). Retinitis pigmentosa. Orphanet J. Rare Dis., 1(1), 40. CrossRefGoogle Scholar
  16. Hanein, S., Perrault, I., Gerber, S., Dollfus, H., Dufier, J.-L., Feingold, J., Munnich, A., Bhattacharya, S., Kaplan, J., Sahel, J.-A., Rozet, J.-M., & Leveillard, T. (2006). Disease-associated variants of the rod-derived cone viability factor (RdCVF) in leber congenital amaurosis. In Retinal degenerative diseases (pp. 9–14). Berlin: Springer. CrossRefGoogle Scholar
  17. Hartong, D. T., Berson, E. L., & Dryja, T. P. (2006). Retinitis pigmentosa. Lancet, 368, 1795–1809. CrossRefGoogle Scholar
  18. Hendrickson, A., Bumsted-O’Brien, K., Natoli, R., Ramamurthy, V., Possing, D., & Provis, J. (2008). Rod photoreceptor differentiation in fetal and infant human retina. Exp. Eye Res., 87, 415–426. CrossRefGoogle Scholar
  19. Jonnal, R. S., Besecker, J. R., Derby, J. C., Kocaoglu, O. P., Cense, B., Gao, W., Wang, Q., & Miller, D. T. (2010). Imaging outer segment renewal in living human cone photoreceptors. Opt. Express, 18, 5257–5270. CrossRefGoogle Scholar
  20. Keener, J., & Sneyd, J. (2008). Mathematical physiology II: systems physiology. Berlin: Springer. MATHGoogle Scholar
  21. Kernan, F., McKee, A. G., Farrar, G. J., & Humphries, P. (2007). On the suppression of photoreceptor cell death in retinitis pigmentosa. In Ophthalmology research: retinal degenerations: biology, diagnostics, and therapeutics. Clifton: Humana Press. Google Scholar
  22. LaVail, M. M., Yasumura, D., Matthes, M. T., Lau-Villacorta, C., Unoki, K., Sung, C.-H., & Steinberg, R. H. (1998). Protection of mouse photoreceptors by survival factors in retinal degenerations. Investig. Ophthalmol. Vis. Sci., 39, 592–602. Google Scholar
  23. Lenhart, S., & Workman, J. T. (2007). Chapman & Hall/CRC mathematical and computational biology series. Optimal control applied to biological models. London: Chapman & Hall/CRC. MATHGoogle Scholar
  24. Léveillard, T., & Sahel, J.-A. (2010). Rod-derived cone viability factor for treating blinding diseases: from clinic to redox signaling. Degener. Retin. Disord., 2, 1–13. Google Scholar
  25. Léveillard, T., Mohand-Saïd, S., Lorentz, O., Hicks, D., Fintz, A.-C., Clérin, E., Simonutti, M., Forster, V., Cavusoglu, N., Chalmel, F., Dollé, P., Poch, O., Lambrou, G., & Sahel, J. A. (2004). Identification and characterization of rod-derived cone viability factor. Nat. Genet., 36(7). Google Scholar
  26. Li, Y., Tao, W., Luo, L., Huang, D., Kauper, K., Stabila, P., LaVail, M. M., Laties, A. M., & Wen, R. (2010). CNTF induces regeneration of cone outer segments in a rat model of retinal degeneration. PLoS ONE, 5, 1–7. Google Scholar
  27. Longbottom, R., Fruttigera, M., Douglasb, R. H., Martinez-Barberac, J. P., Greenwooda, J., & Mossa, S. E. (2009). Genetic ablation of retinal pigment epithelial cells reveals the adaptive response of the epithelium and impact on photoreceptors. Proc. Natl. Acad. Sci. USA, 3, 18728–18733. CrossRefGoogle Scholar
  28. Lukes, D. L. (1982). Differential equations: classical to controlled. San Diego: Academic Press. MATHGoogle Scholar
  29. Malanson, K. M., & Lem, J. (2009). Rhodopsin-mediated retinitis pigmentosa. In Progress in molecular biology and translational science (pp. 1–31). Amsterdam: Elsevier. Google Scholar
  30. McAsey, M., Mou, L., & Han, W. (2012). Convergence of the forward-backward sweep method in optimal control. Comput. Optim. Appl., 53(1), 207–226. CrossRefMathSciNetMATHGoogle Scholar
  31. Mohand-Said, S., Hicks, D., Léveillard, T., Picaud, S., Porto, F., & Sahel, J. A. (2001). Rod-cone interactions: developmental and clinical significance. Prog. Retin. Eye Res., 20(4), 451–467. CrossRefGoogle Scholar
  32. Murakami, Y., Ikeda, Y., Yonemitsu, Y., Onimaru, M., Nakagawa, K., Kohno, R.-i., Miyazaki, M., Hisatomi, T., Nakamura, M., Yabe, T., Hasegawa, M., Ishibashi, T., & Sueishi, K. (2008). Inhibition of nuclear translocation of apoptosis-inducing factor is an essential mechanism of the neuroprotective activity of pigment epithelium-derived factor in a rat model of retinal degeneration. Am. J. Pathol., 173, 1326–1338. CrossRefGoogle Scholar
  33. Oyster, C. W. (1999). The human eye: structure and function. Sunderland: Sinauer. Google Scholar
  34. Pallikaris, A., Williams, D. R., & Hofer, H. (2003). The reflectance of single cones in the living human eye. Investig. Ophthalmol. Vis. Sci., 44, 10. CrossRefGoogle Scholar
  35. Papermaster, D. S. (2002). The birth and death of photoreceptors: Friedenwald lecture. Investigat. Ophthalmol. Vis. Sci., 43(5), 1300–1309. Google Scholar
  36. Phelan, J. K., & Bok, D. (2000). A brief review of retinitis pigmentosa and the identified retinitis pigmentosa genes. Mol. Vis., 6, 116–124. Google Scholar
  37. Punzo, C., Kornacker, K., & Cepko, C. L. (2009). Stimulation of the insulin/mTOR pathway delays cone death in a mouse model of retinitis pigmentosa. Nat. Neurosci., 12(1), 44–52. CrossRefGoogle Scholar
  38. Reichman, S., Kalathur, R. K. R., Lambard, S., Aït-Ali, N., Yang, Y., Lardenois, A., Ripp, R., Poch, O., Zack, D. J., Sahel, J.-A., & Léveillard, T. (2010). The homeobox gene CHX10/VSX2 regulates RdCVF promoter activity in the inner retina. Hum. Mol. Genet., 19, 250–261. CrossRefGoogle Scholar
  39. Ripps, H., Brin, K. P., & Weale, R. A. (1978). Rhodopsin and visual threshold in retinitis pigmentosa. Investig. Ophthalmol. Vis. Sci., 17, 735–745. Google Scholar
  40. Sahel, J.-A. (2005). Saving cone cells in hereditary rod diseases: a possible role for rod-derived cone viability factor (RdCVF) therapy. Retina J. Retin. Vitr. Dis. Suppl., 25(8), S38–39. Google Scholar
  41. Shen, J., Yang, X., Dong, A., Petters, R. M., Peng, Y.-W., Wong, F., & Campochiaro, P. A. (2005). Oxidative damage is a potential cause of cone cell death in retinitis pigmentosa. J. Cell. Physiol., 203, 457–464. CrossRefGoogle Scholar
  42. Shintani, K., Shechtman, D. L., & Gurwood, A. S. (2009). Review and update: current treatment trends for patients with retinitis pigmentosa. Optometry, 80, 384–401. CrossRefGoogle Scholar
  43. Strauss, O. (2005). The retinal pigment epithelium in visual function. Physiol. Rev., 85, 845–881. CrossRefGoogle Scholar
  44. Wenzel, A., Grimm, C., Samardzija, M., & Remé, C. E. (2005). Molecular mechanisms of light-induced photoreceptor apoptosis and neuroprotection for retinal degeneration. Prog. Retin. Eye Res., 24, 275–373. CrossRefGoogle Scholar
  45. Yang, Y., Mohand-Said, S., Danan, A., Simonutti, M., Fontaine, V., Clerin, E., Picaud, S., Léveillard, T., & Sahel, J.-A. (2009). Functional cone rescue by RdCVF protein in a dominant model of retinitis pigmentosas. Molec. Ther., 17, 787–795. CrossRefGoogle Scholar
  46. Young, R. (1971). The renewal of rod and cone outer segments in the rhesus monkey. J. Cell Biol., 49, 303–318. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  • E. T. Camacho
    • 1
  • L. A. Melara
    • 2
  • M. C. Villalobos
    • 3
  • S. Wirkus
    • 1
  1. 1.School of Mathematical & Natural SciencesArizona State UniversityPhoenixUSA
  2. 2.Department of MathematicsShippensburg UniversityShippensburgUSA
  3. 3.Department of MathematicsThe University of Texas-Pan AmericanEdinburgUSA

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