Skip to main content

Advertisement

SpringerLink
Log in

Optimal Control in the Treatment of Retinitis Pigmentosa

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • 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.

    Chapter  Google Scholar 

  • Besharse, J., & Bok, D. (2011). The retina and its disorders. San Diego: Academic Press.

    Google Scholar 

  • Bok, D. (1985). Retinal photoreceptor-pigment epithelium interactions: Friedenwald lecture. Investig. Ophthalmol. Vis. Sci., 26(12), 1659–1694.

    Google Scholar 

  • Bovolenta, P., & Cisneros, E. (2009). Retinitis pigmentosa: cone photoreceptors starving to death. Nat. Neurosci., 12, 5–7.

    Article  Google Scholar 

  • 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 

  • Camacho, E. T., & Wirkus, S. (2013). Tracing the progression of retinitis pigmentosa via photoreceptor interactions. J. Theor. Biol., 317C, 105–118.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

  • 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.

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • Fleming, W. H., & Rishel, R. W. (1975). Deterministic and stochastic optimal control. New York: Springer.

    Book  MATH  Google Scholar 

  • 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 

  • 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 

  • Hamel, C. (2006). Retinitis pigmentosa. Orphanet J. Rare Dis., 1(1), 40.

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • Hartong, D. T., Berson, E. L., & Dryja, T. P. (2006). Retinitis pigmentosa. Lancet, 368, 1795–1809.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Keener, J., & Sneyd, J. (2008). Mathematical physiology II: systems physiology. Berlin: Springer.

    MATH  Google Scholar 

  • 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 

  • 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 

  • Lenhart, S., & Workman, J. T. (2007). Chapman & Hall/CRC mathematical and computational biology series. Optimal control applied to biological models. London: Chapman & Hall/CRC.

    MATH  Google Scholar 

  • 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 

  • 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).

  • 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 

  • 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.

    Article  Google Scholar 

  • Lukes, D. L. (1982). Differential equations: classical to controlled. San Diego: Academic Press.

    MATH  Google Scholar 

  • 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 

  • McAsey, M., Mou, L., & Han, W. (2012). Convergence of the forward-backward sweep method in optimal control. Comput. Optim. Appl., 53(1), 207–226.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Oyster, C. W. (1999). The human eye: structure and function. Sunderland: Sinauer.

    Google Scholar 

  • Pallikaris, A., Williams, D. R., & Hofer, H. (2003). The reflectance of single cones in the living human eye. Investig. Ophthalmol. Vis. Sci., 44, 10.

    Article  Google Scholar 

  • Papermaster, D. S. (2002). The birth and death of photoreceptors: Friedenwald lecture. Investigat. Ophthalmol. Vis. Sci., 43(5), 1300–1309.

    Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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 

  • 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.

    Article  Google Scholar 

  • Shintani, K., Shechtman, D. L., & Gurwood, A. S. (2009). Review and update: current treatment trends for patients with retinitis pigmentosa. Optometry, 80, 384–401.

    Article  Google Scholar 

  • Strauss, O. (2005). The retinal pigment epithelium in visual function. Physiol. Rev., 85, 845–881.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Young, R. (1971). The renewal of rod and cone outer segments in the rhesus monkey. J. Cell Biol., 49, 303–318.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical & Natural Sciences, Arizona State University, Phoenix, AZ, 85306, USA

    E. T. Camacho & S. Wirkus

  2. Department of Mathematics, Shippensburg University, Shippensburg, PA, 17257, USA

    L. A. Melara

  3. Department of Mathematics, The University of Texas-Pan American, Edinburg, TX, 78539, USA

    M. C. Villalobos

Authors
  1. E. T. Camacho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. L. A. Melara
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. C. Villalobos
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. S. Wirkus
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to L. A. Melara.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Camacho, E.T., Melara, L.A., Villalobos, M.C. et al. Optimal Control in the Treatment of Retinitis Pigmentosa. Bull Math Biol 76, 292–313 (2014). https://doi.org/10.1007/s11538-013-9919-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-013-9919-1

Keywords

Search

Navigation