Using physics and chemistry principles, I formulate equations for different processes in physiology. The processes involve transport of substances through different spaces, which are separated by membranes. The formulation yields systems of nonlinear differential equations, which solutions must be approximated by means of numerical methods. The problem structures and the chemical reactions represented in the model, introduce parameters which values are unknown or known with uncertainty. Then I develop optimization problems to study the variation of the model parameters with respect to the problem's expected performances. Resulting in differential equation constrained optimization problems.

1. Mariano Marcano, Amitabha Bose, and Paul Bayman. A one-dimensional map to study multi-seasonal coffee infestation by the coffee berry borer. Mathematical Biosciences, 333, March 2021. DOI: 10.1016/j.mbs.2020.108530 [Preprint]
2. Mónica Nadal-Quirós, Leon Moore, and Mariano Marcano. Parameter estimation for a mathematical model of a non-gastric H+(Na+)/K+(NH4+) ATPase, Am. J. Physiol. Renal Physiol., 309: F434–F446, 2015. DOI: 10.1152/ajprenal.00539.2014
3. Claudia Patricia Ruiz-Diaz, Carlos Toledo-Hernández, Alberto M. Sabat, Mariano Marcano. Immune response to a pathogen in corals. Journal of Theoretical Biology, 332, 141-148, 2013. DOI: 10.1016/j.jtbi.2013.04.028
4. Aniel Nieves-González, Chris Clausen, Mariano Marcano, Anita T. Layton, Harold E. Layton, and Leon C. Moore. Fluid Dilution and Efficiency of Na+ Transport in a Mathematical Model of a Thick Ascending Limb Cell. Am. J. Physiol. Renal Physiol., 304, 6, F634 -F652, 2013. DOI: 10.1152/ajprenal.00100.2012
5. Miguel A. Acevedo, Mariano Marcano, Robert J. Fletcher, Jr. A diffusive logistic growth model to describe forest recovery. Ecological Modelling, 244:13-19, 2012. DOI: 10.1016/j.ecolmodel.2012.07.012
6. Mariano Marcano, Anita T. Layton, and Harold E. Layton. Maximum Urine Concentrating Capability in a Mathematical Model of the Inner Medulla of the Rat Kidney. Bull. Math. Biol. 72(2):314-339, 2010. DOI: 10.1007/s11538-009-9448-0
7. Mariano Marcano, Hun-Mo Yang, Aniel Nieves-González, Chris Clausen, and Leon C Moore. Parameter Estimation for Mathematical Models of NKCC2 Cotransporter Isoforms. Am. J. Physiol. Renal Physiol. 296: F369-F381, 2009.
DOI: 10.1152/ajprenal.00096.2008
8. Mariano Marcano, Anita T. Layton, and Harold E. Layton. An Optimization Algorithm for a Distributed-Loop Model of an Avian Urine Concentrating Mechanism. B. Math. Biol., 68: 1625-1660, 2006. DOI: 10.1007/s11538-006-9087-1
9. M. Marcano-Velázquez and H. E. Layton. An Inverse Algorithm for a Mathematical Model of an Avian Urine Concentrating Mechanism. B. Math. Biol., 65(4):665-691, 2003. DOI: 10.1016/S0092-8240(03)00029-6
10. R. P. Tewarson, W. Toro, and M. Marcano. Preferential interaction and inverse problem algorithms in models of renal concentrating mechanism. Appl. Math. Lett., 11:3, 51-59, 1998. DOI: 10.1016/S0893-9659(98)00033-0
11. R. P. Tewarson and M. Marcano. Use of generalized inverse in a renal optimization problem. Inverse Probl. Eng., 5, 1-9, 1997. DOI: 10.1080/174159797088027649
12. M. Marcano-Velázquez and P. V. Negrón-Marrero. The numerical solution of the von Kármán equations using multigrid methods. Congr. Numer., 94, 17-28, 1993.

• Optimal Mathematical Models of Renal Transport Processes. Principal Investigator. National Institutes of Health (NIH) Grant 1SC1GM084744, $596,000, August 1, 2008-July 31, 2013.
• Mathematical modeling and inverse problem for the urine concentrating mechanism. Principal Investigator of a pilot project in Support for Continuous Research Excellence (MBRS-SCoRE) program through the NIH Grant S06GM08102, $105,000, August 1, 2004-July 31, 2007.
• Mathematical models for the urine concentrating mechanism: Direct and inverse problems, Puerto Rico Experimental Program to Stimulate Competitive Research (PR-EPSCoR) start-up funds through the NSF Grant EPS-0223152, $107,594, January 1, 2004-July 31, 2005.
• PI, Mathematical modeling and inverse problem for the urine concentrating mechanism, UPR Institutional Research Fund (FIPI), $21,200, July, 2003-June, 2005.
• PI, Mathematical Model and Numerical Solution for the Kidney Concentrating Mechanism, FIPI, $7,700, 1999-2000 and $9,695, 2001-2002.
• Numerical Solution of Renal Transport Equations, PI, Reginald P. Tewarson, Research Supplement for Under Represented Minorities for Mariano Marcano, National Institute of Diabetes and Digestive and Kidney Diseases, NIH, number: 3R01DK1759318S1, 1995-1997.