(656g) Optimal Treatment of HIV Primary Infection Via a Stochastic Formulation
The significant impact that human immunodeficiency virus (HIV) infection has on today's society has motivated numerous research efforts addressing diverse aspects of the disease. Drug development is an active research topic with the objective of prolonging life expectancy of infected persons. Mathematical models may provide significant assistant in this endeavor by identifying optimal dosage strategies. This stems from the ability of models to predict the response of the average patient to treatment. Numerous mathematical models have been proposed in the open literature to capture different aspects of disease progression. They include phenomenological estimation of biological parameters through development of an extracellular model , developing an intracellular level model , incorporation of both intracellular and extracellular infection level descriptions , and disease control and optimization of medication schedules , . These models traditionally take the form of deterministic differential/algebraic equation systems; however in , stochastic versus deterministic modeling of intracellular viral kinetics was studied and it was shown that for low concentration systems, they lead to different predictions. As a result, a current need exists for accurate descriptions that account for phenomena at the molecular level. In , a hybrid stochastic method that partitions the system into subsets of fast and slow reactions, approximating the fast reactions as a Markov process and describing slow reactions using the ?next reaction? algorithm was described. A specific stochastic model to describe HIV infection has also been developed in . The focus of this work was on the treatment of the disease at the primary infection stage. HIV is usually diagnosed after a period of time because initially most of the infected people have flu-like symptoms which are indistinguishable. Consequently, the disease is usually diagnosed when the treatment may prolong the life expectancy of the average patient, but the chance of infection eradication is very low . Conversely, if treatment starts early after initiation of virus into the body the probability of infection eradication is higher. Although immediate start of treatment generally is infeasible, in some cases the person might be aware of contamination with virus (e.g. hospital personnel). However, high dosages of medication can not be prescribed for the patients due to drug toxicity. Motivated by this, we focused on identifying a treatment policy that maximizes the probability of infection eradication in the presence of constraints on the available amount of drug. An extracellular stochastic model was initially developed to describe the initial stage of infection. We applied the Gillespie algorithm  and by comparing the predictions of stochastic model with deterministic model, showed that stochastic modeling is capable of accurately studying the dynamics at this early stage. Subsequently, assuming a thirty day treatment period starting at the initial stage of infection, two types of drugs were employed. The first was reverse transcriptase inhibitor, RTI, which blocks the reverse transcription of virus RNA to DNA. The second drug was protease inhibitor, PI which inhibits the cleavage of proteins and consequently noninfectious virions are produced. Subsequently, the sensitivity of HIV dynamics to medication type, efficacy of medication, and treatment initiation time was analyzed. An important problem limiting the therapeutic usefulness of RTI and PI, besides toxic effects, is the production of new mutant viral variants which are resistant to RTI and PI. Resistance has been associated with specific amino acid substitutions at different parts of virus genome. The mutation rate in HIV is high because the transcription of the virus genetic code to DNA is an error-prone process . However, few of all the possible mutations in viral genome will lead to production of resistant virions to medication . Based on the mutation rate and information from , the probabilities of production of resistance virions to each of the two types of medication were calculated and the model was revised to include the development of resistance to the specific medication. The model predictions considering mutation were subsequently compared with initial results. Based on the developed model, the treatment policy problem was formulated as a dynamic optimization problem in the presence of path constraints and set constraints. Specifically, the control vector parameterization approach was used to formulate the problem. The high computational cost of the Gillespie algorithm (for both SSA and ô-leap methods) precludes the direct use of standard search algorithms to solve this formulation. This issue was circumvented by extending the applicability of the in situ adaptive tabulation (ISAT) superstructure to the stochastic simulator. ISAT tabulates the process data and process sensitivities during the simulations; it is computationally less demanding than direct tabulation since only the realizable region (the parameter space traversed during simulation) is tabulated, which is usually a small subset of the whole state space. Employing ISAT enabled us to reduce the computational requirements of the used search algorithm. Consequently, treatment policies for the primary stage of infection for different initial viral loads which maximizes the probability of infection eradication were identified. Subsequently, the policies were analyzed with respect to changes due to resistance and treatment initiation time.
References:  Perelson A. S. and Nelson P. W., ?Mathematical analysis of HIV-1 dynamics in vivo,? SIAM review, vol. 41, no. 1, pp. 3?44, 1999.  Reddy B. and Yin J., ?Quantitative intracellular kinetics of HIV type 1,? AIDS research and human retroviruses, vol. 15, pp. 273?283, 1999.  Yin J. Haseltine E. L., Rawlings J. B., ?Dynamics of viral infections: incorporating both the intracellular and extracellular levels,? Computers and chemical engineering, vol. 29, pp. 675?686, 2004.  Serbin S. Kirschner D., Lenhart S., ?Optimal control of the chemotherapy of HIV,? J. Math. Biol., vol. 35, pp. 775?792, 1997.  Snedecor S. J., ?Comparison of three kinetic models of HIV-1 infection: Implications for optimization of treatment,? J. theor. Biol., vol. 221, pp. 519?541, 2003.  Summers J. Srivastava R., You L. and Yin J., ?Stochastic vs. deterministic modeling of intracellular viral kinetics,? J. theor. Biol., vol. 218, pp. 309?321, 2002.  Salis H. and Kaznessis Y., ?Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions.,? The Journal of Chemical Physics, vol. 122, pp. 054103,1?054103,13, 2005.  Tuckwell H. C. and Le Corfec E., ?A stochastic model for early HIV-1 population dynamics,? J. theor. Biol., vol. 195, pp. 451?463, 1998.  Gillespie D. T. and Petzold L. R., ?Improved leap-size selection for accelerated stochastic simulation,? J. Chem. Phys., vol. 119, no. 16, pp. 8229?8234, 2003.  Hughes S. H. Coffin J. M. and Varmus H. E., Retrovirus, Cold spring harbor laboratory press, 1997.  http://www.mediscover.net/antiviralintro.cfm