Previous work on modeling includes several extracellular deterministic models [1]; such mathematical models have been also employed to control disease and optimize medication schedules [2]. There are also a few intracellular models [3] which describe the dynamics of intracellular events. Stochastic models to describe HIV infection have also been developed, such as a stochastic model for early HIV population dynamics [4]. In [5], Monte Carlo method was employed to estimate the natural variation of HIV infection and CD8 T-cells. These works are based on the model proposed in [6] where it was shown that there is a positive probability that the virus will be eliminated.
HIV is usually diagnosed after the establishment of infection in the body because initially the patients have indistinguishable flu-like symptoms. However, few groups of people, such as medical staff, might be aware of contamination with virus promptly after occurrence. Due to the toxicity and also high cost of the currently used drugs, high dosages of medication should be avoided, and thus it is important to schedule the optimal medication strategy for patients in the primary stage of HIV infection. To achieve this, in [7] we developed an extracellular stochastic model to describe the infection process. We showed that the ability of deterministic models to accurately describe the expected behavior of early HIV infection is limited.
To increase the reliability of the results, developing more accurate models considering infection dynamics at both extra and intracellular levels is necessary. Multi-scale modeling platforms that span several biological levels have been previously developed and employed to study cancerous tumor growth [8]. In this study, a multi-scale model of HIV infection to incorporate both extra and intracellular dynamics is developed. Inside the actively infected cell a sequence of events necessary for production of new virus particles take place: transcription of viral RNA to DNA by reverse transcription enzyme, integration of DNA into the host nucleus, production of viral RNA, modification of proteins, and finally gathering of viral content at the surface of the host. Then, new virus particles bud from the surface of the actively infected host cells. We assume that all the virus particles emerge instantaneously causing lysis of the T-cell. After maturation, which is a result of cleavage of proteins inside the virus particle by the protease enzyme, the new virus particles are able to repeat the infection cycle. The time scale of intracellular events inside the infected host is a few orders of magnitude smaller than the time scale of cell population interactions at the extracellular level. The developed model enhanced the understanding of infection cycle and was useful in predicting the overall infection dynamics while considering the intracellular infection events.
To obtain the optimal treatment strategy for patients in the early stage of HIV infection, two important issues should be considered. First, we are seeking to decrease the infection establishment probability to the lowest possible value given a certain amount of drug. Second, the optimal schedule should utilize the minimum possible amount of drug over the period of treatment. To achieve this, we formulated a dynamic optimization problem where the calculation of the objective function involves KMC simulations describing the unavailable in closed form intracellular/extracellular dynamics. Moreover, due to the stochastic nature of KMC simulations, gradient based optimization algorithms are inapplicable in a straight forward manner. As a result, direct search algorithms, e.g., Hooke-Jeeves, which do not require gradients to compute search direction, are more suitable for the current problem. Since the immediate start of medication is not always possible, the optimal treatment strategy was scheduled for prompt treatment and also for a few days latency in treatment initiation. To show the advantage of optimal treatment, the results were compared with constant treatment strategies. We showed that optimal treatment schedule might be more beneficial for the patients, i.e., either the infection establishment probability or medication dosage was lower than constant medication strategy.
References:
[1] A. S. Perelson and P. W. Nelson. Mathematical analysis of HIV-1 dynamics in vivo. SIAM review, 41(1): 3-44, 1999.
[2] D. Kirschner, S. Lenhart, and S. Serbin. Optimal control of the chemotherapy of HIV. J.Math. Biol., 35: 775-792, 1997.
[3] R. Srivastava, L. You, J. Summers, and J. Yin. Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theor. Biol., 218: 309-321, 2002.
[4] H. C. Tuckwell and E. Le Corfec. A stochastic model for early HIV-1 population dynamics. J. Theor. Biol., 195: 451-463, 1998.
[5] J. M. Heffernan and L. M. Wahl. Natural variation in HIV-1 infection: Monte carlo estimates that include CD8 effector cells. J. Theor. Biol., 243: 191-204, 2006.
[6] W. Tan and H. Wu. Stochastic modeling of the dynamics of CD4 t-cell infection by HIV-1 and some Monte Carlo studies. Mathematical Biosciences, 147: 173-205, 1998.
[7] S. Khalili and A. Armaou. Sensitivity analysis of HIV infection response to treatment via stochastic modeling. Chemical Engineering Science, In Press, 2007.
[8] L. Zhang, C. A. Athale, and T. S. Deisboeck. Development of a three-dimensional multiscale agent-based tumor model: Simulating gene-protein interaction profiles, cell phenotypes and multicellular patterns in brain cancer. J. Theor. Biol., 244: 96-107, 2007.