Final paper acc 2003 denver.doc

Analysis of a sub-optimal scheme of drug dosage in the AIDS treatment
J.A.M. Felippe de Souza
Universidade da Beira Interior, Covilhã, Portugal M. A. L. Caetano
UNESP - Universidade Estadual Paulista, IGCE, Rio Claro, Brazil Takashi Yoneyama
ITA - Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil
KEYWORDS
Remarkably, some cases were related to improvements in CD4+T counts and decrease of the viral load. Optimization, direct method, AIDS, treatment. The intense clinical research has made available now several quantitative descriptions of the dynamics of AIDS as well as more advanced mathematical modeling methods ABSTRACT
These models can be used to optimize the drug doses Here the results for CD4+T cells count and the viral load required in the treatment. Here we use a model that was obtained from HIV sero-positive patients are compared with originally proposed by Tan and Wu [6] and is similar to results from numerical simulations by computer. Also, the standard scheme of administration of drugs anti HIV Caetano & Yoneyama have shown in [1] that it is possible (HAART schemes) which uses constant doses is compared to improve the treatment effectiveness by using closed loop with an alternative sub-optimal treatment scheme which uses variable drug dosage according to the evolution of a They showed that closed loop treatment schemes could quantitative measure of the side effects. The quantitative have advantages when compared to the standard treatment analysis done here shows that it is possible to obtain, using because more information is used for the control of the drug the alternative scheme, the same performance of actual data doses. Also, it is possible to use optimal control theory to but using variable dosage and having fewer side effects. reduce the side effects during a short-term treatment Optimal control theory is used to solve and also to provide a scheme while adequate therapeutic results are obtained (see prognosis related to the strategies for control of viraemia. Caetano & Yoneyama also carried out a comparative study and two optimization methods [3]. Responses corresponding to the actual observed data and simulation data were compared in terms of long-term period and short - Models of dynamical systems have been extensively u sed in term period drug administration strategies. studying biological phenomena. The increase in the number Here the main objective is to analyze the dynamics of the of cases of AIDS have lead to the development of several viral load and the CD4+T cells counts with a model that is new mathematical models which describe the dynamical fitted to match the actual data and considering the sub- behavior of the viral load on CD4+T cells counts as well as optimization problem with dynamic constraints for multi- analyze the effects o f treatment strategies [12, 13, 14]. On the other hand new treatment schemes has helped many The data were provided by Centro de Referência e sero-positives patients to have a normal life. In [9] one can Treinamento em DST-AIDS in São Paulo, Brazil. see reports on success achieved by highly active In the first case the patient went through the treatment antiretroviral therapy (HAART) that has prolonged the life during a period of around 495 days which he received AZT (600 mg) and 3TC (300 mg). Before this period the patient However, long-term use of HAART could have adverse had not received any specific drug for the treatment of effects, such as metabolic abnormalities and irreversible fat The second case is a patient that also received AZT (600 Also, the eventual intolerance against HAART by some mg) and 3TC (300 mg) for around 340 days before the start patients may leads to treatment interruptions. In some instances [8, 9] up to three stops were noted during the treatment. The model adopted consists of four differential equations The x2 cells can be activated to become x3 cells. This representing the CD4+T cells (uninfected, latent infected and actively infected) and also the free viruses. The x3 cells are short living and will normally be killed We construct a performance index that tries to describe the When x3 cells die free viruses x4 are released with rate N(t) 1, x2 cells and x4 (free viruses) also have finite life and The model in Tan [6] describes the HIV pathogenesis under The effects of drugs such as reverse transcriptase inhibitors treatment by antiviral drugs. It has four differential and protease inhibitors are considered via the parameters k equations and stochastic terms in the variable that represent the number of latent infected T cells. It has also stochastic components on infection free HIV and non-infection free The model used here is given below by the differential equations in (1). It is a simplified version of a more general The standard treatment of sero positive patients includes model that includes stochastic terms, as originally presented two classes of drug to block the action of HIV. The first class includes drugs that blocks reverse ( 4 ) + ( 1, 2 , 3 ) 1 − 1 [ 1 + k1( 1 transcriptase enzyme involved in exchange of viral-RNA and viral-DNA. Some of the available reverse transcriptase inhibitors are: zidovudine (AZT), didanosine (ddI), ) 1 ( 1 ) 4 1 + 2 ( 2 ) 2 − 3 x3 lamivudine (3TC), zalcitabine (ddc), stavudine (D4T), Abacavir (Ziagen), Viramune (nevirapine), Rescriptor (delavirdine) and Sustiva (efavirenz), among others. The second class includes drugs that inhibit the protease x1 (t) ≡ uninfected CD4+T cells; enzyme: Invirase (saquinavir), Norvir (ritonavir), Crixivan (indinavir), Viracept (nelfinavir), Agenerase (amprenivir) 2 (t) ≡ latent infected CD4+ T cells; x3 (t) ≡ actively infected CD4+T cells; The administration of these drugs is made according to tables proposed by World Health Organization. The s = the rate of generation of x1; medical staff may change the used drugs when patients r = rate of stimulated growth of x1; develop resistance or present intense side effects. Tmax = maximum T cells population level; In the present work the parameters in Tan’s model takes the µi = death rate of xi; I = 1,2,3,4. 1 = infection rate from x1 to x2 by viruses; k2 = conversion rate from x2 to x3; N = the number of infectious virions produced by an where k 10 and k20 are natural rate for conversion of θ = viral concentration needed to decrease s. uninfected CD4 cells into latently infected CD4 cell and natural rate for conversion of latently infected CD4 cell into 1 and k2 are functions of the drug doses actively infected cell, respectively. Also, the parameters α1 2 are the efficiency of drugs for reverse transcriptase 1 cells are stimulated to proliferate with rate (x , x , x ) in the presence antigen and HIV, that is, The variables m1(t) and m2(t) are the doses from drugs λ(x , x , x ) = r 1− (x + x + x )/T administrated to reverse transcriptase and protease Without the presence of HIV the rate of generation is S(x4), inhibitors respectively, while α1 and α2 are constants. In [7,17] one can see other works dealing with the control of the viral load using a mathematical modeling and In the presence of free HIV (x4), uninfected cells x1 can be infected to become x2 cells or x3 cells, depending on the One possible approach could be the use Pontryagin’s probability of the cells to become actively or latently However, in many actual application problems the respectively. To fit these parameters is very difficult and analytical solutions for optimal control are very difficult to only are possible with a controlled experiment. obtain because of the need to solve the TPBVPs (Two Point In this optimization method (see [4]) the control input m(t) is represented by an expansion over the interval [0 , tf] with This way of solving the optimal control problem is said to be an indirect method. An alternative is to optimize directly the cost functional (known as performance index) using the parameterization of the control (the input functions m(t)). In the present case, this involves a subset of the coefficients in a series expansion employing hyperbolic functions. Those approximations are sub -optimal, in the sense that the cost achieved is generally worse when the higher terms of the series expansion are neglected. However, those sub-optimal inputs were found to be satisfactory in the present where the coefficients cij are to be determined by The numerical method used here was proposed by Jacob in Here, after fitting the parameters the obtained values were: [4] and is available in the form of a computer program called EXTREM. The objective is to find a control input Table I - Parameters fitted, Patient-A. J (m) = h( x t where to and tf are the initial and final instants of time, fixed The functions h and g are constrained by the state equation that in the specific problem is described by the equations Table II - Initial conditions, Patient-A. The chosen performance index tries to make a compromise between the side effects and the therapeutic effects, reflected by the CD4 count and a the viral load and it has Table III - Parameters fitted, Patient-B. ∫  1− e 1 1 + φ 1− e 2 2 + Table IV - Initial conditions, Patient-B. The biological interpretation of the proposed cost functional is that the two first terms out of integral represent the target of maximizing non-infected CD4 cells and also to minimize The values for CD4+T cell counts and the viral load were the viral load after a pre-specified time horizon. obtained from two patients in a sample of 43 patients of The coefficients φ1 and φ2 of the terms in the integrand are medical reports at the Centro de Referência e Treinamento weights that reflect the dose-related side effects of the two em DST-AIDS in São Paulo, Brazil. drugs (m1(t) and m2(t)) which must be adequately balanced. They were patient-039 in the sample, to whom we shall call The two last terms are included to force x1 (uninfected Patient-A, and patient-002 in the sample, to whom we shall CD4+T cells) to increase and x4 (viral load) to decrease with These patients were chosen because they had not used Finally, ε1 and ε2 are sensibilities of the patient with respect specific drugs to combat HIV up to the moment that the to reverse transcriptase inhibitors and protease inhibitors first symptoms of AIDS appeared. Then they used the combination AZT (600 mg) + 3TC (300 mg) (constant doses) for 495 days (Patient-A) and for 340 days (Patient- B) when they switched to the HAART scheme. The results are shown in figures 1-7. During the period presented in Fig-1 and Fig-5 the two patients had manifested symptoms of AIDS. In those figures the CD4 count and the viral load are represented by the “triangles” and the “plus signs”, respectively. region of study
Fig-3 - Viral load (optimal and fitted) for Patient-A. The results of the fitting process are shown in Fig-2 and Fig-6. The parameters were obtained by fitting Tan’s model to the clinical data using several computer simulations until good precision was obtained in terms of CD4 counts and the viral load. It is possible to note (Fig. 3) that the fitting is adequate in Fig-1 - Historical data of Patient-A. The Patient-A had bacterial pneumonia + herpes + hepatitis C and the Patient-B had advanced stages of herpes. The drugs prescribed for the two patients were Zidovudine (AZT) 600 mg plus Lamivudine (3TC) 300 mg each day. Subo ptimal
Fit Curve
Time (Days)
Fig-4 - Performance index for Patient-A. The corresponding parameters that were used in the simulations are presented in Tables I-IV. For this simulations it was used the constant control m1 = 900 (AZT = 600 mg plus 3TC = 300 mg) Fig-2 - Optimal and fit curves for CD4 count and drug region of study
After the fitting of curves to the actual data the direct method described in the last section is used to solve the optimal control problem for the two patients. In Fig-2 it is possible to see that for Patient-A the optimal treatment will yield smaller doses in the initial period, but this should be gradually increased until the final period. He could start the treatment with 865 mg and 495 days later 4.1 - Optimal treatment for Patient-A During this simulated treatment the sub-optimal values of side effects would be less than with the standard treatment. We can observe a comparison between the simulated results and actual values for Patient-A in Fig-4. When the actual values are inserted into equation (9), the performance index, one can see that it has higher values than the sub-optimal Fig-5 - Historical data of Patient-B. 4.2 - Optimal treatment for Patient-B The Patient-A had an increase in CD4+T cells count until mid-period but after that he presented a typical picture of In Fig-6 it is possible to see that for Patient-B the optimal immunological resistance with the viral load tending to be treatment would be improved by starting using around 880 very high, around 32000 copies/ml (Fig-2). After 495 days mg of drugs and gradually decreased until the final period he started a HAART scheme and it is possible to observe the decrease of the viral load and increase of CD4 count. The Patient-B (Fig-5) also has a typical immunological The results for CD4 are similar to the actual data (also in resistance with the viral load increase in this period (340 Fig-6) and to HIV the curves had undetectable differences days). The initial CD4+T cells count was 30 cells/mm3 and of the same type of Fig-3 for Patient-A. in the final period was around 110 cells/mm3 (healthy Also in this case it is possible to observe the better results individuals have CD4 cells counts about 1,000 cells/mm3 ). for sub-optimal control than the actual data (Fig-7). But the solution is almost the same fo r constant use of drugs. Time (Days)
Fig-7 - Performance index for Patient-B. Fig-6 - Optimal and standard treatment for Patient-B. [3] Caetano & Yoneyama. 2002. Short and Long Period Optimization of Drugs Doses in the Treatment of AIDS. Here we have seen an application of the optimal control Anais da Academia Brasileira de Ciências. V. 74. No. 3/4 theory for improving the administration strategy of drugs [4] Jacob, H. G., 1972. An engineering optimization method The work includes comparisons of theoretical solutions with application to STOL aircraft approach and landing obtained using computer simulations based on mathematical model with actual clinical data from two [5] Nowak M. A. & C. R. M. Bangham 1996. Population cases: Patient-A and Patient-B from the Centro de Dynamics of Immune Responses to Persistent Viruses. Referência e Treinamento em DST-AIDS in São Paulo city, [6] Tan, W. Y. & Wu, H. 1998. Stochastic modelling of the The optimal control problem was solved with a numerical dynamics of CD4+T-cell infection by HIV and some Monte direct method. The mathematical solutions show that Carlo studies. Mathematical Biosciences. 147. 173-205. treatments can be improved in terms of side effects, [7] Wein, L. D., D’Amato, R. M., Perelson, A. S., 1998. although further studies are required to establish the Mathematical analysis of antiretroviral therapy aimed at HIV-1 eradication or maintenance of low viral loads. J. An important fact related to the use of the optimal (sub- optimal) strategy lies in the probability of CD4 infected [8] Taffé, M. Rickenbach, B. Hirschel, M. Opravil, H. cells transforming from latent to active forms. For Patient- Furrer. P. Janin, F. Bugnon, B. Ledergerber, T. Wagels, P. Sudre. 2002. Impact of occasiona short interruptions of HAART on the progression of HIV infection: results from a cohort study. AIDS. V. 16. No. 5. 747-755. [9] Garcia, M. Plana, G. M. Ortiz, S. Bonhoeffer, A. It is interesting to observe that the chance of infected CD4 Soriano, C. Vidal, A. Cruceta, M. Arnedo, C. Gil, G. cells to become latent under the optimal control is almost Pantaleo, T. Pumarola, T. Gallart, D. F. Nixon, J. J. Miró, J. M. Gatell. 2001. The virological and immunological When more infected cells become latent their actions may consequences of structured treatment interruptions in have a longer time horizon and hence, more difficult to chronic HIV-1 infection. AIDS. V. 15. No. 9. F29-F40. combat. The initial viral load was 32,473 copies/ml for Patient-A [10] Perelson, A. S., D. E. Kirshner, R. DeBoer. 1993. and 407,000 copies/ml for Patient-B. While the viral load Dynamics of HIV-infection of CD4+T-cells. Mathematical decreases for Patient-A, it (slowly) increases for Patient-B [11] Phillips, A. 1996. Reduction of HIV concentration during acute infection, independence from a specific immune response. Science. V. 271. 497-499. [12] Mittler, J.E., Sulzer, B., Neumann, A. U., Perelson, A. The authors are grateful to the Centro de Referência e S., 1998. Influence of delayed viral production on viral Treinamento em DST-AIDS in São Paulo city, Brazil, in dynamics in HIV-1 infected patients. Mathematical particular to Dr. Mylva Fonsi for her help in providing the [13] Murray, J. M., Kaufmann, G.; Kelleher, A. D., Cooper, D. A., 1998. A model of primary HIV-1 infection. Mathematical Biosciences. 154. 57-85. [14] Wick, D. 1999. On T-cell dynamics and [1] Caetano, M. A. L. & Yoneyama, T. 1999. Comparative hyperactivation theory of AIDS pathogenesis . evaluation of open loop and closed loop drug administration Mathematical Biosciences. 158. 127-144. strategies in the treatment of AIDS. Anais da Academia [15] Kirk, D. E., 1970. Optimal Control Theory; An Brasileira de Ciências. V. 71. No. 4-I. 589-597. Introduction. Prentice-Hall. New Jersey, 452 p. [2] Caetano, M. A. L. & Yoneyama, T. 1999. Optimal [16] Lewis, F. L., 1986. Optimal Control. New York. 362 p. control theory applied to the anti-viral treatment of AIDS. Theory and Mathematics in Biology and Medicine [17} Perelson, A. S., P. W. Nelson. 1999. Mathematical Conference. Amsterdam. Netherlands. Analysis of HIV-1 Dynamics in Vtvo, SIAM Review. V. 41. No. 1. 3-44.

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