**Use of mathematical modeling for optimizing and adapting immunotherapy protocols in HIV-infected patients**

Chloé Pasin (1,2,3), Laura Villain (1,2,3), François Dufour (4,5), Daniel Commenges (1,2,3), Mélanie Prague (1,2,3), Rodolphe Thiébaut (1,2,3)

(1) Univ. Bordeaux, INSERM, Bordeaux Population Health Center, France, (2) INRIA SISTM team, France, (3) Vaccine Research Institute, France, (4) Bordeaux INP, IMB, France, (5) INRIA CQFD team, France

**Objectives: **

In some cases, HIV-infected patients under antiretroviral therapy fail to restore their immune system and especially CD4+ T lymphocytes (CD4 in short) counts. Exogenous interleukin 7 (IL7) has shown beneficial effect to help increasing the number of CD4. A simple mechanistic model based on ordinary differential equations was developed to assess the effect of IL7 on CD4 proliferation, thymopoiesis and survival [1][2]. Then, phase I/II human clinical trials (INSPIRE studies) on 128 HIV-infected patients have shown that repeated cycles IL7 injections help maintaining HIV-infected patients with CD4 levels above 500 cells/µL [3], a level associated with a nearly healthy clinical status. Interestingly, the mechanistic model fitted very well the data although the distribution of the number of injections received was very different across the patients. Following this work, a question of interest was to determine optimal schedule of injections, to conduct the lightest intervention leading to the longest time above 500 cells/µL. We developed two approaches. One is based on the theory of optimal control and the other uses a Bayesian approach. Both methods succeeded in providing an optimal strategy for pseudo patients with different profiles, generated with the maximum a posteriori law obtained through previous parameters estimations on INSPIRE data.

**Methods:**

In the first method, we model the process with a Piecewise Deterministic Markov Process (PDMP), assuming that the patient’s parameters are known and the stochasticity is due to the biological process. All actions realized on the process to modify its trajectory constitute a strategy, characterized by an optimality criterion balancing the time spent with CD4 levels below 500 cells/µL and the number of injections made. This criterion is minimized to determine the optimal strategy and its associated cost. Some theoretical results in [4] have shown that this minimization can be obtained through the iteration of an operator. This construction leads to a natural method of computation and enabled us to develop a numerical tool on Matlab. However, this method does not account for the uncertainty induced by the parameters estimation. The second approach deals with this issue by introducing random effects in a population model and estimating individual parameters with MCMC algorithm each time new information is available. Treatment can be adapted by using the predicted distribution of given criteria related to the CD4 trajectory. Two protocols are proposed: either the decision for a new cycle is based on the risk to fall below 500 cells/µL before the next visit, or the time of control visits are adapted.

**Results:**

We have used the optimal control method to determine the optimal cost and strategy for 50 pseudo patients on a horizon of one year. For all patients, the optimal strategy always gives a lower cost than other possible protocols and achieves a good balance between clinical criteria such as time spent with CD4 levels under 500 cells/µL, mean of CD4 cells/µL and number of injections realized. This strategy consists in first cycles of two injections until the number of CD4 is high enough and then one-injection cycles maintain the CD4 levels over 500 cells/µL. In the Bayesian approach, the different protocols were simulated for 150 pseudo patients on a horizon of two years. All reduce the time spent under 500 CD4 cells/µL without increasing too much the number of visits and injections compared to the original protocol. Altogether these results confirm the possibility to adapt and optimize the strategy of IL7 injections.

**Conclusions: **

Both approaches can be used to adapt schedule of injections while maintaining patient above 500 CD4 cells/µL as long as possible. These positive results are also mainly attributed to the very good prediction ability of the deterministic model used for the dynamics of the CD4 cells. Although the optimal control method considers that the patient’s parameters are known and induces large computing time, it could be more adapted if a deterministic model would not be sufficient to describe the biological process. The Bayesian approach is successful as it can be easily implemented on large horizons of time and accounts for the diversity of response of the patients without needing a long phase of learning. It offers clinical perspective, such as the evaluation of the adaptive strategy on clinical outcomes in larger trials.

**References:**

[1] Thiebaut R, Drylewicz J, Prague M, Lacabaratz C, Beq S, Jarne A, Croughs T, Sekaly RP, Lederman MM, Sereti I, Commenges D. Quantifying and predicting the effect of exogenous interleukin-7 on CD4+ T cells in HIV-1 infection. PLoS computational biology. 2014 May 22;10(5):e1003630.

[2] Jarne A, Commenges D, Villain L, Prague M, Lévy Y, Thiébaut R. Modeling CD4+ T cells dynamics in HIV-infected patients receiving repeated cycles of exogenous Interleukin 7. The Annals of Applied Statistics. 2017;11(3):1593-616.

[3] Levy Y, Sereti I, Tambussi G, Routy JP, Lelievre JD, Delfraissy JF, Molina JM, Fischl M, Goujard C, Rodriguez B, Rouzioux C. Effects of recombinant human interleukin 7 on T-cell recovery and thymic output in HIV-infected patients receiving antiretroviral therapy: results of a phase I/IIa randomized, placebo-controlled, multicenter study. Clinical infectious diseases. 2012 May 1;55(2):291-300.

[4] Costa OL, Dufour F, Piunovskiy AB. Constrained and unconstrained optimal discounted control of piecewise deterministic Markov processes. SIAM Journal on Control and Optimization. 2016 Jun 2;54(3):1444-74.