Optimal Control of HIV-Virus Dynamics
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- Radisavljevic-Gajic, V. Ann Biomed Eng (2009) 37: 1251. doi:10.1007/s10439-009-9672-7
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In this paper we consider a mathematical model of HIV-virus dynamics and propose an efficient control strategy to keep the number of HIV virons under a pre-specified level and to reduce the total amount of medications that patients receive. The model considered is a nonlinear third-order model. The third-order model describes dynamics of three most dominant variables: number of healthy white blood cells (T-cells), number of infected T-cells, and number of virus particles. There are two control variables in this model corresponding to two categories of antiviral drugs: reverse transcriptase inhibitors (RTI) and protease inhibitors (PI). The proposed strategy is based on linearization of the nonlinear model at the equilibrium point (steady state). The corresponding controller has two components: the first one that keeps the system state variables at the desired equilibrium (set-point controller) and the second-one that reduces in an optimal way deviations of the system state variables from their desired equilibrium values. The second controller is based on minimization of the square of the error between the actual and desired (equilibrium) values for the linearized system (linear-quadratic optimal controller). The obtained control strategy recommends to HIV researchers and experimentalists that the constant dosages of drugs have to be administrated at all times (set point controller, open-loop controller) and that the variable dosages of drugs have to be administrated on a daily basis (closed-loop controller, feedback controller).
KeywordsNonlinear controlLinearizationOptimal controlHIV dynamic model
The human immunodeficiency virus (HIV) causes the acquired immune deficiency syndrome, knows as AIDS. Mathematical models of HIV dynamics were derived several years ago.7,8,16,19 Controlling dynamics of HIV has been an interesting and challenging research problem.2,3,5,6,10,12,14,21 In this study, we consider the third-order model of HIV dynamics that captures the time rate of healthy and infected white blood cells (T-cells) and the number of HIV viruses. This model has two control variables corresponding to two major categories of antiretroviral drugs: reverse transcriptase inhibitors (RTI) and protease inhibitors (PI). There are more complex models of HIV dynamics that can be found in the literature.2,7,8,15,18 The methodology presented in this paper can be applied with minor modifications to the other models of HIV dynamics. However, as indicated by an anonymous reviewer of the manuscript, it should be emphasized that the presented analysis is limited in the sense that it “can not take into the account patient factors as physiological/genetic level, physicochemical factors at cell–protein–viral interactions level, and viral factors that relate to the various HIV strains and clades.”
s = 10 per mm3 per day is the constant source of healthy T-cells (thymus);
d = 0.02 per day represents the death rate of healthy T-cells;
μ1 = 0.24 per day represents the death rate of viruses;
μ2 = 2.4 per day represents the death rate of infected T-cells;
β = 2.4 × 10−5 per (mm3 × day) is the infectivity rate of free viruses;
k = 100 per cell is the rate of virons (free virous particles) produced per infected T-cell.
These parameters can be estimated using an adaptive observer.23 When the number of T-cells, which is normally around 1000 per mm3 is under 200 per mm3, a HIV-infected patient is classified as having AIDS.
HIV patients can be well treated with a combination of drugs over a long period of time. One of the reasons for the negative outcome of the treatment is the high dosage of drugs that patients receive over such a long time period. In the next section, we will discuss control theory strategies that minimize the amount of drugs needed, while keeping the patient at the stable “quasi steady state” equilibrium for long time.
Open- and Closed-Loop (Feedback) Control
The controlled dynamic system (5) does not reflect different chemical structures of the RTI and PI drugs and the fact that they target different proteins in human cells. At the present state of knowledge, no mathematical dynamic models for the RTI and PI drugs and the corresponding proteins exist in the scientific literature. However, a new rapidly developing field, named systems biology16,20,21 (developed within bioinformatics), studies dynamics of chemical processes in human cells and develops corresponding mathematical models described by controlled differential equations. Hopefully, such models will be obtained in not very distant future. Once such models become available, it will be very interesting and very challenging to couple those models with the existing mathematical models for HIV dynamic and controls and perform the corresponding analysis at a much more sophisticated level. In our study, we have a simplified model in which the RTI drug affects multiplicatively the virus infectivity rate and the PI drug affects multiplicatively the infected T-cell productivity of free virous particles. Since model (5) uses the normalized values of control variables (RTI and PI drugs) our analysis will have a simplified goal to determine (via optimal feedback control theory) what percentage values of these drugs should be administrated daily over a long period of time.
The concept of system controllability4 was used to make a decision when to initiate HIV therapy13 for the HIV dynamic model very similar to (5). The conclusion reached states: “the therapy is best initiated when the viral load is easier to control because this implies the use of lower drug doses and consequently bearable side effects.”13 This conclusion was achieved by finding the singular values of the corresponding controllability Grammian,4 “a higher singular value indicates an easier to control viral load in the sense that the transition to the treatment steady state is faster. A lower singular value indicates a more difficult to control viral load characterized by a slow transition to the final state, given the same control effort.” Even more, the paper finds that the controllability profiles of both RTI and PI drugs are similar. Several types of controller for HIV dynamics can be found in the literature: classical PID controller1,6 controller based on linear state feedback.5 In addition, a controller based on the backstepping14 technique of nonlinear control technique that can drive state variables to their desired values and assure non-negativity of all states in the closed-loop (feedback) system was considered.10 An optimal controller15 based on dynamic programming was recommended “to reduce medication and establish long-term immune response against HIV-infection.” That paper also studies the controllability property of the HIV model used. A nonlinear model of HIV was controlled via the state-dependent Riccati equation2 approach. That paper considers both the full and partial state feedback cases, and in the case when all state variables are not available for feedback, an observer is designed to reconstruct all state space variables.
Equilibrium Points of the Controlled HIV Model and the Set-Point Controller
Optimal Feedback Controller
Since the viral drugs are administrated based on measurements of healthy T-cells, x1(t), and viruses x3(t), the system output equation can be taken as y1(t) = x1(t) and y2(t) = x3(t). Note that the number of infected T-cells can be determined using an observer.6 Hence, we can assume that all state variables are available for feedback so that information about xi(t) − xiss = Δxi(t), i = 1,2,3, is known.
It can be observed from Fig. 8 that for the optimal feedback strategy, the number of healthy T-cells decays slowly from 560 to 490 per mm3, reaching the target value after roughly 110 days. On the other hand, it can be seen from Fig. 11 that the number of viruses decays very rapidly and in less than one day reaches its desired target value of 30 per mm3. Figure 10 shows that the number of viruses remains in its steady state desired target value of 30 per mm3 for the entire time interval of simulation equal to 500 days. Figure 9 indicates that the number of infected T-cells at steady state is only one per mm3. The proposed optimal feedback strategies are presented in Figs. 12 and 13. The results obtained for the drug treatment schedule are very interesting. They indicate that during the first 110 days the drug RTI should not be administrated at all and after that period it should in a matter of days be increased to the relative value of 0.43. The drug PI should be the only one given during the initial therapy. Its dosage should very quickly go to 0.54 and after 110 days it should be gradually reduced such that after 150 days its dosage is equal to 0.14. That dosage should remain constant during the time interval from 150 to 500 days.
Note that the proposed treatment strategy has two components: the first one that comes from the constant feed-forward set-point controller (open-loop controller) and the second one comes from the time varying optimal feedback strategy (closed-loop controller), see Fig. 7, where these two components are added in the summing block to form the proposed control signal u(t). It is interested to see what the results will look like when the feedback loop is not present and the drug administration is based only on the feedforward controller (open-loop strategy). The feed-forward controller is designed such that the desired target values are reached at the steady state. However, this controller is not able to take care about the transient behavior, that is, it is unable to control the system passage from the given initial state to the desired steady state.
Open-Loop Drug Treatment
We have presented a drug treatment strategy based on the optimal control theory and the control of nonlinear system technique via linearization. It has been shown that such a feedback (closed-loop) strategy regulates the HIV dynamics (represented by the considered model) much more efficiently than the open-loop strategy. Open-loop drug administration is common in medical practice (it prescribes constant levels of medications, for example 10 mg, 20 mg, and patients are checked every month or three or six months to eventually receive new dosages). The closed-loop drug administration requires that dosages take continuum of values and that the state of a patient be monitored at all times, every day. This might be a difficult requirement, but the advantages of the closed-loop over the open-loop drug administration, as shown in our simulation results, are enormous.