Control of the HIV infection and drug dosage

doi:10.1016/j.bspc.2009.05.001

Biomedical Signal Processing and Control

Volume 5, Issue 1, January 2010, Pages 45-52

https://doi.org/10.1016/j.bspc.2009.05.001 Get rights and content

Abstract

An increasing number of sophisticated control algorithms become available in the current literature to optimize the HIV therapy. Unfortunately, the pharmacokinetics and pharmacodynamics of antiretroviral drugs are ignored and these algorithms remain purely theoretic. This issue is investigated explicitly in this paper. An elementary pharmacodynamics model is combined with a non linear feedback control computed from standard engineering methods. It is shown that it results in the design of a realistic dosage regimen which drives the immunological system close to the healthy equilibrium state. Although the problem is dealt as a single input system, it is argued that the procedure can be extended to a multitherapy design or to any available control law.

Introduction

Treatment of human immunodeficiency virus (HIV) remains a major challenge. About three decades ago, HIV began to spread worldwide at an alarming rate. It is reported that in 2007, 33.2 million people were living with HIV/AIDS, 2.5 million people were newly infected and 2.1 million people died due to development of AIDS [1]. Significant progress has been made in the treatment of HIV infected patients, resulting in improved quality of life and greater longevity. Nowadays, thanks to advances in drugs development and their combination in “drug cocktails”, most of the patients maintain undetectable viral load and safely high T-cell count for several years.

Most of the currently available anti-HIV drugs fall into one of two categories: Reverse Transcriptase Inhibitors (RTIs) and Protease Inhibitors (PIs). From a system theoretic approach, these families of drugs represent independent “control inputs”. On one hand, RT inhibitors prevent HIV RNA from being converted into DNA, thus blocking integration of the viral code into the target cell. On the other hand, protease inhibitors affect the viral assembly process in the final stage of the viral life cycle, preventing the proper cutting and structuring of the viral proteins before their release from the host cell. Highly Active Antiretroviral Therapy (HAART) is the most prevalent treatment strategy for HIV infected patient. It uses two or more drugs, typically one or more RTI and one PI. However, these drugs have a number of side effects on vital human functions. Thus, HIV treatment methods have drawn attention from the biomedical and control engineers societies and many control methods have been reported for HIV treatment.

The dynamic multidrug therapy problem is modeled in ref. [3] as an optimal control model that maximizes the inhibition of HIV. In ref. [4], finite horizon open loop control tools are applied to an HIV chemotherapy model using an objective function based on a combination of maximizing T-cells count and minimizing the systemic cost of chemotherapy. Continuous-time feedback control used in refs. [2], [5], [6], [7], [8] suppresses the viral load to undetectable levels while in refs. [9], [10] model predictive control is performed. In ref. [13] gradual reduction of drug dose is shown to boost the immune response. In refs. [11], [12] the possibility of a long term non-progressor (LTNP) is demonstrated.

The previous studies developed several new approaches in the control of the HIV infection. Although sophisticated, these control schemes are not yet ready to be applied in real-life monitoring of patients because they do not provide a discrete-time drug posology but rather a continuous-time optimal control stated in terms of abstract mathematical parameters hardly interpreted by clinicians. This paper includes basic pharmacokinetics (PK) and pharmacodynamics (PD) principles in the design of nonlinear control laws to reduce the gap between purely theoretical control laws and real clinical drug dosing. Furthermore, the majority of existing studies consider the HIV dynamics as a system with two independent control inputs, each input being related to one of the two drug types i.e. RTIs and PIs. To the best of our knowledge, this assumption is not confronted to real clinical data.

This paper has two major contributions:

(1)
First, it states that the basic HIV dynamics model may be approximated as a single input model. This result is derived from real clinical data analysis. To the best of our knowledge, therapy effect on model parameters has been ignored in a wide range of studies. Thus, confrontation of the literature assumptions to real clinical data is needed to demonstrate the feasibility of theoretical algorithms.
(2)
Pharmacological models (PK and PD) are introduced explicitly in the design of the control input. Thus, a dosage regimen that takes into consideration pharmacokinetics and pharmacodynamics of antiretroviral drugs is derived from feedback linearization of the single input model. This makes much more sense for a real life therapy.

In Section 2, the third order basic model of the HIV dynamics is presented. Section 3 describes the identification technique used in the paper. Section 4 presents the first result of this study which states that the HIV dynamics is a single input system based on real clinical data analysis. In Section 5, it is shown that the single input model proposed in Section 4 is accessible and fully linearizable by state feedback. Full and partial linearization are also detailed in this section. Section 6 presents two basic pharmacological concepts: pharmacokinetics (PK) and pharmacodynamics (PD). Section 7 presents the second contribution of this work. It uses the continuous control law derived in Section 5 and the pharmacological concepts of Section 6 to design an equivalent “discrete” dosage regimen. The monotherapy case is detailed first and it is shown that the method is still suitable for multitherapies. Some concluding remarks are given in Section 8.

Section snippets

Modeling the infection dynamics

The basic modeling of the HIV/AIDS dynamics is described by a $3$ -dimensional (3D) continuous-time model ([14], [15]). This model, given by Eq. (1), includes the dynamics of the uninfected CD4+ T-cells, the infected CD4+ T-cells, and the virions. $\{\begin{matrix} \dot{T} = s - δ T - β T V, \\ {\dot{T}}^{*} = β T V - μ T^{*}, \\ \dot{V} = k T^{*} - c V . \end{matrix}$ T (CD4/mm³) represents the amount of uninfected CD4+ T-cells, $T^{*}$ (CD4/mm³) represents the amount of the infected CD4+ T-cells, and V (RNA copies/ml) the free virions. Healthy CD4+ T-cells are produced at a constant source

Parameter identification

According to identifiability theory presented in ref. [16], the 3D model (1) is algebraically identifiable from output measurements, namely the viral load $(V)$ and the total CD4 count $(T + T^{*})$ [17], [18], [19].

The identification of all the parameters of the 3D model from standard clinical data was first introduced in ref. [20]. This approach was based on the nonlinear simplex optimization method. A new estimation method was introduced in ref. [21]. It is based on a Monte-Carlo approach which is

Parameters sensitivity to the therapy: state of the art

Commonly, an HIV/AIDS therapy includes some RTIs and some PIs. The action of RTIs is to block the production of new viral particles since they inhibit the reverse transcription of viral RNA into viral DNA. On one hand, the RTIs slow down the infection process and, according to refs. [15], [23], [24], [9], [25] it is admitted that their action is on parameter $β$ . On the other hand, the PIs affect the natural life cycle of the virus by prohibiting the maturation of new virions. As a result, the

Analysis and control of the single input model

The control design requires a preliminary and mandatory test of accessibility of the system. This section shows that the proposed single input model (4) preserves the accessibility property and demonstrates the feasibility of standard nonlinear control laws when using this model. The accessibility of models with two inputs or with one input affecting parameter $β$ has been demonstrated in ref. [18], [26]. Section 5.1 proves that this property is not lost when using the single input model (4).

Pharmacological concepts

The major drawback in the existing control strategies [2], [5], [13], [6], [8] is that they are too sophisticated to be applied in real life monitoring of patients. In fact, treatment efficacy is usually used to be the control input as it is the case in Section 5. This efficacy varies from 0 (no medication) to 1 (full medication) and is not expressed in terms of drug amounts. The current section addresses the problem of relating this control input to the real drug posology. Thus, incorporation

Dosage regimen computation

Parmacological models (20), (23) are incorporated explicitly in the computation of a realistic dosage regimen based on the feedback linearization presented in Section 5.2.2-ii).

Conclusion and future work

Based on real clinical data, the analysis of the impact of antiretroviral therapy on the HIV dynamics shows that the system is essentially a single input system in the sense that one single parameter is mostly affected by the drugs. After feedback linearization of the single input model, incorporation of pharmacokinetics (PK) and pharmacodynamics (PD) of antiretroviral drugs is introduced. This approach allows the design of a “realistic” dosage regimen which drives the system near the healthy

Acknowledgment

This work was supported by a CNRS-INRIA/MINCyT project. The authors wish to thank the Nantes University Hospital (CHU) for providing clinical data used in this study.

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Control of the HIV infection and drug dosage

Abstract

Introduction

Section snippets

Modeling the infection dynamics

Parameter identification

Parameters sensitivity to the therapy: state of the art

Analysis and control of the single input model

Pharmacological concepts

Dosage regimen computation

Conclusion and future work

Acknowledgment

Journal of Theoretical Biology

Biomedical Signal Processing and Control

Journal of Theoretical Biology

Feedback control of the chemotherapy of HIV

International Journal of Bifurcation and Chaos

Optimal control of the chemotherapy of HIV

Journal of Mathematical Biology

Feedback control of a biodynamical model of HIV-1

IEEE Transactions on Biomedical Engineering

Nonlinear control of a dynamic model of HIV-1