When to Initiate HIV Therapy
A summary and explanation of the paper written by
Annah Jeffrey, Xiaohua Xia and Ian Craig
Overview
- This project explores the application of control theory to human immunodeficiency virus (HIV)/AIDS models. A system of ordinary differential equations describes the relationship of the host-pathogen interaction between the immune system and HIV. Minimum singular value decomposition is used to determine which stage of the disease is most controllable. With this information, the optimal time to begin therapy can be determined.
Background
- Beginning with a standard model for the HIV virus and the corresponding immunological response, Annah Jeffrey, Xiaohua Xia, and Ian Craig attempt to make a more formal mathematical basis for identifying the optimal time frame to begin administering the drug therapies in their paper titled When to Initiate HIV Therapy: A Control Theoretic Approach.1 Beginning with the basic differential equation model, and then exploring the effectiveness of the two standard types of drugs used to reduce the viral load, a control theory approach is then used to help determine the optimum initiation time. Control theory basically deals with determining the relative effectiveness of a control parameter in a nonlinear system as time increases, whether that effectiveness increases or decreases, and the overall effect on the system. The main goal of the research was to determine the differences in drug effectiveness between the early stage of infection and the later stage, compare it with the middle stage when the viral load is lower, and draw a conclusion about the optimal time frame to begin treatment. (For more information on control theory, please see APPM4390:Robustness Analysis of an Observer-Based Controller in a Food Web).
History
- Much study has taken place on when to initiate the treatment of HIV, with no clear consensus on the answer. For example, the International AIDS Society – USA recommends starting treatment earlier than the United States Public Health Service. However, to improve the quality of life of those infected with HIV, it would be ideal to minimize the drug dosages and hence their side effects, while still maximizing their effectiveness. It is not in the best interest of the doctors or patients to increase the dose with no actual benefit, but if the dose is too small, the maximum benefit cannot be obtained. In general, depending on many factors, antiretroviral drugs are administered when the CD4+ T cell count is between 200 and 350 copies/mm^3. If the count is higher, the virus has a greater chance of developing resistance before the drugs can bring it under control, decreasing treatment options later in the course of the disease. If the CD4 count is lower, the resurgence of the immune system may cause problems as it attacks latent infections that have occupied the body.
Type of Model & Biological System Studied
- HIV/AIDS model that demonstrates T cell and viral load dynamics.
- The researchers explore specific components of the immune system. More specifically, they investigate how the interactions between CD4+ T cells and the HIV/AIDS virus impact the CD4+ T cell count and the viral load when the two different treatment options are offered at specific stages of the disease.
- For more information on the limitations, expectations, and future directions of HIV/AIDS modeling, see Mathematical Modeling of Aids Progression.
- Below is a schematic diagram of a typical HIV dynamic model for the mechanisms taking place within a single HIV-infected patient.^{[1]}
Mathematics Used
- Controllability analysis is used to determine the best control strategy to be applied, or whether it is even possible to control or stabilize the system. The viral load is considered to be controlled if the control input (reverse transcriptase inhibitors or protease inhibitors) can reduce the viral load by 90% in eight weeks from the time treatment is initiated and continues to suppress it to below 50 copies per milliliter of plasma in six months.
- In order to utilize controllability analysis, the nonlinear dynamical system specific to either reverse transcriptase inhibitors or protease inhibitors is linearized to develop an approximate analysis. Then a controllability matrix is determined from the system's Jacobian matrices. The system is output controllable if and only if the output controllability matrix has full row rank. Lastly, one can apply minimum singular value decomposition (SVD) to the controllability matrix. Minimum SVD is used to measure the extent to which the different stages in the progression of HIV/AIDS disease are controllable and, consequently, when best to initiate reverse transcriptase inhibition or protease inhibition such that the viral load is necessarily reduced.
- Steady states are also found for the infectious and noninfectious virus particles, viral load without treatment, and viral load with treatment - which is determined by drug efficacy. Therefore, if there is a desired treament steady state, one can determine the drug efficacy that is required.
- Simulations are used in order to analyze the non-linear dynamical system and demonstrate the effect of treatments at various stages of the disease.
Mathematical Model with Parameter Definitions
- The working model for the host-disease interaction characterized by HIV is given by
Where "x_{1}" is the concentration of uninfected CD4+ T cells,
- "x_{2}" is the concentration of infected CD4+ T cells, and
- "x_{3}" is the concentration of free virus particles.
- The parameters are estimated to be
Solving the above problem numerically in Mathematica (Mathematica rules!) produces the following plot for X_{1} which is the number of uninfected CD4+ T cells. The plot shows the general behavior of a few oscillations early in time, before seeming to settle down towards a stable level. The initial condition on the number of uninfected CD4+ T cells was 500/mm^3,
This model, however, does not take into account either of the two common treatment drugs, which are reverse transcriptase inhibitors and protease inhibitors. The reverse transcriptase inhibitors work by blocking the infection of CD4+ T cells. Protease inhibitors, on the other hand, work by rendering the virions produced by infected CD4+ T cells harmless. In practice, treatment is some combination of these two drugs, but individually, the changes to the equations are as follows.
For reverse transcriptase inhibitors, the equations become
where measures the effectiveness of the drug, and in this case is the control input. When this parameter is varied, we are changing how good the drug is at blocking the infection of uninfected CD4 cells, and a higher effectiveness will decrease the viral load faster then a lower effectiveness.
When using protease inhibitors, the equations change slightly, and a 4th equation is added:
Where is the control input linked to the effectiveness of the protease inhibitor drugs at rendering the virions produced by infected CD4 cells harmless
Both of the above models lead to a higher steady state value for uninfected CD4 cells, and the plot below is from the model for protease inhibitors, with a 40% effectiveness and an initial value of 500 uninfected CD4 cells.
Full Model With Results From Paper
- Using the models outlined above, the ultimate goal of the research is to use control theory to help determine the optimal time to administer treatment. Control theory is defined as “the mathematical study of how to manipulate the parameters affecting the behavior of a system to produce the desired or optimal outcome. ”1 Control theory works to determine when a parameter has the greatest effect on the system you're trying to control, and that will tell us when to begin treatment, or phase treatment out if it is ineffective. To do this, the three general steps are 1) find the steady states and evaluate the Jacobian at this steady state, 2) obtain the controllability matrix, and 3) obtain the minimum singular value decomposition of the controllability matrix, which will estimate how controllable the system is at any particular time.
Linearizing a nonlinear system, finding the steady states, and then evaluating the Jacobian at these steady states are elementary exercises for students of dynamical systems, and will not be re-derived here, however, the results for a single drug regimen, consisting of just protease inhibitors, is given below. The Jacobian, when evaluated at a steady state operating point given by (x^{0}, u^{0}_{2} is then A_{CO} and B_{CO}, where
with the parameter k_{4} defined by
To obtain the controllability matrix, note that this matrix is defined as
where B and A are the Jacobians from linearizing the system and evaluating at the steady states, and for the protease inhibitor model, these matrices are given above. Then, the controllability matrix for this treatment is given by
File:Koepke 70.png and in its full form is
Where and
For a combination of reverse transcriptase and protease inhibitors, the Jacobians are A_{CO} and B_{CO} for the combined therapy, designated with a CO subscript, where
with new parameter k_{1} defined by
The crux of control theory relates to the row rank of the control matrix C. If the row rank is less than full, then the system is less than fully controllable. If the matrix has full row rank, however, then it it considered controllable, which is a good thing. In our case, the row rank of C is only less than full either when the viral load is zero or when the CD4 count is zero, both cases excluded from the study. Hence, for all cases of interest for this research, the system of governing equations is going to be controllable.
Following on the fact that we have a controllable system, the next step is to determine how the level of controllability varies with time. To this end, the minimum singular value decomposition was used, which, when combined with the matrix C, will give an accurate measure of how controllable the system is. By definition, a singular value decomposition of a matrix is given by M = U where is an nxm diagonal matrix, and has values on the diagonal which are the singular values of the matrix M. This matrix is a direct measure of how controllable the system is at any particular time. The lower the singular values, the less control we have over the system, and conversely, higher values mean we have increased control over the system.
On the next page is a plot taken from the paper, When to Initiate HIV Therapy: A Control Theoretic Approach by Annah Jeffrey, Xiaohua Xia, and Ian Craig.[1] I can recreate everything except the curves representing the controllability of the system, labeled pi and rt in the plot.
In summary, the larger the singular values of are, the more control these drugs have on the system. Control over HIV is taken to be possible when the viral load is reduced by 90% over an 8 week period, and kept below a threshold number of viruses per mm^3. The obvious correlation from the graph is between the controllability and the viral load, since the plots virtually mimic each other. However, no connection can be seen between the controllability and the CD4+ T cell count.
Analysis/Interpretation
The main conclusions drawn in this paper are that,
- the controllability of the viral load varies with time
- the early acute and the more advanced stages are the most difficult to control
- when the viral load is easier to control, ie in the middle asymptotic stage, is the best time to initiate treatment, since a given amount of drug will have more of an effect then that same amount given when the level of controllability is lower. What this means is that a lower dose, amplified by the level of controllability will have a greater effect, while reducing the severity of any side effects that would correspond to a higher dose.
One item that the authors of this paper did not mention, and that may be of interest, is that the controllability matrix has a close relative called the observability matrix. This matrix indicates how well the system under consideration can be observed, just as the controllability matrix provides information on how well the system can be controlled. It would be an interesting project to add this component to the research already in place and illustrated by the paper When to Initiate HIV Therapy: A Control Theoretic Approach by Annah Jeffrey, Xiaohua Xia, and Ian Craig, and see if any new insight can be gained. My guess is that if the system is controllable like in this case, then it is also completely observable, but I have no proof of that yet.
Further Reading
For more recent information and a different approach on determining the optimal treatment for HIV, please read the following summary of a paper by Neri et al. (2007). This paper cites the paper by Jeffrey et al. (2003), which is discussed in the above sections.
- In "An Adaptive Multimeme Algorithm for Designing HIV Multidrug Therapies," the authors present methods for creating multidrug treatments which inflict such a powerful immune response that the HIV is suppressed to a state in which no further medication is needed after the initial treatment. Although many Highly Active AntiRetroviral Therapy (HAART) medications have been successful, they still present many disadvantages. These include developing resistance to a medication, unpleasant side effects, as well as being unaffordable.
- The authors focus on a dynamic system of multidrug therapy. Structured Treatment Interruption (STI) medications are studied, which involve a maximum dose of medicine being given to the patient, or no medication at all. This type of treatment is modeled as a binary valued function. The goal of the authors is to determine STI medications which are easier to administer and have a lower risk of developing drug resistance. The optimization problem for the STI therapies has a high dimensionality with a given parameterization. Also, the optimization of multidrug HIV therapies results in a nonlinear integer programming problem, also with a high dimensionality. The problem is also difficult to solve due to the fairly flat fitness landscape. This means that many of the solutions of the decision space have very similar fitness values. In addition, the basin of attraction encompassing the optimal solution is very narrow. All of these situations result in the stagnation of many optimization algorithms. A Computational Intelligence Approach is proposed in order to find the optimal HIV multidrug therapy. The authors present an optimization algorithm, Adaptive Multimeme Algorithm (AMmA). The AMmA is composed of evolutionary framework which has dynamic properties, as well as three different local searchers which are adaptively used to explore the decision space from complementary perspectives. This allows the available candidate solutions to be exploited. The optimal solution determined by the AMmA presented promising results. The algorithm leads to an immune response which reaches a fairly healthy steady state in a significantly shorter time than presented in previous studies. Also, the proposed treatment plan has a relatively low number of medication days. This is essential in attempting to avoid harmful side effects and mutations of HIV to drug-resistant strains.
The following summary of a paper entitled Using mathematical modeling and control to develop structured treatment interruption strategies for HIV infection by Rosenberg et. al. (2007) offers a simulation-based application of the control theoretic methods developed by Jeffery et. al. (2003).
- In an attempt to avoid the numerous complications associated with antiretroviral therapies (particularly in patient adherence to complex regimens), the researchers study in an alternative HIV/AIDS treatment method known as structured or supervised treatment interruption (STI). This therapy is based on adaptive cycles of antiretroviral treatment, withdrawal, and re-initiation. They then go on to create mathematical models depicting the interaction over time between HIV and a “virtual patient’s” immune system. These models are seen as an important tool for suggesting adaptive treatment strategies for clinical studies.
- The adaptive treatment strategy develops a treatment process that is based on information on the patient in the form of tailoring variables, such as previous treatments received, response to those treatments, and adherence of the patient up to that point. An application of the mathematical control principles established by Jeffrey et. al. to HIV dynamic models along with simulations can determine these decision rules, so that the desired outcomes are achieved.
- Treatment decisions are largely based on CD4+ T cell counts and viral load measurements. Dynamics between these elements are incorporated into a system of ODEs. Further analysis via computer simulations builds “virtual patient” time profiles for various STI scenarios. These mathematical tools could lead to an iterative approach to the clinical study of HIV infection and the development of STI treatment strategies. By combining existing data with control theoretic methods, new adaptive STI strategies can be evaluated by simulated application to the “virtual” target population.
- For more information on how mathematical modeling informs the design of STI HIV treatment strategies, click here.
Links and Citations
- ↑ E.R. Rosenberg, M. Davidian, and H.T. Banks. Using mathematics modeling and control to develop structured treatment interruption strategies fro HIV infection, Drug Alcohol Depend. 2007 May ; 88(Suppl 2): S41-S51.