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
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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.
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[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
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[7] Wein, L. D., D’Amato, R. M., Perelson, A. S., 1998.
although further studies are required to establish the
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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-
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It is interesting to observe that the chance of infected CD4
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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
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particular to Dr. Mylva Fonsi for her help in providing the
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