A Model for Treatment Strategy in the Chemotherapy of AIDS
Catherine
Lloyd
Bioengineering Institute, University of Auckland
Model Status
This model contains partial differentials and as such can not currently be solved by existing CellML tools.
Model Structure
Optimal treatment strategies for patients with the human immunodeficiency virus (HIV) are needed for improving the effectiveness of chemotherapy for aquired immune deficiency syndrome (AIDS). AZT is an approved method of chemotherapy for the treatment of HIV infection. The drug works as an inhibitor of the reverse transcriptase enzyme, which catalyses the conversion of RNA to DNA during gene transcription. HIV is an RNA virus. The presence of AZT in a cell prevents the viral RNA from being transcribed into DNA, thus halting cellular infection and hence, viral spread.
Unfortunately AZT is not a cure for HIV, it only serves to control the virus and prevent its further progress. This often has the effect of prolonging the life of a HIV positive patient, and sometimes, AZT treatment also makes them less infectious to their sexual partners. There is debate as to what stage during the disease the drug should be administered (early or late?), and at what dose (high or low?). Further problems concerning AIDS therapy involve the possibility of negative side effects and the evolution of viral resistance.
To date, there have been several mathematical models published which describe the dynamics of HIV and different therapies. These include:
Perelson et al., Modelling HIV-1 Dynamics In Vivo, 1996
Mittler et al., Influence of Delayed Viral Production on Viral Dynamics in HIV-1 Infected Patients, 1998, and
Wodarz and Nowak, Modelling the Interaction Between HIV and the Immune System, 1999.
In the publication decribed here, Kirschner and Webb present a mathematical model which describes the interactions of chemotherapy with the HIV infected immune system, as shown in below. They define three populations:
1) non-infected CD4+ (helper) T cells;
2) CD4+ T cells which are infected with HIV; and
3) the population of HIV that is free living in the blood.
The effects of AZT chemotherapy are also included in the model. AZT reduces the ability of the virus to infect the CD4+ T cells.
The 1996 Kirschner and Webb publication present two mathematical models. In order to provide a mechanistic description of chemotherapy, age structure is incorporated into the infected CD4+ T cells in the second model. Introducing a time scale allows the model to mimic the effects of several drug doses, administered over a defined time period (e.g. 24 hours). The models have been described here in CellML, (the raw CellML descriptions of the Kirschner and Webb 1996 models can be downloaded in various formats as described in ).
The complete original paper reference is cited below:
A Model for Treatment Strategy in the Chemotherapy of AIDS, Denise Kirschner and G. F. Webb, 1996,
Bulletin of Mathematical Biology
, 58, 367-390. PubMed ID: 8713663
reaction diagram
Schematic diagram of a mathematical model of the interaction between HIV and the immune system.
T
non-infected CD4+ T cells
$\frac{d T}{d \mathrm{time}}=s-\mathrm{muT}+\mathrm{kv}TV+rT\frac{V}{C+V}+\int_{0}^{\mathrm{a1}} \mathrm{gamma}\mathrm{Ti}\,d as=5.0+\frac{5.0}{1.0+V}$
Ti
infected CD4+ T cells
$\frac{\partial^{1}\mathrm{Ti}}{\partial \mathrm{time}}=-(\mathrm{muTi}\mathrm{Ti}+r\mathrm{Ti}\frac{V}{C+V}+\mathrm{gamma}\mathrm{Ti}+\mathrm{dTi\_da})\frac{\partial^{1}\mathrm{Ti}}{\partial a}=\mathrm{dTi\_da}$
V
concentration of virus that is free living in the blood
$\frac{\partial^{1}V}{\partial \mathrm{time}}=Nr\frac{V}{C+V}\int_{0}^{\mathrm{amax}} \mathrm{Ti}()\,d a-\mathrm{kT}TV+\frac{\mathrm{gV}V}{b+V}$
gamma
treatment function
$\mathrm{gamma}=\begin{cases}\frac{cp}{1.0-e^{-(kp)}}e^{-(k\mathrm{time})} & \text{if $(a\ge 0.0)\land (a\le \mathrm{a1})\land (\mathrm{time}\ge 0.0)\land (\mathrm{time}\le p)$}\\ \frac{cp}{1.0-e^{-(kp)}}e^{-(k(\mathrm{time}-p))} & \text{if $(a\ge 0.0)\land (a\le \mathrm{a1})\land (\mathrm{time}\ge p)\land (\mathrm{time}\le 2.0p)$}\\ 1.0 & \text{if $a> \mathrm{a1}$}\end{cases}$
immunology
pharmacology
T cell
hiv-1
t cell
aids
AIDS
G
Webb
F
2004-03-28
Catherine
Lloyd
May
This is the CellML description of Kirschner and Webb's 1996 mathematical model for treatment strategy in the chemotherapy of AIDS.
The University of Auckland
The Bioengineering Institute
c.lloyd@auckland.ac.nz
Denise
Kirschner
keyword
Catherine Lloyd
Kirschner and Webb's 1996 mathematical model for treatment strategy in the chemotherapy of AIDS.
T lymphocyte
The University of Auckland, Bioengineering Institute
8713663
1996-03-01
A Model For Treatment Strategy In The Chemotherapy Of AIDS
58
367
390
Bulletin of Mathematical Biology