Theoretical Approach
Scientific Paper
This analysis was based on the scientific paper “Modeling the drug therapy for HIV infection” by P. K. Scrivastava, M. Banerjee and Peeyush Chandra [1] and in a work exploration of the same, done by Baptiste Enes and Joana Rita Santos [2] in Computer Models of Physiological Processes, a subject of the Integrated Masters in Biomedical Engineering of the University of Coimbra. The last one is available for downloading, in the “Download” tab.Glossary
Human Immunodeficiency Virus (HIV) – A lentivirus – a subgroup of retrovirus - composed by two single chains of Ribonucleic Acid (RNA) and envelope proteins. HIV attacks, gradually, the immune system, destroying a type of white blood cells, CD4 + T cells, by making copies of itself inside them [3] [4] [5] [6].
Immune System – The immune system is a collection of cells, tissues and organs that protect the body against diseases and infections. When functioning properly, it detects threats – as viruses, bacteria and parasites – distinguishing them from the body’s tissue – and fights this pathogen agents [7].
Acquired Immunodeficiency Syndrome (AIDS) – It is a syndrome caused by HIV virus, when a person’s immune system is too weak to fight and the organism becomes vulnerable to many diseases and infections. When the number of CD4 + T cells drops below 200 cells per millimeter of blood, they are said to have AIDS [3].
CD4+ T Cells – CD4+ T cells are a type of white blood cells that are vital to human immune responses as they coordinate the activity of other T-cells subsets, B-cells and innate immune responses. CD4+ T cells have specific membrane receptors; after they recognize pathogen agents, bound to macrophages, dendritic cells or B cells, they stimulate the activity, differentiation and expansion of other immune cells, through the production and release of chemical compounds [2] [6] [8] [9].
Reverse Transcriptase (RT) – Enzyme encoded in retroviruses that catalyzes the transcription of the retrovirus ribonucleic acid (RNA) into deoxyribonucleic acid (DNA). This is the reverse process of what occurs in cellular transcription, in which DNA is transcribed into RNA [10].
Literature Review
Immunity can be defined as our body’s way to protect us from infections and diseases. When our body is infected by pathogen agents, such as viruses and bacteria, the immune system fights those organisms and substances [2] [11] As a consequence, the following time our body meets the same organism, we will be immune to that infection, which means that we will be less likely to get that disease or, if we do, it will be less intense [12].
CD4+ T cells, also known as T-helper cells, are possibly the most important cells in adaptive immunity, once they are required for all immune responses [13]. This type of white blood cells plays a major role in protecting our body, by sending signals to activate our body’s immune system, when invaders are detected, controlling other T-cell subsets, B-cells and innate immune response [9] [14]. Consequently, our CD4+ T cells count – number of CD4+ T cells per cubic millimeter of blood - provides an indication of the health of our immune system [15] [16]. A normal CD4+ T cells count range from 500 to 1600 cells per cubic millimeter of blood [14].
HIV attacks the immune system, especially the CD4+ T cells. AIDS is the final stage of HIV infection, when the immune system becomes badly vulnerable. When the number of CD4+ T cells drops below 200 cells per cubic millimeter of blood, a person is considered to have AIDS [17]. There is still no definitive cure for HIV, but there are ways to manage it, such as antiretroviral therapy [17].
Once HIV is inside the cells, the virus releases its RNA and RT turns the virus’ RNA into DNA, becoming this DNA into the cell [18].
Figure 1: HIV replication cycle. Taken from [19].
RT can be blocked by RT inhibitors [20]. Thus, drugs such as RT inhibitors have been used to control HIV, once they can prevent the reverse transcription of the virus [21].
Computational Simulation
Algorithm Analysis
The following analysis is based on what is referred and reported in [1] and explorated in [2]. The symbols used in the equations, their meaning and their value are in Table 1, which was transcribed from [1].
Table 1: List of parameters used in the simulation. Transcribed from [1].
| Parameter | Definition | Value |
|---|---|---|
| s | Inflow rate of CD4+ T cells | 10 mm-3 days-1 |
| k | Interaction-infection rate of CD4+ T cells | 0.000024 mm3 days-1 |
| µ | CD4+ T cells natural death | 0.01 days-1 |
| η | Effectiveness of reverse transcriptase inhibitor | 0.6 |
| α | Transition rate from pre-RT infected CD4+ T cells class to productively infected class (post-RT) | 0.4 days-1 |
| b | Reverting rate of infected cells to uninfected class due to non-completion of reverse transcriptase | 0.05 days-1 |
| µ1 | Death rate of infected cells | 0.015 days-1 |
| δ | Death rate of actively infected cells | 0.26 days-1 |
| N | Total number of viral particles produced by an infected cell | 1000 |
| c | Clearance rate of virus | 2.4 days-1 |
This mathematical model corresponds to the dynamics of HIV on the effect of RT inhibitor. The equations to be simulated are [1]:
dT/dt = s - kVT - µT + (ηα + b) T1* (1)
dT1*/dt = kVT - (µ1 + α + b) T1* (2)
dT*/dt = (1 - η) αT1* - δT* (3)
dV/dt = NδT* - cV (4)
Three types of CD4+ T-cells populations: Equation 1 represents the susceptive CD4+ T-cells; Equation 2 represents the infected CD4+ T-cells, before reverse transcriptase – pre-RT cells; Equation 3 represents the infected CD4+ T-cells, after reverse transcriptase – post-RT cells, and that are able to produce the virus; and Equation 4 represents the virus density [1] [2].
The initial conditions presented in the scientific paper for the mathematical model, and used in the simulation, are: T(0) = 300 mm-3, T1*(0) = 10 mm-3, T*(0) = 10 mm-3 and V(0) = 10 mm-3. Some of the constants can be changed in the simulation, according to what was performed in the article.
Computational Simulation
Effectiveness of the Reverse Transcriptase Inhibitor
Transition rate between pre-RT and post-RT CD4+ T-cells
Rate of infected to uninfected CD4+ T-cells due to non-completion of the reverse transcriptase
Results Interpretation
The graphics obtained are identical to those presented in [1].
Through the simulation, we see that the greater the effectiveness of RT inhibitor (η), more pre-RT infected cells become uninfected and less pre-RT cells become post-RT cells. Consequently, the greater η is, the more effective is the treatment [1] [2].
The lower is the transition rate between pre-RT and post-RT CD4+ T cells (α), less pre-RT cells become post-RT cells. Consequently, the lower α is, the more effective is the treatment [1] [2].
When the rate of infected to uninfected CD4+ T-cells due to non-completion of the reverse transcriptase (b) is zero, no cell is converted from infected to uninfected. However, the greater b is, more infected cells become uninfected. Consequently, the greater b is, the more effective is the treatment [1] [2].
References
[INTRO IMAGE] Stocktrek Images via Getty Images, “These 12 Viruses Look Beautiful Up Close But Would Kill You If They Could (PHOTOS),” The Huffington Post, 23 January 2014. [Online]. Available: http://www.huffingtonpost.com/2014/01/13/deadly-viruses-beautiful-photos_n_4545309.html. [Accessed 29 August 2016].
[1] P. K. Srivastava, M. Banerjee, Peeyush Chandra, “Modeling the drug therapy for HIV Infection,” Journal of Biological Systems, vol. 17, nº 02, pp. 213-223, 2009.
[2] Baptiste Enes, Joana Rita Santos, “Projecto de Modelos Computacionais de Processos Fisiológicos acerca de: Modelação Matemática do tratamento da infecção do HIV,” University of Coimbra, 2011/2012.
[3] “WHAT ARE HIV AND AIDS?,” AVERT, [Online]. Available: http://www.avert.org/about-hiv-aids/what-hiv-aids. [Acedido em 2016 August 25].
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[19] Emanuele Fanales-Belasio, Mariangela Raimondo, Barbara Suligoi, Stefano Buttò, “HIV virology and pathogenetic mechanisms of infection: a brief overview,” Annali dell'Istituto superiore di sanita, vol. 46, nº 1, pp. 5-14, 2010.
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