The aim of this study is to develop two-dimensional cellular automata model of HIV infection that depicts the dynamics involved in the interactions between acquired immune system and HIV infection in the peripheral blood stream. The appropriate biological rules of cellular automata model have been extracted from expert knowledge and the model has been simulated with determined initial conditions. Obtained results have been validated through comparing with the accepted AIDS reference curve. The new rules and states were added to the proposed model to show the effects of applying combined antiretroviral therapy. Our results showed that by applying RTI and PI drugs with maximum drug effectiveness, comparing with cases in which no treatment was applied, the steady state concentrations of healthy (infected) CD 4 +T cells were increased (decreased) 53% (41%). Also, the use of cART with maximum drug effectiveness led to a 69% reduction in the steady state level of viral load. At this time, obtained results have been validated through comparing with available clinical data. Our results showed good agreement with both reference curve and the clinical data. In the second phase of this study, by applying genetic algorithms, a therapeutic schedule has been provided that its use, while maintaining the quality of the treatment, leads to a 47% reduction in both drug dosage and the side effects of antiretroviral drugs.
Human Immunodeficiency Virus (HIV) is a subgroup of retrovirus that causes the Acquired Immune Deficiency Syndrome (AIDS), a condition in humans in which progressive failure of the immune system allows life-threatening opportunistic infections and cancers to thrive. HIV infects vital cells in the human immune system such as helper T cells (specifically CD4+T cells), macrophages and dendritic cells [
By finding an appropriate model for a biological system, experts will be needless to do many relevant clinical trials. If we also consider the effects of drug therapy in such models, we will be able to introduce the best treatment by simulating the effects of different therapeutic methods. Although antiretroviral therapy has significant effects on prolonging survival of patients, the side effects of long-term applying of these drugs may put the patient at risk [
Most of researchers have used models with Ordinary Differential Equations (ODEs) or Partial Differential Equations (PDEs) to simulate the dynamics of HIV infection [
The Cellular Automata (CA) modeling approach is a perfect choice for this type of modeling techniques. Cellular Automata (CA) is discrete dynamical systems for which its behavior is based entirely on local communications [
In the previous mentioned literature, the dynamics of HIV infection have been studied in the lymph node. However, whenever most physicians try to assess the progress of disease, they would prefer to use patient’s blood data because such data can be accessed more easily. Moreover, they can find all types of cells in the peripheral bloodstream. Because of these reasons, some of the authors selected peripheral bloodstream to model the dynamics of HIV infection using CA method [
drugs was not considered. Khabouze et al., developed the study done by Jafelice by taking into account the role of humoral response as one of the basic part of adaptive immune system by considering a new type of cell called antibody but the effects of drug treatment in their model was not studied [
In this study we model the interactions between HIV enters the human’s body and the immune system response, in the peripheral bloodstream, using two-dimensional CA technique. We also consider the effects of antiretroviral therapy in our model rules. Actually our CA model is an extended version of model proposed by Khabouze [
In the next step, we used an intelligent optimization technique to provide an optimal therapeutic schedule that simultaneously keeps disease progression under control and allows the restoration of immune system. So far, in often models that have been used to depict the dynamic behavior of HIV, the time-continuous equations have been applied, and in this case, using optimal control methods (despite the high computational volume) were the best treatment strategy. However, since the CA model that used in this study is described by biological rules and there is no analytical equation, applying the classic optimal control methods is impossible. Therefore, in order to optimize treatment protocol and to provide optimal therapeutic schedule in terms of number and order of days of applying antiretroviral drugs and drug dosage, utilizing of algorithms which are search-based in discrete-time space, is recommended. Since the genetic algorithm is a good candidate of such algorithms, in this study we used genetic algorithms to provide the optimal therapeutic schedule.
After this introduction, this paper is organized as follow: in the next section, we introduce the proposed CA model by incorporating the antiretroviral therapy process into the rules of this model. Then we present the simulation results and we compare our obtained results with available clinical. After ensuring the validity of proposed model, in the next step, by utilizing the genetic algorithms, the optimal therapeutic schedule is obtained. In order to prove the effectiveness of the proposed treatment program, the results obtained from applying this optimum schedule is compared with the results obtained from two other therapeutic programs named “Full therapy” and “Random therapy”, respectively. In the last section we have concluded our work and suggestions for future works are expressed.
The structure of proposed model in this study is two-dimensional CA model with Moor neighboring. For developing such a structure, we used two-dimen- sional cell grids with size of 250 × 250 to depict the bloodstream toroidal system.
Since CA models belong to subset of agent based modeling methods, defining the agents of proposed CA model is the first step of organizing this model. In this study, following agents have been defined:
1) Healthy cells (H): this is the state of a normal healthy cell which has not absorbed any type of antiretroviral drugs. The cell in this state is capable of becoming infected and this happens, it can infect other healthy cells in own neighboring.
2) Healthy cells resistant to infection (HR): this is the state of healthy cells in which due to the absorption of both RTI and PI antiretroviral drugs, they will not become infected even if they are exposed to infectious factors.
3) Active infected cells (A1): this state refers to the active infected cells which are able to infect the normal healthy cells in their own neighboring [
4) Inactive infected cells (A2): this state corresponds to the infected cells that after Ƭ1 time delay, the immune system has developed own cellular response against it. With considering such time delay we can depict the mutation rate of HIV. In fact we consider that since each infected cell contains a new strain of HIV, thus the immune system needs time to identify each of them. On the other hand, as a result of cellular response of immune system, such infected cells alone are not able to infect the normal healthy cells in their own neighboring but local aggregation of them will cause the neighbor healthy cells to become infected. It means that if, at least, R numbers of A2 cells have been located in the neighboring of a normal healthy cell, the healthy cell will become an active infected cell [
5) Latently infected cells (A0): this is the state of infected cells which have kept the latent infection in to themselves for a time and they are not capable of spreading the infection during this period of time. But they may be re-activated after a time delay Ƭ2 and if it happens, they will be able to infect their own adjacent healthy cells [
6) Inactivated infected cells in effect of PI drug absorption (API): this state corresponds to the infected cells which have arisen due to infecting healthy cells that have absorbed PI antiretroviral drugs. The cells in this state are not able to spread the infection and they will not produce any new HIV strain.
7) HIV Infectious: this is the state of active and infectious HIV particles which are capable of infecting their own adjacent normal healthy cells [
8) HIV non-Infectious: when a healthy cell that has absorbed PI antiretroviral drugs becomes infected, this type of infected cell produces free viruses that are often defective and are not capable of infecting other healthy cells. This state is corresponds to such detective and non-infectious viruses [
9) HIV Inactive: this state corresponds to viruses that the immune system has developed own humoral response against them. Therefore, due to the performance of immune system, the ability of such viruses is decreased for infecting other healthy cells [
10) Dead cells (D): this state corresponds to both of the healthy cells and free viruses that are dead due to natural death. This state also contains the infected cells that will die in next step due to the performance of immune system against them [
11) CTLs: the state of immune system cells that is responsible for identifying killing the infected cells [
12) Anti bodies (AB): the state of anti bodies that are responsible for killing the viruses [
In second step, we considered following percentages as an initial number of each of cell types in our model: 25% ± 10% of the entire cell grid was initially considered as normal healthy CD4+T lymphocytes [
As previously mentioned, in this study we have developed a CA model for modeling the HIV infection in the peripheral bloodstream with taking into account the effects of applying antiretroviral therapy. For this purpose, the effectiveness of RTI and PI drugs was considered by PRTI and PPI, respectively, and the value of them was selected between 0 to 0.9 (the value of 0 indicates no use of drug of interest and the value 0.9 indicates maximum effectiveness of the drug). We defined following rules for our proposed CA model based on neighboring. It is worth noting that in the defined rules where ever the phrase “in the presence of infecting factors” is used, it means that, at least, one active infected cell (A1) or one infectious HIV or R numbers of inactivated infected cells (A2) exist in the neighborhood of target cell.
Rule 1) The rules for updating the state of a normal healthy cell (H):
a) We have defined a lifespan limit for normal healthy cells and a random initial age (between 0 and the lifespan limit) has been assigned to each of H cells. At the beginning of each of iterations, if an H cell reaches its lifespan then it will die, otherwise the following updating rules will be used [
b) The H cell will choose randomly an empty place in the neighborhood and moves there for the next iteration. If there is no empty place then it remains at own place [
c) For the next iteration, each of H cells, becomes an HR cell with probability of PRTI×PPI in the presence or absence of infecting factors―it shows that when a normal healthy cell absorbs both of RTI and PI drugs simultaneously, it becomes a resistant healthy cell which none of the infecting factors are not able to infect that and it is protected from infection for longer period of time than the normal healthy cells which have absorbed only RTI drug;
Or remains an H cell with probability of PRTI × (1 − PPI) in the presence or absence of infecting factors and its age will be increased one unit?this shows that if a normal healthy cell only absorbs RTI drug, in the effect of this type of drug the virus prevents from entering to healthy cell. So the healthy cell will be protected from infection, but since it has not absorbed both RTI and PI drugs, this protection lasts only until the next iteration-;
Or with probability of (1 − PRTI) × PPI, two modes may occur: 1) in the absence of infecting factors, it remains an H cell and its age will be increased one unit, 2) in the presence of infecting factors, it becomes an AP cell with age zero?it shows that since the healthy cell has not absorbed RTI drug, it will not protected from entering virus and, therefore, it will be infected in presence of infecting factors. But since it has absorbed PI drug, even if it becomes infected by infecting factors, the risen infected cell (API) will not be able to infect other healthy cells-;
Or with probability of (1 − PRTI) × (1 − PPI) two modes may occur: 1) in the absence of infecting factors it remains at H state and its age will be increased one unit for the next iteration, 2) in the presence of infecting factors it becomes an active infected cell (A1) with age of zero with probability of Pinf, or it becomes a latently infected cell (A0) with age of zero with probability of (1 − Pinf) [
d) The normal healthy cells are proliferating with a constant rate during the simulation [
Rule 2) The rules for updating the state of a resistant healthy cell (HR):
Since a resistant healthy cell has absorbed both of RTI and PI drugs, therefore, even if the infecting factors exist in its neighborhood, it will not become infected. This type of healthy cell becomes a normal healthy cell for the next iteration [
Rule 3) The rules for updating the state of active infected cells (A1):
If an A1 cell is not identified by the immune system cells, it releases a new infectious HIV from within itself with probability of PHIV-PR [
The state of A1 cell is updated according to the following rules:
a) Until three weeks after infection, an A1 cell produces HIVInfectious with probability of PHIV-PR or produces HIVInactive with probability of (1-PHIV-PR) for the next iteration. The new released HIV is placed at one of empty sites in A1’s neighborhood. Also the age of an A1 cell is incremented one unit for the next iteration [
b) Between three until five weeks after infection, following modes may occur:
1) If there is, at least, one CTL in the neighborhood, for the next iteration, an A1 cell becomes A2 with probability of Pidentification; or produces HIVInfectious with probability of (1 − Pidentificatin) × PHIV-PR; or produces HIVInactive with probability of (1 − Pidentificatin) × (1 − PHIV-PR). Also the age of an A1 cell is incremented one unit for the next iteration [
2) If there is no CTL in the neighborhood, for the next iteration, an A1 cell produces HIVInfectious with probability of PHIV-PR; or produces HIVInactive with probability of (1-PHIV-PR). Also the age of an A1 cell is incremented one unit for the next iteration [
c) If due to mutations, an A1 cell is not detected by the CTLs up to five weeks after infection, due to releasing new viruses from within it, finally the cell membrane of this A1 cell will tear and it will die for the next iteration [
d) Before an infected cell dies due to membrane rupture, if it is not detected at each of iterations by CTLs then for the next iteration it will select randomly an empty place in its neighborhood to move there and the age of this A1 cell is incremented one unit. But if there is no empty place in its neighborhood then it remains at an own place and its age is incremented one unit for the next iteration [
Rule 4) The rules for updating the state of an inactivated infected cell (A2):
An A2 cell becomes a dead cell for the next iteration [
Rule 5) The rules for updating the state of a latently infected cell (A0):
a) Each of A0 cells after a time delay TRe-activation will be reactivated with probability of PRe-activation and becomes a new A1 cell with the age of zero for the next iteration; or remains an A0 with probability of (1 − PReactivation) and dies for the next iteration [
b) Before reaching the time of TRe-activation, for the next iteration each of A0 cells will randomly select an empty place in its neighborhood to move there and if there is no empty place, it remains at own place. Also the age of A0 cell is incremented one unit for the next iteration.
Rule 6) The rules for updating the state of an API cell:
As we mentioned before, whenever a healthy that has absorbed PI drugs becomes infected by infecting factors in its neighborhood, it becomes an API cell with the age of zero for the next iteration. We also considered a lifespan limit for API cells similar to what we had previously defined for A1 cells (5 iterations). We defined following rules for updating the state of API cells:
a) If an API cell reaches its lifespan limit, it becomes a dead cell for the next iteration.
b) Before reaching the lifespan limit, following modes may occur for the next iteration:
1) It remains an API cell and produces HIVnoninfectious with probability of PPI × PHIV-PR; or it remains an API cell and produces HIVInactive with probability of PPI × (1 − PHIV-PR). The new released HIV is placed at one of the empty sites in API’s neighborhood. In this case, the API cell will randomly choose an empty place in the own neighborhood to move there and if there is no empty place, it remains at own place. Also the age of API cell is incremented one unit for the next iteration.
2) It becomes an A1 cell with the last age of API cell and produces HIVInfectious with probability of (1 − PPI) × PHIV-PR; or it becomes an A1 cell with the last age of API and produces HIVInactive with probability of (1 − PPI) × (1 − PHIV-PR). In this case, as we mentioned before, the arisen A1 cell continues its life with the last age of corresponding API cell and from now on, the rules that were defined for updating the state of A1 cell will be used for it.
The rules defined for updating the state of API cell is inspired from [
It is worth mentioning that the viruses which are produced by API cells or by A1 cells will be placed randomly in one of the neighboring places of infected cell producing their own. Depending on the type of the selected place, following modes may occur:
I. If this place is an empty place then the produced virus will be located there for the next iteration.
II. Since only the normal healthy lymphocytes are hosts for viruses, thus if this place be relevant to any state except H, the virus will not have no effect on the state of the place.
III. If this place is relevant to a normal healthy cell, since both of HIVnon-Infectious and HIVInactive are not capable enough to infect other cells, thus only HIVInfectious particles can infect the H cell.
Rule 7) The rules for updating the state of dead cells:
A dead cell can be replaced by a new healthy cell for the next iteration. The probability of this replacement is shown by Preplication. In fact, Preplication depicts the ability of the immune system for reconstructing the destroyed cells. Among new produced healthy cells, a very small fraction of them contain the infection. The new produced cell may be infected with probability of Pnew-infection [
a) Each of dead cells becomes a new normal healthy cell (H) with probability of Preplication × (1 − Pnew-infection) or becomes a new infected cell (A1) with probability of Preplication × Pnew-infection or remains a dead cell with probability of (1 − Preplication) for the next iteration [
Rule 8) The rules of HIV proliferation and updating its state:
As mentioned before, HIVInfectious is a virus particle that is able to infect own adjacent healthy cells. It means if there is, at least, one HIVInfectious in the neighborhood of a normal healthy cell, this virus transcribes own RNA in to the DNA of host cell through reverse transcriptase enzyme and infect the healthy cell in this way. On the other hand, the rules numbers 3a, 3b and 6b show how one A1 or API cell causes to the proliferation of this type of HIV. We also defined following rules for updating the state of HIVInfectious:
a) We defined a lifespan limit for HIVInfectious particles that is shown by the parameter “V_LSL”. If one HIVInfectious particle reaches the V_LSL, this virus particle will die due to the natural death for the next iteration [
b) Before the HIVInfectious reaches V_LSL, following modes may occur:
1) At each iteration, if there is, at least, one anti body cell (AB) in the neighborhood of an HIVInfectious and detects this HIV particle with probability of Pidentification then the HIVInfectious will lose its ability for spreading the infection due to the performance of humoral response of immune system. Therefore, the HIVInfectious becomes HIVinactive for the next iteration [
2) If the HIVInfectious is not identified by the AB cell in its adjacent with probability of (1 − Pidentification), or if there is not any AB cell in the neighborhood of this HIVInfectious, then for the next iteration the HIVInfectious will randomly select one of the empty places in own neighborhood to move there and if there is no empty place, it will remain at own place. Also, the age of this HIVInfectious is incremented one unit for the next iteration [
As mentioned before, whenever a normal healthy cell that has absorbed PI drugs, becomes infected by infecting factors, the arisen infected cell (API) produces the HIV particles which are often faulty and are not able to infect other healthy cells. We called this type of HIV particles HIVnon-infectious. The rule 7b shows how API cells produce this type of HIV. We also defined following rule for HIVnon-infectious:
c) Each of HIVnon-infectious becomes HIVinactive for the next iteration [
Also we defined following rule for HIVinactive:
d) Each of HIVinactive becomes a dead cell for the next iteration [
Rule 9) The rules for updating the state of CTLs:
The immune system develops own cellular response against infection through CTLs. This type of immune system cell is responsible for killing the infected cells. The rule number 3 shows how CTLs detect the infected cells and kill them. Following rules has been defined for updating the state of CTLs:
a) A lifespan limit for CTLs (CTL_LSL) has been defined. If a CTL reaches the lifespan limit then it becomes a dead cell for the next iteration. Otherwise, for the next iteration it will randomly select an empty place in the own neighborhood to move there and if there is no empty place then it remains at own place. Also, the age of CTL is incremented one unit for the next iteration [
b) At each of iterations, CTLs are being reproduced by a constant rate (CTL_PR).
Rule 10) The rules for updating the state of anti bodies:
The main role of antibodies (AB) is identifying virus particles and killing them. In rule 8b, we described how AB cells do this task. Following rules has been defined for updating the state of AB:
a) A lifespan limit for AB cells (AB_LSL) has been defined. If an AB reaches the lifespan limit then it becomes a dead cell for the next iteration. Otherwise, for the next iteration it will randomly select an empty place in the own neighborhood to move there and if there is no empty place then it remains at own place. Also, the age of AB is incremented one unit for the next iteration [
b) At each of iterations, AB cells are being reproduced by a constant rate (AB_PR).
In order to simulate the proposed model, we used a set of parameters some of which have been extracted from reference articles and the rest of them are derived from expert knowledge and experimental findings. The expert knowledge includes medical reference books on AIDS [
Parameter | Definition | Value | Reference |
---|---|---|---|
L | Lattice size | 250 | Ad hoc |
H_PR | Production rate of normal healthy CD4+T cell | 18 cells/iteration | [ |
CTL_PR | Production rate of CTLs | 30 cells/iteration (in primary infection phase) & 3 cell every 14 iterations (after the end of primary infection phase) | [ |
AB_PR | Production rate of Anti Bodies | 50 cells/iteration (in primary infection phase) & 5 cell every 14 iterations (after the end of primary infection phase) | [ |
H_LSL | Life span limit of normal Healthy CD4+T cells | 4 iterations | [ |
CTL_LSL | Life span limit of CTLs | 15 iterations | [ |
AB_LSL | Life span limit of Anti Bodies | 15 iterations | [ |
V_LSL | Life span limit of HIVInfectious | 3 iterations | [ |
R | Number of infected A2 cells in the neighborhood of a healthy CD4+T cell which makes converting a healthy cell into an A1 infected cell | 4 cells | [ |
TRe-activation | Delay time needed to re-activate the A0 latently infected cells | 30 iterations | [ |
Pinf | The probability of converting a healthy cell into an A1 cell in the presence of infecting factors | 0.99 | [ |
Preplication | The probability of replacing a dead cell by a normal healthy cell | 0.99 | [ |
Pnew-infection | The probability of replacing each new healthy cell by an A1 infected cell | 10-5 | [ |
Pidentification | The probability of identifying an infected A1 cell by CTLs | 55%, 75% and 95% respectively 3, 4 and 5 weeks after infection | Ad hoc |
PAB-identification | The probability of identifying an HIVInfectious by Anti Bodies | 0.95% | Ad hoc |
PHIV-PR | The probability of releasing an HIVInfectious particle from an A1 infected cell | 0.99 | [ |
PRe-activation | The probability of reactivating an A0 cell after a specified period of time. | 0.0005 | [ |
PPI & PRTI | The effectiveness of PI and RTI antiretroviral drugs | Varies between 0 to 0.9 | [ |
By using the antiretroviral drugs, the concentration of healthy CD4+T cells is increased and both concentration of infected CD4+T cells and viral load are decreased. But at the same time, due to occurrence of virus’s mutation and due to appearance of new strains of HIV, after a period of time from start of treatment, the patient becomes drug-resistant and the effectiveness of these drugs will be reduced. Due to the reduction of drug effectiveness, trend of increasing (reducing) of concentration of healthy cells (concentration of infected cells and viral load) stops and finally the concentration of these cells reaches to steady state level. The saturation of drug activity has been modeled through a negative exponential function by Caetano Marco [
In the above equation, PPI (t) and PRTI (t) denote PI and RTI drug effectiveness at iteration number t, respectively. P0PI and P0RTI denote the initial value of PI and RTI drug effectiveness, respectively. V(t − 1), I(t − 1) and H(t − 1) denote the viral load, concentration on infected cells and the concentration of healthy cells at iteration number t − 1, respectively. V(ts), I(ts) and H(ts) denote viral load, concentration of infected cells and concentration of healthy cells at time of onset of treatment, respectively.
The RTI and PI antiretroviral drugs were considered as the inputs of our model. To consider these inputs, we used a parameter named effectiveness as PRTI and PPI for RTI and PI antiretroviral drugs, respectively [
After ensuring the validity of proposed CA model (according to the results obtained in Section 3.2), an optimal therapeutic schedule was presented by using Genetic Algorithms (GA). An introduction of GA can be found in [
In GA, each of the therapeutic schedules was defined by a chromosome with m bits, that m defines the number of weeks of therapeutic schedule [
a) If the ith bit of a chromosome is set equal to zero, then it means that during the ith week of therapeutic schedule, no treatment is applied.
b) If the ith bit of a chromosome is set equal to one, then it means that during the ith week of therapeutic schedule, cART method with maximum drug effectiveness is applied.
Therefore, one of the values 0 or 1, was assigned randomly to each of bits of chromosome. Then, corresponding to the assigned value, the values of drug effectiveness were adjusted in CA model. Accordingly, if the assigned value was equal to zero, then in CA model, the values of both P0RTI and P0PI was set equal to zero (i.e. P0RTI = P0PI = 0). And if the assigned value was equal to one, then in CA model, the values of both P0RTI and P0PI was set equal to 0.9 (i.e. P0RTI = P0PI = 0.9).
In the GA optimization, the population size was set equal to 32 × 10 = 320. The reason of this choice was that each of chromosomes has 32 bits, and 10 different states can be assigned to each of bits. Therefore, with setting the population size equal to 320, many different therapeutic schedules will be evaluated by GA and obtained optimal therapeutic schedule will be more reliable. The GA optimization procedure ran on 50 generations. The operations and parameters of genetic algorithms that were applied in our simulations are shown in
Parameter/Operator | Value/Type | Reference |
---|---|---|
Generations | 50 | Ad hoc |
Population Size | 320 | Ad hoc |
Population Type | Bit string | [ |
Number of design variables | 2 | Ad hoc |
Selection Fcn | Selection tournament | [ |
Crossover Fcn | Crossover scattered | [ |
Crossover Fraction | 0.8 | Ad hoc |
Mutation Fcn | Mutation uniform | [ |
Mutation rate | 0.01 | Ad hoc |
Elite Count | 2 | Ad hoc |
The drug dosage and concentrations of healthy CD4+T cells and viral load at steady state due to applying a therapeutic schedule are the most important factors in evaluating the quality of that therapeutic schedule. Therefore, in order to evaluate the quality of each of chromosomes (therapeutic schedules), the following functions were calculated for each of chromosomes [
In the above equations, ts indicates the time of onset of treatment; te indicates the time of end of treatment.
A desirable therapeutic schedule is a schedule that by applying it, with minimum drug dosage, the concentration of healthy CD4+T cells and viral load at steady state reaches to maximum and minimum level, respectively. Considering this description and according to functions described above, that is more worthy chromosome for which the calculated value of the cost function is smaller (i.e.
Since the fitness function is defined as the inverse of cost function, therefore, the highest calculated value of cost function is related to the fittest chromosome. According to GA optimization procedure, all of the chromosomes of first generation are sorted according to the value of cost function that was calculated for each of them. Chromosomes with smaller calculated value of cost functions were transferred to the next generation as the fittest chromosomes of first generation.
In order to simulate the proposed CA model, a cell grid with size of 250 × 250 was considered. According to the defined initial numbers for each type of cells, an initial numbers of normal healthy cells, infected cells (A1), infectious viruses, CTLs and antibodies were distributed randomly in the grid space. According to the defined rules in Section 2.1.1 and with considering the values of
For validating the results obtained from proposed model, we compared our results with the standard reference AIDS curve which has been shown in
In the next step, we studied the effects of applying medication on the concentrations of healthy and infected cells. As we mentioned before, due to occurrence of drug-resistance and as a result of HIV mutation phenomenon, combined antiretroviral therapy (cART) method (the therapy method in which a combination of both of RTI and PI drugs is used) will be more effective in control of progression of AIDS. Therefore, in this study, the effects of applying cART method, with consideration of different values for drug effectiveness, are studied. The concentrations of healthy CD4+T cells and infected CD4+T cells (A1 + A2)
obtained from simulating our CA model have been shown in
As
As
Since the viral load plays an important role in spread of infection, the level of infectious HIV particles was studied under following conditions: a) without treatment, b) with applying cART method with minimum, medium and maximum drug effectiveness.
Clinical data collection in the field of AIDS is not a simple task. Gonzalez et al.,
compared the results obtained from their CA model with clinical data which had been published in some of the authentic articles [
happened at 220th week. 2) The simulations were ran nine times (to show nine patients) and at each time, the cART was started at 220th week. During a period of 24 weeks, every three weeks, the concentration of healthy CD4+T cells obtained from our model was recorded. 3) This procedure was done once with consideration of minimum and once again with maximum drug effectiveness.
In [
All steps of GA optimization were repeated for 50 generations.
In order to prove desirable performance of optimal therapeutic schedule obtained from GA optimization, three different therapeutic schedules were defined and results obtained from applying these schedules were compared with each other. These therapeutic schedules are as below. In each of the therapeutic schedules the treatment was started from 160th week.
a) Full therapy with maximum drug dosage: in this case the cART with maximum drug dosage (i.e. PPI = PRTI = 0.9) was applied at every week.
b) Best therapy with optimal drug dosage: in this case the optimal therapeutic schedule obtained from GA optimization was applied.
c) Random therapy: in this case, the number of weeks in which cART with minimum drug dosage is applied, was set equal to the number of therapy weeks in the case of best therapy, but a random therapeutic schedule was applied during these weeks.
As
Therefore, by concluding our results we find that by applying full therapy schedule, although the quality of treatment is maintained, due to the high level of drug dosage, the patient will suffer from side effects of antiretroviral drugs. On the other hand, by applying random therapy schedule, although the level of drug dosage has been decreased, the quality of treatment also decreased. But by
applying optimal therapy schedule that obtained from GA optimization, while maintaining the quality of treatment, leading to reduction in both, level of drug dosage and the side effects of antiretroviral drugs.
In this study, by using genetic algorithms (GA), we found an optimum therapeutic schedule for HIV infected patients based on combined antiretroviral therapy (cART) method. In order to simulate the effects of therapy, two-dimen- sional cellular automata (CA) model was used. This CA model depicts the dynamics involved in the interactions between acquired immune system and HIV, in the peripheral blood of HIV infected individuals. This model was developed taking into account the following points: a) occurrence of mutation phenomena, b) existing time delay in the immune system, c) the role of latently infected cells in spread of infection, d) the effects of antiretroviral drugs such as reverse transcriptase inhibitors (RTI) and protease inhibitors (PI), e) occurrence of drug resistance in patients who are under antiretroviral therapy.
For this purpose, first, we collected appropriate rules related to HIV infection in patient’s peripheral blood according to reference articles and base on expert knowledge including books and medical journals in the field of AIDS and we developed our CA model using these rules. At this step, by comparing the results obtained from our CA model with reference HIV curve, we showed that our results are in a good agreement with the existing reality in the interactions between acquired immune system and HIV. At next step, we studied the effects of cART by adding some new states and parameters to the proposed CA model. For this purpose, we considered a parameter named drug effectiveness for each of RTI and PI drugs that were shown by PRTI and PPI, respectively. Occurrence of drug resistance was considered by a negative exponential equation. The concentrations of healthy and infected CD4+T cells over the weeks and the level of viral load were obtained by consideration of different values for PRTI and PPI. Our results showed that by applying maximum effectiveness of RTI and PI drugs, compared with cases in which no treatment was applied, the steady state concentrations of healthy (infected) CD4+T cells were increased (decreased) 53% (41%). Also, the use of cART with maximum drug effectiveness led to a 69% reduction in the steady state level of viral load. In order to validate the results obtained from our proposed CA model in the cases in which cART was applied, we used clinical data which are provided in authoritative articles. By comparing results obtained from our model with clinical data, we prove that the best compliance between our results and clinical data arises when we consider minimum effectiveness for antiretroviral drugs.
In the second phase of this study, we used GA optimization to provide an optimal therapeutic schedule of cART with maximum drug effectiveness based on structured interruptions. To prove the effectiveness of obtained therapeutic schedule from GA optimization, three different therapy schedules were defined as following: a) Full therapy, b) Best therapy with optimum drug dosage, c) Random therapy. According to each of these therapy schedules, cART with minimum drug effectiveness was applied or interrupted at each of weeks in the proposed CA model. Then the steady state concentrations of healthy and infected CD4+T cells, and the amount of drug dosage obtained from applying each of these schedules were presented as our results. Our results showed that by applying the optimum therapeutic schedule obtained from GA optimization, the quality of treatment (i.e. high steady state concentration of healthy cells and low steady state concentration on infected cells) is very close when the full therapy schedule was applied. But the main point is that applying the optimum therapeutic schedule, compared with full therapy schedule, led to 47% reduction in the amount of drug dosage. Therefore, the optimum therapeutic schedule provides the high quality treatment, low amount of drug dosage and low side effects of antiretroviral drugs, simultaneously.
The CA model that was developed for AIDS in this study, involves the biological facts of this disease and it could depict all three patterns of progression of HIV infection properly. Most importantly, the effects of applying different antiretroviral drugs were also considered in this model. Therefore, as an application of the model developed in this study can be said, by using this model, physicians can evaluate the effects of applying different therapeutic methods on virtual patients through model, without requiring costly clinical trials on real patients. Also, by using Genetic Algorithms (GA), the best therapeutic schedule that provides the high quality treatment with low amount of drug dosage, can be obtained through this model. Therefore, results obtained from this model can be useful for physicians. However, this model is still developing. Finding newer and more accurate rules, more accurate estimating of model parameters, adding new states to show different aspects of disease, investigating more varied therapeutic methods and adjusting the drug dosage in the therapeutic schedule can be considered as future works.
Golpayegani, G.N., Jafari, A.H. and Dabanloo, N.J. (2017) Providing a Therapeutic Scheduling for HIV Infected Individuals with Genetic Algorithms Using a Cellular Automata Model of HIV Infection in the Peripheral Blood Stream. J. Biomedical Science and Engineering, 10, 77-106. https://doi.org/10.4236/jbise.2017.103008
HIV: Human Immunodeficiency Virus
AIDS: Acquired Immunodeficiency Syndrome
RTI: Reverse Transcriptase Inhibitors
PI: Protease Inhibitors
cART: Combined Antiretroviral Therapy
CA: Cellular Automata
ODE: Ordinary Differential Equations
PDE: Partial Differential Equations
AB: Anti Bodies
GA: Genetic Algorithms