Indian Journal of Experimental Biology Vol. 42, February 2004. pp. I3I-I3 7

Review Article

Cellular automata and its advances to drug therapy for HIV infection

M A Peer*', N A Shah & K A Khan P G Departments of Computer Sciences, Electronics & Instrumentation Technology. University of' Kashmir, Srinugar I90 006, Indiu

This paper gives an over view of the usc of cellular automata (CA) model of drug therapy for HIY infection. Non­uniform CA is employed to simulate drug treatment of HIY infection, where each computational domain muy contain different CA rules, in contrast to normal uniform CA models. Ordinary (or partial) differential equation models arc insufficient to describe the two extreme time scales involved in HIY infection (days and decades), as well us the implicit spatial heterogeneity . Zorzcnon and Coutinho [Phy Rev Leu, 16 (2001) II reported a cellular automata approach to simulate three-phase patterns of human immunodeficiency virus (HIY) infection consisting of primary response, clinical latency und onset of acquired immunodeficiency syndrome (AIDS). But here we present a related model , bused on non-uniform CA to study the dynamics of drug therapy of' HIY infection. The muin aim in this model is to simulate the four phases (acute, chronic, drug treatment responds and onset of AIDS). The results shown here indicate thut both simulations (with and without treatments) evolve to the relatively same steady state (characteristics of Wolfrum's class. II behavior) . Different kinds of drug therapies can also be simulated in this model, which can be found usefu l for deve loping a proper drug thcrupy.

Keywords : Cellular uutomutu, Drug therapy. HIY-AIDS

Cellular automata are a class of very powerful models originated in computer science. They are well known extensions of the classical automata. They provide mathematical models for a wide variety of complex natural phenomena from growth of pattern in biological systems to turbulence in fluid dynamics1

'4

•

ow a days a cellular automaton (CA) is a kind of digital computer. It is discrete and deterministic. It is made up of cells like the points in a lattice or like the squares of checkerboard. Jt is called automaton because it follows a simple digital (di crete) rule. A well-known CA is the game of life invented by John Conwa/' 6

.

Building cellular automata The cell - The basic element of n CA is the cell. A

cell is a kind of a memory element and stores-to say it with easy words-states. In the simplest case, each cell can have the binary states I or 0. In more complex simulation, the cells can have more different states7

·K.

It is even thinkable, that each cell has more than one property or attribute and each of these properties or attributes can have two or more state .

The lattice- The arrangement of cells in a spatial web is called a lattice. The simplest one is the one­dimensional "lattice", meaning that all cells are

"'For correspondence : Phone: 0194-2427280 Fax: 0194-2421357 E-mail: pccr@kashmirunivcrsity.nct drmapccr@ hotmai !.com

arranged in a line like a string of perils. The most common CA 's are built in one or two dimensions.

Neighbourhoods- Up to now, these cells arranged in a lattice represent a static state. To introduce dynamic9

'11 into the system, we have to add niles. The

"job" of these rules is to define the state of the cel ls for the next time step. In cellular automata a rule defines the state of a cell in dependence of the neighbourhood of the cell.

The three types of neighbourhoods are shown in Fig. I.

Basic concepts of HIV infection and drug therapy Biological background of HIV infection and the

effect of drug therapy for possible treatment are explained briefly.

Biological background of HIV il!f'ection- The infection of human immunodeficiency virus (HIV), causing AIDS (acquired immunodeficiency syndrome), is almost invariably a progressive, lethal disease with insidious time course. Currently, clinicians identified two common laboratory markers for detection of disease progression, the amount of virus (HIV-RNA) and the number of T helper cells (CD4 T cells) in blood. Immune response for typical virus infection varies from days to weeks, but HIV infection typically follows a three-phase pattern 19 (Fig. 2, Bryan Christie, July 1998, Scientific American).

During few weeks (varying from two to six weeks), a transient and dramatic jump of plasma virion level is present with a marked decrease of immune cell count (CD4 T helper cells), following by a sharp decline.

132 INDIAN J EXP BIOL, FEBRUARY 2004

In the subsequent chronic phase (varying from 1-10 or more years, on average 8-10 years), the immune system partially eliminates the HIV virus and the rate of viral production reaches a lower, but relatively steady state that varies great ly from patient to patient. Their apparent good health continues because CD4 T cell levels remain high enough to preserve defensive responses to other. pathogens. But over time, CD4 T cell concentrations gradually fall. .

An outbreak of the virus (varying from one to two years), together with constitutional symptoms and onslaught by opportunistic di seases, cause death 12

.

Biological background of drug therapy of HIV infection-A vaccine would certainly be ideal for preventing infection by HIV and thus for avoiding AIDS when immunity is severely impaired. The near­term prospects for a vaccine are poor due to en·or occurrence during each transcription of HIV . Therefore, for the immediate future, many scientists

. . . h h 13 14 are concentratmg on unprovmg t e t erapy · . Currently , there are 15 drugs licensed for treatment

of individual s infected with HIV. These drugs belong to two classes, one inhibiting the viral enzy me reverse transcriptase and the other inhibiting the viral

(a) Von Neumann

Neighbourhood

(b) Moore

Neighbourhood

(c) Extended Moore

· Neighbourhood

Fig. I - The red cell is the center cell ; the blue cells are the Neighbourhood cells. The tales of these cells are used to calculate the next state of the (red) center cell according to the defined rule. Figs I a, I b and I e represent three types of nei ghbourhood known as Von­

eu mann, Moore and Ex tended Moore Neighbourhoods.

ACUI'E PHASE

CHRONIC PHASE AIDS

1,200 4 ~4

§ 1,000 Ill

"' 0

F-o~ 800 z,... ~:s

8~ 600 ....Ju ,J_

tJ~ F-o~ 400

ere ~

200 Ill 8.

·~ 107

< A

l(f> ~ s: "' 0

"" 10S Q~

<:j s~

ut ....J"' ~:: >~

0 0 < ~

CD4 T CELL COUNT

>

0 Jol g

6 11 WOKS

Fig. 2 - Common pattern o f HIV infec tion in a typical untreated patient indicates a three-phase evolution. The two lines represent CD4 T cell count and viral load respectively (Bryan Christie, July 1998 Scientific American).

PEER el. al.: CELLULAR AUTOMATA & DRUG THERAPY FOR HIY INFECTION 133

protease. These drugs are used in combination therapy to maximally inhibit viral replication and decrease HIV-RNA to below level s of detection levels (currently defined as below 50 copies per ml ) in blood. In one class, the nucleoside analogues resemble the natural substances that become building blocks of HIV-DNA; and when reverse transcriptase tries to add the drugs to a developing strand of HIV­DNA, the drugs prevent completion of the strand . The other agent in this class, non-nucleoside reverse transcriptase inhibitors, composed of other kinds of substances, constitute the second class of anti­retroviral. The other class, the protease inhibitors, and blocks the active, catalytic site of the HIV protease, thereby preventing it from cleaving newly made HIV proteins.

HIV therapy is classified into three classes: mono­therapy, combined therapy and triple therapy . Mono­therapy (such as based on reverse transcriptase inhibitor) or combined drug therapy (reverse transcriptase and protease inhibitors) are considered to suppress the viral mu ltiplication. Because of incompletely blocking the replication pathway and occasionally creation of a resistant virus strain , the CD4 T counts will come back to the pre-treatment base line within many weeks. The problem of drug res istance in the tr~atment has become an increas ing signi ficant barrier in the effectiveness of AIDS immune-therapy.

Currently, there is no sing le class of drug that can completely prevent HlV from replicating. Treatment with drug combinations is in only 50% of the cases successful in inhibiting vira l replication undetectable levels. In the remaining 50% of cases viruses can be detected with a reduced sensiti vity to one or more drugs from the patients regimen. Theory and c linical trails indi cate that the best way to achieve maximum

(a) (b)

viral suppression is through highly active anti ­retroviral therapy (HAART), which consists of triple therapy including two nucleoside analogues and a protease inhibitor.

Computational task and related work Modeling the population dynamics of cells in

immune response relevant to HIV has recently attracted a considerab le interest. Currently, the only two ways to model this dynamics of the immune response with respect to the pathology and therapy of HIV infection are analytic partial differential equation models, ordinary differential equation models and cellular automata models . Analytical approaches are successful to describe different aspects of HIV infection dynamics 15

. But they have strong limitations to describe the two times scales observed in the time course of infection in term of weeks and years and have serious difficulties in exploiting spatial information. Cellular automata are recently regarded as a good strategy to model spati al temporal dynamics with emphasis on local interactions. Mielke and Pande/ 6 developed a fuzzy interaction model for mutating HIV with a fuzzy set of 10 interactions for macrophages, helper cells, cytotoxic cells and virion. [Using a microscopic simulation , that the time course of AIDS is determined by the interactions of the virus and the immune cells in the shape space of antigens, and that it is the virus's ability to move more rapidly in thi s space (its high mutab ility) which cause the time course and eventual "v ictory" of the disease 17

].

Thi s model clearly showed the three stages of the disease. A simple set of CA rules was used to model the evolution of HIV infection as per Zorzenon and Coutinho 18

. The three phase patterns were also presented and the results indicated that the infected cells organize themselves into spatial structures,

64 128 256

32 I 2

16 8 4

(c)

20CA grid 20 transition dynamics 20 rule description

Fig. 3-A cell at the centre and its possible transition along different directions in a 2DCA is shown in Fig. 3a, 3b and 3c represent the lattice, a cell and the rule description respecti vely.

134 INDIAN J EXP BIOL, FEBRUARY 2004

which are responsible for the decrease in the concentration of un-infected cells, leading to AIDS. This CA model inspires the drug therapy simulation presented here. In this model we can investigate the HIY infection dynamics with therapy using microscopic simulations 19

• The main ingredients in our model are destruction of previously emerged spatial patterns (wave-like and solid-like structure) and reconstruction of new spatial patterns (wave-like structures) due to incorporation of the drug therapy concept. In the sequel we will refer to the Zorenon dos Santos's model as the HIY infection model (HI model) and our model as the drug therapy of HIY infection model (OTHl model).

The HI model The founding element of any cellular automata

model is the cell (Fig. 3b). It is surrounding by neighbouring cells in Moore Neighbourhood of 20CA lattice.

In 20 cellular automata (Fig. 3a), the cel ls are

~ o4(a)

Ill

~ 1,000

n 100

8~ 600

""~ 400

~ ~ 200

0

2 4 6 I 10 12

! 1,000 ~

n 100

600

s~ 400

""~ ~a 200

0

2 4

arranged in a two-dimensional grid with connections among the neighbouring cells. Consider a 20 CA comprising of mn cells organized as an m x n array with m rows and n columns. The state of CA at any time instant can be represented by an m x n binary matrix . The neighbourhood function specifying the next state of a particular cell of the 20 CA is affected by the current state of itself and eight cells in its nearest neighbourhood. Mathematically, the next state q of the (i, j) lh cell of a 20 CA is given by

[

Q l-l,j-1 (t), Q I-I,J (t ), q l- l , j+ l (t ), l q iJ (t +I)= f q t.J-I (t), Q 1.J (t), Q 1,J•I (t),

• q i+l.j-1 (t), q i+ l. j (t), q l+l,j+l (t)

where (f) is the Boolean function of nine variables.To express a transition rule of 20 CA, specific rule convention is given in (Fig. 3c).

The central box represents the current cell (that is, the cell being considered) and all other boxes represent the eight nearest neighbours of that cell. The

6

Wttlts

... ----· ., ......

I 10 12 Yean

;ig . 4 - Three ph usc dynmnics wi th two time scules (weeks unci years) shown in Fig. 4a und 4b where obtained. It represents primary esponse, clinicnllnthency und AIDS (steady state after ten years). The solid, hash and hash-dot lines represent healthy, infected (A I and ~2 ) and dead cells respectively. The density of dead cells shifted one time step, compared with result of HI model. The evolution speed f cells with corrected rule wus consistent with the result of HI model, but wus slower than the result without correction.

PEER et. a/.: CELLULAR AUTOMATA & DRUG THERAPY FOR HIV INFECTION 135

number within each box represents the rule number associated with that particular neighbour of the current cell-that is, if the next state of a cell is dependent only on its present state, it is referred to as rule 1. If the next state of a cell is dependent on the present state of its right neighbour, it is referred to as rule 2. Similarly, if the neKt state of a cell is dependent on the present state of its top-left cell it is referred to as rule 64 and so on . These are called primary rules . In case, the next state of a cell depends on the present state of itself and/ or its one or more neighbouring cells (including itself), the rule number will be the arithmetic sum of numbers of the relevant cells. For example, rule 3 ( = 1 + 2). These are called secondary rules9

•

However, in the present work with Moore neighbourhood and periodic boundary conditions, we limit the range of HIV infection (Infected AI and A2) by giving a rank level R (2 < R < 8). It has been clearly mentioned that A 1 and A2 cells (or neighbours of considered cell) are infected cells i.e. those cells against which the immune system has developed a response, hence the ability to spread the infection is reduced. While as CD4T (cluster of differential) cells are those cells, which are non-infected cells and help the immune system to fight against the foreign body and consequently in recovery . Clinicians called them helper cells and the relation can be obtained as follows:

Pco4T = 1- (Pinfected +Pdcnd)

where Pc0 4r is probability (or % quantity) of healthy

Acute phase Chronic phase

~~

§' IXl 1,000 ~ 0

~~

~~ 800

O:s

j~ 600

~Sl u!§ 400 ~(..)

a~ 200 u~ ~ 0

--Jj~2cJ:!- 100 200

or helper cells, P;,rcctcd is probability (or % quantity) of infected cells, and P dend is probability (or % quantity) of dead cells.

The biological description of HIV infection model (HI) with Moore neighbourhood and periodic boundary 18

·20 for simulation process is explained as

under: [Rule 1] Update of a healthy cell.

(a) If it has at least one infected-A I neighbour, it becomes an infected-A! cell.

- The spread of the HIV infection by contact before the immune system had developed its speoific response against the virus.

(b) If it has no infected-A! neighbour but does have at least R(2<R<8) infected-A2 neighbours, it becomes infected A-I.

-Before dying, infected-A2 cells may contaminate a healthy cell if their concentration is above some threshold.

(c) Otherwise it stays healthy . [Rule 2] An infected-A! cell becomes an infected-A2 cell after t time steps.

-An infected cell is the one against which the immune response has developed a response hence its ability to spread the infection is reduced. The t represents the time required for the immune system to develop a specific response to kill an infected cell. A time delay is requested for each cell because each new infected cell carries a different lineage (strain) of the virus . This is the

AIDS

Start up of treatment

II Jt(' Infected cells

I .. ··· ·············· ............... ······· ' ..... ··

Dead cells

400 600 Weeks

Fig. 5- Four-phase dynamics with two time scales (weeks and years) were obtained, which were qualitatively comparable with clinical data. The solid, hash and hash-dot lines represent healthy, infected (A I and A2) and dead cells respectively . The vertical hash li ne indicates the starting point of the therapy . The profile indicated that after primary response, the CD4 T cells decreased gradually in the latency period. Once the therapy started, CD4 T cell count increased due to the drug therapy. Finally they evolved into AlDS state due to the resistance against drug.

136 INDIAN J EXP BIOL, FEBRUARY 2004

way to incorporate the mutation rate of the virus in thi s model. On the average, o ne mutation is produced in one generation due to the error occurrence during HIV transcriptio n. Assume that mutation in each tri a l is varied in thi s model due to the stochastic characteri stic.

[Rule 3] Infec ted-A2 cell s become dead cell s . - The depletion of the infected ce ll s by the immune

response. rRule 4]

(a) Dead cell s can be repl aced by healthy ce ll s with probability Prep! in the next step cPrcpl = 99%) or remain dead with probability 1- P rcp l. ·

- The repleni shment of the depl eted cells mimics the hi gh abili ty of the immune system to recover from the immuno-suppress ion generated by infection. As a consequence, it will also mimic some diffus ion of the ce ll s in the ti ssue.

(b) Each new healthy cell introduced, may . be replaced by an in fected-A 1 cell with probability P;nrec (P;nrec = 1 o·\

- The introduction of new infec ted cell s in the system, e ither coming fro m other compartments of the immune sys tem or from the activation of the latent in fec ted ce ll s will increase the diffusion rate of the cell s in the ti ssue.

The time is defined on real bases, there is one to one corTespondence between the defined time and real time shown in Fig. 4. The Fig . 4 is for delay parameter t =4 (t represents the time requ ired for an immu ne system to deve lop a specific response to kill

the in fec ted cell ). T he de lay parameter t. may vary from 2 to 6 week which is shown by dashed lines.

Based on the model summarized above, we incorporate the drug therapy process into CA model for drug therapy. All approved anti-HIV, or anti ­retrov ira l, drugs attempt to bl ock vira l replicatio n within ce ll s by inhibiting either reverse transcriptase or the HIV protease. In addition to the delayed infection modelled in rule 1 b and the latent in fection in rule 4b, the main source of HJ V infec tion in the Hl model is rule Ia. We limit the range o f HIV infection (infected Al cell s) by giving a rank level N (O ~ N ~ 7 ) . This mimics the principle that the drug prevents the virus from replication, resulting in less efficient infection. N represents the effec tiveness of each drug. The bigger N, the less effi cient the drug. Different drug therapies are modelled by different response functions cPresp) over the time. P resp represents function for each drug therapy, which have effects on the infected Al cells after the start ing of a drug therapy. This models the fact that the

drug therapy will not immediately influence all of infected A I cells, but rather it will affect part of them at each time step. Overtime these effect s of drug therapy can (and will) decay. At the same time, this also mimics the concept of drug resistant virus strains. Using this model the drug therapy of HIV in fection (DTHI), without treatment is simulated and the three phase dynamics is shown in Figs 4a and 4b for weeks and years respecti vely.

Introduction of drug therapy into DTHI model ­In this secti on we introduce the respon' e of the system to drug therapy using HI model. The simulation results in Fig. 5 are in time scale o f weeks but can also be observed in time scale of years along with drug therapy effect. The data shown were averaged over 500 simulations. The first acute phase ind icates the fas t pro liferation of the original HIV strains before the actual immune system response. This phase ends when specific immune response occurs for these strains. The nex t phase, the chronic phase that takes years, is the phase where the viral load increases slowly and CD4 counts decrease slowly. When CD4 T counts drop to a certa in level (normal 200 to 500 counts per rnl ), the drug therapy is start ed. In this phase, virus replication is blocked and C D4 T counts increase. Once resistant strains against the drugs evolve, the last phase of the disease occurs disrupting the whole immune system.

The simul ation results indicate that the extension of lo ng-term surviva l is dependent on the drug effecti veness (N) and the response fu ncti on ( P resp).

The hi gh quality of the drug effi c iently prevents the virus fro m replication and thus few res istant new viruses are generated . As a consequence, a re latively prolonged lo ng-te rm survi va l can be obta ined.

Conclusions and outlook The main success of the present model is the

adequate modeling of the four-phase of HIV in fec tio n with different time scales into one model. The s imul ations show a qualitati ve correspondence to c linical data. During the phase o f drug therapy response, temporal fluctu atio ns where observed , this is due to the re lati ve simple form of the response distribution functi on cPrcsp) applied to the drug effectiveness parameter (N) at each time-step. Simulation results indicate that, in contrast to ordinary di fferentia l equation or parti a l di fferential equati on, this model supports a more flexible approach to mimic different therapies through the use of mapping

f I. . I d ? I ?? the parameter space o Prcsp to c lll iCa ata- ·--. However, analytical approach like ODE/POE models are also success ful to describe diffe rent aspects of Hlv . c . d . 15 ?3 24 1111 eCtron ynamrcs ·- · .

PEER et. al.: CELLULAR AUTOMATA & DRUG THERAPY FOR HlV INFECTION 137

There is ample room to incorporate biologically more relevant response functions into the model. This future work requires in depth investigation of the parameter space of Presp· Theory and clinical trials indicate that the best way to achieve maximum viral suppression is through highly active anti-retroviral therapy (HAART), which consists of triple therapy including two nucleoside analogues and a protease inhibitor. By selecting a suitable response function HAART can be simulated using the proposed model.

The results from Fig. 4 with respect to the amount of CD4T counts are not completely supported by clinical data. The number of CD4 T cells should completely go down at least to a level that is , undetectable. The chosen value of PH 1v in thi s work (PHtv = 0.05) is too large with respect to data known for clinic. A more realistic value would be 1 infected cell per 102 to 103 cells, resulting in PH1v =0.005. This effect can also be investigated . Clinical data indicate an increased sensitivity ofT-cells over time, probably due to activation of the immune system. This will be modelled by making P;nrec a function of the number of infected cells. Finally, it is known that in the early stages of infection, virus replication is confined to mono-cytic white blood cells. Only in latter stages , CD4T-cells will become the new target cells. This trophysm effect can be studied in the future.

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