Multiple Latticed Cellular Automata: HIV Dynamics in Coupled Lymph Node and Peripheral Blood Compartments

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1 Muliple aiced Cellular Auomaa: HIV Dynamics in Coupled ymph Node and Peripheral Blood Comparmens S. MOONCHAI 1,3, Y. ENBURY 2,3 *, W. TRIAMPO,5 1 Dep of Mahemaics, Faculy of Science, Chiangmai Universiy, THAIAND 2 Dep of Mahemaics, Faculy of Science, Mahidol Universiy, THAIAND 3 Cenre of Excellence in Mahemaics, PERDO, CHE, THAIAND R&D Group of Biological and Environmenal Physics, Dep of Physics, Faculy of Science Mahidol Universiy, THAIAND 5 Cener of Excellence for Vecor and Vecor-Borne Diseases, Mahidol Universiy Thailand Cener of Excellence in Physics * Corresponding auhor: scylb@mahidol.ac.h Absrac: Cellular auomaa simulaion approach has become well known as a useful echnique o invesigae complex biomedical sysems in siuaions where radiional mehodologies are difficul or oo cosly o employ. So far, relaively simple cellular auomaa models have been proposed o simulae he dynamics of HIV infecion in human. Mos cellular auomaa models only considered viral proliferaion in he lymph node. However, mos clinical indicaions of AIDS progression are based on blood daa, because hese daa are mos easily obained. Since viral populaion circulaes beween lymph node and plasma, viral load in he wo comparmens are imporan for he descripion of HIV infecion dynamics. We presen here cellular auomaa simulaions of a wo-comparmen model of HIV proliferaion wih delay. Key-Words: Cellular Auomaa, HIV proliferaion, double comparmens simulaion. 1. Inroducion Cellular Auomaa (CA) modeling has been widely used in modeling complex sysems. Despie of is simple srucure, CA is well suied o describe he propagaion phenomena, such as rumor spreading, paricle percolaion, and disease spreading [1-]. In some pracical applicaions, muliple laices are needed o simulae in parallel mulicomparmenal sysems. In paricular, Acquired Immune Deficiency Syndrome (AIDS) has become indispuably he mos devasaing epidemic so far. The Human Immunodeficiency Virus (HIV) causes AIDS, so ha he sudy of he dynamics of he virus-immune sysem in he human body is necessary in order o discover a proper herapy for HIV infecion and how i migh be conrolled. Many researchers have used mahemaical models o describe he populaion dynamics of cells involved in he immune response sysem relevan o HIV proliferaion [5-10]. Mos of hese models are based on sysems of ordinary differenial equaions (ODES) and parial differenial equaions (PDES) [11-1]. However, since he developmen of he disease ypically exhibis hree phases of infecion, ha is, an acue phase (measured in days), a chronic phase (measured in weeks), and full blown AIDS (measured in years), he ODES and PDES are insufficien o describe he hree differen ime scales and hard o consruc o simulae he enire course of he HIV infecion. In recen years, cellular auomaa (CA) models have been used in modeling HIV infecion model in he lymph node [1, 2]. In 2001, a simple CA model was used o simulae he evoluion of HIV infecion in he lymph node by Zorenon dos Sanos and Couinho (SC-model). The resul of heir model was capable of simulaing he hree phase paern of HIV dynamics observed in criical daa. aer, Sloo e al. [2] proposed a CA model o sudy he dynamics of drug herapy for HIV infecion. The CA model was a modificaion of SC-model. Recenly, Veronica Shi e al. [3] also formulaed a CA model based on SC-model for HIV dynamics and drug reamen. Viral load, is effec on infecion rae in he lymph node was included in heir CA model. Mos of hese CA models only considered he dynamics in he lymph node. However, mos clinical indicaions of progression are based on blood daa, because hese daa are mos easily obained. Since viral populaion circulaes beween he lymph node and plasma comparmens, viral ISSN: ISBN:

2 load in he wo comparmens are imporan for he descripion of he dynamics of HIV infecion. In his paper, by modifying he CA rules based on SC-model, we illusrae he use of double laiced CA simulaion o invesigae he dynamics of HIV infecion in boh he lymph node and blood comparmens while he viral loads in he wo comparmens are coninuously updaed hroughou he simulaion. Our model also akes ino accoun a delay τ in he ransformaion of a newly infeced CD + T cell ha is free o spread he infecion, ino a final saged infeced cell. 2. CA Model of HIV Proliferaion A cellular auomaon (CA) is a discree dynamic sysem in which space is divided ino regular spaial cells, and ime progresses in discree seps. Each cell in he sysem has one of a finie number of saes. The sae of each cell is updaed according o local rules, ha is, he sae of a cell a a given ime depends on is own sae and he saes of is neighbors a he previous ime sep []. In his work, he CA model is defined on wo coupled square laices of sizes. The Moore neighborhood is adoped o define he rules. The saes of he cells in each of he laices are updaed a each ime sep in parallel according o he rules, wih each ime sep corresponding o one week. Each sie on he laice is occupied by a cell which is assigned one of he five saes ha describe he possible saes in which hose cells may be found: non-acivaed cells, healhy cells (represening CD + T-cells which are he main arge of he HIV), infeced A 1 cells (corresponding o infeced cells ha are free o spread he infecion), infeced A 2 cells (infeced cells in he final sage before dying due o he acion of he immune sysem) or dead cells (infeced cells killed by he immune response). In his work, we consruc he CA model of HIV infecion in wo coupled comparmens; he lymph node comparmen and he peripheral blood comparmen. For each comparmen, he simulaion seps sar wih N 0 non-acivaed or non-proliferaing cells, H 0 healhy acive cells, and a small fracion P of infeced A 1 cells ( A 10 ), HIV such ha A10 = PHIV H0, disribued randomly. These numbers depend on he iniial viral load V 0. A each ime sep, all cells are updaed using he rules described below. The updaing rules (modified from SCmodel ) are as follows. Rule 1: Updaes of non-proliferaing cells. (a) If a non-proliferaing cell has nonproliferaing cells as neighbors, i may become an acive healhy cell a he probabiliy P op, accouning for opporunisic infecion, or i remains he same a he probabiliy 1- P op. (b) If i has a neighbor which is A 1 or A 2 infeced, i becomes an acive healhy cell, by which he body ries o figh he infecion. Rule 2: Updae of healhy cells. (a) A healhy cell ges infeced by coming in conac wih a virus a he probabiliy * av Pv = Pv f ( V ) = P (1 v e ). (1) (b) If i has a leas one infeced A 1 neighbor, i becomes an infeced A 1 cell a he probabiliy * * P1 = r1 (1 P ) v. (2) (c) If i has no infeced A 1 neighbor, bu has a leas R (2 < R < 8) infeced A 2 neighbors, i becomes an infeced A 1 cell a he probabiliy * * P2 = r2 ( 1 r1 )(1 P v ). (3) (d) Oherwise, i remains a healhy cell a he probabiliy * * * 1 P1 P2 Pv where * * * 0 < P1 + P2 + P v < 1. Rule 3: Updae of infeced A 1 cells. An infeced A 1 cell becomes an infeced A 2 cell afer τ ime seps. Thus, infeced A 1 cells become infeced A 2 cells a differen ime wih a delay of τ. Rule : Updae of infeced A 1 cells. Infeced A 2 cells become dead cells, corresponding o he depleion of infeced cells by he immune response. Rule 5: Updaes of dead cells (a) Dead cells can be replaced by healhy cells wih he probabiliy (1 Pi n f e c ) Pr e p l in he nex sep, or by an infeced A 1 cell wih he probabiliy P P. infec repl Oherwise, i remains a dead cell a he probabiliy 1 P rep l. (b) Afer sep (a), a dead cell can be replaced by an inacivaed cell wih he probabiliy P nona. Oherwise, i remains a dead cell a he probabiliy1 P. nona ISSN: ISBN:

3 3. Viral oad Simulaion In his CA model, he viral load influences he dynamics of he healhy and infeced cells hrough he probabiliy P v *. Afer all of he five cell saes are updaed in he wo laices, he viral load in each comparmen is calculaed using Eq. (1)-(2) and he following difference equaions which represen he evoluion of viral load in he lymph node comparmen (wih V = V ) and peripheral blood comparmen (wih V = B V ) a ime. In he lymph node comparmen B V 1 V = ps I + ( α V Vɶ ) + ch H V cv () where I = virus-producing infeced cells = A1 + A2 In he blood comparmen B B B B B B B V 1 V = psbi + ( Vɶ B V ) + cbhh V cv (5) I = virus-producing infeced cells B = B B A1 + A2 = 1 week ( ime sep) Here, A1 and A 2 are he numbers a ime of A 1 and A 2 infeced cells in he lymph node, B B respecively, while A 1 and A 2 are he corresponding amouns in he blood comparmen. H and H are he numbers of healhy cells in he B ( α ) Vɶ = e V + V B ( β ) Vɶ = e V + V B respecive comparmens a ime, p is he average viral producion rae per infeced cell, e represens he circulaion of virus beween he wo comparmens, and c is he deah rae of free virus. Since he amoun of virus in peripheral blood is measured in milliliers and ha in he lymph node is measured in oal virus couns, we use conversion facors α (blood o lymph node) and β (lymph node o blood) in Eq. () and (5) o describe he exchange beween he wo comparmens. We model immune aciviy in he lymph node and blood wih ch and c BH, respecively. In healhy individuals, oal CD + T- cell couns are abou cells in he lymph node, and 1000 cells/mm 3 in he blood [15]. Since each sie in he laice represens one of he five saes of he cells in he wo comparmens, we conver he numbers of he cells from he CA models o numbers of cells in he body by using he conversion facors S 1 and S 2 for he lymph node and blood, respecively. To simulae he CA model, all parameers are se up as shown in Tables 1-3. Table 1. Model parameers in he CA model in he lymph node comparmen. Symbol P HIV Definiion Value [reference] aice size 500 N 0 Number of non-acivaed or 250,000 non-proliferaing cells a = 0 H 0 Number of healhy acive 150,000 cells a = 0 Probabiliy or percenage of 0.05 iniial infeced cells P op Probabiliy for a nonproliferaing cell o be replaced wih an acive healhy cell P V a r 1 r 2 τ Pinfec Consan in probabiliy for a healhy cell o come in conac wih a virus Consan in probabiliy in Eq. (1) Consan in probabiliy in Eq. (2) Consan in probabiliy in Eq. (3) Time delay for an infeced A 1 cell o become an infeced A 2 cell Probabiliy for a healhy cell o be replaced wih an infeced A 1 cell P Probabiliy for a deah cell repl o be replaced wih a healhy cell Pnona Probabiliy for a deah cell o be replaced wih nonacivaed cells R Number of infeced A 2 cells in a cell s neighborhood o induce a healhy cell o become an infeced A 1 cell ISSN: ISBN:

4 Table 2. Model parameers in he CA model in he blood comparmen. Symbol P HIV Definiion Value [reference] aice size 100 N 0 Number of non-acivaed or non-proliferaing cells a = 0 H 0 Number of healhy cells a = Probabiliy or percenage of 0.05 iniial infeced cells P op Probabiliy for a nonproliferaing cell o be replaced wih an acive healhy cell P V Consan in probabiliy for a healhy cell o come in conac wih a virus a Consan in probabiliy in Eq. (1) r 1 Consan in probabiliy in Eq. (2) r 2 Consan in probabiliy in Eq. (3) τ Time delay for an infeced A 1 cell o become an infeced A 2 cell P Probabiliy for a healhy infec cell o be replaced wih an infeced A 1 cell P Probabiliy for a deah cell repl o be replaced wih a healhy cell Pnona Probabiliy for a deah cell o be replaced wih a nonacivaed cell R Number of infeced A 2 cells in he neighborhood of a cell o induce a healhy cell o become an infeced A 1 cell Simulaion Resuls The resul of model simulaions shows he evoluion of he numbers of he non-acivaed cells, healhy cells, infeced A 1 cells, infeced A 2 cells, dead cells, viral load in he lymph node and viral load in he peripheral blood. Table 3. Model parameers in viral load simulaion. Symbol B V 0 V 0 p S S B Definiion Palsma virus concenraion a = 0 Value [reference] 10 [16] (can vary) 0 Virus concenraion in he lymph node a = 0 Average virion producion rae per [17] 80 infeced cell Scaling facor in he lymph node / H 0 Scaling facor in he blood 1000 / H c Clearance rae of free H virus in he lymph node c Clearance rae of free BH virus in he blood c Deah rae of free virus e Circulaion fracion of virus beween lymph node and blood β Scaling facor: lymph node blood α Scaling facor: blood lymph node [17] 0.02 [18] [16] 5 10 [16] In Fig.1, he graphs exhibi hree phases of he infecion, acue, chronic, and progression o AIDS. The ime evoluion of healhy cells in he wo comparmens and he viral load in peripheral blood closely simulae clinical daa [19]. In addiion, o simulae he dynamics of HIV infecion, he model reproduces he wo ime scales of weeks and years in he graphs. The simulaion shown in Fig. 1 uses he parameric values given in Tables 1-3 which correspond o he case in which a paien evenually succumbs o AIDS in approximaely 5-6 years. Wih differen parameric values, he chronic phase may be lenghened. In Fig. 2, we use r 1 = r 2 = 0.9 and he model in his case simulaes he progression of a paien who succumbs o AIDS more or less righ away. 6 ISSN: ISBN:

5 (a) (a) (b) (b) (c) (c) (d) Fig. 1. The evoluion of HIV infecion wihou reamen indicaes a hree phase evoluion using he parameric values in Tables 1-3. (a), (b) Plos of he numbers of non-acivaed cells, healhy cells, infeced A 1 cells, infeced A 2 cells, and dead cells in he lymph node and blood comparmens, respecively, (c) viral load in he lymph node comparmen, and (d) viral load in he peripheral blood comparmen. (d) Fig. 2. Model simulaion using probabiliies r 1 = r 2 = 0.9. (a), (b) Plos of numbers of non-acivaed cells, healhy cells, infeced A 1 cells, infeced A 2 cell, and dead cells in he lymph node and blood comparmens, respecively, (c) viral load in he lymph node comparmen, and (d) viral load in peripheral blood comparmen. 5. Conclusion The model simulaion has been carried ou by updaing each sie in wo coupled laices in ISSN: ISBN:

6 parallel. The viral load in each comparmen is updaed a each ime sep. The virus is allowed o circulae from he lymph node comparmen o he blood comparmen whenever he viral load in he blood falls below a fracion e of he combined viral load. A delay τ for a highly infecious A 1 cell o become a final saged A 2 cell has been aken ino accoun o make he model more realisic. Moreover, opporunisic infecion is considered here o play an essenial role in allowing he model o simulae differen cases of infecion exhibiing differen chronic phase duraions and evenual progression o full blown AIDS ha closely simulae general clinical daa. 6. Acknowledgemens Appreciaion is exended oward he Cener of Excellence in Mahemaics, and Mahidol Universiy for financial suppor. References R. M. Zorzenon dos Sanos, S. Couinho, Dynamics of HIV infecion: A cellular auomaa approach, Phys. Rev. e., 2001, Vol. 87(16), pp [2] P. Sloo, F. Chen, C. Boucher, Cellular auomaa model of drug herapy for HIV infecion, ecure Noes in Compuer Science, 2002, Vol Springer, Berlin, pp [3] V. Shi, A. Tridane, Y. Kuang, A viral loadbased cellular auomaa approach o modeling HIV dynamics and drug reamen, J. Theor. Biol., 2008, Vol. 253(1), pp [] S. Wolfram, Cellular auomaa as models of complexiy, Naure, 198, Vol. 311, p [5] U. Hershberg, Y. ouzoun, H. Alan, S. Solomon, HIV ime hierarchy: Winning he war while, loosingall he bales, Physica., 2001, pp [6] D. E. Kirschner, G. F. Webb, A mahemaical model of combined drugherapy of HIV infecion, J. Theore. Med., 1997, pp [7] A. Mielke, R. B. Pandey, A compuer simulaion sudy of cell populaion in a fuzzy ineracion model for muaing HIV, Physica A, 1998, Vol. 251, pp [8] A. S. Perelson, Modellinghe ineracion of he immune sysem wih HIV. In:C. Caasillo-Chavez (ed): Mahemaical and Saisical Approaches o AIDS Epidermiology, ecure Noes in Biomahemaics,1989, Vol. 83, pp [9] N. Silianakis, C.A.B. Boucher, M.D. De Jong, R. Van eeuwen, R. Schuurman, R.J. De Boer, Clinical daa ses of HIV-1 reverse ranscripase-resisan muans explained by a mahemaical model, J. of Virol., 1997, pp [10] D. Wodarz, K. M. Page, R. A. Arnou, A. R. Thomsen, J. D. ifson, M. A. Nowak, A new heory of cyooxic -lymphocye memory: implicaions for hiv reamen, Philos. Trans. R. Soc. ond. (B Biol. Sci.), 2000, Vol. 355(1395), pp [11] R.J. De Boer, A.S. Perelson, Targe cell limied and immune conrol models of HIV infecion: a comparison, J. Theor. Biol. 1989, Vol. 190, pp [12] N.M. Dixi, A.S. Perelson, HIV dynamics wih muliple infecions of arge cells, Proc. Nal Acad. Sci., 2005, Vol. 102 (23), pp [13] D. Kirschner, Using mahemaics o undersand HIV immune dynamics, Na. Am. Mah. Soc., 1996, Vol. 3 (2), pp [1] D. Wodarz, M.A. Nowak, Mahemaical models of virus dynamics and resisance, J. HIV Ther., 1998, Vol. 3, pp [15] A.T. Haase, Populaion biology of HIV-1 infecion: viral and CD+ T cell demographics and dynamics in lymphaic issues, Annu. Rev. Immunol., 1999, Vol. 17, pp [16] S. H. Bajaria, G. Webb, M. Cloyd, D. Kirschner, Dynamics of naive and memory CD+ T lymphocyes in HIV-1 disease progression, J Acquir. Immune Defic. Syndr., 2002, Vol. 30, pp [17] P.W. Nelson, J.D. Murray, A.S. Perelson, A model of HIV-1 pahogenesis ha includes an inracellular delay, Mah. Biosci., 2000, Vol. 163, pp [18] D. J. Seckel, J. Theor. Biol. 186, 91 (1997); D. J. Seckel e al., Immunol. Today. 18, 216 (1997) [19] A. S. Fauci, G. Panaleo, S. Sanley, D. Weissman, Immunopahogenic mechanisms of HIV infecion, Ann. Inern. Med., 1996, Vol. 12, pp ISSN: ISBN: