ANALYSIS OF TUMOR GROWTH AND IMMUNE SYSTEM INTERACTION MODEL INTRODUCTION

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1 ANALYSS OF TUOR GROWTH AND UNE SYSTE NTERACTON ODEL NOR AZRAN AWANG a & NORAH AAN b a,b Departmet of athematical Scieces, Faculty of Sciece Uiversiti Teologi alaysia, 830 Johor Bahru, alaysia Correspodig ormahmaa@utmmy azira83@gmailcom Abstract mmue system plays a vital role i cotrollig the tumor growth Therefore, this paper proposes a ew mathematical model that describes tumor-immue iteractio, focusig o the role of atural iller (NK) cell ad CD8 + T cell The tumor populatio is subdivided ito two differet phases, amely iterphase ad mitosis This model used Ordiary Differetial Equatios (ODEs) ad the fuctios ivolved i the model represets tumor-immue growth, resposes ad iteractio betwee the cells The stability ad aalysis of the model are carried out From the aalysis, it show that the stability curve limits tumor growth regio The curve from the model lie below the curve of the model with sigle immue respose (CD8 + T cell) This result cocluded that the proposed model with ivolvemet of NK cell suppressio will lower the tumor growth regio Keywords: tumor growth; atural iller (NK); CD8 + T cell PACS: Replace this text with PACS umbers; choose from this list: NTRODUCTON Tumor represets a real crisis for public health system worldwide Primary tumor growth is a complex process, ivolvig may iteractio betwee the tumor ad surroudig tissues Treatmets traditioally used to fight agaist tumor are surgery, radiotherapy ad chemotherapy Ufortuately, these treatmets ill ormal cells as well as tumor cells The, immuotherapy is itroduced [-6] t is based o the geerally-accepted hypothesis that the immue system is the best tool humas have for fightig disease [3,0-3] recet years, there is evidece that immue system is capable of recogizig ad elimiatig tumor cells Therefore, some researchers itesively cotiue to develop ad ivestigate the iteractio betwee growig tumor ad immue system [-3] ajority of the research o immue cells focused o NK cell, T cell ad macrophages [3,5-7,0] The experimetal result have show that these immue cells are associated with good tumor progosis ad also beig ivolved i the lysis of tumor cells [0] Each of them uses a differet mechaism ad plays a differet role i cell lysis NK cells are lymphocytes i iate immuity that have ability to ill tumor cells directly without activatio [4-6] t also play ey role by destroyig abormal cells before they replicate ad grow [0] By cotrast, T cells are part of the adoptive immue system ad also have capability to fight the tumor cells Both iate ad adoptive immue cells actively prevet eoplastic developmet i a process called cacer immuesurveillace So, the effectors of the adaptive ad iate immuity eed to actively cooperate i order to reject tumor cells [7] Early 990s, Kuzetsor [7] proposed a mathematical model of the cytotoxic T lymphocyte respose to the tumor growth The model described the ietic growth ad regressio of lymphoma They foud that the cytotoxic effector cells are resposible for the lysis of tumor cells 998, Kirscher ad Paetta defie a ODE model for three mai populatio: effector cell, tumor cell ad cocetratio L- Effector cell or also called as immue-system cell are stimulated to grow based o two terms: recruitmet term ad proliferatio term Logistic growth fuctio is used as the growth term i this model de Pillis [0] foud that the combiatio effect of the NK cells ad T cells ca elimiate larger tumors compared to idividual effect of immue cell (the depletio of NK cells have differet impact to depletio of T cell) ost of the tumor-immue models are usig logistic fuctio, gompertz ad expoetial fuctio as the growth term [0,7-8] Gompertz have a serious limitatio which is ot appropriate i describig the dyamics of very small tumors [9-0] while expoetial fuctio oly fixed for small populatio of tumor cells [7] The, Villasaa [] come out with the mitosis as the growth term i tumor populatio She itroduced tumor-immue system model which ivolved cell-cycle The populatio of tumor tae placed durig iterphase ad mitosis ( phase) However, Villasaa oly cosider T cell as the immue cell that ca fight tumor cells The system is simple ad does ot cotai may importace iteractios A followed up study of Villasaa s model was recetly coducted by Liu [] Sila garisa ama pembetag

2 The author modified the model by addig quiescet phase ito the model The result showed the restig phase (quiescet phase) is the most importace compartmet for tumor treatmet Our cocer is to use Villasaa s model by addig aother immue respose (NK cell) sice the combiatio of T cell ad NK cell will give a great impact to tumor growth However, this wor will oly discuss o Ordiary Differetial Equatios (ODE) The model ivolves cell-cycle which populatio of tumor will based o iterphase ad mitosis without cosiderig quiescet phase ODEL Cell cycle is the alteratig growth ad divisio activities of the cell Cell must accomplish two basic thigs durig the cell-cycle which are copyig cellular compoets ad dividig the cell so that compoets are distributed evely to the daughter cells The cycle ca be divided ito two phases: terphase ad phase (itosis phase) The divisio activity correspods to phase while growth activity correspods to iterphase The iterphase is divided ito three further phases: G phase (Gap phase) is a restig phase called pre-sythetic phase G correspods to the iterval betwee mitosis ad iitiatio of DNA replicatio This phase may last up to 48 hours ad it is the logest phase of the cycle The ext phase is the S phase or sythesis period This is the phase whe DNA replicatio occurs S phase could last betwee 8 ad 0 hours The cell will eter aother gap period G called as post-sythetic phase after completig the DNA replicatio G ad late S are preparig phase for mitosis phase (itosis phase) is the last phase where the actual divisio of the cell occur This is the shortest phase of the cycle, lastig up to oe hour t is sigificat to ote that i the 4 hours average duratio of cell cycle of the huma cell, cell divisio proper lasts for oly oe hour The iterphase lasts more tha 95% of the duratio of the cell cycle The divisio to proceed from oe part of the cell cycle to aother depeds o the variety of factors There are growth, DNA replicatio, DNA itegrity ad cellular itegrity f the cell cycle detect the abormality activities, the it will prevet the cell from completig the cycle This situatio is called cell cycle checpoits Whe checpoits are activated, sigals are relayed to the cell-cycle progressio machiery These sigals cause a delay i cycle progressio util the dager of mutatio has bee averted Curretly, there are three ow checpoits: G checpoit, also ow as the restrictio or start checpoit At this poit, the cells have optios either to divide, delay divisio or exist the cell cycle The followed by G checpoit (ed of G) ad metaphase checpoit The cell will ot proceed with mitosis if DNA replicatio is ot complete at the G ed poit The iability of the cell to complete the cycle ca cause cacer Cacer is a disease where regulatio of the cell cycle goes wrog ad ormal cell growth is lost this sectio, a ew mathematical model is proposed based o Villasaa [] ad de Pillis s [0] model to provide a descriptio of the iteractio betwee growig tumor ad immue system The model will focused o tumor growth at differet stages of cell cycle which is iterphase ad mitosis phases NK cells ad CD8 + T cells are icluded i the mathematical model because both of these cell ca lyse tumor cells To describe this iteractio, the model is comprised by four ordiary differetial equatios of tumor populatio durig iterphase ad mitosis ad NK ad CD8 + T cell populatios, where G+S+G T T NK TC, populatio of tumor cells durig iterphase, tumor populatio durig mitosis t, populatio of NK cells t, populatio of CD8 + T cells t t To costruct the iitial mathematical, the assumptios based o owledge of the immue system ad some assumptios state i [0-] These assumptios are listed as follows Both NK cell ad CD8 + T Cell ca ill the tumor NK cell- ate immuity, NK cells are always preset ad active i the system, eve i the absece of tumor cells 3 Each NK cell ad CD8 + T cell will evetually become iactivated after some umber of ecouters with tumor cell 4 As part of specific immuity, tumor-specific CD8 + T cell are recruited oce tumor cells are preset

3 Usig these assumptios, the model taes the form Terms a ad a 4 T t a T c T d T a T c T 4 TC 7 NK from mitosis destroyed by T cell atural death releasig to mitosis destroyed by NK cell T t a T d T a T c T 3 4 from iterphase atural death releasig to mitosis 3 TC 8 NK destroyed by T cell destroyed by NK cell TC T T TC TC costat growth source T T destroyed with tumor i iterphase t c T 4 TC destroyed with tumor i mitosis growth due to stimulus c T d r T T TC NK atural death NK simulatio c T NK T T NK 5 NK costat growth source T T destroyed with tumor i it erphase t s c T growth due to stimulus c T d 6 NK 4 NK destroyed with tumor i mitosis atural death represet the differet rates at which cells cycle or reproduce ad the ecouters of tumor cells withi cells The terms dt, atural cells death ad the terms T T TC T T ad dt 3, dt TC T T NK T T ad dt 4 NK c i terms represet losses from i the model represet proportios of represet the oliear growth of the immue populatio due to stimulus by the tumor cells ichaelis-ete form have bee chose for this term followig other i literature This form is reasoable, because proliferatio of cacer-specific effector cells is stimulated by the presece of cacer cells but reaches a saturatio level at cacer populatio The saturatio level depeds o the health of the immue system, specifically o its ability to produce certai cytoies the absece of cacer cells T T 0, the immue cells grow at costat sources rate ad s Hece, the recruitmet fuctio should be zero whe there is o tumor cells, ad should icrease mootoically towards a horizotal asymptote the case of the CD8 + T cells, i additio to beig recruited by iteractios with T cell processed tumor cells through a ichaelis-ete dyamics, additioal CD8 + T cells are stimulated by the iteractio of NK cells with tumor cells This NK simulatio is represeted by the term r T T This term represets iteractio of NK cells with tumor cells NK The parameters, ad deped o the type of tumor ad the health of the patiets immue system; more specifically their ability to produce certai id of cytoies The term represets the icrease of immue cells due to a stimulus ad parameter represets the half value for the immue respose Whe the tumor level is equal to the immue respose is half way to its maximum value Larger values of mea that it taes the immue system a loger time to recogize the tumor The growth of tumor cell populatio is obtaied through mitosis ad is give by the costats a ad The term T TC, T NK, T TC ad T NK are stadard competitio that i our model will represet losses due to ecouters amog the differet cell types reality, all parameters deped o the tumor type ad the health of the perso Table () summarizes parameter values by Villasaa [] ad Table () summarized estimated parameters by de Pillis [0] a 4

4 TABLE () Estimated parameters Parameter Estimated value day - c c a a 4 d d day day day - d 004 day - c 6 x 0-7 cell - day - 34 x 0-0 cell - day - 3 c 4 3 x 0 4 cell day day- (03 x 0 6 cell) 3 TABLE () Estimated parameter Parameter Estimated value 004 day - c c d 4 c x 0-7 cell - day - c 33 x 0-7 cell - day - 3 x 0 4 cell day - s r x 0-7 cell - day - STABLTY ANALYSS We first determie the dyamics of the system i order to uderstad the behavior of the model We begi by aalyzig the tumor model without immue resposes Tumor-odel i the Absece of mmue Respose this subsectio, we shall study the tumor model case without immue respose Necessary ad sufficiet coditios that guaratee the stability of the cacer-free equilibrium are obtaied this case the equatios are a simple set of ordiary differetial equatios: T a T d T a T 4 T a T d T a T 3 4 () with iitial values T T 0 0 The oly fixed poit for this liear system () is E0 T, T 0,0 The Jacobia matrix of the system is

5 J T, T d a a4 a d3 a4 ad the characteristic equatio is The eigevalues associated with the matrix are give by d a d a d a d a a a tr tr 4 The trace of the correspodig matrix represetatio of the system is tr d a d3 a4, is give by d a d a a a Clearly, the trace, tr d a d a f ustable fixed poit f , the tr 4 0 ad a stable ode 4 0 ad the determiat tr will always egative values, tr ad the fixed poit result i a saddle which will result i 4 0, the we are i the preset of a stable fixed poit We will have a stable spiral tr 0 The, the ecessary coditio for tumor to grow is, such that d a d a a a which meas that the growth rate domiate the death rate Otherwise, the tumor will ot grow As the result, we have the followig propositio PROPOSTON The tumor-free equilibrium 0 E 0 of the system () is locally asymptotically stable if ad oly if Biomedical terpretatio Note that phase acts as a cotrol ceter to determie the rate of proliferatio from the mitosis ito the iterphase ad a a 4 represets the rate of cell reproduce represets the rate of cell release from iterphase eter ito mitosis ( phase) The atural cell death for each phases (iterphase ad phase) represet by ad Assume such that a tumor is growig We cosider the parameters are beeficial for tumor elimiatio Propositio, the oly coditio for cacer decay is 0 Hece for tumor to grow, the ecessary coditio is d a d a a a, which meas that the rate of cell growth is domiate the rate of atural cell death f d a d a a a, the tumor will ot grow d d 3 Tumor-odel i the Preset of mmue Respose Now we will add the effect of immue suppressio to study how CD8 + T cell ad NK cell will chage the dyamical behavior of tumor cells New coditios for tumor growth or extictio that ivolve the immue suppressio parameter terms will be obtaied

6 Tumor-model i the preset of sigle immue respose Firstly, we study the iteractio betwee tumor growths with a sigle immue respose, which is CD8 + T cell So, the system will be T t a T c T d T a T 4 TC T t a T d T a T c T TC TC T T TC TC 4 TC TC t c T c T d T T () ET, T, TC E0,0, is the fixed poit of the system which represet a tumor-free state with positive d immue level The Jacobia matrix about the system is d a c a4 0 d,, 0 c c4 d d d J T T TC a d3 a4 c3 d Clearly, we have d is a eigevalue ad the remaiig eigevalues are give by the solutio to the characteristic equatio d a c d3 a4 c 3 aa4 0 d d Solve the equatio The, we will have 0 0 where d a c c d a d 0 d a c d3 a4 c3 aa4 d d is always positive By Routh-Hurwitz criteria, sufficiet ad ecessary coditios for to become egative real part is 0 0 such that d a c d3 a4 c3 aa4 d d As the result, we have the followig

7 PROPOSTON For system (), the equilibrium E is locally asymptotically stable if ad oly if 0 0 Note that 0 Biomedical terpretatio A immue respose ca cotrol tumor growth this model it occurs whe d a c d3 a4 c3 aa4 The parameter d d represets the growth of T cell (CD8 + T cell), ad represet the rate at which the immue cell destroyed the tumor cell at iterphase ad mitosis, ad, ad represet the atural death of T cell at every phase However if, the chace for tumor cell to grows is icreased This occurs because the reproductio of tumor 0 0 cell is higher tha atural death cell ad immue respose, d a c d3 a4 c3 aa4 d d d d d 3 c c 3 Tumor-model i the preset of double immue respose Now, we proceed with the iteractio betwee tumor growth with two immue respose that is CD8 + T cell ad NK cell The system is T t a T c T d T a T c T 4 TC 7 NK T t a T d T a T c T c T TC 8 NK T T T T t c T c T d r T T T TC TC TC 4 TC TC NK T T t s c T c T d T NK NK 5 NK 6 NK 4 NK (3) Agai the system is assumed i tumor-free state poit with positive immue level J T, T,, TC NK s d a c c7 a4 0 0 d d 4 s a d3 a4 c3 c8 0 0 d d 4 c c4 d 0 d d s s c5 c6 0 d4 d4 d4 Clearly, d, d4 are two eigevalues The remaiig eigevalues are the same as the solutio to characteristic equatio, that is where 0 0

8 Obviously, become egative real part are s d a c c c c d a d d4 s s 0 d a c c7 d3 a4 c3 c8 aa4 d d4 d d4 will always positive By Routh Hurwitz criteria, the ecessary ad sufficiet coditios for to 0 ad 0 0 The s s d a c c d a c c a a d d4 d d4 As the result, we have the followig PROPOSTON 3 For system (3), the equilibrium E is locally asymptotically stable if ad oly if 0 0 Biomedical terpretatio Note that 0 0 As the result before by icludig immue respose i the model, the survival of tumor s s growth is decreased, d a c c7 d3 a4 c3 c8 aa4 d d4 d d4 Figure showed the regio of tumor growth with ad without immue resposes From the graph, we ca see that the size of tumor growth is reduced whe immue respose is icluded The regio become smaller compared to the case without immue respose FGURE Settig our parameter to a4 05, d 03, 0, d 06, c c 005, s 0, c8 c9 0 ad 4 04 d, we obtai a stability map i a ad d plae The regio of tumor growth with o immue suppressio is limited by the upper curve Below this curve, there is tumor growth ad above this curve there is tumor decay The correspodig curve whe immue suppressio is added is show to lie cosistetly below the first

9 CONCLUSON The model preseted represet the basic iteractio betwee tumor cells ad immue resposes The model started with the tumor populatio without immue suppressio The tumor populatio is divided ito two phases related to the cell cycle, iterphase ad mitosis ( phase) From Propositio, the tumor growth maily deped o the atural death rate ad the productio rate of the cell i the cell cycle The tumor will grow if the reproductio rate of the cell is domiate the atural death rate that case, the tumor will eep growig ad evetually become maligat to the body By addig immue suppressio i the model, NK cell ad CD8 + T cell, it greatly helps to stabilize the system ad ihibit the further growth of tumor cells (Propositio ) This is reasoable because some tumor cells are destroyed by T cell ad NK cells our body, NK cell ad T cell are iate ad adaptive immuity i immue system The compariso of tumor regio betwee immue respose ad without immue resposes showed that the ivolvemet of immue resposes have lower the tumor growth regio mmue system will react aturally to fight the tumor cells So, there is importat to mae sure that the immue cells i our body have ability ad capability to ill tumor cells NK cell is exhibit spotaeous illig agaist a variety of tumor cell without the eed for atige specific activatio as required by T cell (CD8 + T cell)the cotributio from both NK cell ad T cell are effective defese agaist tumor growth T cell are particularly efficiet at illig tumor cells However, T cell will tae several days to recogize tumor cells, whereas NK cells are spotaeously active agaist ifected cells So, NK cell is helped T cell i miimizig the tumor growth Therefore, the probability for tumor cells to reach maturity become lower This is ogoig research There are a few sigificat features that have ot bee icluded i this paper For example, the restig phase (quiescet phase) is used to be a importat phase for tumor growth [] Tumor cells will escape the cell cycle ad eter the quiescet phase if they ca escapig the immue cells attaced The survivig quiescet cells ca cotribute to further tumor growth Besides, the iclusio delay term i the model will represet the real situatio of tumor growth i the body The delay term appear aturally whe oe cosider the cell cycle These features will be icluded i future research ACKNOWLEDGENT The authors are thaful to Research aagemet Ceter (UT) ad iistry of Higher Educatio (OHE), alaysia for fiacial support through research grats of vote 06H49 REFERENCES Kirscher, D, & Paetta, J C, odelig immuotherapy of the tumor immue iteractio Joural of mathematical biology, 37(3), pp35-5 De Pillis, L G, & Radusaya, A, A mathematical tumor model with immue resistace ad drug therapy: a optimal cotrol approach Computatioal ad athematical ethods i edicie, 3(), 00, pp errill, Stephe J, "A model of the role of atural iller cells i immue surveillace " Joural of mathematical biology 3, (98), pp Kolev,, Kozłowsa, E, & Lachowicz,, A mathematical model for sigle cell cacer mmue system dyamics athematical ad computer modellig, 4(0), 005, pp Pedroza-Pacheco,, adrigal, A, & Saudemot, A, teractio betwee atural iller cells ad regulatory T cells: perspectives for immuotherapy Cellular & molecular immuology, 0(3), 03, pp-9 6 Eguizabal, C, Zearruzabeitia, O, oge, J, Satos, S, Vesga, A, aruri, N & Borrego, F, Natural iller cells for cacer immuotherapy: pluripotet stem cells-derived NK cells as a immuotherapeutic perspective Frotiers i immuology, 5, 04 7 Fruci, D, Lo, E, Cifaldi, L, Locatelli, F, Tremate, E, Beevolo,, & Giacomii, P, T ad NK cells: two sides of tumor immuoevasio J Trasl ed,, 03, pp30 8 Eladdadi, A, & Radusaya, A E, odelig cacer-immue resposes to therapy Joural of pharmacoietics ad pharmacodyamics, 4(5), 04, pp Bachma, J W, & Hille, T, athematical optimizatio of the combiatio of radiatio ad differetiatio therapies for cacer Frotiers i ocology, 3, 03

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