Paintings & Photography

A cellular automata model for avascular solid tumor growth under the effect of therapy

Description
"Tumor growth has long been a target of investigation within the context of mathematical and computer modeling. The objective of this study is to propose and analyze a twodimensional stochastic cellular automata model to describe avascular solid
Published
of 12
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
     U   N   C   O   R   R   E   C    T   E   DP   R   O   O   F PHYSA: 11782 Model 3G pp. 1–12 (col. fig: nil ) ARTICLE IN PRESS Physica A xx (xxxx) xxx–xxx   Contents lists available atScienceDirect Physica A  journal homepage:www.elsevier.com/locate/physa A cellular automata model for avascular solid tumor growth under theeffect of therapy E.A. Reis, L.B.L. Santos, S.T.R. Pinho ∗ Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, Brazil a r t i c l e i n f o  Article history: Received 25 September 2008Received in revised form 23 November2008Available online xxxx PACS: 87.17.Aa05.10.-a87.17.Ee Keywords: Tumor growthCellular automataParameter spaceNecrosisTherapy a b s t r a c t Tumor growth has long been a target of investigation within the context of mathematicaland computer ∧ modeling. The objective of this study is to propose and analyze a two-dimensional stochastic cellular automata model to describe avascular solid tumor growth,taking into account both the competition between cancer cells and normal cells fornutrients and/or space and a time-dependent proliferation of cancer cells. Gompertziangrowth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular borders, arecaptured. The parameter space is studied in order to analyze the occurrence of necrosisand the response to therapy. Our findings suggest that transitions exist between necroticand non-necrotic phases (no-therapy cases), and between the states of cure and non-cure(therapy cases). To analyze cure, the control and order parameters are, respectively, thehighest probability of cancer cell proliferation and the probability of the therapeutic effecton cancer cells. With respect to patterns, it is possible to observe the inner necrotic coreand the effect of the therapy destroying the tumor from its outer borders inwards. © 2008 Published by Elsevier B.V. 1. Introduction 1 Neoplastic diseases are the cause of 7 million deaths annually or 12% of deaths worldwide [1]. Mathematical and 2 computer ∧ modelingmay lead toagreater understanding of the dynamics of cancer progression in the patient [2,3]. These 3 techniques may also be useful in selecting better therapeutic strategies by subjecting available options to computer testing 4 ( in silico ). Continuous models have been proposed to describe the stages of tumor growth since the middle of the 20th 5 century[4]. Initially, a tumor grows exponentially (linear rate). After this transient stage, the growth rate decreases and a 6 steadystateisattained,duetoseveralfactorsincludingalackofnutrientsandhypoxia.Thisnonlinearbehaviorcharacterizes 7 avascular tumor growth when ∧ neovascularisationhas not yet been triggered. The decelerating avascular growth may be 8 guidedbydifferentrulessuchas,forexample,Gompertzianandlogisticfunctions.Gompertziangrowthhasbeenoneofthe 9 most studied decelerating tumor growth over the past 60 years[5–7]. It is found, for example, in some solid tumors such as 10 breast carcinomas [8,9]. It is also observed in tumors in vitro [10,11]. 11 Although continuous models are capable of describing the behavior of tumor growth, it would appear more reasonable 12 to adopt a discrete approach when describing the prevascular stage. Due to the fact that the angiogenic process has not 13 triggered early tumor growth, few cancer cells are present and growth depends predominantly on the interactions of these 14 cellswithadjacentcellsandwiththeenvironment[12].Inaddition,inthediscreteapproach,itiseasiertocapturethetime- 15 spatialpatterngeneratedbythemodelinordertocompareitwithactualpatterns[13].Somecellularautomatamodels[14– 16 16] and hybrid cellular automata [12,17,18]have been proposed to study tumor growth. 17 ∗ Corresponding author. Tel.: +55 71 32836658; fax: +55 71 32357730. E-mail addresses: eareisphysics@gmail.com(E.A. Reis),santosl@ufba.br(L.B.L. Santos),suani@ufba.br(S.T.R. Pinho). 0378-4371/$ – see front matter © 2008 Published by Elsevier B.V.doi:10.1016/j.physa.2008.11.038Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038     U   N   C   O   R   R   E   C    T   E   DP   R   O   O   F PHYSA: 11782 ARTICLE IN PRESS 2 E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx Anotherimportanttopicthatisanalyzedinmathematicalmodelsistheresponsetotherapy,includinghowtumorgrowth 1 changesundertheeffectofadrug[19].Thefocusisdirectedtowardsidentifyingtheoptimaltherapytomaximizetheeffect 2 on cancer cells and minimize the effect on normal cells [20–22]. There are various continuous chemotherapy models[23– 3 26], with this aim. To the best of our knowledge, the majority of the models cited in the literature, that ∧ analyzeboth the 4 Gompertzian growth and the response to therapy, are continuous models [27,28]. 5 In this work, Gompertzian growth, necrosis and the response and therapy effect are ∧ analyzedusing a discrete model for 6 avascular tumor. It is a two-dimensional cellular automata model consisting of 4 states (empty site, normal cell, cancer cell 7 and necrotic tumor cell) to describe avascular tumor growth. It should be emphasized that in this study the term tumor 8 growth is used to refer to the number of cancer cells rather than the volume of the tumor; in other words, it is assumed 9 that the tumor volume is proportional to the number of cancer cells[4]. Assuming that the angiogenic process has not yet 10 beentriggered,thereisnoincreaseinnutrients,whichareuniformlydistributedoverthelattice.Inthissimplemodel,some 11 relevant processes involved in the prevascular phase of tumor growth are assessed: a dynamic proliferation of cancer cells 12 and the competition between normal cells and cancer cells for nutrients and/or space. Since necrosis is often present in the 13 prevascular stage of tumor growth[10,29], the possibility of necrosis in the model must also be taken into consideration. 14 Finally, the effect of therapy is included in order to investigate whether the system evolves to a state of cure. 15 This paper is organized as follows. In Section2,the model is presented, together with its local rules, parameters and the 16 scopeofthealgorithm.Section3describesthesimulatedtimeseriesofcelldensityinthepresenceorabsenceoftreatment, 17 and shows the features of the time-spatial pattern of simulated solid tumors. In Section4,the parameter space related to 18 the process of necrosis and the effect of therapy is analyzed. Finally, in Section5,our results are discussed from the point of  19 view of the phenomenon and some concluding remarks are made. 20 2. The model 21 We propose a stochastic two-dimensional ( L × L ) cellular automata model [30] under periodic boundary conditions, 22 usingaMooreneighborhoodwitharadiusof1.Attheinitialcondition( t  = 0),thereisonlyonecancercell(toensurebetter 23 visualization, this was taken from the center of the lattice). Since the intention is to model a non-viral tumor, normal cells 24 would not be transformed into cancer cells with the exception of the cancer cell that triggers tumor growth at t  = 0[31]. 25 The lattice represents a tissue sample; there is a cell in each site that may be in one of four states: normal cell (NoC), cancer 26 cell (CC), necrotic cell (NeC) or empty site (ES). We assume that the nutrients are uniformly available over the lattice. In this 27 respect, lack of space is identified with lack of nutrients in our model. 28 A local measure, that we call the growth potential, is associated with each normal cell ( P  noc ) or cancer cell ( P  cc  ). The 29 growth potential of a cancer cell, P  cc  , corresponds to the total number of cancer cells, i.e., in addition to the cancer cell on 30 the site of the lattice, the cancer cells generated by these cells. 31 The mitotic probability of cancer cells, that we assume as a global measure, is responsible for the proliferation of cancer 32 cells. Its initial value is given by a parameter p 0 that measures the available resources at the beginning of the tumor. After 33 that, it decreases by a factor   p mitot until reaching the null value: 34   p mitot = exp  −  n noc ( t  ) n cc  ( t  )  2  (1) 35 in which n noc ( t  ) and n cc  ( t  ) are the number of normal and cancer cells at time t  , respectively. As shown,   p mitot ( t  ) depends 36 only on the dynamics, setting up a feedback inhibition mechanism [13]: as the tumor grows,   p mitot ( t  ) decreases because 37 ofthe combinedeffect ofthe decreaseinthe numberof normalcells andthe increaseinthe numberof cancercells. Inorder 38 to intensify the effect of this mechanism [32], an exponent 2 in Eq.(1)is considered. Since there is no new available source 39 of nutrients and/or space, it decreases as the density of cancer cells increases because the available nutrients and/or space 40 arereduced.Theeffectoftheproliferationofcancercellsisthattheirgrowthpotentialincreasesbyaunitateachtimestep. 41 The local rules are such that: 42 (i) Thecancercellscompetewithnormalcellsfortheemptysites,dependingonthepotentialgrowthofneighboringcells. 43 According to the majority rule, a normal cell is displaced by a cancer cell following local battles occurring between 44 healthy and cancerous cells [14]. 45 (ii) Ifthegrowthpotentialofacancercellreachesathresholdvaluethatisafraction  f  ofthelatticesize L anditsneighbors 46 are also cancer cells, it becomes necrotic and its growth potential falls to zero. 47 (iii) Dependingontheirgrowthpotentialsandtheirneighborhood,bothnormalandcancercellsmaydie,withprobabilities 48  p drugn and p drugc , respectively, due to the continuous infusion of a drug that is applied after t  ap time steps; in this case, 49 the site becomes empty. 50 (iv) If there are no cancer cells in the neighborhood of a dead cell (empty site), regeneration of normal cells occurs; if the 51 cancer cells in the neighborhood of a necrotic cell die as a result of the therapy, the necrotic cell is eliminated. 52 The algorithm was computationally implemented in FORTRAN 77 in accordance with the following steps: input data; 53 calculate   p mitot ( t  ) ;identifythestateofthecell chooseoneofthesubroutines:normalcell(NoC),cancercell(CC),necrotic Q1 54 cell (NeC), empty site (ES); update the cells of the lattice; after N iterations, output data. A framework of the algorithm is 55 represented inFig. 1. 56 Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038     U   N   C   O   R   R   E   C    T   E   DP   R   O   O   F PHYSA: 11782 ARTICLE IN PRESS E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx 3 Fig. 1. A block diagram of the mode with the general framework, and the change of states (i-xii) for the subroutines. The input data are the following cellular automata (CA) parameters: 1 (1) The spatial parameters: lattice size L ; necrosis threshold fraction f  of lattice size; 2 (2) The temporal parameters: the length of the time series t  final and the initial time of therapy infusion t  ap ; 3 (3) The probabilities of: the initial proliferation of cancer cells p 0 ; the effect of therapy on normal cells and cancer cells 4 (  p drugn and p drugc ). 5 The output data are the time series of the density of each type of cell and the final configuration of the lattice at any 6 time step. In addition, the time-spatial configurations, controlled by a package denominated g2 [33] whose commands are 7 inserted into the computer program in FORTRAN, are generated in ‘real time’. This package may be used in C, PYTHON and 8 PERL too. 9 The following is a description of each subroutine based on the local rules. The change of states (i)–(xii) associated ∧   with 10 the subroutines described below, are also indicated inFig. 1. 11 (a) Normal Cell (NoC) - a random number y is compared ∧ with p drugn . If  y < p drugn , the growth potential P  noc is checked: if  12 P  noc = 0, the cell dies and the site becomes empty (i); otherwise ( P  noc = 1), it remains occupied by a normal cell but 13 P  noc = 0. If  y ≥ p drugn , t  > t  ap and if there is at least one neighboring cancer cell, then the normal cell is ’dislocated’, 14 P  noc = 0 and the site becomes empty (ii); otherwise it remains a normal cell (iii). 15 (b) Cancer Cell (CC) - if all of its neighbors are cancer cells and its growth potential reaches a fraction f  of lattice size L , the 16 cancer cell becomes necrotic (iv). Otherwise, a number y is randomly chosen. If  y < p drugc and t  > t  ap , the potential P  cc  17 is checked: if  P  cc  > 0, it is reduced by a unit and the cell remains a cancer cell (v); otherwise the cell dies and the site 18 becomes empty (vi). Finally, if  y ≥  p drugc , the cell remains a cancer cell; however, its growth potential P  cc  increases by a 19 unit. 20 (c) NecroticCell(NeC) -ifatleastoneofitsneighborsisneitheracancercellnoranecroticcell,itiseliminatedandthesite 21 becomes empty (vii); otherwise, it continues necrotic (viii). 22 (d) Empty Site (ES) - if there are cancer and normal cells in its neighborhood, the local battle between cancer cells and 23 normal cells is such that if the sum of the potential growth of its neighboring cancer cells is greater or equal to 24 the sum of the potential growth of its normal cell neighbors, then the empty site is occupied by a cancer cell that 25 diffuses from one of the randomly chosen neighbors (ix); otherwise, it is occupied by one of the randomly chosen 26 normal cells that were previously dislocated (x). If there are only normal cells in its neighborhood, it becomes a normal 27 cell through a process of regeneration (xi). Finally, if none of its neighbors ∧ iscancer cells or normal cells, it remains 28 empty (xii). 29 3. Results: Simulated time series and time-spatial patterns 30 Computational simulations of the model were performed in order to analyze two classes of behavior: cases in which no 31 treatment was given and treatment cases. In each subsection, the time series of the simulated tumor as well as time-spatial 32 patterns are shown. 33 Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038     U   N   C   O   R   R   E   C    T   E   DP   R   O   O   F PHYSA: 11782 ARTICLE IN PRESS 4 E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx Fig. 2. Nonnecrotic tumor: The average of simulated time series (black color) with error bars (grey color) of the density of: (a) normal cells, (b) cancercells. We consider M  samples = 100 and the following parameter values: L = 225, p 0 = 0 . 9, f  = 0 . 4, p drugn =  p drugc = t  ap = 0 . 0. Fig.3. Non-necroticcase: assuming M  samples = 100andthe sameparametersvaluesof Fig.2,the averageofsimulatedtime series(blackcolor)with error bars (grey color) of: (a) the number of cancer cells; (b) the sum of growth potential of cancer cells. The Gompertzian fitting (triangle symbol) is applied to(a) those in which parameters of time series are α 0 = 4 . 050 ± 0 . 043, β = ( 1 . 818 ± 0 . 011 ) × 10 − 2 and n 0 = ( 1 . 336 × 0 . 058 ) × 10 2 . 3.1. No treatment case 1 In the case of no treatment, the following parameters are considered: p drugn = p drugc = t  ap = 0. The average of the 2 different samples is considered, corresponding to different seeds of random numbers. They do not differ very much ∧   from 3 eachother(theerrorbarsareverysmall).Forfixedvaluesof  L , t  final and  p 0 ,butdifferentvaluesof   f  ,inFigs.2and4,thetime 4 series of the density of cells for nonnecrotic and necrotic tumors, respectively, are shown. In both cases, the cell densities 5 reach saturated values due to the effects of the competition between normal cells and cancer cells, and the time-dependent 6 mitotic probability. ComparingFigs. 2(b) and4(b), the stationary value of cancer cell density is clearly greater in necrotic 7 tumorsthaninnonnecroticones.Thisisaconsequenceofthefactthat   p mitot ( t  ) assumessmallervaluesinnecrotictumors 8 compared to nonnecrotic ones because it does not depend on the density of necrotic cells. 9 Figs. 3(b) and5(b) show that the sum of potential growth of cancer cells, both in the necrotic and non-necrotic cases, 10 reproduces the decelerating process of tumor growth. 11 Figs. 3(a) and5(a) show that the tumor growth obeys the Gompertzian function both in nonnecrotic and necrotic cases. 12 This behavior is observed with respect to the number of cancer cells for a range of values of  p 0 . In Gompertzian growth [4], 13 the specific growth rate of the number of cancer cells decreases logarithmically: 14 1 n cc  d n cc  d t  = α 0 − β log  n cc  n 0  (2) 15 where 16 (a) n 0 is the initial population of cancer cells; 17 (b) α 0 is the specific growth rate of  n 0 cells at t  = 0; 18 Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038     U   N   C   O   R   R   E   C    T   E   DP   R   O   O   F PHYSA: 11782 ARTICLE IN PRESS E.A. Reis et al. / Physica A xx (xxxx) xxx–xxx 5 Fig. 4. Necrotic case: The average of simulated time series (black-color) with error bars (grey color) of density of (a) normal cells, (b) cancer cells. Weconsider M  samples = 100 and the following parameter values: L = 225, p 0 = 0 . 9, f  = 0 . 25, p drugn =  p drugc = t  ap = 0 . 0. Fig. 5. Assuming M  samples = 100 and the same parameters values of Fig. 4,the average of simulated time series (black color) with error bars (grey color) of:(a)thenumberofcancercells;(b)thesumofgrowthpotentialofcancercells.TheGompertzianfitting(trianglesymbol)isappliedon(a)thoseinwhichparameters of time series are α 0 = 3 . 827 ± 0 . 039, β = ( 1 . 601 ± 0 . 010 ) × 10 − 2 and n 0 = ( 1 . 870 ± 0 . 074 ) × 10 2 . (c) β measureshowrapidlythecurvedepartsfromasingularexponentialandcurvesover,assumingitscharacteristicshape. 1 The solution of (2)is 2 n cc  ( t  ) = n 0 exp  α 0 β [ 1 − exp ( − β t  ) ]  . (3) 3 Itisevidentthatthestationaryvalueof  n cc  is n cc  ∞ = n 0 exp (α 0 /β) .AccordingtotheGompertzianfittingrepresentedby 4 Eq.(3),the results of the simulations shown inFigs. 3(a) and5(a) correspond respectively to the following parameters: 5 (I) The nonnecrotic tumor: α 0 = 4 . 050 ± 0 . 043, β = ( 1 . 818 ± 0 . 011 ) × 10 − 2 and n 0 = ( 1 . 336 × 0 . 058 ) × 10 2 6 (II) The necrotic tumor: α 0 = 3 . 827 ± 0 . 039, β = ( 1 . 601 ± 0 . 010 ) × 10 − 2 and n 0 = ( 1 . 870 ± 0 . 074 ) × 10 2 . 7 Comparing the above parameters of (I) and (II), the behavior of nonnecrotic and necrotic tumors was found to be very 8 similar. It was also found that the number of cancer cells obeys the Gompertzian fitting. 9 An important validation of our model is shown by comparing the Gompertzian fitting parameters of simulated tumors 10 withthecorrespondingparametersofactualtumors[6,7].Forinstance,inthecaseofthetesticulartumorsshowninTable3 11 of Ref. [6], the β values are in the range of  [ 0 . 005 ; 0 . 016 ] day − 1 . If we consider the time step of our simulations to be one 12 day, our simulated β value is within the above range for necrotic simulated tumor, and very close to the maximum actual 13 value of  β for non-necrotic simulated tumor. 14 In relation to the parameters α 0 and n 0 , the simulated values are not comparable ∧   withthe actual values, since no 15 information on the initial size of the tumor was included in our model. Both α 0 and n 0 are strongly dependent on that 16 information. 17 Please cite this article in press as: E.A. Reis, et al., A cellular automata model for avascular solid tumor growth under the effect of therapy, Physica A(2008), doi:10.1016/j.physa.2008.11.038
Search
Similar documents
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks