Comparing System Dynamics and Agent-Based Simulation for Tumour Growth and its Interactions with Effector Cells

Comparing System Dyna m ics and Agent-Based Simulation f or T umour Gr owth and its Interactions with Effector Cell s Grazziela P . Figueredo, U we Aickelin IMA Research Gr oup, Computer Science School, Nottingham University , W ollaton Road, Nottingham, NG8 1BB UK gzf@cs.no tt.ac.uk, uxa@cs.not t.ac.uk Keywords: system dynamics simulation, agent-based sim - ulation, immune sy ste m simulation, co mparison of system dynamics and agent-based si mulation. Abstract There is little research concern ing comp arisons and com- bination o f System D y nami cs Simul ation (SDS) an d Agent Based Simul ation (ABS). AB S is a p aradigm used in many lev els o f abstraction, including th ose lev els covered by SDS. W e be lieve t hat the establishment of fr amew orks for the choice between these t w o si mulation ap proaches would con- tribute to the simulation re search. Hence , our work aims for the establis hment of direct ions for th e c hoice betwe en SDS and ABS approache s for immune s y stem-rel ated probl ems. Previously , w e co mpared the u se o f ABS an d SDS for mod- elling agent s’ behaviour i n an environment with no movement or int eractions between t hese agent s. W e conclud ed t hat for these t y pes of agents it is preferable to use SDS, as it takes up less computational resou rces and prod uces t he s ame results as t hose o btained by t he AB S mo del. In or der to m ov e this resear ch forward , our next resear ch q uestion is: if we i ntro- duce inte ractions betw een these agents will SDS still b e the most app ropriate paradigm to be u sed? T o answe r t his qu es- tion for immune s y stem s imulation proble ms, we will use, as case st udies, models i n v olving i nteraction s bet ween tum our cells and immune e f f ector cells. Experime nts show that there are cas es where SDS an d ABS can not b e used interchange- ably , and the refore, t heir comparison is not straightforward. 1. INTR ODUCTION The current scenario in the simulation fi eld presents paradigms that allow us to build simul ation models for var - ious domains. Some of t he important simulation approaches are Sy st em Dy nami cs Simulation (SDS), Agent-Based Simu- lation ( ABS), Di scr ete E vent Simul ation (DES) and Dy namic Systems (DS)[ 3]. N e w research als o combines th ese meth- ods and d efines framew orks for the usage of ea ch par adigm. There i s already work co mparin g SDS/D ES, DES/ABS as well as their combin ations. Howe ver , there is fe w research on the com parison and combin at ion of SDS an d ABS [8]. Hence, our research aims a t establishing a frame work for the dev elopment of simulations inv o lving the choice b etween SDS and ABS approaches and their combination for immune system-rel ated problems . ABS is a paradigm used in many l e vels of abstraction, in- cluding those l ev el s covered b y SDS. As th ere is this inte r - section , som e range o f simulation pr ob lem s can be s olved b y either SDS or ABS. In prev ious work [1], we compared the use of ABS an d SDS for modelli ng static agents’ behaviour in a n immun e s y stem ageing problem. By static we mean that ther e is no movement or interactions between the agent s. W e conclud ed th at for these t ypes of agents, it is preferable to use SDS instead of ABS. Whe n co ntrasting the resul ts of both simulatio n appr oaches, we see th at SDS is less complex and takes up less computational resourc es, producing t he same re- sults as those obtained by t he ABS model . T o advance thi s stud y , our next inquiri ng is: Once we have est ab lished t hat SDS i s more suitabl e for st at ic agents t han ABS, i f we i ntroduce interactions b etween th ese agen ts will SDS still be the most appropriate paradigm to be used? T o answer this question for immune sy stem simulation problems , we u se models as case studi es, which include i n- teractions between t umour cells a nd i mmune effector cells . Ef fect or cells are responsibl e for eliminatin g tum our cells in the organism. Our goal is to d etermine w hich s cenarios for immun e system simul ations inside the SDS/ABS int ersection would benefit fro m SDS res ources and th ose that are more suitable for ABS. For SDS we need to est ablish math ematical equations that determine th e flows. Therefore w e us e the models r e viewe d in [ 4]. The a uthors of t his study expl ored existing sp atially homogen eous mechanistic mathemati cal model s describing the interactions bet ween a mali gnant tumor and the immune system. They begi n with the simpl es t (single equation) mod- els fo r tu mor gro wth and proceed t o co nsider great er im- munolo g ica l detai l (and correspondin gly m ore equ ations) in st ep s. For our simul ation, we intend to build SDS and ABS for two of the most important m athematical models described in [ 4]. W e intend to use t he mat hematical model als o as ba- sis for the A BS. The i dea is to c heck if the re sults would be simil ar and if we can us e SDS and ABS fo r our case studies interchan geably . This work is o r ganized as foll ow s. In Section 2, there is the r elated work rel ev an t t o our stu dies. Section 3 presents the m athematical models used for our simulations, as well as the s imulations we ca rried o ut and t heir results. Fin ally , in Section 4 we draw the conclusions and present fu tur e work. 2. RELA T ED WORK The theoretical work pr esented by [ 3] compares ABS and SD conceptu ally , and discusses the potential synergy bet ween them in order to solve probl ems of t eaching decision-making processes. The authors ex plore the conceptual frameworks for ABS an d SD t o model group lea rn ing. They show the con- ceptual differences between these two paradigms a nd propose their use in a complement ary way . They outs tand t he l ack of knowledge in multi-paradi gm si mulation i n v olving SD and ABS. In [6] and [8], the authors p resent a cross -study of SD and ABS. They d efine their features an d ch arac teristi cs and con- trast t he two methods. Moreov er , they al so present ideas of how t o int egra te both approaches. As a conti nuation of this work, i n [7] they present an approach to integrate t he SD and ABS tech niques for sup ply chain management problems. They pr esent s ome preliminar y results , which were not t he same as the SD simulation alone. Theref ore, they propos e new tests as fut ure work. In th eir case st udy , t hey were no t able to reach the same results with both simulations. In [9], the authors sh ow t he ap plication of both S D and ABS to si mulate non-equili br ium ligand-rece ptor dy namics over a broad r ange of co ncentrations. They conc luded that both appro aches are po werful tools and are also complemen- tary . In th eir case s tudy , they di d not i ndicate a preferr ed paradigm, altho ugh intuitively SD is an obvious choice when stud y ing s y stems at a high lev el of aggregation and abstrac- tion. O n the oth er hand , SD is not capabl e of si mulating re- ceptors and m olec ule s and their i ndi vidual inte ract io ns, which can be done with ABS. Rahmandad and Sterman [5] c ompare the d ynamics of a stochasti c ABS m odel wit h those of the analogous determin- istic compartment differential equation model for conta gious diseas e spread. The authors conv ert t he ABS i nto a di f fer - ential eq uation model and examine the impact o f individual heterogenei ty and different net work topologie s. T he deter- ministi c mode l yields a single trajectory for each param ete r set, while stochastic models y ield a dist ribu tion of ou tcomes. Moreover , the differential equati on model and ABS d ynam- ics dif fer for severa l metrics re lev ant to public health. The re- sponse of the models to policies can also d iffer w hen the base case b eha viour i s similar . Under so me conditions, howev er , the dif ferences in mean s are small, comp ared t o variability caused by stoch as ti c ev ents, param eter uncertainty and model boundar y . In our previous w ork [1], we compared SDS a nd A BS for a naive T cell output model . In that study , we conclu ded that for that case stud y SD S is m ore suita ble. W e had a scenari o where the agents had no int eractions and SDS and ABS prod uced simil ar outputs . Therefore, we decided that, in this cas e, it i s prefer able t o choose th e SDS, as it takes up l ess computa- tion al resources. In order to conti n ue the inv estigation, whi ch compares th ese two simulation approaches, we will add inter - actions betwe en agents and comp are the results. The descrip- tion of the pr oblem, as well as t he mat hematical equations can be se en in the nex t section. 3. MA T HEMA TICAL MODELS In this secti on, we present mathe matical models used as basis for ou r s imulati ons. The simplest ones inv o lve only one equation and d efines m athematical r ules for tumour grow th (Sectio n 3 .1.). The se cond mathematic al model addres ses the intera ct ions betw een tumour cells and immune effector cells. This is shown in Section 3.2.. 3.1. One-Equation Models: T umour Gro wth In t his section, we present the s implest mathematical m od- els. They hav e onl y one equ ation that d escribes how tumours grow . There are three models of tumour gro wth from [4] con- sidere d in this stud y : th e logisti c model, the von Bertalanffy model a nd the Gompertz model. Accordin g to [4 ], th e mo st general equation des cr ibin g the dynamics of tumor growth can be written as : d T dt = T f ( T ), (1) where: • T i s the tumour ce ll popul at ion at time t , • T (0) > 0, • f ( T ) specifies the density depend ence in p roliferation and death o f the tumour cell s. Th e densit y dep endence factor can be written as : f ( T ) = p ( T ) − d ( T ), (2) where: • p ( T ) defines tumour cells proliferation, • d ( T ) define tumou r cells death. The expressions for p ( T ) an d d ( T ) are generall y d efined by power laws: p ( T ) = aT , (3) d ( T ) = bT , (4) For our experiments, we defined th e values for and using the three well established models: Logistic Model: = 0 and = 1 ( a , b > 0 and b < a for growth), will be use d for our next simulatio n set o f ex periments using a two-equatio n model. von Bertalanffy Mo de l: = a for growth), 1 and = 0 ( a , b > 0 and b < As the outcomes for ABS are stoch astic, we ran each sim - ulation for 5 0 times and t he me an simul ation output is p re- sented. For both s imul ations, we defined th e in itial v alues for Gompertz Mode l: p ( T ) = a and d ( x ) = b ln( T ) ( a , b > 0 and e b > a for g rowth). 3.1.1. SDS for the One-Equation M odel W e hav e implement ed the one-equation m odels using SDS. Figure 1 sh ow s the stock and flow diagram us ed fo r modelling the mathematical equations. From the SDS pers pectiv e, t he amount of t umour cells i s a stock that will be modified by proliferati on and death flo ws. Figure 1. SDS diagram for the general one-equatio n math e- matical model. 3.1.2. ABS for the One-Equation M odel Figure 2. On e equation agent. W e hav e als o impl emented the on e-equation model using ABS. In this case we ha ve a tumour cell agent that will repro- duce and die, as it can be seen in Figure 2. 3.1.3. Experiments W e car ried out t wo experiments to comp are the SDS and ABS simul at ion o utputs. F or the first experiment, we hav e est ab lished a v ariable c , which rep resents the ratio b etween a and b . The pur pose of c is to obs erve t he impact of a and b on th e tumour growth curve. Therefore, we set c as 5, 2.5, 1.7 and 1.25. In t he second experiment, we d efined a = 1.636 and b ∈ { 0 . 00 2 , 0.005 } . T hese v alues we re d etermined in [2] and they tumour cells eq ual to 1. 3.1.4. Results For t he fi rst experiment, as is s ho wn in Figure 3 (middle and bottom ), the ou tputs fo r both si mulation ap proaches are simil ar for the v on Bertalanffy and Gompertz models. For the logistics model, on the other hand , the results of th e ABS did not match the SDS ones (Figure 3 top). This can be explained by the stochastic and individual behav io ur of the agents i n the ABS model . In the experiments, t here a re f ew agen ts o n the lo gistics model and m ost of th em die befo re t hey reprod uce. In this ABS mo del, t he reproducti on rat e i s given b y p ( T ) = a , be- cause al ph a = 0 for t he logistics model. In t he case where a = 1 and b = 0.2, fo r example, when the number of agents becomes bigger than 5, d ( T ) get s greater than p ( T ). As for ABS the deat h rate is defined for t he agents in di vidually , all the ne wborn tumour cells after 5 agents i n the system will hav e de ath rates bigger t han reprodu ction rates. Hence, the agents popul ation di sappears over the first days. This differ - ence on the SD S and ABS r esults lead us to conclude t hat ther e are cases using stat ic agents wh ere SDS and ABS o ut- puts w ill not be the same. Ne vertheless, if we increas e the difference between the pa- ramet ers a a nd b , as is shown in Fi gure 4, we av oid th e pre- mature d eath of the tumour agents and therefore the ABS and SDS r esults become similar again. From what we have found on t he li terature, the Logisti cs model is o ne of th e most used for average tum ours whereas the von Bert alanf fy and Gom- pert z models are us ed for more aggressive t umours. There- fore, a s we increase in th e difference between a and b , the Gompertz and von Bertalanffy model s growth will p resent a consid erable incre ase on t he proliferat ion of tumour cells , as illustrat es Figure 5. Therefore, we could not carry o ut exp erim en t two u sing ABS fo r Gomp ertz an d v on Ber talanffy mo dels . Fro m the SDS r esults, we can s ee th at the n umber of tu mour cells by- passes the magnitu de o f 10 64 in t he Gompertz m odel (Fig- ure 3). T o run the same experi ment with ABS we w ould need more comp utational resou rces and it w ould take up more pro- cessin g time. In our case, as each agen t t ak es u p around 1 megabyte of m emory , we would n eed a memor y capacit y of 10 64 megabytes. Theref ore, in this case it is prefe rable to run the simulation using SDS, ev en tho ugh suc h big nu mber of tumour cells also seems to be unrealistic in tumour biology . c = c = c = c = Tumour g rowth using the Logisti cs model for the SDS 6 Tumour g rowth using the Logistics m odel for the ABS 6 4 4 2 2 0 0 20 40 60 80 100 Days Tumour gr owt h u sin g the von Bertalanffy model for the SDS 0 0 20 40 60 80 100 Day s Tumour gr owt h u sin g the von Bertalanffy model for the ABS 100 100 50 50 0 150 0 20 40 60 80 100 Days Tumour gr owt h u si ng the Gompert z model for the SDS 0 150 0 20 40 60 80 100 Day s Tumour gr owt h u si ng the Gompert z model for the ABS 100 100 50 50 0 0 20 40 60 80 100 Days 0 0 20 40 60 80 100 Day s Figure 3. Res ults for the one-equation model usin g SDS an d ABS. Tumour gr owth using the Logis tics m odel for the SDS Tumour gr owth using the Logis tics m odel for the ABS 800 600 a = 1.636, b = 0.002 a = 1.636, b = 0.004 800 600 400 400 200 200 0 0 10 20 30 40 50 60 70 80 90 100 Days 0 0 10 20 30 40 50 60 70 80 90 100 Days Figure 4. Res ults for the second experiment of one-equati on Logistics model using SDS an d ABS. 3.2. T wo-Equation Models: In teraction Be- tween T umour Cells and Generic Ef fector d E = p E ( T , E ) − d E ( T , E ) − a E ( E ) + ( T ), (6) Cells T o add complexity t o our model, we are going t o consider tumour cells growth together with th eir interactions w ith gen- eral i mmune effector cells . In t his s tage we are n ot y et con - siderin g s pecific t ypes of i mmune cell s. Effector cells are re- sponsible for kil ling t he tumour cel ls. T hey proli ferate and die per apoptosis, which is a programm ed cellular death . In the mod els, cancer treatment is also considered. The interactio ns between t umour cells and i mmune effec- tor cells can be defined by the equations: d T d t = T f ( T ) − d T ( T , E ) (5) d t where: • T i s the number of tu mour cells, • E i s the number of ef fector cells, • f ( T ) is t he gro wth of tumour cells, • d T ( T , E ) is the number o f t umour cells kill ed b y ef fect or cells, • p E ( T , E ) is the proliferati on of ef fector cells, • a E ( E ) i s the death (apoptosis) of ef fector cells. Scenario 1 0.002 0.1908 0.318 2 0.004 0.318 3 0.002 0.3743 0.1181 4 0.002 0.3743 4 x 10 15 10 Tumour g rowth using t he von Bertalan ffy model for the SDS 5 0 0 20 40 60 80 100 Days Tumour growth using the Gompertz model for th e SDS 70 60 50 40 30 20 10 0 0 20 40 60 80 100 Days Figure 5. Results for the second experim en t of one -e quat io n Gompertz and V on Bertalanf fy models usi ng SDS. • ( T ) is the treatment or influx of cells. For our models, we used the Kuznetsov model [2]: f ( T ) = a (1 − bT ), (7) d T ( T , E ) = nT E , (8) pT E p E ( E , T ) = g + T , (9) d E ( E , T ) = m T E , (1 0) a E ( E ) = d E , (1 1) ( t ) = s . (12) As it can be s een, t he Logistic model was adopted for tu- mour growth. It seems to be the most common model of tu- mour grow th used in the mathemati cal m odels in v olving can- cer and the immune sy stem . The v a lues f or the parameters on the equations can be seen in T able 1. W e got these v alues fro m [4]. In the first three s cenarios we cons ider can cer treatm ent. The fourt h case does not consider any t reatment. 3.2.1. SDS for the T wo-Equation Mode l W e have conv er ted t he mathemati cal mo del into a SDS model. Fi gure 6 shows the stock and flow diagram we hav e defined. W e consider two stocks, the tu mour cells and t he immu n e effector cells. The tumour cell stock is changed by p rolifera- tion and natural death (as defined in the one-equation model in Secti on 2.1); and d eath caused by the immune ef fe ctor T able 1. Simulation p arameters for different scenarios . For the othe r paramete rs, the va lues are t he s ame in all experi- ments, i.e, a = 1.636, g = 20.19, m = 0.00311 , n = 1 a nd p = 1 . 13 1. cells. The i mmune cells stock is changed by deat h, apo pto- sis, proli feration and injection of new cells as treat ment. The numbe r of tumour cells i n the organism also sti mulates t h e proliferatio n of immune cells. Figure 6. SDS diagram for the two-equation math ematical model. 3.2.2. ABS for the T wo-Equatio n Mo de l For the ABS mo del, we defi ned two agents that will in - teract with each other: t he tumour cell agent and the effector cell ag ent. Figure 7 show s the s tate ch arts for the ef fector ce lls and the tumour cells. Th e state ch art for t he ef fector cells (left hand side of Figure 7) has t w o states. Eit her the cell is aliv e and able to reproduce or is dead. Ef fector cells can die by nat- ural means or by damage. T he t umour cells st ate chart also has two st ates. T umour cells can repr oduce, die with age or die killed by effector cells. Th e rates defi ned in the t ransitions are the sam e o f the mathematical m odel. 3.2.3. Experiments As we me ntioned before, we carried out four experiments for the two-equati on model. The param eter var iation for the experiments i s shown i n T able 1 . In t he four scenarios, it is considered differences in t he death rat e of tum our cells (de- fined by the parameter b ), effector cells apoptosis rate (de- fined b y the parameter d ) and treatment (paramete r s ). Similar t o the one-equatio n model, we r an the simulation Figure 7 . ABS d iagram fo r t he two-equation mathematical model. On t he l eft hand side we h a ve the stat e chart for the effector cell agents and o n the right hand si de we hav e t he st ate c hart for the tumour cell ag ent s. for th e ABS 50 times and d isplay t he mean values as the re- sults. 3.2.4. Results The result s co mparing SDS an d ABS for the f our experi- ments can be seen in Figure 8 (first and second columns). For th e first scenario, th e behaviour of t he tumour cell s is very si milar for the SDS a nd ABS resu lts. Howe ver , when we ran th e W ilcoxon test for the tumour cells outcome, it re- jected the simila rity h y pothe sis for b oth outcomes, as shown in T able 2. This m ight be due to the fact th at the tumour cells for t he SDS m odel decrease in an asymptotic way to- wards to zero, while ABS beh a viour is discrete and hence, it reaches ze ro. The numb er of effector cel ls f or b oth sim ula- tions follows the same pattern, alt hough the n umbers are not the same. The variances in the ABS curve were expected due to its stochastic character . The results f or th e s econd scenario s eem to be sim ilar for effector cells, althou gh the W ilcoxon test rejects this hy poth- esis. The re sults for the tumour cells are not the same. For scenarios 3 and 4 , the results are c ompletel y d iffer - ent for both simu lation approaches. Moreover , when we look at the t umour cells curve, the d if ferences are even more evi- dent. In scenario 3 u sing SDS, tumour cells decrease a s effec- tor cells increase, follow in g a p redator-prey trend cur v e. This output is what wo uld be expected by th e mathematical model . On t he other hand, for the AB S, t he number of effector cells decreases until a v alue close t o zero while t he t umour cells numbe rs vary in time differently from th e SDS results. The predator-prey beha viour is not observed in this ex periment. In scenari o 4, althou gh effec tor ce lls se em to deca y i n a simil ar trend for both app roaches, the results for tumour cells are compl ete ly diffe rent. In the SDS sim ulat io n, the num- bers of tum our cells reac h a v alue clos e t o zero aft er twenty days and then i ncreases again . For t he ABS simulation , on the other hand , t umour cell s reach zero a nd n e ver increase again. It h appens because SDS deals with continuou s st ocks and ABS has a discrete number of agents. The results i n these experiments show t hat th ere are sim- ulation cases where SDS and ABS deriv ed from the same math ematical model do not hav e t he same outp ut. Therefore, it is not possible to co mpare whic h appro ach would b e more suitable for these cases. One alternative would be th e development of an A BS so- lution, w hich is not b ased on th e rat es defin ed in th e math- ematical model. Howe v er , it seems t hat for each output (or parameter ch ange o n the mathemati cal m odel), there sh ould be a different ABS impl ement ation. For example, if we determi ne that the number of tum our cells should always be b igger t han zero in the ABS, fo r t he second an d fo urth scenarios t he outp uts become closer to the SDS, as shown in Figure 8. On the oth er hand, this constraint also changes the first s cenario results, wh ich seemed satisfac- tor y before . For scenario 4 , although t he outcome with t his fix do n ot seem ver y similar on the b eg inning of the simulation, the stead y state has closer v alues. T o achiev e simil ar results for scena rio 3, w e also had t o determine t hat th e number of effector cells s hould be bigger than zer o. The results are s ho wn in line 4 of Figure 8. For scenari o 3 th e fix did not work p erfectl y . It s eems that onl y the stead y st ate of the simul ation has closer results . W e also tried to randomly add some effector cells over time, bu t t he results did not looked similar as well. 4. CONCLUSIONS The current scenario in t he simulation f ield presents paradigms that allow us to bui ld simul ation models for var - ious domains. New resear ch comp ares s imulation methods and defi nes frame works for the u sage of each p aradigm. How- ev er , ther e is few research on the comparison and combin a- tion of SDS and ABS [ 8]. W e ai m at contribu ting to this area by stu dy in g i mmune s y stem s imulation probl ems. Therefore, in this stud y , we use d case studies which includ e interactions betwe en tumour cells and immune ef fector cells. Our goal was to det ermine which scenarios for i mmune system si mu- lations i nside t he SDS/ ABS int ersection would benefit from SDS resou rces and those that are more suitable for ABS. The models we used were rev iewed i n [4]. W e b egan with the s implest (single equation) m odels for tumor grow th and proceed t o consider t ow- equatio n models in v olving effector and tumo u r cells. W e u sed mathe matical models as basis for both ABS an d SDS. The idea was t o ch eck i f the re sults would be similar and i f we can use SDS and ABS for our case stu dies interchan geably . W e carr ied out two experiments to com pare the outputs for the o ne-equation model. F o r the fir st experiment , the out- puts for bo th simulation appr oaches are similar for two cases. There was an example, t hough, where t he results of the ABS did not m atch the SDS’ s. This i s explained by the s tochastic and in d ividual behav iour of the agents in the ABS model . T o add complexity to our tests, we considered tumour cells Implementa tion Cells ABS T umour Effector ABS - F ix 1 T umour Effector 0.0103 0.4441 ABS - F ix 2 T umour Effector T able 2 . W ilcoxon test for tumour cells and ef f ector cells. Compari ng th e results between S DS and ABS. 100 80 60 40 20 0 Scenario 1 using SDS 10 Tumour cells 8 Effect or cells 6 4 2 0 100 50 0 Scenario 1 using ABS 10 200 5 100 0 0 Scenario 1 using ABS − fix 1 10 5 0 100 50 0 Scenario 1 using ABS − f ix 2 10 5 0 0 50 100 Scenario 2 using SDS 0 50 100 Scenario 2 using ABS 0 50 100 Scenario 2 using ABS − fix 1 0 50 100 Scenario 2 using ABS − fix 2 10 10 10 10 200 8 200 8 200 8 200 8 150 6 150 6 150 6 150 6 100 4 100 4 100 4 100 4 50 2 50 2 50 2 50 2 0 0 0 0 0 0 0 0 0 50 100 0 50 100 0 50 100 0 50 100 Scenario 3 using SDS Scenario 3 us ing ABS Scenario 3 us ing ABS − fix 1 Scenario 3 using ABS − fix 2 100 10 100 10 600 10 600 10 80 8 80 8 480 8 480 8 60 6 60 6 360 6 360 6 40 4 40 4 240 4 240 4 20 2 20 2 120 2 120 2 0 0 0 0 0 0 0 0 0 50 100 0 50 100 0 50 100 0 50 100 Scenario 4 using SDS Scenario 4 us ing ABS Scenario 4 using ABS − fix 1 Scenario 4 u sing A BS − fix 2 600 10 600 10 600 10 600 10 480 8 480 8 480 8 480 8 360 6 360 6 360 6 360 6 240 4 240 4 240 4 240 4 120 2 120 2 120 2 120 2 0 0 0 0 0 0 0 0 0 50 100 0 50 100 0 50 100 0 50 100 Days Days Days Days Figure 8. Results for t he two -equation ma thematical model using SDS an d A BS. On the first columns we hav e the SDS r esults for the fo ur scenarios. The second column has the ABS res ults. The third and fourth columns sho ws th e fixes we tried to make the outco mes simil ar . Fix 1 adds t he constr aint t umour cel l s > 0. Fix 2 ad ds t he c onstraint t umour cel l s > 0 an d e f f ect or cel l s > 0. For each graph we hav e the results for tumour cells (continu o us line) and effector cells ( dotted lines). T he s cale for the tumour cells results is on the left y axis of each graph while the scale for effect or cells is in the right y axis. growth toget her with their inter actions with general imm une effector cells. W e d efined four scena rios and, for only on e of them, t he result s were similar using the mathemati cal m odel. The d if ferences in t he output are due to th e f act that effector cells numbers c hange continuousl y o n t he SDS, whil e for the ABS, they change in a discret e pat tern. The results in these experiments s how th at t here are sim- ulation cases where SDS and ABS der i ved from the same math ematical model do not hav e t he same outp ut. Therefore, it is not possibl e to comp are which approach would b e more suitable for these cas es. One alternative would be th e development of an A BS so- lution, w hich is not b ased on th e rat es defin ed in th e math- ematical model. Howe v er , it seems t hat for each output (or parameter ch ange o n the mathemati cal m odel), there sh ould be a different ABS impl ement ation. As future work w e intend t o work wi th models wi th th ree an four eq uations and compare the results of SDS and ABS without u sing the mathematic al equation as baselin e. REFERENC ES [1] Grazz ie la P . Figueredo and Uwe Aic kelin. 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