Volume 14 Supplement 6
Selected articles from the 10th International Conference on Artificial Immune Systems (ICARIS)
Investigating mathematical models of immunointeractions with earlystage cancer under an agentbased modelling perspective
 Grazziela P Figueredo^{1}Email author,
 PeerOlaf Siebers^{1} and
 Uwe Aickelin^{1}
DOI: 10.1186/1471210514S6S6
© Figueredo et al.; licensee BioMed Central Ltd. 2012
Published: 17 April 2013
Abstract
Many advances in research regarding immunointeractions with cancer were developed with the help of ordinary differential equation (ODE) models. These models, however, are not effectively capable of representing problems involving individual localisation, memory and emerging properties, which are common characteristics of cells and molecules of the immune system. Agentbased modelling and simulation is an alternative paradigm to ODE models that overcomes these limitations. In this paper we investigate the potential contribution of agentbased modelling and simulation when compared to ODE modelling and simulation. We seek answers to the following questions: Is it possible to obtain an equivalent agentbased model from the ODE formulation? Do the outcomes differ? Are there any benefits of using one method compared to the other? To answer these questions, we have considered three case studies using established mathematical models of immune interactions with earlystage cancer. These case studies were reconceptualised under an agentbased perspective and the simulation results were then compared with those from the ODE models. Our results show that it is possible to obtain equivalent agentbased models (i.e. implementing the same mechanisms); the simulation output of both types of models however might differ depending on the attributes of the system to be modelled. In some cases, additional insight from using agentbased modelling was obtained. Overall, we can confirm that agentbased modelling is a useful addition to the tool set of immunologists, as it has extra features that allow for simulations with characteristics that are closer to the biological phenomena.
Introduction
Advances in cancer immunology have been facilitated by the joint work of immunologists and mathematicians [1–3]. Some of the knowledge regarding interactions between the immune system and tumours is a result of using mathematical models. Most existing mathematical models in cancer immunology are based on sets of ordinary differential equations (ODEs) [2]. This approach, however, has limitations pertaining problems involving spatial interactions or emerging properties [4, 5]. In addition, the analysis of ODE models is conducted at a high level of aggregation of the system entities. An alternative to ODE modelling that overcomes these limitations is systems simulation modelling. It is a set of methodologies and applications which mimic the behaviour of a real system [6–8]. Systems simulation modelling has also benefits compared to realworld experimentation in immunology, including time and cost effectiveness due to the resourceintensiveness of the biological environment. Furthermore, in a simulation environment it is possible to systematically generate different scenarios and conduct experiments. In addition, the "whatif" scenarios studied in such an environment do not require ethics approval.
Agentbased modelling and simulation (ABMS) is an objectoriented system modelling and simulation approach that employs autonomous entities that interact with each other [9–11]. The focus during the modelling process is on defining templates for the individual entities (agents) being modelled and establishing possible interactions between these entities. The agent behaviour is described by rules that determine how these entities learn, adapt and interact with each other. The overall system behaviour arises from the agents' individual behaviour and their interactions with other agents and the environment. For cancer immunology, it can amalgamate in vitro data on individual interactions between cells and molecules of the immune system and tumour cells to build an overview of the system as a whole [12]. Few studies, however, apply ABMS to cancer research. Although there are examples showing the success of simulation aiding advances in immunology [13–18], this set of methodologies is still not popular. There are several reasons for this: (1) ABMS is not well known in the immunology research field; (2) although simulation is acknowledged as being a useful tool by immunologists, there is no knowledge of how to use it; and (3) there is not enough trust in the results produced by simulation.
 1.
Is it possible to obtain an equivalent ABMS model based on the mathematical equations from the ODEs (i.e. can we create an object oriented model by reusing ODEs that have been created for modelling behaviour at an aggregate level)?
 2.
Do we get equivalent simulation outputs for both approaches?
 3.
What benefits could we gain by reconceptualizing a mathematical model under an ABMS view?
Case studies
Case studies considered
Case Study  Number of populations  Population size  Complexity 

1) Tumour/Effector  2  5 to 600  Low 
2) Tumour/Efector/IL2  3  10^{4}  Medium 
3) Tumour/Effector/IL2/ TGFβ  4  10^{4}  High 
The first case study considered is based on an ODE model involving interactions between tumour cells and generic effector cells. The second case study adds to the previous model the influence of IL2 cytokine molecules in the immune responses of effector cells towards tumour cells. The final case study comprises an ODE model of interactions between effector cells, tumour cells, and IL2 and TGFβ molecules. For all case studies, the mathematical model as well as the ABMS model are presented, the outcomes are contrasted and the benefits of each approach are assessed. The models differ in terms of complexity of interactions, population sizes and the number of agents involved in the interactions (Table 1).
The remainder of this paper is organized as follows. We start with a literature review of works comparing ODEs and ABMS for different simulation domains. First, we show general work that has been carried out and then we focus on research concerned with the comparison for immunological problems. Finally, we discuss gaps in the literature regarding cancer research. In the methodology section, we introduce our agentbased modelling development process and the methods used for conducting the experimentation. In the following section, we present our case studies, comparison results and discussions. In the last section, we finally draw our overall conclusions and outline future research opportunities.
Related work
In this section we describe the literature concerned with the comparison between ODE and ABMS modelling for different simulation domains. We start our review by showing general work that has been carried out to assess the differences of both approaches. Subsequently, we focus on research concerned with the comparison of the strategies for immunological problems. We found that there is a scarcity of literature comparing the two approaches for immune simulations. Furthermore, to our knowledge there is no research contrasting the approaches for the immune system and cancer interactions.
Over the past years several authors have acknowledged that little work has been done to compare both methods. In one of the pioneer studies in this area, Scholl [19] gives an overview of ODE and ABMS, describes their areas of applicability and discusses the strengths and weaknesses of each approach. The author also tries to identify areas that could benefit from the use of both methodologies in multiparadigm simulations and concludes that there is little literature concerned with the comparison of both methodologies and their cross studies. Pourdehnad et al. [20] compare the two approaches conceptually by discussing the potential synergy between them to solve problems of teaching decisionmaking processes. The authors explore the conceptual frameworks for ODE modelling (using Systems Dynamics (SD)) and ABMS to model group learning and show the differences between the approaches in order to propose their use in a complementary way. They conclude that a lack of knowledge exists in applying multiparadigm simulation that involves ODEs and ABMS. More recently, Stemate et al. [21] also compare these modelling approaches and identify a list of likely opportunities for crossfertilization. The authors see this list as a starting point for other researchers to take such synergistic views further.
Studies on this comparison for Operations Research were also conducted. For example, Schieritz [22] and Scheritz et al. [23] present a crossstudy of SD (which is implemented using ODEs) and ABMS. They define their features and characteristics and contrast the two methods. In addition, they suggest ideas of how to integrate both approaches. Continuing their studies, in [24] the authors then describe an approach to combine ODEs and ABMS for solving supply chain management problems. Their results show that the combined SD/ABMS model does not produce the same outcomes as ODE model alone. To understand why these differences occur, the authors propose new tests as future work.
In an application in health care, Ramandad et al. [25] compare the dynamics of a stochastic ABMS with those of the analogous deterministic compartment differential equation model for contagious disease spread. The authors convert the ABMS into an ODE model and examine the impact of individual heterogeneity and different network topologies. The deterministic model yields a single trajectory for each parameter set, while stochastic models yield a distribution of outcomes. Moreover, the ODE model and ABMS dynamics differ in several metrics relevant to public health. The responses of the models to policies can even differ when the base case behaviour is similar. Under some conditions, however, the differences in means are small, compared to variability caused by stochastic events, parameter uncertainty and model boundary.
An interesting philosophical analysis is conducted by Schieritz [26] analyses two arguments given in literature to explain the superiority of ABMS compared with ODEs: (1) "the inability of ODE models to explain emergent phenomena" and (2) "their flaw of not considering individual diversity". In analysing these arguments, the author considers different concepts involving simulation research in sociology. Moreover, the study identifies the theories of emergence that underlie the ODE and ABMS approaches. The author points out that "the agentbased approach models social phenomena by modelling individuals and interactions on a lower level, which makes it implicitly taking up an individualist position of emergence; ODEs, on the other hand, without explicitly referring to the concept of emergence, has a collectivist viewpoint of emergence, as it tends to model social phenomena on an aggregate system level". As a second part of the study, the author compares ODEs and ABMS for modelling species competing for resources to analyse the effects of evolution on population dynamics. The conclusion is that when individual diversity is considered, it limits the applicability of the ODE model. However, it is shown that "a highly aggregate more ODElike model of an evolutionary process displays similar results to the ABMS". This statement suggests that there is the need to investigate further the capabilities and equivalences of each approach.
Similarly, Lorenz [27] proposes that three aspects be compared and that this helps with the choice between ODE and ABMS: structure, behaviour and emergence. Structure is related to how the model is built. The structure of a model in ODE is static, whereas in ABMS it is dynamic. In ODE, all the elements, individuals and interactions of the simulation are developed in advance. In ABMS, on the other hand, agents are created or destroyed and interactions are defined through the course of the simulation run. The second aspect (behaviour) focuses on the central generators of behaviours in the model. For ODE the behaviour generators are feedback and accumulations, while for ABMS they are micromacromicro feedback and interaction of the systems elements. Both methodologies incorporate feedback. ABMS, however, has feedback in more than one level of modelling. The third aspect lies in their capacity to capture emergence, which differs between the two methodologies. In disagreement with [26] mentioned earlier, the author states that ABMS is capable of capturing emergence, while the onelevel structure of ODE is insufficient in that respect.
In this work we discuss the merits of ODEs and ABMS for problems involving the interactions with the immune system and earlystage cancer that can benefit from either approach. To our knowledge (and as the gap in the literature shows) such a study has not been conducted before. The differences between ODEs and ABMS when applied to classes of problems belonging to different levels of abstraction are well established in the literature [23]. However, we believe there is a range of problems that could benefit from being solved by both approaches. In addition, in many cases such as for example molecular and cellular biology, it is still not possible to use the full potential of ABMS as only the higher level of abstraction of the system is known. Another reason to investigate problems that can interchangeably benefit from both approaches is that, as many realworld scenarios, such as biological systems, constantly gather new information, the corresponding simulations have to be updated frequently to suit new requirements. For some cases, in order to suit these demands, the replacement of the current simulation approach for new developments needs to be considered. Our case study investigation seeks to provide further understanding on these problems and fill some of the gaps existing in simulations for earlystage cancer research.
Methodology
In this section we outline the activities and methods necessary to realise our objectives. We examine case studies of established mathematical models that describe some immune cells and molecules interacting with tumour cells. These case studies were chosen by considering aspects such as the behaviour of the entities of the model, size (and number) of populations involved and the modelling effort. The original mathematical models are built under an agentbased approach and results compared.
ABMS is capable of representing space; however, as we chose mathematical models which do not consider spatial interactions, our corresponding ABMS models do not regard space (distance) and how it would affect the simulation outcomes. The outcome samples obtained by ODEs and ABMS were statistically compared using the Wilcoxon ranksum test to formally establish whether they are statistically different from each other. This test is applied as it is robust when the populations are not normally distributed; this is the case for the samples obtained by the ODEs and ABMS. Other approaches for assessing whether the two samples are statistically different, such as the ttest, could provide inaccurate results as they perform poorly when the samples are nonnormal.
The agentbased model development
The agentbased models were implemented using (AnyLogic™ 6.5 [28]). For the agent design we follow the steps defined in [29]: (1) identify the agents (cells and molecules), (2) define their behaviour (die, kill tumour cells, suffer apoptosis), (3) add them to an environment, and (4) establish connections to finally run the simulations, as further discussed next:
1. Identify the possible agents. For this purpose, we use some characteristics defined in [9]. An agent is: (1) selfcontained, modular, and a uniquely identifiable individual; (2) autonomous and selfdirected; (3) a construct with states that varies over time; and (4) social, having dynamic interactions with other agents that impact its behaviour. By looking at the ODE equations, therefore, the variables that are differentiated over time (their disaggregation) will either be corresponding to agents or states of one agent [29, 30]. The decision whether the stock is an agent or an agent state varies depending on the problem investigated. Based on our case studies, however, we suggest that: (1) these variables preferably become states when they represent accumulations of elements from the same population; or (2) they become agents when they represent accumulations from different populations. For example, if you have an ODE $\frac{dx}{dt}=y$, x should either be an agent or an agent state, depending on the problem context.
2. Identify the behaviour and rules of each agent. In our case, the agent's behaviours will be determined by mathematical equations converted into rules. Each agent has two different types of behaviours: reactive and proactive behaviours. The reactive behaviour occurs when the agents perceive the context in which they operate and react to it appropriately. The proactive behaviour describes the situations when the agent has the initiative to identify and solve an issue in the system.
3. Implement the agents. Based on step 2 we develop the agents. The agents are defined by using state charts diagrams from the unified modelling language (UML) [31]. With state charts it is possible to define and visualize the agents' states, transitions between the states, events that trigger transitions, timing and agent actions [4]. Moreover, at this stage, the behaviours of each agent are implemented using the simulation tool. Most of our transitions occur according to a certain rate. For our implementation, the rate is obtained from the mathematical equations.
4. Build the simulation. After agents are defined, their environment and behaviour previously established should be incorporated in the simulation implementation. Moreover, in this step we include parameters and events that control the agents or the overall simulation.
where:

T is the tumour cell population at time t,

T (0) > 0,

f (T ) specifies the density dependence in proliferation and death of the tumour cells. The density dependence factor can be written as:$f\left(T\right)=p\left(T\right)d\left(T\right)$(2)
where:

p(T ) defines tumour cells proliferation

d(T ) define tumour cells death
Agents' parameters and behaviours for the tumour growth model
Parameters  Reactive behaviour  Proactive behaviour 

a, alpha, b and beta  Dies if rate < 0  Proliferates if rate > 0 
The transition connecting the state alive to the branch is triggered by the growth rate. In the state charts, the round squares represent the states and the arrows represent the transitions between the states. Arrows within states indicate the agent actions (or behaviours) and the final state is represented by a circle.
Our agents are stochastic and assume discrete time steps to execute their actions. This, however, does not restrict the dynamics of the models, as most of our agents state transitions are executed according to certain rates  this will go in parallel with steps execution, as defined in AnyLogic [28]. The rate triggered transition is used to model a stream of independent events (Poisson stream). In case more than one transition/interaction should occur at the same time, they are executed by AnyLogic in a discrete order in the same timestep. In the next section we apply the methodology to study our case studies and compare the outcomes.
Case 1: interactions between tumour cells and generic effector cells
For the first case, a mathematical model of tumour cells growth and their interactions with general immune effector cells defined in [32] is considered. Effector cells are responsible for killing the tumour cells inside the organism. Their proliferation rate is proportional to the number of existing tumour cells. As the quantities of effector cells increase, the capacity of eliminating tumour cells is boosted. These immune cells proliferate and die per apoptosis, which is a programmed cellular death. In the model, cancer treatment is also considered. The treatment consists of injections of new effector cells into the organism. The details of the mathematical model are given in the following section.
The mathematical model
where

T is the number of tumour cells,

E is the number of effector cells,

f(T ) is the growth of tumour cells,

d_{ T }(T, E) is the number of tumour cells killed by effector cells,

p_{ E }(T, E) is the proliferation of effector cells,

d_{ E }(T, E) is the death of effector cells when fighting tumour cells,

a_{ E }(E) is the death (apoptosis) of effector cells,

Φ(T ) is the treatment or influx of cells.
The agentbased model
Agents' parameters and behaviours for case 1
Agent  Parameters  Reactive behaviour  Proactive behaviour 

Tumour Cell  a and b  Dies (with age)  
a and b  Proliferates  
m  Damages effector cells  
n  Dies killed by effector cells  
Effector Cell  m  Dies (with age)  
d  Dies per apoptosis  
p and g  Proliferates  
s  Is injected as treatment 
Transition rates calculations from the mathematical equations for case 1
Agent  Transition  Mathematical equation  Transition rate 

Tumour Cell  proliferation  aT (1  Tb)  a  (TotalTumour.b) 
death  aT (1  Tb)  a  (TotalTumour.b)  
dieKilledByEffectorCells  nTE  n.TotalEffectorCells  
causeEffectorDamage  mTE  m  
Effector Cell  Proliferation  $\frac{pTE}{g+T}$  $\frac{p.TotalTumourCells}{g+TotalTumourCells}$ 
DieWithAge  dE  d  
DiePerApoptosis  mTE  message from tumour 
Experimental design for the simulations
Simulation parameters for different scenarios of case 1.
Scenario  b  d  s 
1  0.002  0.1908  0.318 
2  0.004  2  0.318 
3  0.002  0.3743  0.1181 
4  0.002  0.3743  0 
Results and discussion
Wilcoxon test with 5% significance level comparing case 1 simulation results
Implementation  Cells  Scenario (pvalue)  

1  2  3  4  
ABS  Tumour Effector  0 0.3789  0 0.6475  0.8591 0  0 0 
ABS  Fix 1  Tumour Effector  0 0  0 0.3023  0 0  0.0011 0 
ABS  Fix 2  Tumour Effector  0 0  0 0  0 0  0 0 
In scenario 4, although effector cells appear to decay in a similar trend for both approaches, the results for tumour cells vary widely. In the ODE simulation, the numbers of effector cells reached a value close to zero after twenty days and then increased to a value smaller than one. For the ABMS simulation, however, these cells reached zero and never increased again.
Similar to scenarios 2 and 3, the continuous ODE simulation outcomes contrasted with discrete agents caused the different outcomes. Furthermore, as occurred in scenario 2, the individual behaviour and rates attributed to the cells seemed to have an impact in the growth of tumours.
Summary
An ODE model of tumour cells growth and their interactions with general immune effector cells was considered for reconceptualization using ABMS. Four scenarios considering small population numbers were investigated and, for only one of them, the ABMS results were similar to the mathematical model. The differences observed were explained by the way each simulation approach is implemented, which includes their data representation and processing. ODE simulations deal with continuous values for the entities whereas ABMS represents discrete agents. Furthermore, the stochastic behaviour of the ABMS and how it affects small populations is not present in the ODEs. It also appears that the individual interactions between populations in the ABMS leads to a more chaotic behaviour, which does not occur at a higher aggregate level. The result analysis also reveals that conceptualizing the ABMS model from the mathematical equations does not always produce the same outcomes. One alternative to obtain better matching results would be the development of an agentbased model, which is not based on the rates defined in the ODE model, but using real data (available or collectable) or some form of parameter calibration.
Case 2: interactions between tumour cells, effector cells and cytokines IL2
The second case study investigated is concerned with a mathematical model for the interactions between tumour cells, effector cells and the cytokine IL2. This is an extension of the previous study, since it considers IL2 as molecules that will mediate the immune response towards tumour cells. They will interfere on the proliferation of effector cells according to the number of tumour cells in the system. Treatment is now applied in two different ways, by injecting effector cells or injecting cytokines.
The mathematical model
The IL2 population dynamics is described by Equation 15. $\frac{{p}_{2}ET}{{g}_{3}+T}$ determines IL2 production using parameters p_{2} and g_{3}. μ_{3} is the IL2 loss. s 2 also represents treatment. The treatment is the injection of IL2 in the system.
The agentbased model
Agents' parameters and behaviours for case 2
Agent  Parameters  Reactive behaviour  Proactive behaviour 

Effector Cell  mu 2  Dies  
p 1 and g 1  Reproduces  
c  Is recruited  
s 1  Is injected as treatment  
p 2 and g 3  Produces IL2  
aa and g 2  Kills tumour cells  
Tumour Cell  a and b  Dies  
a and b  Proliferates  
aa and g 2  Dies killed by effector cells  
c  Induces effector recruitment  
IL2  p 2 and g 3  Is produced  
mu 3  Is lost  
s 2  Is injected 
Transition rates calculations from the mathematical equations for case 2
Agent  Transition  Mathematical equation  Transition rate 

Effector Cell  Reproduce  $\frac{{p}_{1}.{I}_{L}E}{g1+IL\text{\_}2}$  $\frac{{p}_{1}.TotalIL\text{\_}2.TotalEffector}{g1+TotalIL\text{\_}2}$ 
Die  μ _{2} E  mu 2  
killTumour  $\frac{{a}_{a}ET}{g2+T}$  $aa\frac{TotalTumour}{g2+TotalTumour}$  
ProduceIL2  $\frac{p2ET}{g3+T}$  $\frac{p2.TotalTumour}{g3+TotalTumour}$  
Tumour Cell  Reproduce  aT(1  bT )  a  (TotalTumour.b) 
Die  aT(1  bT )  a  (TotalTumour.b)  
DieKilledByEffector  $\frac{{a}_{a}TE}{g2+T}$  message from effector  
IL2  Loss  μ _{3} I _{ L }  mu 3 
 1.
TreatmentS 1, which adds effector cell agents according to the parameter s 1
 2.
TreatmentS 2, which adds IL2 agents according to the parameter s 2
Experimental design for the simulation
Parameter values for case 2
Parameter  Value 

a  0.18 
b  0.000000001 
c  0.05 
aa  1 
g2  100000 
s1  0 
s2  0 
mu2  0.03 
p1  0.1245 
g1  20000000 
p2  5 
g3  1000 
mu3  10 
Results and discussion
Wilcoxon test with 5% significance level comparing the results from the ODEs and ABMS for case 2
Population  p 

Effector  0.7231 
Tumour  0.5710 
IL2  0.4711 
Summary
A mathematical model for the interactions between tumour cells, effector cells and the cytokine IL2 was considered to investigate the potential contribution of building the model under an ABMS perspective. Experimentation shows that results are very similar, which is explained by the large population sizes considered in the experiments. In further experiments, the same model was also run under small population sizes and the results for the simulations were different due to stochasticity and the approaches particularities, as discussed in the previous case study. Regarding the use of computational resources for larger data sets, ABMS was far more time and memoryconsuming than the ODEs.
Case 3: interactions between tumour cells, effector cells, IL2 and TGFβ
The third case study is based on the mathematical model of Arciero et al. [34], which consists of a system of ODEs describing interactions between tumour cells and immune effector cells, as well as the immunestimulatory and suppressive cytokines IL2 and TGFβ. According to Arciero et al. [34] TGFβ stimulates tumour growth and suppresses the immune system by inhibiting the activation of effector cells and reducing tumour antigen expression. The mathematical model, together with further details on the interactions studied is introduced in the following section.
The mathematical model
Equation 19 describes the rate of change of the suppressor cytokine, TGFβ. According to Arciero et al. [34], "experimental evidence suggests that TGFβ is produced in very small amounts when tumours are small enough to receive ample nutrient from the surrounding tissue. However, as the tumour population grows sufficiently large, tumour cells suffer from a lack of oxygen and begin to produce TGFβ in order to stimulate angiogenesis and to evade the immune response once tumour growth resumes". This switch in TGFβ production is modelled by term $\frac{{p}_{4}{T}^{2}}{{\theta}^{2}+{T}^{2}},\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\mathsf{\text{where}}\phantom{\rule{0.3em}{0ex}}{p}_{4}$, is the maximum rate of TGFβ production and τ is the critical tumour cell population in which the switch occurs. The decay rate of TGFβ is represented by the term μ_{3}S.
The agentbased model
Agents' parameters and behaviours for case 3
Agent  Parameters  Reactive behaviour  Proactive behaviour 

Effector Cell  mu 1  Dies  
p 1, g 1, q 1 and q 2  Reproduces  
c  Is recruited  
aa and g2  Kills tumour cells  
Tumour Cell  a  Dies  
a  Proliferates  
aa and g2  Dies killed by effector cells  
g3 and p2  Has growth stimulated  
p4 and tetha  Produces TGFβ  
c  Induces effector recruitment  
IL2  alpha, p 3 and g 4  Is produced  
mu 2  Is lost  
TGFβ  p 4 and tetha  Is produced  
mu 3  Is lost  
p 2 and g 3  Stimulates tumour growth 
Transition rates calculations from the mathematical equations for case 3
Agent  Transition  Mathematical equation  Transition rate 

Effector Cell  Reproduce  $\frac{p1IE}{g1+I}\times \left(p1\frac{q1S}{q2+S}\right)$  $\frac{p1\times TotalIL\text{\_}2}{g1+TotalIL\text{\_}2}\times \left(p1\frac{q1\times TotalTGF\phantom{\rule{0.3em}{0ex}}Beta}{q2+TotalTGF\phantom{\rule{0.3em}{0ex}}Beta}\right)$ 
Die  μ _{1} E  mu 1  
ProduceIL2  $\frac{p3TE}{\left(g4+T\right)\left(1+alphaS\right)}$  $\frac{p3.TotalTumour}{\left(g4+TotalTumour\right)\left(1+alpha.TotalTGF\right)}$  
KillTumour  $\frac{{a}_{a}.TE}{g2+T}$  $\frac{aa\times TotalTumour\times TotalEffector}{g2+TotalTumour}$  
Tumour Cell  Reproduce  $\left(aT\left(1\frac{T}{1000000000}\right)\right)$  $\left(TotalTumour.a\phantom{\rule{2.77695pt}{0ex}}\left(1\frac{TotalTumour}{1000000000}\right)\right)$ 
Die  $\left(aT\left(1\frac{T}{1000000000}\right)\right)$  $\left(TotalTumour.a\phantom{\rule{2.77695pt}{0ex}}\left(1\frac{TotalTumour}{1000000000}\right)\right)$  
DieKilledByEffector  $\frac{{a}_{a}.TE}{g2+T}$  message from effector  
ProduceTGF  $\frac{p4{T}^{2}}{tet{a}^{2}+{T}^{2}}$  $\frac{p4TumourCells}{tet{a}^{2}+TumourCell{s}^{2}}$  
EffectorRecruitment  $\frac{cT}{1+\gamma S}$  $\frac{c}{1+gamma.TotaltGF}$  
IL2  Loss  μ _{2} I  mu 2 
TGFβ  Loss  μ _{3} S  mu 3 
Stimulates TumourGrowth  $\frac{p2T}{g3+S}$  $\frac{p2.TotalTGF}{g3+TotalTGF}$ 
Experimental design for the simulation
Parameter values for case 3
Parameter  Value 

a  0.18 
aa  1 
alpha  0.001 
c  0.035 
g1  20000000 
g2  100000 
g3  20000000 
g4  1000 
gamma  10 
mu1  0.03 
mu2  10 
mu3  10 
p1  0.1245 
p2  0.27 
p3  5 
p4  2.84 
q1  10 
q2  0.1121 
theta  1000000 
k  10000000000 
Results and discussion
For most ABMS runs the pattern of behaviour of the agents is the same as that obtained by the ODEs. For a few runs, however, the populations decreased to zero, indicating that it is not always possible to obtain similar results with both approaches.
The differences observed occur for two reasons: (1) the ABMS stochasticity and (2) the agents individual behaviour and their interactions. While ODEs always use the same values for the parameters over the entire population aggregate, ABMS rates vary with time. Each agent therefore is likely to have distinct numbers for their probabilities. The agents individual interactions, which give raise to the overall behaviour of the system, are also influenced by the scenario determined by the random numbers used. By running the ABMS multiple times with different sets of random numbers, the outcomes vary according to these sets. For the ODEs, on the contrary, multiple runs always produce the same outcome, as random numbers are not considered.
In addition, the unexpected patterns of behaviour found the the ABMS results are the consequence of the agents individual interactions and their chaotic character. We believe that these unexpected patterns obtained with ABMS should be further investigated by specialists to determine if they are realistic and plausible to happen in biological experiments.
Regarding the TGFβ outcomes, the ODEs results reveal numbers smaller than one, which is not possible to achieve with the ABMS. The results for the simulations regarding these molecules are therefore completely different and the ABMS results are always zero.
Summary
The third case study comprised interactions between effector cells, tumour cells and two types of cytokines, namely IL2 and TGFβ. There were two important aspects observed in the ABMS outcomes. The first observation is that the TGF β population was not present in the simulation when using the mathematical model's parameters, as its numbers are real values smaller than one. This indicates that there is the need of further model validation with real data in order to check which paradigm outcome is closer to reality. The second aspect observed is that ABMS produces extra patterns of population behaviour (extreme cases) distinct from that obtained by the mathematical model. This could in turn lead to the discovery of other realworld patterns, which would otherwise not be revealed by deterministic models.
Conclusions
Summary of findings
Case Study  Outcome of the comparison  Explanation  Population size 

1) Tumour/Effector  • Most results were different  • It appears that variabilities in small populations have major impacts in the outcomes  Varied from 5 to 600 
2) Tumour/Efector  • Results were statistically/IL2 the same  • Large populations • Less variability in the agents' populations  10^{4} 
3) Tumour/Effector/ IL2/ TGFβ  • Different runs with outcome variations • Simulations produced alternative scenarios • The behaviour of the curves is less erratic for agents  • Agentbased stochasticity • New scenarios need further investigations to assess their feasibility • Large numbers of agents  10^{4} 
Case study 1 was concerned with the use of ODEs to model interactions with general immune effector cells and tumour cells. The objective of this model is to observe these two populations evolving overtime and to evaluate the impacts of cancer treatment in their dynamics. Four different scenarios regarding distinct sets of parameters were investigated and in the first three scenarios treatment was included. The ABMS produced very different results for most scenarios. The outcomes from ODEs and ABMS only resembled for Scenario 1. It appears that two major characteristics of this model influenced the differences obtained: (1) The small quantities of individuals considered in the simulations (especially regarding the effector population size, which was always smaller than ten) that significantly increase the variability of the ABMS; and (2) The original mathematical model considers fractional population sizes (smaller than one) which is impossible to be considered in ABMS. In addition to this particular model's characteristics, for any mathematical model considering cyclic intervals of growth or decay of populations observed in our studies, the corresponding curves in the ABMS outcomes are more accentuated, given the fact that ODEs changes quantities continuously whereas ABMS varies discretely. Small numbers do not allow to recreate predatorprey patterns in stochastic models, as such models need to be perfectly balanced to work. Stochasticity in small models does not allow such balance and therefore might produce chaos.
Case study 2 referred to the investigation of the interactions between effector cells, cytokines IL2 and tumour cells, and only one scenario was considered. ODEs and ABMS simulations also produced very similar results. As populations' sizes had a magnitude of 104 individuals, the ABMS erratic behaviour in the outcomes was not evident, which contributed to the outcomes similarity. The differences observed in the curves were explained by the continuous numbers produced by the ODEs versus discrete values from ABMS.
Case study 3 added complexity to the previous case study by establishing a mathematical model including the influence of the cytokine TGFβ in the interactions between effector cells, cytokines IL2 and tumour cells. The simulation outcomes for the ABMS were mostly following the same pattern as that produced by the ODEs; however there were some alternative outcomes where the patterns of behaviour demonstrated a total extermination of tumour cells by the first two hundred days. This indicates that for this case study the ABMS results are more informative, as they illustrate another set of possible dynamics that should be validated through further immune experimentation.
In response to our research questions, we conclude that not everything modelled in ODEs can be implemented in ABMS (e.g. no half agents); however  this does not matter if population sizes in the original model definition are large enough. In addition, population size has a positive impact on result similarity. The bigger the population, the closer the simulation outputs. Finally, ABMS can contribute additional insight, as due to its stochastic nature it can produce different results (normal and extreme cases). Further, variability in the graph is closer to the real world, although knowing the underlying pattern might be more useful. ODEs therefore also have an advantage as they show more clearly underlying patterns in the output (as for example predatorprey pattern).
In the future, we want to investigate new case studies and systematically determine when phenomena such as agentbased stochasticity mostly influences on the outcome differences and in which circumstances extreme cases occur. In addition, with regard to extreme cases, it is necessary to gain additional insights of (1) how frequent these extreme cases occur and (2) wether there is any relation between the frequency of occurrence of these cases in the simulation and in the realworld. For example, we could count the appearance of these unusual cases (as a measure of system stability or robustness of the solution) when running the experiments 10, 000 times. This could help immunologists defining vaccination strategies and appropriateness of cancer treatments by making them aware of the possible outcome scenarios and how frequently they occur.
List of Abbreviations
 ODE :

Ordinary differential equation
 ABMS :

Agentbased modelling and simulation.
Declarations
Declarations
This article has been published as part of BMC Bioinformatics Volume 14 Supplement 6, 2013: Selected articles from the 10th International Conference on Artificial Immune Systems (ICARIS). The full contents of the supplement are available online at http://www.biomedcentral.com/bmcbioinformatics/supplements/14/S6.
Authors’ Affiliations
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