- Research
- Open Access
A multiscale mathematical model of cancer, and its use in analyzing irradiation therapies
- Benjamin Ribba^{1}Email author,
- Thierry Colin^{2} and
- Santiago Schnell^{3}
https://doi.org/10.1186/1742-4682-3-7
© Ribba et al; licensee BioMed Central Ltd. 2006
- Received: 28 September 2005
- Accepted: 10 February 2006
- Published: 10 February 2006
Abstract
Background
Radiotherapy outcomes are usually predicted using the Linear Quadratic model. However, this model does not integrate complex features of tumor growth, in particular cell cycle regulation.
Methods
In this paper, we propose a multiscale model of cancer growth based on the genetic and molecular features of the evolution of colorectal cancer. The model includes key genes, cellular kinetics, tissue dynamics, macroscopic tumor evolution and radiosensitivity dependence on the cell cycle phase. We investigate the role of gene-dependent cell cycle regulation in the response of tumors to therapeutic irradiation protocols.
Results
Simulation results emphasize the importance of tumor tissue features and the need to consider regulating factors such as hypoxia, as well as tumor geometry and tissue dynamics, in predicting and improving radiotherapeutic efficacy.
Conclusion
This model provides insight into the coupling of complex biological processes, which leads to a better understanding of oncogenesis. This will hopefully lead to improved irradiation therapy.
Keywords
- Cell Cycle Regulation
- Cell Cycle Phase
- Multiscale Model
- Quiescent Cell
- Tumor Control Probability
Background
Mathematical models of cancer growth have been the subject of research activity for many years. The Gompertzian model [1, 2], logistic and power functions have been extensively used to describe tumor growth dynamics (see for example [3] and [4]). These simple formalisms have been also used to investigate different therapeutic strategies such as antiangiogenic or radiation treatments [5].
The so-called linear-quadratic (LQ) model [6] is still extensively used, particularly in radiotherapy, to study damage to cells by ionizing radiation. Indeed, extensions of the LQ model such as the 'Tumor Control Probability' model [7] are aimed at predicting the clinical efficacy of radiotherapeutic protocols. Typically, these models assume that tumor sensitivity and repopulation are constant during radiotherapy. However, experimental evidence suggests that cell cycle regulation is perhaps the most important determinant of sensitivity to ionizing radiation [8]. It has been suggested that anti-growth signals such as hypoxia or the contact effect, which are responsible for decreasing the growth fraction, may play a crucial role in the response of tumors to irradiation [9].
Nowadays, computational power allows us to build mathematical models that can integrate different aspects of the disease and can be used to investigate the role of complex tumor growth features in the response to therapeutic protocols [10]. In the present study we propose a multiscale model of tumor evolution to investigate growth regulation in response to radiotherapy. In our model, key genes in colorectal cancer have been integrated within a Boolean genetic network. Outputs of this genetic model have been linked to a discrete model of the cell cycle where cell radiosensitivity has been assumed to be cycle phase specific. Finally, Darcy's law has been used to simulate macroscopic tumor growth.
The multiscale model takes into account two key regulation signals influencing tumor growth. One is hypoxia, which appears when cells lack oxygen. The other is overpopulation, which is activated when cells do not have sufficient space to proliferate. These signals have been correlated to specific pathways of the genetic model and integrated up to the macroscopic scale.
Methods
Gene level
Five genes are commonly mutated in colorectal cancer patients, namely: APC (Adenomatosis Polyposis Coli), K-RAS (Kirsten Rat Sarcoma viral), TGF (Transforming Growth Factor), SMAD (Mothers Against Decapentaplegic) and p53 or TP53 (Tumor Protein 53). These genes belong to four specific pathways, which funnel external or internal signals that cause cell proliferation or cell death (see [15] and [16, 17] for more details).
The anti-growth, p53, pathway is activated in the case of DNA damage [18, 19]. This is particularly relevant during irradiation [20]. p53 pathway activation can block the cell cycle and induce apoptosis [21, 22]. The K-RAS gene belongs to a mitogenic pathway that promotes cell proliferation in the presence of growth factors [23]. Activation of the anti-growth pathways TGFβ/SMAD and WNT/APC inhibits cell proliferation. The SMAD gene is activated by hypoxia signals [24, 25], while APC is activated through β-catenin by loss of cell-cell contact [26–30]. Moreover, it has recently been hypothesized that overpopulation of APC mutated cells can explain the shifts of normal proliferation in early colon tumorigenesis [31].
Apoptotic activity. Apoptotic activity induced by two 20 Gy radiotherapy protocols applied to APC-mutated tumor cells.
Apoptotic activity | ||||
---|---|---|---|---|
Total dose (Gy) | Scheduling | Apoptotic fraction – mean – (%) | Apoptotic fraction – max – (%) | |
Standard protocol | 20 | 2 Gy daily | 2.59 | 4 |
Heuristic | 20 | 2 Gy Repeated 10 times before hypoxia | 3.14 | 4.25 |
Genetic model. Boolean (logical) functions used in the genetic model depicted Figure 1. For APC, SMAD and RAS, Boolean values are set to 0, 0 and 1 respectively when genes are mutated.
Boolean model | |
---|---|
Node | Boolean updating function |
APC ^{ t } |
APC ^{ t+1}= 0 if mutated |
βcat ^{ t } | βcat ^{ t+1}= ¬APC ^{ t } |
cmyc ^{ t } | cmyc ^{ t+1}= RAS ^{ t }∧ βcat ^{ t }∧ ¬SMAD ^{ t } |
p27^{ t } | p27^{ t+1}= SMAD ^{ t }∨ ¬cmyc ^{ t } |
p21^{ t } | p21^{ t+1}= p53^{ t } |
Bax ^{ t } | Bax ^{ t+1}= p53^{ t } |
SMAD ^{ t } |
SMAD ^{ t+1}= 0 if mutated |
RAS ^{ t } |
RAS ^{ t+1}= 1 if mutated |
p53^{ t } |
p53^{ t+1}= 0 if mutated |
CycCDK ^{ t } | CycCDK ^{ t+1}= ¬p21^{ t }∧ ¬p27^{ t } |
Rb ^{ t } | Rb ^{ t+1}= ¬CycCDK ^{ t } |
Cell level
For each spatial position (x, y), we assume that:
- If the local concentration of oxygen is below a constant threshold Th _{ o }and if SMAD is not mutated, hypoxia is declared and causes cells to quiesce (G _{ 0 }) through SMAD gene activation (see Figure 2);
- If the local number of cells is above a constant threshold Th _{ t }and if APC is not mutated, overpopulation is declared and leads cells to quiesce (G _{ 0 }) through the APC gene (see Figure 2);
- Otherwise, if the conditions are appropriate, cells enter G _{ 2 } M and divide, generating new cells at the same spatial position.
Induction of apoptosis through p53 gene activation is discussed later.
Tissue level
We use a fluid dynamics model to describe tissue behavior. This macroscopic-level continuous model is based on Darcy's law, which is a good model of the flow of tumor cells in the extracellular matrix [38–40]:
v = -k∇p (1)
where p is the pressure field. The media permeability k is assumed to be constant.
We study the evolution of the cell densities in two dimensions. We formulate the cell densities in the tissue mathematically as advection equations, where n _{ φ }(x, y, t) represents the density of cells with position (x, y) at time t in a given cycle phase φ. Assuming that all cells move with the same velocity given by Eq. (1) and applying the principle of mass balance, the advection equations are:
where P _{ φ }is the cell density proliferation term in phase φ at time t, retrieved from the cell cycle model.
The global model is an age-structured model (see Section 2.7). Initial conditions for n _{ φ }are presented in Section 2.6.
The pressure is constant on the boundary of the computational domain.
In our model, the oxygen concentration C follows a diffusion equation with Dirichlet conditions on the edge of the computation domain Ω:
C = C _{max} on Ω_{ bv } (5)
C _{ ∂Ω}= 0 (6)
D is the oxygen diffusion coefficient, which is constant throughout the computation domain. In this equation, Ω_{ bv }stands for the spatial location of blood vessels, α _{ φ }is the coefficient of oxygen uptake by cells at cell cycle phase φ and C _{ max }is the constant oxygen concentration in blood vessels.
Therapy assumptions
Cell sensitivity depends on cell cycle phase [8]. We assume that only proliferative cells are sensitive to the treatment. In addition, we assume that DNA damage is proportional to the irradiation dose. This is known as the 'single hit' theory, which is governed by the expression
n _{ dsb }= R _{ φ } d (7)
Table of parameters Table of numerical parameters used for simulations.
Model parameters | ||||
---|---|---|---|---|
Parameter | Description | Unit | Value | Reference |
| Duration of G _{ 1 }phase | h | 20 | [35,44] |
T _{ S } | Duration of S phase | h | 10 | [35,44] |
| Duration of G _{ 2 }M phase | h | 3 | [35,44] |
| Duration of G _{ 0 }phase | h | 5 | Estimated |
T _{ Apoptosis } | Duration of the apoptotic phase | h | 5 | Estimated |
C _{ max } | Oxygen in blood | mlO _{2} | 10^{-2} | Estimated |
α _{ φ } | Oxygen consumption in phase φ | mlO _{2} s ^{-1} | 5 – 10 × 10^{-15} | Estimated |
Th _{ o } | Hypoxia threshold | cell ^{-1} | 5 × 10^{-15} | Estimated |
Th _{ t } | Overpopulation threshold | cell | 2000 | Estimated |
R _{ φ } | Cell Radio-sensitivity in phase φ | Gy ^{-1} | 0.2 – 2 | [41-43] |
k | Media permeability | m ^{2} | 0.2 | Estimated |
We set a constant treatment threshold Th _{ r }such that if n _{ dsb }due to the irradiation dose is above Th _{ r }at any time, p53 is activated and the cells are labeled as 'DNA damaged cells'. DNA damaged cells are identified at the R point of the cell cycle and are directed to apoptosis. They die and disappear from the computational domain after T _{ Apoptosis }, i.e. the duration of the apoptotic phase.
The standard radiotherapy protocol used in the simulations consists of a 2 Gy dose delivered each day, five days a week, and can be repeated for several weeks. The radiotherapeutic dose is assumed to be uniformly distributed over the spatial domain.
According to the radiosensitivity parameters found in the literature [41–43], only a fraction of mitotic cells are assumed to be sensitive to the standard 2 Gy dose.
Model parameters
Cell cycle kinetic parameters were retrieved from flow cytometric analysis of human colon cancer cells [35, 44]. Table 3 summaries the quantitative parameters used in our model.
Computational domain and initial conditions
The number of cells in each tumor is the same, and they are uniformly distributed. The number of cells in each phase of the cell cycle is proportional to the duration of the phase. For instance, the G _{ 1 }phase contains twice as many cells as the S phase because the G _{ 1 }phase is twice as long as the S phase. It is important to emphasize that the cell cycle phases are discrete (see Section 2.7).
Simulation technique
The model is fully deterministic. Cell cycle phases durations τ _{ φ }have been discretized in several elementary age intervals a ∈ {1, ..., N _{ φ }} where N _{ φ }is an integer such as τ _{ φ }= dt × N _{ φ }. Here dt is the time step of the cell cycle model. The cell density n _{ a, φ }at age a in phase φ is governed by:
In this equation, φ ∈ {G _{1}, S, G _{2} M, G _{0}, Apoptosis} and a ∈ {1, ..., N _{ φ }}. P _{ a,φ }is the cell density proliferation term in phase φ at age a retrieved from the cell cycle model. In the simulations, the intracellular and extracellular conditions were identified for cells at the end of G _{ 1 }phase. These were used as initial conditions for the gene level model. The genetic model was computed until it reached steady state (this is of the order of 10 iterations).
The computer program starts from an initial distribution of cells in each state {a, φ}. The computations are performed using a splitting technique. First we run the cell cycle model for one time-step dt, then retrieve new values for n _{ a,φ }and compute P _{ a, φ }. Pressure is retrieved by solving Eq. (9) and velocity is computed using Darcy's law (see Eq. (1)). Since the contribution of the source term has been taken into account by the cell cycle model at the first stage of the splitting technique, Eqs. (8) are solved continuously and without second members:
which can also be written [using (9)]:
This equation is then solved using a splitting technique. The advection parts of Eq. (11) are solved by sub-cycling finite different scheme computations, with time-step dt _{ adv }being smaller than dt (for stability reasons). We set n _{ a,φ }= 0 on the part of the boundary where v·υ < 0, υ denoting the outgoing normal to the boundary. For the pressure p, we set p = 0 on the boundary.
All simulations (except the ones shown in Figure 7) were run for 320 h with time step dt = 1 h in a discrete computational domain composed by 100 × 100 elementary spatial units.
Results and discussion
We divide our results and discussion into three parts. The first section concerns simulations of the model without therapeutic interactions (Sections 3.1–3.2). The second part deals with the interactions between tumor growth and the effect of therapeutic protocols (Section 3.3). Finally, we investigate the sensitivity of the results to model parameters and initial conditions (Section 3.4). Genetic mutations are simulated by running the model, having set the Boolean values of particular genes constant (see Table 2). Since the genetic model is run until steady state is reached, simulation of mutated cell growth is equivalent to simulation of cells that are not sensitive to particular anti-growth signals. In the following, we will refer to cells with at least one mutation as 'cancer cells'. Cells with no mutations are called 'normal cells'.
Gene-dependent tumor growth regulation
The simulation results reproduce the evolution of colorectal cancer [16, 45]. Indeed, APC has been shown to promote shifts in pattern of the normal cell population in early colorectal tumorigenesis, and SMAD/RAS mutations promote evolution from early adenoma to adenocarcinoma.
Features of anti-growth signals and effect on tumor growth
APC-dependent growth regulation
SMAD/RAS-dependent growth regulation
Figure 7 shows the time courses of total cell number and quiescent cell number. In this figure, cells are APC mutated and the growth regulation is controlled by SMAD/RAS signaling, which has been activated by hypoxia. Before hypoxia, cell population growth is exponential and becomes more linear as the anti-growth signals start.
Influence of gene-dependent growth regulation on the response to irradiation protocols
Simulated irradiation protocols on APC and SMAD/RAS mutated tumor cells
Simulated irradiation protocols and APC-dependent tumor growth
Simulated irradiation protocols on APC-mutated (SMAD/RAS-dependent) tumor growth regulation profiles
Sensitivity to model parameters and initial conditions
We study the potential influence of the choice of parameters values on the model's results. The most critical parameters to account for include:
• cell-specific radiosensitivity parameters (α _{ φ });
• anti-growth signals, i.e. hypoxia and overpopulation, activation thresholds above which cells go into quiescence (Th _{ o }and Th _{ t });
• initial conditions, i.e. initial number of cells and spatial configurations of oxygen sources.
Decreasing the initial number of cells has the same effect as increasing Th _{ t }, while decreasing the number or the initial strength of the oxygen sources has the same effect as increasing Th _{ o }. The initial configuration of tumor cells and oxygen sources is important for spatial propagation of the hypoxia signal. Indeed, Figure 9 shows a particular hypoxia propagation in the tumor cell masses that is correlated with the locations of the oxygen sources. Since Th _{ t }and Th _{ o }are merely constants, it seems that we may change the spatial configuration and size of the initial cell population and vary the oxygen sources and yet produce the same qualitative results.
Conclusion
We have presented a multiscale model of cancer growth and examined the qualitative response to radiotherapy. The mathematical framework includes a Boolean description of a genetic network relevant to colorectal oncogenesis, a discrete model of the cell cycle and a continuous macroscopic model of tumor growth and invasion. The basis of the model is that the sensitivity to irradiation depends on cell cycle phase and that DNA damage is proportional to the radiation dose. Anti-growth regulation signals such as hypoxia and overpopulation activate the SMAD/RAS and APC genes, respectively, and inhibit proliferation through cell cycle regulation.
Simulation results show the different features of the antigrowth signal activation and propagation within the tumor (see Figure 8). The overpopulation signal mediated by the APC gene initially induces oscillatory growth owing to a combination of proliferating and quiescent cells (see Figure 6). Because of its non-local effect, the hypoxia signal mediated by genes SMAD/RAS appears later but develops quickly within the tumor masses, and leads the mitotic fraction to collapse (see Figures 11 and 14). These features make the evolution of the number of quiescent cells and thus the efficacy of irradiation protocols depend on the type of anti-growth signals to which the tumors are exposed. Figure 11 and Table 1 show that efficacy could be improved, without increasing radiation doses, by planning schedules that take account of the features of tumor growth through cell cycle regulation.
The proposed framework emphasizes the significant role of gene-dependent cell-cycle regulation in the response of tumors to radiotherapy. Clinical studies have recognized p53 status as a major predictive factor for the response of rectal cancer to irradiation. Nevertheless, some results encourage investigation of other different factors [46]. In particular, it has been suggested that macroscopic factors such as hypoxia and tumor volumes are important [47]. The present modeling framework integrates these factors through cell cycle regulation and allows consideration of other factors at the genetic, cellular or tissue level.
Some modeling assumptions must be discussed. We chose a continuous approach that provides cell density rather than actual cell number. This assumes that the region of interest is large since we have restricted our analysis to late-stage tumor development. We have not considered cell shape, which has been shown to be important for the correct description of growth control processes [48]. Individual-based models of cell movement, e.g. the Potts model [49, 50] and the Langevin model [51], would improve our approach. We reduced the system to two dimensions. A three-dimensional tumor model could reveal new factors in the dynamics.
The aim of this study is to understand the qualitative effect of therapeutic protocols on colorectal cancer. Our analysis raises some interesting points about the influence of anti-growth regulation signals and genetic pathways on the efficacy of the standard protocol. Efforts have been made to improve the LQ model by taking into account multiple factors such as tumor volume and repopulation between treatment cycles [52]. However, we have produced a multiscale model that is more realistic and demonstrated its use in comparing efficacy of treatment protocols.
Declarations
Acknowledgements
BR is funded by the ETOILE project: "Espace de Traitement Oncologique par Ions Légers dans le cadre Européen". Part of this work was carried out during the "Biocomplexity Workshop 7" held at Indiana University (Bloomington Campus) in May 9–12, 2005. The workshop was sponsored by the National Science Foundation (Grant MCB0513693) and the National Institute of General Medical Science/National Institutes of Health (Grant R13GM075730). BR is very grateful for the hospitality of the Indiana University School of Informatics and the Biocomplexity Institute during his visit May 8–14. The authors wish to acknowledge particularly the two referees for their useful comments; Professor Jean-Pierre Boissel and François Gueyffier for manuscript review; Professor Emmanuel Grenier, Dr Didier Bresch, and Nicolas Voirin for their valuable advice regarding model implementation; and Dr Ramon Grima and Edward Flach for critical comments.
Authors’ Affiliations
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