Department of Radiology, The Methodist Hospital Research Institute, Weil Cornell Medical College, Houston, TX, 77030, USA

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P R China

College of Computer and Information Science, Southwest University, Chongqing, 400715, P R China

Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

College of Global Change and Earth System Science, Beijing normal University, Beijing, 100875, P R China

Computation-based Science and Technology Research Center, The Cyprus Institute, Nicosia, 1645, Cyprus

Abstract

Background

The epidermal growth factor receptor (EGFR) signaling pathway and angiogenesis in brain cancer act as an engine for tumor initiation, expansion and response to therapy. Since the existing literature does not have any models that investigate the impact of both angiogenesis and molecular signaling pathways on treatment, we propose a novel multi-scale, agent-based computational model that includes both angiogenesis and EGFR modules to study the response of brain cancer under tyrosine kinase inhibitors (TKIs) treatment.

Results

The novel angiogenesis module integrated into the agent-based tumor model is based on a set of reaction–diffusion equations that describe the spatio-temporal evolution of the distributions of micro-environmental factors such as glucose, oxygen, TGFα, VEGF and fibronectin. These molecular species regulate tumor growth during angiogenesis. Each tumor cell is equipped with an EGFR signaling pathway linked to a cell-cycle pathway to determine its phenotype. EGFR TKIs are delivered through the blood vessels of tumor microvasculature and the response to treatment is studied.

Conclusions

Our simulations demonstrated that entire tumor growth profile is a collective behaviour of cells regulated by the EGFR signaling pathway and the cell cycle. We also found that angiogenesis has a dual effect under TKI treatment: on one hand, through neo-vasculature TKIs are delivered to decrease tumor invasion; on the other hand, the neo-vasculature can transport glucose and oxygen to tumor cells to maintain their metabolism, which results in an increase of cell survival rate in the late simulation stages.

Background

Brain cancer is a very complex and deadly disease. Traditional diagnoses and treatments of this disease are from

In this paper, we presented a novel multi-scale agent-based model to describe tumor growth with angiogenesis and study the response of brain cancer to EGFR tyrosine kinase inhibitors (TKIs)

In order to integrate an angiogenesis module into the existing agent-based tumor growth models

The simulation results demonstrate that we can investigate the response of brain cancer to tyrosine kinase inhibitors (TKIs) and also can use the model to reveal the dual role of angiogenesis.

Implementation

Our model encompasses four biological scales (Figure

Flow chart of multi-scale agent-based cancer modeling

**Flow chart of multi-scale agent-based cancer modeling.** The model encompasses four biological scales: the molecular scale, the cellular scale, the micro-environmental scale and the tissue scale. The molecular scale consists of the EGFR signaling and cell cycle pathways (Figure

EGFR signaling pathway connected to cell cycle pathway with TKIs to block the EGFR signaling pathway

**EGFR signaling pathway connected to cell cycle pathway with TKIs to block the EGFR signaling pathway.** EGFR signaling pathway is connected to the cell cycle pathway, and TKIs block the EGFR signaling pathway.

Illustrations of phenotype switch "decision" of tumor cell

**Illustrations of phenotype switch "decision" of tumor cell.** Please see details in the main text (Cellular scale: Phenotype switch of tumor cell as "agent").

Probability of migration of tip endothelial cell

**Probability of migration of tip endothelial cell.** The tip endothelial cell probabilistically moves up, down, right, or left, or stays at its current position.

We performed our simulations on a two-dimensional square lattice. The lattice size was set to L = 200 representing a 4 ~ 5 mm length of a brain tissue slice. The lattice spacing is 20

**Table A1.** Kinetic equations describing the reactions between the components of the simplified EGFR signaling pathway.

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**Table A2.** Coefficients of the simplified EGFR signaling pathway.

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**Table A3.** Kinetic equations describing the reactions between the components of the cell-cycle.

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**Table A4.** Parameter in cell-cycle pathway.

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**Table A5.** Parameters of microenvironmental PDEs in the model.

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Molecular scale: signaling pathway

The concentration of each component (Figure

where _{+} represents the production rate of _{
i
} and

Cellular scale: phenotype switch of tumor cells as "agents"

At every simulation step, each "agent" (i.e. a brain tumor cell) switches its phenotype according to the following rules:

1. First, each agent evaluates the concentration of glucose at its current location. If the concentration is greater than the cell active threshold, the agent becomes active and uses its EGFR signaling pathway to determine its phenotype. If the concentration of glucose is less than the dead threshold, the cell dies. If the concentration is between active and dead thresholds, the agent enters into a reversible quiescent state

2. Each active agent evaluates its migration potential (

where _{PLCγ,} the average rate of change of PLCγ concentration, the agent will choose the migration phenotype.

3. If _{PLCγ}, the agent starts to proliferate. If the concentration of CDh1 is less than a threshold _{1} and the concentration of cycCDk is greater than the threshold _{
2
}, the cell divides. After that, the cell chooses the most attractive free site

(detailed in equation (3)) in the neighborhood to deliver its offspring. If there is no empty neighborhood, the cell turns into a reversible quiescent state until free space becomes available.

4. Each agent chooses the "most attractive" location mentioned above according to the following probability:

where _{
j
} is the fibronectin concentration at _{
j
} ~N(0,1) is a normally distributed error term, the parameter ψ ∈(0, 1) represents the extent of the search precision, which is set to 0.7

Microenvironmental scale: extracellular chemotaxis

Five extracellular micro-environmental factors, glucose, oxygen, TGFα, VEGF and fibronectin are included in this model. A set of reaction–diffusion equations describe the diffusion, penetration and uptake of glucose, oxygen and VEGF.

Glucose first penetrates blood vessels, and then diffuses in the extracellular microenvironment. After that, it is consumed by the tumor cells. This process is modeled by the following equation:

where _{
G
} is the diffusivity of glucose. _{G} = 2πr_{
G
}, where _{
G
} is the vessel permeability for glucose and ^{
blood
} is the glucose concentration in blood and _{G} is the cell’s glucose uptake rate. The time dependent characteristic function _{
ves
} (_{
tum
} (_{
ves
} and _{
tum
} are updated at each simulation step according to the developing profile of the tumor and its micro-vascularity.

Oxygen also permeates the blood vessels' walls, diffuses in the surrounding and is consumed by tumor cells. This process is modeled by the following equation:

where _{
C
} is the oxygen diffusivity, _{
C
} is the vessel permeability for oxygen, and _{
C
} is a cell’s uptake rate of oxygen.

TGFα, an analogue of EGF, is secreted by tumor cells and can be paracrine and juxtacrine

where _{
T
} is its diffusivity, _{
T
} is vessel permeability to TGFα. _{
T
} is a cell’s net production rate of TGFα and _{
T
} is the natural decay rate of TGFα.

We applied homogeneous Neumann boundary conditions for all the above equations by assuming zero flux along the boundary of the considered domain. Additional file **.** We solved these equations numerically with the finite difference method

**Text A1.** Equations describing the Initial distribution of glucose, oxygen, TGFα, VEGF and fibronectin.

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Tissue scale: angiogenesis

Tumor induced angiogenesis is due to the secretion of VEGF by the tumor cells. VEGF diffuses into the surrounding corneal tissue and is also consumed by the endothelial cells

where _{
V
} is the diffusivity of VEGF, _{
V
} is the vessel permeability for VEGF and _{
V
} is a cell’s VEGF secretion rate. _{
V
} is the natural decay rate of VEGF.

Fibronectin is a component of the corneal tissue secreted by endothelial cells. In addition, tumor cells can consume fibronectin. This process is described by the following equation:

where

We assume that the motion of individual endothelial cell (EC) located at the tip of a capillary sprout governs the motion of the whole sprout. Chemotaxis in response to VEGF gradients and haptotaxis in response to fibronectin are the major factors that influence the motion of the endothelial cells at the capillary sprout tip

We defined the probability of migration of a tip endothelial cell (see Figure

where _{
k
} is the directional vector along the _{
v
} is a positive constant controlling the weight of VEGF concentration in chemotactic sensitivity. The term

For a given tip of endothelial cells at location (

The un-normalized probability _{
5
}, for a tip cell to remain stationary is the average of _{1}, _{2}
_{3} and _{4.} After normalization, the above equations give the likelihood of the tip endothelial cell to move up, down, right, or left, or stay at its current position. The probability, _{
5
}, for a tip cell to remain stationary is the average of _{1}, _{2}
_{3} and _{4}.

The algorithm related to angiogenesis is as follows:

1. Calculate the migration probabilities of ECs:

1.1 Solve the equations (7) and (8), and then calculate _{1}
_{5} from (10);

1.2 Normalize the above numbers:

2. For every sprout tip cell, we check whether the age of vessel is greater than 18 hours and whether there are any free sites in its nearest neighborhood.

2.1 Sprout branching: If the above conditions are satisfied, two random numbers _{1} and _{2} between 0 and 1 are generated. If _{1} ∈ _{2} and _{2} ∈ _{3}, then we move two endothelial cells one below and one to the right of the spout tip endothelial cell.

2.2. Sprout migrating: If the above branching conditions are not satisfied, we generate another random number _{3}, we move the tip endothelial cell to the right of spout tip endothelial cell.

3. Anastomosis: If two sprouts encounter each other, a new sprout continues to grow.

TKI treatment

This study uses TKIs, particularly gefitinib, as inhibitors delivered by capillary vessels to treat brain cancer. The TKIs are modeled as a continuous concentration field. These molecules permeate the blood vessels and diffuse continuously in the microenvironment to produce an accumulative inhibitive effect on the tumor growth. We describe the evolution of the concentration of TKIs by the following equation:

where _{
TKi
} is the diffusivity of the TKIs, _{
TKi
} is the vessel permeability for TKIs, ^{
blood
} is the blood TKIs concentration, _{
TKi
} is a cell’s uptake rate of TKIs, and _{
Tki
} is the natural decay rate of TKIs.

The EGFR signaling pathway controls a cell’s phenotypic switch by binding TGFα molecules to the EGF receptors. Because the TKI molecules inhibit the autophosphorylation receptors, the downstream EGFR pathway responsible for a cell's phenotypic switch remains inactivated _{
b
} and _{
u
}, respectively, are described by the following equation:

Since the chemical reaction rate is much faster than the cells' phenotypic switch

where _{
m
} is the Michaelis constant, _{
m
} ≈ _{
b
} / _{
u
}, and [EGFR]_{0} is the initial concentration of the EGFR. We can then derive the effective amount of EGFR for the activation of downstream factors as follows:

Because of the TKI treatment, the effective amount of EGFR of some tumor cells will decrease. The decrease of the amount of effective EGFR results in a slow rate of change of PLCγ concentration. This in turn inhibits tumor progression by reducing the migration potential of these tumor cells (see equation 2 and detailed phenotype change of tumor cells in Figure

Finally, we summarize our computing algorithm at each step across multi-scales (Figure

Results

We have implemented the above model into software "ABM-TKI" in the Matlab programming environment. “ABM-TKI” is a tool employing agent-based model (ABM) to simulate brain tumor growth. It includes an EGFR signaling pathway, a related cell-cycle, angiogenesis and TKIs treatment. We can employ this tool to predict the responses of brain cancer and reveal the dual roles of angiogenesis under TKI treatment.

Regarding software usage, the user can download and decompress the package from the project home page (

The output includes: (a) the vascular tumor growth pattern with or without TKI treatment; (b) the tumor growth visualization with the background of fibronectin; (c) the spatio-temporal evolution of the concentration of glucose, oxygen, TGFα and/or TKI; (d) various tumor cell numbers such as active cells, apoptotic cells, migratory cells, proliferative cells, quiescent cells and the number of endothelial cells; (e) the average change rate of PLCγ with or without TKI treatment.

Vascular tumor growth patterns

The vascularized tumor patterns at 60, 90, 120 and 150 hours are shown in Figure

**Figure A1.** Tumor induced angiogenesis and vascular tumor growth without TKI treatment.

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**Figure A2.** Vascular tumor growths with concentration change of fibronectin.

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**Figure A3.** The concentration change of glucose, oxygen, TGFα and VEGF without TKI treatment.

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Simulation of vascular tumor growth without TKI treatment

**Simulation of vascular tumor growth without TKI treatment.** (**a**) The evolution of vascular tumor patterns at 60, 90, 120 and 150 hours. (**b**) The distribution of glucose, oxygen, TGFα and VEGF in microenvironment at 150 hours. (**c**) Different types of cell numbers from 0 hour to 150 hours. (**d**) Proliferation rate of tumor cells derived from simulations (**red line)** and from **blue line**) for t=0-96 hours.

Figure

**Figure A4.** Various tumor cell numbers without TKI treatment.

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In Figure

TKI treatment response

In our study we chose gefitinib as the EGFR TKI to treat brain cancer. In this section the term TKI refers to treating cell by gefitinib. Figure

**Figure A5.** Vascular tumor growth with TKI treatment.

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**Figure A6.** Vascular tumor growth in the presence of fibronectin with TKIs treatment.

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Simulation of vascular tumor growth with TKI treatment

**Simulation of vascular tumor growth with TKI treatment.** (**a**) Vascular tumor growth patterns with TKI treatment at 60, 150, 240 and 300 hours; (**b**) The distribution of TKI in tumor vasculature at 300 hours; (**c**) Average change rate of PLCγ from hour 0 to 300 hours.

Figure

**Figure A7.** The concentration change of glucose, oxygen, TGFα and VEGF with TKI treatment.

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**Figure A8.** The concentration change of TKIs shown at 60, 150, 240 and 300 hours.

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Figure

**Figure A9.** Various tumor cell numbers with TKI treatment.

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Figure

Prediction and validation of cell survival percentage during TKI treatment

**Prediction and validation of cell survival percentage during TKI treatment.** (**a**) Prediction of cell survival percentage during TKI treatment from 0 hour to 300 hours; (**b**) Cell survival percentage from simulations (**purple line**) and from in vitro experimental observations (**black line**) for t=0-96 hours.

Figure

Sensitivity analysis and model robustness

Parameter sensitivity analysis is to quantitatively discover sensitive parameters in the system. And robustness analysis is to examine whether the system is stable to modest fluctuations of these sensitive parameter values.

In this work, local parameter sensitivity analysis

where P is the parameter that is varied and MP is tumor cell's migration potential; Δ_{2}) and TGFα (_{1}). The sensitivity analysis also demonstrates that kinetic rates, _{1}, _{2}, _{3}, _{5}, _{4} and _{4}, are more critical than others in the system. Furthermore, it turns out that the developed intracellular pathway system is rather robust since all of the sensitivity coefficients are less than 1.8%.

Sensitivity analysis of EGFR signaling pathway

**Sensitivity analysis of EGFR signaling pathway.** The result shows that the migration potential of brain cancer cells is most sensitive to the initial concentration of EGFR (_{2}) and TGFα (_{1}). The result also shows us forward kinetic rates (_{1}, _{2}, _{3}, _{5}) are more important in the system and demonstrate that the developed intracellular pathway system is robust.

Based on above sensitivity analysis results, we selected 3 pathway components, TGFα (_{1}), EGFR (_{2}) and PLCγ (_{6}), and 6 kinetic rates including _{1}, _{2}, _{3}, _{5}, _{4} and _{4} as key parameters to investigate the robustness of the model to much larger ranges of parameter variations. For each of these parameters, we varied its value by 0.1-fold, 2.0 fold, 5.0 fold, 10.0 fold, 50.0-fold, and 100.0-fold of its original reference value respectively. Each time, only one of the above parameters was varied with a corresponding fold change, and all other parameters were kept fixed. We chose active cell number (

where _{
p
} is the active cell number with the varied parameter _{
0
} is the average active cell number from the 100 simulations with unvaried reference parameters.

Figure _{1}). Figure _{1} at variations between 5.0-fold to 10.0-fold. The results show that regarding TGFα and _{1}, at the variations between 0.1-fold and 7.0-fold, the absolute values of the robustness indexes are less than 0.1894. Furthermore, for some parameters, such as PLCγ, _{2}, _{3}, _{5}, _{4}, the robustness index is very small even with the variations up to 100.0-fold. Then we calculated the coefficient of variation (CV) of system outcomes which is defined as the ratio of the standard deviation to the mean of the above variational active cell numbers. At the parameter variations between 0.1-fold and 10.0-fold, the CV of system outcomes is 0.2007 which is much less than 1, indicating that the system is relatively stable to the parameter variations. These results reveal that our model is comparably robust with respect to relatively large ranges of parameter variations.

Model robustness analysis

**Model robustness analysis.** TGFα (_{1}), EGFR (_{2}), PLCγ (_{6}), and _{1}, _{2}, _{3}, _{5}, _{4} and _{4} were varied their values by 0.1-fold, 2.0 fold, 5.0 fold, 10.0 fold, 50.0-fold, and 100.0-fold of original reference values each time. The results show that the absolute values of the robustness indexes are less than 0.1894 with respect to most parameters (except TGFα and _{1}) for variations between 0.1-fold and 10.0-fold.

Detailed robustness analyses for TGFα and _{1}

**Detailed robustness analyses for TGFα and **_{1}**.** TGFα (_{1}), and _{1} were varied by 5.0-fold, 6.0 fold, 7.0 fold, 8.0 fold, 9.0-fold, and 10.0-fold of original reference values each time. The results show the absolute values of the robustness indexes are less than 0.1894 with respect to TGFα and _{1} for variations between 0.1-fold and 7.0-fold.

Discussion

We developed a multi-scale model by integrating a novel angiogenesis module into an agent-based tumor model based on a set of reaction–diffusion equations that describe the spatio-temporal evolution of the distributions of micro-environmental factors such as glucose, oxygen, TGFα, VEGF and fibronectin. These molecular species regulate tumor growth during angiogenesis. Each tumor cell is equipped with an EGFR signaling pathway linked to a cell-cycle to determine its phenotype.

Our simulations show several interesting findings. The first is that tumor cells tend to move towards blood vessels and gradually developed to a fan-shape as shown in Figure

The second interesting finding is that blood vessels tend to migrate to tumor and form a dense tree-branching vascular network. The reason is that a high VEGF gradient close to the tumor attracts endothelial cells, which in turn lead to branching of vessels in these regions.

The third interesting finding is that TKI treatment can inhibit tumor progression. The binding of TKI molecules to EGFR decreases the amount of effective EGFR, which results in low expression of PLCγ and low cell’s migration potential (Figure

The fourth interesting result is that the tumor cells' survival rate does not always decrease. This is due to the dual role of angiogenesis. Newly formed capillaries delivers a substantial amount of TKI molecules to tumor cells and blocks the EGFR signaling pathway, which lead to an inhibition of tumor growth. This in turn results in a decreased cell survival rate in the early stage of the tumor development. On the other hand, new capillaries transported a lot of glucose and oxygen to tumor cells which results in an increased cell survival rate at later stages (Figure

The sensitivity analysis reveals sensitive parameters in the EGFR signaling pathway. The robustness study confirms that our model is relatively robust and stable to fluctuations of these sensitive parameters.

Herein we used the two-dimensional

We are going to extend the model to three dimensions to simulate

The potential of our model will further increase, by incorporating more realistic biological and physical features, such as blood flow and tumor growth-induced pressure

Conclusions

This work presents a novel multi-scale agent-based brain tumor model encompassing an EGFR signaling pathway together with a related cell-cycle, an angiogenesis module and TKI treatment. It incorporates four relevant biological scales: the molecular scale, the cellular scale, the microenvironment scale and the tissue scale. At the molecular scale, a system of ordinary differential equations simulates the dynamics of the EGFR signaling pathway and the cell cycle to determine the cells' phenotypic switch at the cellular scale. We employed a set of partial differential equations to simulate the concentration changes of five extracellular chemical cues (glucose, oxygen, TGFα, VEGF and fibronectin) in the tumor micro-environmental scale. Angiogenesis was coupled into tumor growth through VEGF secreted by the tumor cells and through the glucose and oxygen permeated from the neo-vasculature at the tissue scale. Moreover, we integrated TKI treatment into EGFR signaling pathway to block the activation of EGFR.

Our simulations demonstrate that the entire tumor growth profile is a collective behaviour of its cells regulated by the EGFR signaling pathway and the cell cycle. We also discovered that angiogenesis has dual effects on TKI treatment: on one hand, neo-vasculature can deliver TKIs to decrease the tumor invasion, whereas on the other hand, it can transport a lot of nutrients ( glucose and oxygen) to tumor cells to maintain their metabolism, which results in an increase of cell survival rate at late simulation stage. There is a great similarity between the simulation results and existing

Availability and requirements

**Project name:** multi-scale agent-based brain tumor modeling project **Project home page: **
**Operating system(s):** Platform independent **Programming language:** Matlab (R2007b) **Other requirements:** None **License:** GNU GPL, FreeBSD etc. **Any restrictions to use by non-academics:** license needed.

Abbreviations

EGFR: Epidermal Growth Factor Receptor; VEGF: Vascular Endothelial Growth Factor; TKIs: Tyrosine Kinase Inhibitors; EC: Endothelial Cell; MP: Migration Potential.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

XS participated to study conception, carried out the model programming, carried out the analysis of the model and drafted the manuscript. XZ participated to study conception and improved the manuscript. LZ participated to study conception, model analysis and improved the manuscript. HT participated to study conception and helped to initiate the model programming. JB helped to study conception. CS helped to improve the manuscript. All authors read and approved the final manuscript.

Acknowledgements

This work was supported by Funding: NIH R01LM010185-03 (Zhou), NIH U01HL111560-01 (Zhou), NIH 1R01DE022676-01 (Zhou) and DoD TATRC (Zhou).

We would like to thank the members of Translational Biosystems Lab of Cornell Medical School for the valuable discussions.