Multiscale agentbased modeling on melanoma and its related angiogenesis analysis
 Jun Wang†^{1},
 Le Zhang†^{1, 2}Email author,
 Chenyang Jing†^{1},
 Gang Ye^{3},
 Hulin Wu^{2},
 Hongyu Miao^{2},
 Yukun Wu^{4} and
 Xiaobo Zhou^{5}Email author
DOI: 10.1186/174246821041
© Wang et al.; licensee BioMed Central Ltd. 2013
Received: 28 April 2013
Accepted: 18 June 2013
Published: 21 June 2013
Abstract
Background
Recently, melanoma has become the most malignant and commonly occurring skin cancer. Melanoma is not only the major source (75%) of deaths related to skin cancer, but also it is hard to be treated by the conventional drugs. Recent research indicated that angiogenesis is an important factor for tumor initiation, expansion, and response to therapy. Thus, we proposed a novel multiscale agentbased computational model that integrates the angiogenesis into tumor growth to study the response of melanoma cancer under combined drug treatment.
Results
Our multiscale agentbased model can simulate the melanoma tumor growth with angiogenesis under combined drug treatment. The significant synergistic effects between drug Dox and drug Sunitinib demonstrated the clinical potential to interrupt the communication between melanoma cells and its related vasculatures. Also, the sensitivity analysis of the model revealed that diffusivity related to the microvasculatures around tumor tissues closely correlated with the spread, oscillation and destruction of the tumor.
Conclusions
Simulation results showed that the 3D model can represent key features of melanoma growth, angiogenesis, and its related microenvironment. The model can help cancer researchers understand the melanoma developmental mechanism. Drug synergism analysis suggested that interrupting the communications between melanoma cells and the related vasculatures can significantly increase the drug efficacy against tumor cells.
Keywords
Microenvironment Drug synergism Agentbased model Multiscale Melanoma AntiangiogenesisBackground
Melanoma is the most malignant skin tumor, causing the majority (75%) of deaths related to skin cancer[1]. Conventional diagnoses and treatments of Melanoma consist of surgical removal, chemotherapy, immunotherapy, and radiation therapy. The interferon can elongate the lifetime of the patient, but it can neither greatly increase the survival rate nor be a standard adjuvant treatment for melanoma[2].
Recently, several molecular drugs, such as doxorubicin and etoposide, were developed to treat melanoma cancer[3]. However, because of absorption, distribution, metabolism, or toxicity (ADME) problems, most molecular drugs did not work as well in vivo as in vitro. Many synergistic drug delivery methods have been developed to increase the drug effect in vivo, but it is difficult to quantitatively evaluate their performance. Thus, one of the aims of this study is to develop such indexes and tools that can estimate drug effects on melanoma cells.
It is known that angiogenesis[4–7] is a significant transforming phase in tumor growth. A drug’s distribution inside a tumor is highly heterogeneous due to the tumor vasculature’s tortuous, chaotic structure compared to fine, nearly parallel blood vessels in normal tissue. Drugs delivered to tissues will not only change the behavior of melanoma cells (secretion of cytokines, proliferation, differentiation, apoptosis, or migration) in the intracellular drugtriggered cell division process, but also inhibit the development of new capillary sprouts by preventing sprouts from receiving vascular endothelial growth factors (VEGF). In turn, inadequate glucose and oxygen transported from the blood vessel will drive even more melanoma cells towards apoptosis. Therefore it is of great necessity to take tumorinduced angiogenesis into consideration and simulate the irregular vasculature inside tumor in order to further study the drug distribution and drug therapeutic effects.
Many mathematical models[8–19] have been proposed to address the current challenges mentioned above. These models studied one or more phases of cancer progression, including tumor growth, angiogenesis, and drug treatment, with the purpose of better understanding the pathophysiology of cancer, mechanisms of drug resistance, and the optimization of treatment strategies. Although biologists have already obtained many experimental data sets at the molecular, cellular, microenvironmental and tissue levels, only a few scientists have integrated these data into a multiscale platform to investigate the tumor progression with regard to its related angiogenesis and drug treatment. Studies on the antiangiogenesis drug effects and the drug combination treatment responses are still rare.
Hence, this research presents a 3D multiscale agentbased model to investigate the role of the tumorangiogenesis interactions in melanoma tumor progression by extending our previously welldeveloped 2D agentbased tumor growth models[20–22]. The multiscale system is comprised of intracellular, intercellular, and tissue levels to describe the melanoma growth with angiogenesis. As a rule based model, this study developed a set of rules to determine the melanoma cell’s phenotypic switch. These rules not only underline the migration of endothelial cells and the branching of vessel sprouts, but can also be more easily integrated into the agent based tumor growth model than previous Hybrid DiscreteContinuum (HDC) rules[23]. The model also can be employed as the test bed to predict the in vivo tumor responses to the combined drug pair: one for antiangiogenesis and the other for the tumor.
In general, the multiscale model can not only simulate melanoma tumor expansion with related angiogenesis, but also explore the best drug combination for tumor treatment and the dual role of angiogenesis (transporting both nutrient and drugs).
Mathematical models
In order to describe tumor growth with angiogenesis and study melanoma’s response to given drug pairs, our model defines two types of agents: the melanoma cell and the endothelia cell. The melanoma cell and the endothelia cell agents represent the progression of tumor and vasculature, respectively. The aforementioned multiscale model consists of three biological levels: the intracellular, intercellular, and tissue levels. The intracellular level describes the fundamental mechanism for cell’s phenotypic switch. The intercellular level bridges the tissue and intracellular scale as follows. (a) The vasculature delivers oxygen, cytokine, and glucose to the tumor microenvironment in the tissue level; (b) The melanoma cells uptake the glucose for metabolism as well as switch the phenotype under the stimulation of specific cytokine in the intercellular scale; (c) In turn, the inadequate glucose and oxygen will stimulate the tumor cell to secrete the VEGF in the intercellular level to induce angiogenesis. In the tissue level, blood vessel sprouts migrate and branch via tip endothelial cells’ migration in response to the diffused VEGF and drugs.
Initialization
Intracellular level: the phenotypic switch of melanoma cells
Apoptosis
where A_{ sDox }and A_{ sglucose } denote the average drug (Dox, which is a cytotoxic drug directly to melanoma cell) and glucose concentrations on the current site and its Von Neumann neighbors, respectively. w_{ 1 }, w_{ 2 } are the regulatory factors.
where λ_{ 2 } is the normal proliferation rate of the melanoma cell equal to the reciprocal of average proliferation time of the cell.
with a die C_{ rand } ∈ [0, 1), if the die C_{ rand } falls into the interval [0, p_{ prol }), the cell enters the cell cycle and starts to proliferate.
Migration
If the cell is neither in the cell cycle nor dividing, it will migrate. The detail of migration will be discussed in the next section.
Quiescence
After the cell determines its phenotype, it will look for a free place that is of least resistance, most permission, and highest attraction[24] to divide or migrate. The cell will enter a reversible quiescent state in the absence of a free space.
Intercellular level
Three major extracellular microenvironmental factors, such as glucose, VEGF, and drugs, are discussed in this model. A set of reaction–diffusion equations describes the diffusion, degrading, and uptake of these factors.
where G_{ ijk }(t + 1) is the glucose concentration on the location P_{ ijk } in the (t + 1) time step, λ_{ g } is the diffusion constant of glucose. are the glucose concentrations of P_{ ijk }’s at its six immediate neighbors (Von Neumann neighbors) in the current time step. The time dependent characteristic functions χ_{ endo } (t, P_{ ijk }) and χ_{ tumor }(t, P_{ ijk }) relate to the occurrences of an endothelia cell or a melanoma cell at P_{ ijk }. If a related cell is located at P_{ ijk }, the value of the function χ equals to 1; otherwise 0. Pe_{ g } is the vessel permeability for glucose. U_{ g } represents the glucose uptake rate of melanoma cell. D_{ g } represents the natural decay rate of glucose.
Where V_{ ijk }(t + 1) is the VEGF concentration at the location P_{ ijk } in the (t + 1) time step, λ_{ v } is the diffusion constant of VEGF. are the VEGF concentrations of P_{ ijk }’s at its six immediate neighbors in the current time step. Se_{ v } is the secretion rate for VEGF. Pe_{ v } represents the vessel permeability rate of VEGF. D_{ v } represents the natural decay rate of VEGF.
where DR_{ ijk }(t + 1) is the drug (Dox or Sunitinib ) concentration at the location P_{ ijk } in the (t + 1) time step, λ_{ d } was the diffusion constant of drug. are the drug concentrations of P_{ ijk }’s at six immediate neighbors in the current time step. Pe_{ d } is the vessel permeability for drug. U_{ d } represents the drug uptake rate.
 1)
Since the tumor cell always looks for a place with more nutrition to migrate to or to deliver its offspring to, we use the mean (M _{ g }) and the standard deviation (σ_{g}) of the glucose concentrations on the place and its Moore neighbors [27] to locate candidate locations. Here, G(P ^{ l } _{ ijk }) represents the glucose concentration of the lth Moore neighbors of the current site. If G(P ^{ l } _{ ijk }) − M _{ g } > 3σ _{g,}, we consider it as an abnormally high nutrition location for a tumor cell to migrate to or deliver its offspring to.
 2)If G(P ^{ l } _{ ijk }) − M _{ g } ≤ 3σ _{ g }, the model needs to evaluate all candidate locations nearby. All candidate locations were ranked through Equation 8.(8)where A^{ l }_{ mglucose } is the average glucose concentration of the lth candidate site and its Moore neighbors of this site. A^{ l }_{ mDox } is the average Dox drug concentration of the lth candidate site and its Moore neighbors of this site. w_{ 3 } is the regulator factor. The tumor cell always prefers a location that has a high nutrition concentration (A^{ l }_{ mglucose }), a low drug concentration (A^{ l }_{ mDox }), and few neighborhoods (V_{ l }). The preference of neighborhoods (V_{ l }) is denoted by Equation 9(9)Ranks of candidates were normalized as Equation 10.(10)Normalized ranks formed the scale as Equation 11(11)
S is an ordered set of S_{ l }. Each S_{ l } is a region in the [0,1] and relates to the lth candidate site. The die casting generates a random valued ∈ [0, 1). If d falls in S_{ l }, the candidate location relates to the will be chosen as the next migration or proliferation stop.
 3)
If no space is available, the cell will become reversible quiescent.
Tissue level
The starving melanoma cells secrete VEGF to induce angiogenesis and the induced vasculature transports nutrient for the tumor growth in the tissue level. Here, we employ the motion of the tip individual endothelial cell (“EC agent”) to represent vasculature progression.
 1)
At each time step, each EC agent evaluates the VEGF concentration in its surrounding tissue. If there is no VEGF, the EC agent becomes quiescent.
 2)Degeneration: The purpose of the drug Sunitinib is to inhibit tip endothelia cell’s ability to receive the VEGF signal as well as increase the apoptosis rate of tip endothelia cell. The threshold of a tip endothelial cell’s apoptosis rate (Apop _{ endorate }) is computed by Equation 12.(12)where λ_{ 3 } is the normal death rate of the endothelia cell and λ_{ 4 } denotes the impact of the Sunitinib which is described by Equation 13.(13)
where A_{ sSuni }and A_{ sVEGF } denote the average Sunitinib drug and VEGF concentrations, respectively, on the current site and its Von Neumann neighbors. w_{ 4 },w_{ 5 } are the regulatory factors. At each time step, a uniformly random number is generated by the die function. If it is less than the apoptosis threshold, the endothelia cell becomes apoptotic and its parent cell is set as the tip endothelia cell.
Progression (migration): The living tip endothelia cell will proliferate or branch. The tip cell usually looks for the location with higher VEGF to branch to or proliferate to. The mean value m_{ svegf } and the standard deviation σ_{ svegf } of the VEGF concentration on the cell’s current site and its von Neumann neighbors are used to determine behavior of the tip endothelia cell P(i,j,k). Here, V(P^{ l }_{ ijk }) represents the VEGF concentration of the l th von Neumann neighbor of the current site. If V(P^{ l }_{ ijk }) − m_{ svegf } > 3σ_{ svegf }, we consider the VEGF is so strong that the blood vessel will directly grow toward this direction. If there are more than one candidate directions that meet the condition, the blood vessel will randomly select a direction to grow toward.
If the V(P^{ l }_{ ijk }) − m_{ svegf } ≤ 3σ_{ g }, the blood vessel tended to search valid spaces to branch.
 3)Branching: For each EC agent, which tends to branch, their Moore neighbors are employed as candidate locations and ranked by Equation 14.(14)
where A^{ l }_{ mvegf } and A^{ l }_{ mSuni } are the average VEGF and average Sunitinib drug concentrations of the lth candidate site and its Moore neighbors of this site. w_{ 6 } is the regulator factor. The endothelia cell always moves to a location with high VEGF, low drug concentration, and low crowdedness (V_{ l }) as described in Equation 9. All R_{ lendo } were normalized by Equation 10. All normalized ranks were incorporated to form a scale S as specified by Equation 10. The die casting generates two random value sd_{1} ∈ [0, 1), d_{2} ∈ [0, 1). If d_{ 1 }, d_{ 2 } fall in S_{ l1 }, S_{ l2 }, the candidate location which relate to the , will be chosen as the branching sites. If both d_{ 1 }, d_{ 2 } fall in the same region, the algorithm will repeat the die casting process.
 4)
If no space is available, cells would remain in a reversible quiescent state and try again in the next round.
This multiscale agent based melanoma cancer model with angiogenesis is summarized as follows (Figure 4). At the intracellular level, it employs exponential functions (Equations, 1–4) to describe the phenotypic (migration, proliferation, or apoptosis) switch of the cancer and endothelia cells. At the intercellular level, a set of reaction–diffusion equations (Equations, 5–7) is employed to describe the spatial concentration changes of glucose, VEGF, and drugs. Cancer cells compete for the best location in the 3D extracellular matrix in order to migrate or proliferate depending on the gradient of glucose, drugs, and cell density (Equations, 8–11). At the tissue level, the spatial concentration distributions of VEGF and drug concentrations play an important role to impact the tip endothelial cells’ migration and sprout branching (Equations, 12–14). In turn, the dynamic vasculature at the tissue level remodels the tumor microenvironment by changing the important factors (spatial concentration distributions of glucose and drugs) in the intracellular level. And the behaviors of melanoma cells (secretion of cytokines, proliferation, migration, or apoptosis) are greatly influenced by these changes at the intracellular level. The parameters of the model are listed in Table 1.
The parameters of the ABM
Symbol  Variable  Initial value  Reference 

λ _{ 0 }  Normal death rate of melanoma cell  0  [28] 
w _{ 1 }  Regulatory factor of Equation 1  0.01  Estimated 
w _{ 2 }  Regulatory factor of Equation 1  1  Estimated 
λ _{ 2 }  Initial proliferation rate of melanoma cell  32 h (16.9 h47.3 h)  [28] 
λ _{ g }  Diffusion constant of glucose  6.7 (5.27.2) × 10^{7} cm^{2} /s  
Pe _{ g }  Permeability for glucose  3 × 10^{5} (0.27)cm/s  
U _{ g }  Glucose uptake rate of melanoma cell  0.28 mmol/h  [22] 
D _{ g }  Natural decay rate of glucose  0.2 (0.10.4)  Estimated 
λ _{ v }  Diffusion constant of VEGF  2.9 (1–9.4) × 10^{7} cm^{2}/ s  
Se _{ v }  Secretion rate for VEGF  0.6 (0.61) nM/h  [32] 
Pe _{ v }  Permeability for VEGF  0.1 × 10^{4} cm/s  
D _{ v }  Natural decay rate of VEGF  0.2 (0.10.4)  Estimated 
λ _{ d }  Diffusion constant of drug  5.18 (1–10) × 10^{7} cm^{2} /s  
Pe _{ d }  Permeability for drug  3 (0.27) × 10^{5} cm/s  
U _{ d }  Drug uptake rate  0.2 (0.10.4)  Estimated 
w _{ 3 }  Regulator factor of Equation. 8  0.01  Estimated 
λ _{ 3 }  Normal death rate of endothelia cell  0  [31] 
w _{ 4 }  Regulatory factor of Equation 13  0.01  Estimated 
w _{ 5 }  Regulatory factor of Equation 13  1  Estimated 
W _{ 6 }  Regulatory factor of Equation 14  0.01  Estimated 
Results
We have implemented the above model in the VC++ programming environment. It includes a 3D melanomaangiogenesis interaction model and its related drug combination treatment. We can employ this tool to predict the responses of melanoma and its related angiogenesis under drug combination treatment.
Volumetric growth dynamics
We measured the tumor system’s (total) volume by counting the number of the lattice sites occupied by a tumor cell regardless of its phenotype, hence lumping together both proliferative and migratory expansion.
Tissue scale behavior
Phenotypic behavior
Combined drug effects on melanoma treatment
Cell dynamics change under drug treatment
Drug effect test
Parameter sensitivity analysis
Spearman rankorder correlations and pvalues between model parameters and simulation outcomes
Total melanoma cells  Active melanoma cells  Endothelia cells  

Parameter  Spear ρ  pvalue  Spear ρ  p value  Spearρ  pvalue 
λ _{ 2 }  −0.1000  0.7698  0.9091  1.0559 E4  0.1091  0.7495 
λ _{ g }  0.1818  0.5926  0.1455  0.6696  −0.3545  0.2847 
Pe _{ g }  0.9818  8.4031 E8  0.9818  8.4031 E8  0.9000  1.5997 E4 
D _{ g }  −0.7273  0.0112  −0.9613  2.4489 E6  −0.1636  0.6307 
λ _{ v }  0.3455  0.2981  −0.0182  0.9577  0.3818  0.2466 
Se _{ v }  0.5550  0.0770  0.6320  0.005  0.6640  0.0260 
D _{ v }  −0.1273  0.7092  0.31434  0.3465  0.1182  0.7293 
λ _{ d }  0.9818  8.4031 E8  0.9431  1.3526 E5  0.9727  5.1422 E7 
Pe _{ d }  −0.0909  0.7904  −0.0365  0.9153  0.6182  0.0426 
U _{ d }  0.9909  3.8406 E9  0.9677  1.0925 E6  1  0 
Discussion and conclusions
This study proposed a 3D multiscale agentbased cancer model by integrating a novel angiogenesis module into a tumor growth module. The major aims of the research are investigating the relationship between the melanomainduced angiogenesis, melanoma development, as well as exploring the optimum synergistic drug combinations to treat melanoma cancer.
The inadequate nutrition in the microenvironment will decrease the tumor proliferation rate (Figure 8) and make the melanoma cell undergo apoptosis (Figure 5 and Figure 7). Starving melanoma cells will release VEGF to induce the angiogenesis for nutrition. In turn, the angiogenesis will promote the tumor growth (Figure 6). This study developed such a platform that can estimate optimum drug dose combinations for tumor treatment. Figure 9 and Figure 10 intuitively showed that the antiangiogenesis drug (Sunitinib) has much better effect than the tumorspecific cytotoxic drug (Dox), which directly kills the melanoma cells. Moreover, the synergistic use of both Sunitinib and Dox can significantly decrease melanoma progression and inhibit tumorinduced angiogenesis, since the drug combination therapy can not only kill the cancer cells by increasing the apoptosis rate, but also inhibit the cancer cells’ ability to obtain the enough nutrients by interrupting communication between the cancer cells and the vasculature. The Drug effect test based on the Loewe drug combination analysis also strongly suggested that combination of cytotoxic drug (Dox) and the angiogenesis inhibitor (Sunitinib) are of high clinical potentials. Classical Loewe combination analyses often use isobole at effect level 50 (E50) as standard index, which is convenient to monitor in animal models and clinical chemotherapy evaluations. Due to the advantages of simulations, the E100 isobole was employed to evaluate the performance of drug combinations in killing all melanoma cells in a given treatment time. Simulations of the combination effects indicated the drug combinations could successfully kill almost all melanoma cells in the given treatment time (the red lines in Figure 11a and b). When it was evaluated by the E100 isobole against the melanoma cells, the simulations also suggested strong synergistic effects (the red line in Figure 11b is lower than the Loewe additivity criteria dashed line). As shown in Figure 11b, without the aid of Sunitinib, Dox alone cannot extinguish all melanoma cells and thus enabling the disease to relapse soon. Taken together, angiogenesis is highly targetable during melanoma treatment, and inhibitors of the interactions between cancer initiating cells and their related angiogenesis are promising codrugs for traditional cytotoxic agents. The sensitivity analysis not only explored the high correlation between simulation outcomes and blood vessel delivery rates of glucose and drugs, but also demonstrated that interrupting communication between the tumor and its related angiogenesis can significantly amplify the effect of the treatment.
This is the first time a 3D multiscale agentbased cancer model was employed to describe the communication between the melanoma and the vasculature around the tumor and investigate how to employ antiangiogenesis drugs to cure melanoma by breaking this communication. This study also indicated that angiogenesis plays a very important role in the transportation of nutrients for the tumor growth. Drug synergism analysis indicated that inhibiting communications between melanoma cells and their related vasculature could increase the efficacy of the treatment, decrease the tumor progression, and finally reduce the cancer cell survival rate. In the distant future, we are going to develop a predictable cancer model by considering more realistic biological and physical data and features, such as blood flow, the influence of focal adhesion kinases, complicated signaling pathway, and the oxygen pressure[41].
Notes
Abbreviations
 ABM:

Agent based model
 ECM:

Extracellular matrix
 VEGF:

Vascular endothelial growth factor
 Dox:

Doxorubicin.
Declarations
Acknowledgements
This work has been supported by the Natural Science Foundation of China under Grant No.61101234 and the Chinese Recruitment Program of Global Youth Experts, in part by USA NIH grants P30AI078498, HHSN272201000055C and U01 CA16688601.
Authors’ Affiliations
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