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# A 3D multiscale model of cancer stem cell in tumor development

- Fuhai Li
^{1}, - Hua Tan
^{1, 2}, - Jaykrishna Singh
^{1}, - Jian Yang
^{1}, - Xiaofeng Xia
^{1}, - Jiguang Bao
^{2}, - Jinwen Ma
^{1, 3}, - Ming Zhan
^{1}Email author and - Stephen TC Wong
^{1}

**7 (Suppl 2)**:S12

https://doi.org/10.1186/1752-0509-7-S2-S12

© Li et al.; licensee BioMed Central Ltd. 2013

**Published:**17 December 2013

## Abstract

### Background

Recent reports indicate that a subgroup of tumor cells named cancer stem cells (CSCs) or tumor initiating cells (TICs) are responsible for tumor initiation, growth and drug resistance. This subgroup of tumor cells has self-renewal capacity and could differentiate into heterogeneous tumor cell populations through asymmetric proliferation. The idea of CSC provides informative insights into tumor initiation, metastasis and treatment. However, the underlying mechanisms of CSCs regulating tumor behaviors are unclear due to the complex cancer system. To study the functions of CSCs in the complex tumor system, a few mathematical modeling studies have been proposed. Whereas, the effect of microenvironment (mE) factors, the behaviors of CSCs, progenitor tumor cells (PCs) and differentiated tumor cells (TCs), and the impact of CSC fraction and signaling heterogeneity, are not adequately explored yet.

### Methods

In this study, a novel 3D multi-scale mathematical modeling is proposed to investigate the behaviors of CSCsin tumor progressions. The model integrates CSCs, PCs, and TCs together with a few essential mE factors. With this model, we simulated and investigated the tumor development and drug response under different CSC content and heterogeneity.

### Results

The simulation results shown that the fraction of CSCs plays a critical role in driving the tumor progression and drug resistance. It is also showed that the pure chemo-drug treatment was not a successful treatment, as it resulted in a significant increase of the CSC fraction. It further shown that the self-renew heterogeneity of the initial CSC population is a cause of the heterogeneity of the derived tumors in terms of the CSC fraction and response to drug treatments.

### Conclusions

The proposed 3D multi-scale model provides a new tool for investigating the behaviors of CSC in CSC-initiated tumors, which enables scientists to investigate and generate testable hypotheses about CSCs in tumor development and drug response under different microenvironments and drug perturbations.

## Keywords

- Cancer Stem Cell
- Tumor Development
- Gompertz Curve
- Tumor Angiogenic Factor
- Matrix Degrade Proteolytic Enzyme

## Background

The mechanisms of tumor initiation, progression, metastasis and drug resistance remain elusive due to the complex system of tumors. Recent studies have shown that a sub-population of tumor cells, named cancer stem cells (CSCs) or tumor initiating cells (TICs) tumor are responsible for tumor development and drug resistance [1–3]. The CSC concept is still controversial, as it is difficult to discover and validate cancer stem cells, particularly their unlimited self-renewal and differentiation capabilities [1]. However, CSCs have been being isolated from more and more cancers [2], since first discovered in the acute myeloid leukemia (AML) by using CD34^{++}/CD38^{-} biomarkers [3]. Recently, the breast CSCS were identified by using CD44^{+}CD24^{-/low} biomarkers in [4], and the colon CSCS were also reported [5]. CSCs are believed to have strong self-renewal capacity, could differentiate into heterogeneous tumor populations through asymmetric division [6, 7], and are responsible for drug resistance and metastasis [8, 9]. Reportedly, CSCs are heterogeneous with different self-renewal and tumor formation abilities, which might be caused by varying activation intracellular signaling (e.g., Wnt, Shh and Stat3) due to the diverse concentrations of external mE factors [10, 11].

However the roles of CSCs in tumor development remain unknown because of tumor complexity in multiple levels, including signaling transduction, cell-cell communication, and cell-microenvironment interactions. The mathematical simulation models have been powerful tools for understanding the tumor systems [12]. In general, the existing mathematical models of the tumor development can be grouped into three major categories: discrete, continuous and hybrid. The discrete models, e.g., cellular automata [13] and Glazier and Graner model [14], simulate cell behaviors individually with a group of rules. The continuous models employ ordinary or partial differential equations to simulate the behaviors of tumor cell populations and dynamics of mE factors [15, 16]. The hybrid models are the combination of the discrete (for modeling cells) and continuous (for modeling mE factors) models [12]. A few mathematical simulation studies have been developed to study CSCs functions in tumor development [17–20]. Whereas, the interactions between mE factors and CSCs, PCs and TCs, the impact of the heterogeneity and fraction of CSCs, have not been adequately considered in the mathematical modeling.

## Methods

### The PDE system characterizing reaction-diffusion process of mE factors

*n*), tumor angiogenic factor (TAF) (

*c*), matrix degrading proteolytic enzyme (MDE) (

*m*), extracellular matrix (ECM) (

*f*), and tissue pressure (

*p*). A system of PDEs is used to delineate the diffusion and reactions mE factors. The

*χ*

_{ Ω }is defined as follows.

where *D*_{
n
} is the diffusion rate of nutrient molecules, and ${\lambda}_{pp}^{n}$, ${\lambda}_{pa}^{n}$ denote the nutrient molecules transferring rates from pre-existing and neo-vasculature vessels. The ${\lambda}_{b}^{n}$ is the nutrient molecule binding rate to fibronectin; ${\lambda}_{u}^{n}$ is the uptake rate by cells, and it is different for specific types of cells. The χ_{Σc} is an indicator function that equals to 1 at the new generated vessels. The term (1-*p*) is used to indicate the difference of nutrient molecule transfer with different pressure, and (1-*n*) is to reflect nutrient molecules' saturation effect.

where *D*_{
c
} is the diffusion rate of TAF, $\partial {\mathrm{\Omega}}_{N}$ is the boundary of necrotic and viable regions, the $\overrightarrow{n}$ is the unit outer normal direction, ${\lambda}_{pN}^{c}$, ${\lambda}_{pV}^{c}$ are the TAF secretion rates by dying and viable tumor cells, respectively; ${\lambda}_{u}^{c}$ is the uptake rate by endothelial cells, and ${\lambda}_{d}^{c}$ is the rate of natural degradation.

Where the ${\lambda}_{p}^{f}$, ${\lambda}_{sp}^{f}$ denote fibronectin production rates by tumor and endothelial cells, respectively, and ${\lambda}_{d}^{f}$ is the ECM degradation rate by MDE.

In this equation, *D*_{
m
} is the diffusion coefficient, ${\lambda}_{p}^{m}$ and ${\lambda}_{sp}^{m}$ denote MDE secretion rates by the endothelial cells and viable tumor cells, and ${\lambda}_{d}^{m}$ is the MDE decay rate.

*λa*and

*λ*

_{ N }are the volume loss rates caused by cellular apoptosis. The first term on the right of equation (6) is the source effect, while the second term can be considered as the sink effect. The diffusion of pressure is obtained by taking the divergence operation on both sides of equation (5), and combined with equation (6):

^{nd}order total variation Runge-Kutta method [29]. The time interval, $\Delta t$, is calculated to keep the stability of the PDEs:

where $\Delta l=0.1$ is the spatial interval, ${V}_{i}$ is the function defined on the TAF, ECM, and cell velocity in [25], and $\stackrel{\u20d7}{{u}_{i}}$ is the cell velocity at *i*-th spatial point [25].

### 3D Cellular Automata

*k*:

1) Check available (empty) neighbor locations.

2) If there is no available neighbor location, go to step 8).

3) Calculate the motion probabilities to the *m (m <= 6)* available neighbor locations as: *q*_{
i
} *= n*_{
i
}*/f*_{
i
}*, i = 0, 1, ..., m*, where *q*_{
0
} means the probability of staying at the same location.

4) Denote ${q}_{k}^{\prime}={\sum}_{i=1}^{k}{q}_{i}$, and normalize them as: ${q}_{i}^{\u2033}={q}_{i}^{\prime}/{q}_{m}^{\prime}$.

6) Define ${R}_{1}=\left[0,{q}_{1}^{\u2033}\right]$, and ${R}_{i}=\left[{q}_{i-1}^{\prime},{q}_{i}^{\u2033}\right]$, *i = 1, 2, ..., m*.

7) Pick up a number *r* randomly from [0, 1], then move the cell to the *i*-the neighbor location where r belongs to.

8) Update the cell age, and then check if the cell is mature to divide. If yes, add one new cell with right cell type to a neighbor location with the above rules. Check the mE conditions to determine if the cell should enter the quiescent or death status.

### Simulation of chemotherapy

In simulation of chemotherapy, all cancer cells (CSCs, PCs, TCs) can be killed by chemo drugs with different doses. Since the diffusion and reaction processes of the drug molecules resemble that of nutrients, we use the same diffusion-reaction equation for drugs. We assume that TCs proliferation is reduced due to the effects of drugs, while CSC proliferation is accelerated as activated by the volume loss due to the drug effect, which is parallel to the normal stem cell functions [30]. We adjust the proliferation age of each cell subtype in such a way to stimulate the reaction of tumor cells to chemotherapy.

### The Gompertz curve fitting

*k*,

*b*denote the axis displacement and growth speed. The least square is used to determine the optimal parameters as follows:

Here, $\u0177\left(t\right)$ is the tumor volume growth function defined in equation (9), ${y}_{i}$ is the measured volume of tumor at time point *i*, obtained either from biological experiments or by computer simulation, and *N* is the number of available measurements.

### Measurements of tumor development

*PP*), time to reach potential (

*TtP*), average aggressive index (

*AAI*), and average fitting error (

*AFE*).

*PP*is defined as (9):

*PP*value could not predict the final volume of a tumor because of many unforeseen contributing factors when tumors grow large. However, it can be used to compare the potential volume of tumors in a relative sense.

*TtP*is estimated by solving an inverse problem of (9), that is, by searching the time when $\u0177\left(t\right)$ reaches

*PP*for the first time:

*inf*' is the

*infimum*operation.

*AAI*is represented as:

*i*, which are represented by the number of cells on the surface and the total number of cells composing the tumor respectively. The performance of the curve fitting is assessed by AFE, estimated by the least square method for (10):

where ${f}_{obj}={\displaystyle \sum _{i=1}^{N}}{\left(\u0177\left({t}_{i}\right)-{y}_{i}\right)}^{2}$.

## Results

### Simulation of tumor development under different CSCs contents

### Important parameters to tumor growth

_{c}are sensitive to the tumor growth, and some parameters proliferation abilities parameters, i.e., K

_{CCP}and K

_{PP}are also sensitive to tumor growth.

Model parameters.

Parameter Symbol | Parameter Annotation | Parameter Value | Reference |
---|---|---|---|

| Diffusion rate of Nutrient | 1.0 | P. Macklin et al. (2009) |

| Diffusion rate of TAF | 100 | Estimated |

| Diffusion rate of MDE | 1.0 | P. Macklin et al. (2009) |

${\lambda}_{u}^{n}$ | Uptake rate of Nutrient | [0.2, 0.5, 0.33, 0.67, 1, 1] | X. Zheng et al. (2005); Estimated |

${\lambda}_{b}^{n}$ | Binding rate of nutrient | 2.5·e-3 | Estimated |

${\lambda}_{pa}^{n}$ | Nutrient transfer rate from neo-vasculature | 0.05 | X. Zheng et al. (2005); Estimated |

${\lambda}_{pp}^{n}$ | Nutrient transfer rate from existing vessel | 0.01 | Estimated |

${\lambda}_{pN}^{c}$ | TAF secretion rate by dying cells | 0.05 | Estimated |

${\lambda}_{pV}^{c}$ | TAF secretion rate by viable cells | 0.004 | Estimated |

${\lambda}_{d}^{c}$ | TAF degradation rate | 0.01 | P. Macklin et al. (2009) |

${\lambda}_{u}^{c}$ | TAF uptake rate by endothelial cells | 0.025 | P. Macklin et al. (2009) |

${\lambda}_{p}^{m}$ | MDE secretion rate by viable cells | {50, 100, 150} | P. Macklin et al. (2009); Estimated |

${\lambda}_{sp}^{m}$ | MDE secretion rate by endothelial cells | 1.0 | P. Macklin et al. (2009) |

${\lambda}_{d}^{m}$ | MDE degradation rate | 10 | P. Macklin et al. (2009) |

${\lambda}_{p}^{f}$ | ECM secretion rate by viable cells | 0.1 | P. Macklin et al. (2009) |

${\lambda}_{sp}^{f}$ | ECM secretion rate by endothelial cells | 0.01 | Estimated |

${\lambda}_{d}^{f}$ | ECM degradation rate | 0.01 | P. Macklin et al. (2009) |

| Volume loss rate due to apoptosis | 0~0.00013 | Estimated |

| Volume loss rate due to necrosis | 0.25 | X. Zheng et al. (2005) |

| Nutrient dose for cell survival | {0.1, 0.17, 0.25} | X. Zheng et al. (2005); Estimated |

| Maximum drug concentration for cell survival | {0.25, 0.27, 0.375} | Estimated |

[ | CSC proliferation probabilities | {0.6, 0.25, 0.1, 0.05} | Estimated |

[ | PC proliferation probabilities | {0.25, 0.75} | Estimated |

[ | TC proliferation probability | [1- | Estimated |

| Relative proliferation ages | [1, 0.4, 1, 0.2] | Estimated |

| Maximum generations a cell can divide | [250, 50, 25] | Estimated |

| Constant for cell size scaling | 10·e-5 | Estimated |

### Tumor response to drug treatment

### Simulation of CSC self-renewal heterogeneity in tumor development

*μ*and variance ${\sigma}^{2}$; and 3) the newly formed CSCs during tumor development will obtain a mean value randomly sampled from the $G\left({\mu}_{p},\sigma \right)$, where ${\mu}_{p}$ is the mean of its parent CSC. In our simulation, we set $\mu =0.25$, 0.5, and 0.75, respectively, to indicate the relative low, medium, and high self-renewal ability of initiating CSCs (with fixed $\sigma =0.1$). The simulation starts from a single CSC to 30 days, then the same chemo-drug treatment is applied. The virtual chemo-drug treatment will be stopped when 85% of tumor cells are killed. Figure 8 shows the simulation results for the size and fraction of CSCs of tumors initiated from CSCs with different self-renewal abilities. As shown, before chemo-drug treatment, the tumor initiated from CSCs with high self-renewal ability has an average bigger tumor volume (about 1.5 fold), whereas the difference of the CSC fraction is about 2 fold. After chemo-drug treatment, the difference in the CSC fraction is significantly increased to about 3 fold, though the CSC fractions of all tumors are increased significantly comparing to those before chemo-drug treatment. The results indicated that though the pure chemo-drug treatment could reduce the size of tumor, it might also increase the aggressiveness of tumor due to the increased fraction of CSCs, especially for the tumor with the CSCs that have high self-renewal ability. Therefore, the combinations of chemo-drugs and anti-CSCs drugs are needed to achieve better treatment outcomes.

## Discussion & conclusions

Here, a multi-scale and multi-factorial computational model is established in 3D space to study the behaviors and roles of CSCs in leading tumor development. The model is implemented at three hierarchical scales (molecular, cellular, tissue scales). The molecular subsystem characterizes the diffusion and reaction processes of mE factors by using PDEs. The cellular level subsystem simulates the proliferation and migration of all cancer cells and endothelial cells, considering the availability of mE factors, with a 3D cellular automaton. The tissue level subsystem evaluates the temporal and spatial variations of tumor morphology by, four indices. The model can be conveniently expanded to a particular application to generate testable hypotheses.

The simulation studies based on the multi-scale model could provide important insights into tumor development and treatment. For example, the simulation indicated that tumors in mice model initiated by the sorted CSC population had stronger aggressiveness and proliferation potential comparing to tumors from unsorted cancer cells. Also the simulation demonstrated that the neo-vasculature could grow into the interior of a tumor, suggesting a possibility of delivering drugs via neo-vasculature to target the CSCs in the interior of tumors, besides the anti-angiogenic therapy that elicits increased local invasion and distant metastasis of tumors [32, 33]. In addition, the simulation indicated that pure chemo-drug treatment may increase the fraction of CSCs significantly, especially for the tumor with CSCs of high self-renewal ability, and consequently, the tumor residual will be more chemo-resistant and aggressive. Thus, a combination of chemo-drugs with CSC inhibition drugs would be more effective in cancer treatment without increasing tumor drug-resistance and aggressiveness.

Many parameters in our model were defined based on general understanding of tumor development through literature mining in this study, as in many other modeling studies. Despite limited experimental data used in defining the parameters, the model we proposed is still valid. It enables us to identify important CSCs behavior and interactions with interested mE factors, and to virtually test hypotheses that cannot be done in an animal model. The simulation studies based on the model can lead to new insight of CSCs in tumor development and shed light on the treatment. As more experimental data become available through our studies, the parameters can be better defined and calibrated, and the resulting model will be better predictable.

Several improvements of the proposed 3D tumor growth model will be conducted in the future work. Cancer is a complex and heterogeneous disease. CSCs from different types of cancer might have different functions and regulatory signaling pathways. With more data of cancer-specific signaling pathways, cell differentiation lineages, stromal cells, and detailed cell-cell interactions becoming available, the proposed model could be extended to study the CSCs in specific cancer types. Also the cell shape could be taken into account as it plays an important role in interactions between cells and mE factors, particularly when the cell density is high and causes shape deformation and cell-cell interaction through cell surface markers. On the other hand, relevant regulatory or signaling pathways could be integrated to refine the modeling. In addition, specific drug effects on different cell cycles and cell types could be considered.

## Declarations

### Acknowledgements

We would like to thank colleagues of the NCI-ICBP Center for Modeling Cancer Development (CMCD) at The Methodist Hospital Research Institute and Baylor College of Medicine for helpful discussions. Preliminary results of this study were published in the proceedings of IEEE ISB2012.

**Declarations**

The publication of this article has been funded by NIH U54CA149169.

This article has been published as part of *BMC Systems Biology* Volume 7 Supplement 2, 2013: Selected articles from The 6^{th} International Conference of Computational Biology. The full contents of the supplement are available online at http://www.biomedcentral.com/bmcsystbiol/supplements/7/S2.

## Authors’ Affiliations

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