Developing a multiscale, multi-resolution agent-based brain tumor model by graphics processing units
- Le Zhang†^{1, 2}Email author,
- Beini Jiang†^{1},
- Yukun Wu^{3},
- Costas Strouthos^{4},
- Phillip Zhe Sun^{5},
- Jing Su^{6} and
- Xiaobo Zhou^{6}Email author
DOI: 10.1186/1742-4682-8-46
© Zhang et al; licensee BioMed Central Ltd. 2011
Received: 1 November 2011
Accepted: 16 December 2011
Published: 16 December 2011
Abstract
Multiscale agent-based modeling (MABM) has been widely used to simulate Glioblastoma Multiforme (GBM) and its progression. At the intracellular level, the MABM approach employs a system of ordinary differential equations to describe quantitatively specific intracellular molecular pathways that determine phenotypic switches among cells (e.g. from migration to proliferation and vice versa). At the intercellular level, MABM describes cell-cell interactions by a discrete module. At the tissue level, partial differential equations are employed to model the diffusion of chemoattractants, which are the input factors of the intracellular molecular pathway. Moreover, multiscale analysis makes it possible to explore the molecules that play important roles in determining the cellular phenotypic switches that in turn drive the whole GBM expansion. However, owing to limited computational resources, MABM is currently a theoretical biological model that uses relatively coarse grids to simulate a few cancer cells in a small slice of brain cancer tissue. In order to improve this theoretical model to simulate and predict actual GBM cancer progression in real time, a graphics processing unit (GPU)-based parallel computing algorithm was developed and combined with the multi-resolution design to speed up the MABM. The simulated results demonstrated that the GPU-based, multi-resolution and multiscale approach can accelerate the previous MABM around 30-fold with relatively fine grids in a large extracellular matrix. Therefore, the new model has great potential for simulating and predicting real-time GBM progression, if real experimental data are incorporated.
Background
Glioblastoma multiforme (GBM) is the most common and aggressive brain cancer [1, 2]. Statistics show that it has the worst prognosis of all central nervous system malignancies [3, 4]. However, with the resolution of functional magnetic resonance imaging (fMRI) [5, 6], currently limited to around 2-3 mm, even the most experienced clinical personnel cannot accurately forecast GBM progression. The difficulties of making such forecasts motivated computational biologists to develop multiscale mathematical models to explore the expansion and invasion of GBM[7–9].
Cancer behaves as a complex, dynamic, adaptive and self-organizing system [10], and agent-based models (ABM) are capable of describing such a system as a collection of autonomous and decision-making agents, which represent the cells. Therefore, computational biologists hope that with the ABM approach they can surpass the current limitations of imaging technology and predict tumor progression [11–16]. Our previous studies [15, 16] developed various multiscale ABMs to simulate GBM progression. In these models, a cell's intracellular epidermal growth factor receptor (EGFR) signaling pathway is stimulated by a chemoattractant (such as transforming growth factor α (TGFα)), which diffuses at the tissue level. We also assumed that the transient rate of change of phospholipase Cγ (PLCγ ), an important molecule in the EGFR pathway, will result in cancer cell migration, whereas a smooth rate of change of PLCγ will result in cancer cell proliferation [11, 12, 15, 16]. At the intercellular scale, the behaviors of cells (such as the autocrine or paracrine secretion of chemoattractants and migration or proliferation phenotypes) remodel the tumor microenvironment and affect the overall tumor dynamics at the tissue level.
An important advantage of multiscale agent-based modeling (MABM) [15, 16] is that we can employ multiscale analysis to investigate the incoherent connections among various scales. For example, we can depict the intracellular (molecular) profiles that lead to phenotypic switches at any cell's dynamic cross points (migration cell number crosses with proliferation cell number) [15] or in the interesting tumor regions [16]. Thus, MABM models [15–17] can be used as tools for generating experimentally testable hypotheses. The consequent validation experiments may reveal potential therapeutic targets.
Though MABM approaches have a great potential for investigating GBM progression, their complexity necessitates immense computational resources [15, 17], which becomes forbidding for real-time simulations of spatio-temporal GBM progression. In fact, two problems prevent MABM doing real-time simulation. The first is that the computation time required for intracellular pathway computing for cancer cells will become huge, since a real cancer system may consist of millions of cells. The second is that it is impossible to employ a conventional sequential numerical solver to model the real-time diffusion of chemoattractants in a large extracellular matrix (ECM) with relatively fine grids.
To overcome the computation time problems, this study incorporates a graphics processing unit (GPU)-based parallel computing algorithm [18] into a multi-resolution design [16] to speed up the previous MABM[15, 17]. The multi-resolution design [16] classified the cancer cells into heterogeneous and homogeneous clusters. The heterogeneous clusters consisted of migrating and proliferating cancer cells in the region of interest, whereas the homogeneous clusters comprised dead or quiescent cells. The limited computational resource was concentrated on the heterogeneous clusters to investigate the molecular profiles of migrating and proliferating cancer cells, while the quiescent and dead cells in the homogeneous clusters were treated with less of the resource. The GPU-based parallel computing algorithm can not only model the diffusion of chemoattractants in a large ECM with relatively fine grids in real time, but also process computing queries concerning the intracellular signaling pathways of millions of cancer cells in a real cancer progression system.
The results presented in this paper demonstrate that the GPU-based multi-resolution MABM has certain novel features that can help cancer scientists to explore the mechanism of GBM cancer progression. First, it is able to simulate real-time cancer progression in a large ECM with relatively fine grids. Second, since multiscale analysis [15, 17] can reveal the correlations between GBM tumor progression and molecular concentration changes, we can tell which molecular species are the important biomarkers that impact tumor progression. Third, a multi-resolution design [16] not only allows us to visualize cancer progression by displaying all the cancer cell clusters in the tissue, but also enables us to track each cancer cell's trajectory.
In the following sections, we will introduce the previously-developed multiscale and multi-resolution ABM, describe how to use GPU to accelerate the simulation of the model, and finally illustrate the advantages of the model that can be used to analyze important biomarkers to inhibit GBM expansion and predict GBM progression.
Mathematical model
Multiscale perspective
Intracellular scale
[16] - (1) Components of the EGFR gene-protein interaction network, (2) Kinetic equations employed to describe the reactions between the EGFR species, (3) Coefficients of the EGFR gene-protein interaction network taken from the literature
(1) | |||
---|---|---|---|
Symbol | Molecular variables | Initial Condition | |
X _{0} | Glucose | 25mM | |
X _{1} | TGFα | 9010.55nM | |
X _{2} | EGFR | 100nM | |
X _{3} | TGFα -EGFR | 0nM | |
X _{4} | (TGFα -EGFR)^{2} | 0nM | |
X _{5} | TGFα -EGFR-P | 0nM | |
X _{6} | PLCγ | 10nM | |
X _{7} | TGFα-EGFR-PLCγ | 0nM | |
X _{8} | TGFα-EGFR-PLCγ-P | 0nM | |
X _{9} | PLCγ-P | 0nM | |
X _{10} | PLCγ-P-I | 0nM | |
(2) | |||
dX_{1}/dt=-v_{1} | (1) | v_{1 =} k_{1}X_{1}X_{2}-k_{-1}X_{3} | (11) |
dX_{2}/dt=-v_{1} | (2) | v_{2 =} k_{2}X_{3}X_{3}-k_{-2}X_{4} | (12) |
dX_{3}/dt = v_{1}-2v_{2} | (3) | v_{3 =} k_{3}X_{4}-k_{-3}X_{5} | (13) |
dX_{4}/dt = v_{2}+v_{4}-v_{3} | (4) | v_{4 =} V_{4}X_{5}/(K_{4}+X_{5}) | (14) |
dX_{5}/dt = v_{3}+v_{7}-v_{4}-v_{5} | (5) | v_{5 =} k_{5}X_{5}X_{6}-k_{-5}X_{7} | (15) |
dX_{6}/dt = v_{8}-v_{5} | (6) | v_{6 =} k_{6}X_{7}-k_{-6}X_{8} | (16) |
dX_{7}/dt = v_{5}-v_{6} | (7) | v_{7 =} k_{7}X_{8}-k_{-7}X_{5}X_{9} | (17) |
dX_{8}/dt = v_{6}-v_{7} | (8) | v_{8 =} V_{8}X_{9}(K_{8}+X_{9}) | (18) |
dX_{9}/dt = v_{7}-v_{8}-v_{9} | (9) | v_{9 =} k_{9}X_{9}-k_{-9}X_{10} | (19) |
dX_{10}/dt = v_{9} | (10) | ||
(3) | |||
Forward rate (s^{-1}) | Reverse rate (s^{-1}) | Michaelis constants (nM) | Maximal enzyme rates (nM s^{-1}) |
k_{1} = 0.003 | k_{-1} = 0.06 | K_{4} = 50 | V_{4} = 450 |
k_{2} = 0.01 | k_{ - }_{2} = 0.1 | K_{8} = 100 | V_{8} = 1 |
k_{3} = 1 | k_{-3} = 0.01 | ||
K_{5} = 0.06 | k_{-5} = 0.2 | ||
K_{6} = 1 | k_{-6} = 0.05 | ||
K_{7} = 0.3 | k_{-7} = 0.006 | ||
k_{9} = 1 | k_{-9} = 0.03 |
where $\frac{d\left(PLC\gamma \right)}{dt}$ denotes the rate of change of phosphorylated PLCγ concentration, and Avg describes the average rate of change of phosphorylated PLCγ of cells switching phenotype at the time step.
Intercellular scale
where T_{ ij } denotes the attractiveness of location (i,j), E_{ ij } is the concentration of TGFα at location (i,j), and ε_{ ij } ~N[μ,σ^{ 2 } ] is an error term that is normally distributed with mean μ and variance σ^{ 2 } . The parameter Ψ takes on a positive value between zero and one and represents the precision of search. Here we choose Ψ = 0.7 on the basis of previous works [9, 13, 15, 17].
Tissue scale
where Y is the concentration of chemoattractant, D is the diffusivity of chemoattractant, t is the time step, and U and S are respectively the cell's chemoattractant uptake and secretion rates.
In general, the multiscale approach incorporates three different scales: intracellular, intercellular and tissue. The intracellular gene-protein interaction pathway affects the intercellular scale by determining a cell's phenotype. In turn, the chemoattractants diffusing at the tissue level affect both the intracellular and tissue scales by stimulating a cell's molecular pathway and remodeling the tumor cells' microenvironment. An important advantage of the multiscale ABM approach is that it can be used to analyze and expose the incoherent relations among the different scales. Such analysis may result in experimentally testable hypotheses. However, owing to the complexity of these types of models, real-time simulations of systems with realistic sizes are extremely difficult because forbiddingly huge computation is required. For example, it took approximately seven computing hours on a high performance CPU (IBM Bladecenter machine, dual-processor, 32-bit Xeons ranging from 2.8-3.2 GHz, 2.5 GB RAM, and Gigabit Ethernet) to simulate approximately twenty thousand cells (final state) in a 100*100*100 extracellular matrix with relatively coarse grids (around 20 μm) for 20 days [15, 17]. Therefore, a realistic in vitro tumor simulation with millions of cells on relatively fine grids would require an immense simulation time. To minimize the simulation time and simulate real-time cancer progression, a multi-resolution design [21] was incorporated into the multiscale ABM.
Multi-resolution perspective
A multi-resolution design is used to relieve the huge computational resource demand of MABM and visualize tumor progression at various resolutions. In this approach, more computational resource is allocated to heterogeneous regions of the cancer and less to homogeneous regions. In summary, the aim of the multi-resolution approach is to reduce the simulation computing time by sacrificing the accuracy of the simulated results compared with the original MABM.
GPU-based parallel computing algorithm
A modern GPU is essentially a massively-parallel, explicitly programmable co-processor consisting of hundreds of programmable processors with a natural programming hierarchy [23]. This hierarchy can mimic the bottom-up organization of ABM models by setting the intracellular and intercellular scale computations at the bottom (communicating locally via the fast shared memory of the GPU) of the hierarchy on the GPU while coordinating the logic control module of the model on the CPU. Modern GPU programming is sufficiently flexible to take advantage of the multi-resolution design by dynamically focusing GPU computing resources on the currently heterogeneous regions of the cancer. Fermi GPUs (GTX 480) have up to 480 processors, which can be bundled together to provide thousands of individual GPU processors. This system can provide significant benefits towards scaling feasible MABM model computations, which help us to approach the target of simulating realistic tumor growth problems [23]. To speed up the current multi-resolution MABM, we parallelized both the chemoattractant diffusion module and the intracellular EGFR pathway module.
Speeding up the computation of the intracellular EGFR molecular pathway module
Speeding up the diffusion module
Results
Our code was written in Microsoft Visual Studio C++[34, 35] and NVCC[36] programming languages, We ran the simulation 10 times with different random number seeds (1-100 time steps, one time step being equivalent to one hour) on a Dell workstation with Fermi GeForce GTX 480[37–39] and obtained the average result. The initial condition is described in Table 1.
Multiscale analysis
Advantages of the multi-resolution approach
Visualization of cancer progression at various resolutions
Speed-up of the multiscale and multi-resolution ABM by GPU
GPU-based MABM versus sequential MABM
GPU-based multi-resolution MABM versus GPU-based MABM
The GPU- based MABM is accelerated further when the multi-resolution design is incorporated into it. Figure 12(b) shows that the GPU-based multi-resolution MABM has a better performance than the GPU-based MABM.
Speeding up the multi-resolution MABM model with a large cell population
As indicated in Figure 12(b), the GPU- based parallelized ODE solver cannot exhibit its advantage in significantly increasing the performance of the code when the cell population is small, because the diffusion module consumes most of the computational resource. However, Figure 12(c) demonstrates that as the tumor cell number increases on a 514 by 514 high-resolution lattice, the GPU- based parallelized ODE can significantly increase the performance of the model.
Discussion and Conclusions
Recently, a variety of cancer research reports have indicated that the EGFR pathway plays an important role in the directional motility [40–42], mitogenic signaling [43, 44] and phenotypic switching of cancer cells [20, 45]. In particular, Dittmar et al. [20] demonstrated that PLCγ , a molecular species in the EGFR downstream pathway [46, 47], is transiently activated in breast cancer cells to a greater extent during migration. In addition, experimental observations of GBM suggested that at the same time interval, migrating tumor cells seldom proliferate and proliferating cells seldom migrate [19]. On the basis of these experimental results, Athale et al. [11] assumed that if the percentage rate of change of the phosphorylated PLCγ concentration exceeds a pre-specified threshold, GBM cells will migrate; otherwise, they will proliferate. Using this assumption, Athale et al. [11, 12] and Zhang et al. [15] developed several in silico 2D and 3D MABMs to investigate how perturbations in the intracellular EGFR gene-protein network affect the progression of the entire tumor at the intercellular and tissue scales.
However, the above works [11, 12, 15] were limited by the available computational resources. As indicated by previous research [16], simulating 3D cell growth with an ABM model is very time consuming. Scale-up analysis showed that one such simulation would take about 40 days with an IBM Bladecenter machine (dual-processor, 32-bit Xeons ranging from 2.8-3.2 GHz, 2.5 GB RAM, and Gigabit Ethernet), which is practically impossible. This limitation prevents simulation using MABMs from modeling more realistic large cancer systems. Therefore, the present research incorporated GPU-based parallel computing algorithms combined with a multi-resolution design into a multiscale ABM to simulate real-time actual GBM cancer progression. The in silico results demonstrated that our GPU- based multi-resolution MABM can be used not only to investigate the incoherent relationships among various scales during cancer progression and visualize tumor progression at different resolutions, but also to overcome the computational resource shortage problem and simulate actual cancer progression in real time.
As is well known, computer simulations of complex agent-based systems result in various emergent behaviors due to non-linear interactions among the agents, which in our case are the cancer cells. Similarly, the multiscale analysis of our simulation results revealed various emergent findings. First, the molecular profiles of cells switching phenotypes from proliferation to migration (PM) and from migration to proliferation (MP) have very similar patterns (Figure 8). Second, we found that X_{8} (TGFα-EGFR-PLCγ-P) and X_{10} (PLCγ-P-I) correlated strongly with the rate of change of X_{9} (PLCγ-P), which determined the cell's phenotypic switch (Equation 2), whereas X_{1} (TGFα)_{,}X_{2} (EGFR)_{,}X_{3} (TGFα -EGFR) and X_{6} (PLCγ) were independent of the rate of change of X_{9} (PLCγ-P). Third, at early time stages, a high percentage rate of change of PLCγ caused the cell's phenotype to switch from proliferation to migration and a comparatively low percentage rate of change in PLCγ caused a switch from migration to proliferation; but the difference in PLCγ between these two molecular profiles (MP and PM) was very small in the final simulation stage. It is noted that the simulation data are from a four day experiment, so we set the simulation duration at 100 hours. These findings imply that the external input (TGFα), the major stimulator of the EGFR pathway, cannot change the concentration of PLCγ substantially at the end stage of simulation.
The multi-resolution design allowed us to visualize the tumor progression at various resolutions. Our simulated results revealed that the heterogeneous clusters consisting of cells with various phenotypes were always on the outer regions of the tumor. In addition, we were able to explore the cells' behavior in the heterogeneous clusters. Using a high resolution lattice we investigated the cells' positions and phenotypes at different time steps. Moreover, the multi-resolution design enabled us to track a cell's trajectory.
We also showed that the performance of the model was significantly improved by employing GPU-based parallel computing algorithms. We showed that the parallelized algorithm (PSGMG) is much better than the sequential algorithm on large lattices or when the cell population is large.
In summary, the simulation results demonstrated that the GPU-based multi-resolution MABM has great potential for simulating actual GBM tumor progression in real time. In the near future, we plan to incorporate more parameters from experiments into the model, which will enable us to simulate GBM progression patterns at various resolutions in a more realistic way. Such simulations will enable us to investigate molecular biomarkers that play an important role in inhibiting cancer expansion and predict real GBM progression. Subsequently, we plan to work with experimentalists to use actual data to validate the effectiveness of the model.
Notes
Declarations
Acknowledgements
This work has been supported by a start-up grant from Michigan Tech University to Prof. Le Zhang.
Authors’ Affiliations
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