Computer simulation of glioma growth and morphology

Extreme protraction for low-grade gliomas: theoretical proof of concept of a novel therapeutical strategy

Grade II gliomas are slowly growing primary brain tumours that affect mostly young patients and become fatal after a variable time period. Current clinical handling includes surgery as first-line treatment. Cytotoxic therapies (radiotherapy RT or chemotherapy QT) are used initially only for patients having a bad prognosis. Therapies are administered following the 'maximum dose in minimum time' principle, which is the same schedule used for high-grade brain tumours. Using mathematical models describing the growth of these tumours in response to radiotherapy, we find that an extreme protraction therapeutical strategy, i.e. enlarging substantially the time interval between RT fractions, may lead to better tumour control. Explicit formulas are found providing the optimal spacing between doses in a very good agreement with the simulations of the full 3D mathematical model approximating the tumour spatiotemporal dynamics. This idea, although breaking the well-established paradigm, has biological meaning since, in these slowly growing tumours, it may be more favourable to treat the tumour as the tumour cells leave the quiescent compartment and move into the cell cycle.

Integrative models of vascular remodeling during tumor growth

Malignant solid tumors recruit the blood vessel network of the host tissue for nutrient supply, continuous growth, and gain of metastatic potential. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature that is in many respects different from the hierarchically organized arterio-venous blood vessel network of the host tissues. Integrative models based on detailed experimental data and physical laws implement in silico the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. In this review, we give an overview over the current status of integrative models describing tumor growth, vascular remodeling, blood and interstitial fluid flow, drug delivery, and concomitant transformations of the microenvironment. WIREs Syst Biol Med 2015, 7:113-129. doi: 10.1002/wsbm.1295 For further resources related to this article, please visit the WIREs website.

CONFLICT OF INTEREST

The authors have declared no conflicts of interest for this article.

Simulation predicts IGFBP2-HIF1α interaction drives glioblastoma growth

Tremendous strides have been made in improving patients’ survival from cancer with one glaring exception: brain cancer. Glioblastoma is the most common, aggressive and highly malignant type of primary brain tumor. The average overall survival remains less than 1 year. Notably, cancer patients with obesity and diabetes have worse outcomes and accelerated progression of glioblastoma. The root cause of this accelerated progression has been hypothesized to involve the insulin signaling pathway. However, while the process of invasive glioblastoma progression has been extensively studied macroscopically, it has not yet been well characterized with regards to intracellular insulin signaling. In this study we connect for the first time microscale insulin signaling activity with macroscale glioblastoma growth through the use of computational modeling. Results of the model suggest a novel observation: feedback from IGFBP2 to HIF1α is integral to the sustained growth of glioblastoma. Our study suggests that downstream signaling from IGFI to HIF1α, which has been the target of many insulin signaling drugs in clinical trials, plays a smaller role in overall tumor growth. These predictions strongly suggest redirecting the focus of glioma drug candidates on controlling the feedback between IGFBP2 and HIF1α.

Current treatment for glioblastoma patients is limited to nonspecific methods: surgery followed by a combination of radio- and chemotherapy. With these methods, glioma patient survival is less than one year post-diagnosis. Targeting specific protein signaling pathways offers potentially more potent therapies. One promising potential target is the insulin signaling pathway, which is known to contribute to glioblastoma progression. However, drugs targeting this pathway have shown mixed results in clinical trials, and the detailed mechanisms of how the insulin signaling pathway promotes glioblastoma growth remain to be elucidated. Here, we developed a computational model of insulin signaling in glioblastoma in order to study this pathway’s role in tumor progression. Using the model, we systematically test contributions of different insulin signaling protein interactions on glioblastoma growth. Our model highlights a key driver for the growth of glioblastoma: IGFBP2-HIF1α feedback. This interaction provides a target that could open the door for new therapies in glioma and other solid tumors.

Strategies for efficient numerical implementation of hybrid multi-scale agent-based models to describe biological systems

Biologically related processes operate across multiple spatiotemporal scales. For computational modeling methodologies to mimic this biological complexity, individual scale models must be linked in ways that allow for dynamic exchange of information across scales. A powerful methodology is to combine a discrete modeling approach, agent-based models (ABMs), with continuum models to form hybrid models. Hybrid multi-scale ABMs have been used to simulate emergent responses of biological systems. Here, we review two aspects of hybrid multi-scale ABMs: linking individual scale models and efficiently solving the resulting model. We discuss the computational choices associated with aspects of linking individual scale models while simultaneously maintaining model tractability. We demonstrate implementations of existing numerical methods in the context of hybrid multi-scale ABMs. Using an example model describing Mycobacterium tuberculosis infection, we show relative computational speeds of various combinations of numerical methods. Efficient linking and solution of hybrid multi-scale ABMs is key to model portability, modularity, and their use in understanding biological phenomena at a systems level.

Effects of anti-angiogenesis on glioblastoma growth and migration: model to clinical predictions

Glioblastoma multiforme (GBM) causes significant neurological morbidity and short survival times. Brain invasion by GBM is associated with poor prognosis. Recent clinical trials of bevacizumab in newly-diagnosed GBM found no beneficial effects on overall survival times; however, the baseline health-related quality of life and performance status were maintained longer in the bevacizumab group and the glucocorticoid requirement was lower. Here, we construct a clinical-scale model of GBM whose predictions uncover a new pattern of recurrence in 11/70 bevacizumab-treated patients. The findings support an exception to the Folkman hypothesis: GBM grows in the absence of angiogenesis by a cycle of proliferation and brain invasion that expands necrosis. Furthermore, necrosis is positively correlated with brain invasion in 26 newly-diagnosed GBM. The unintuitive results explain the unusual clinical effects of bevacizumab and suggest new hypotheses on the dynamic clinical effects of migration by active transport, a mechanism of hypoxia-driven brain invasion.

Discrimination between glioblastoma multiforme and solitary metastasis using morphological features derived from the p:q tensor decomposition of diffusion tensor imaging

The management and treatment of high-grade glioblastoma multiforme (GBM) and solitary metastasis (MET) are very different and influence the prognosis and subsequent clinical outcomes. In the case of a solitary MET, diagnosis using conventional radiology can be equivocal. Currently, a definitive diagnosis is based on histopathological analysis on a biopsy sample. Here, we present a computerised decision support framework for discrimination between GBM and solitary MET using MRI, which includes: (i) a semi-automatic segmentation method based on diffusion tensor imaging; (ii) two-dimensional morphological feature extraction and selection; and (iii) a pattern recognition module for automated tumour classification. Ground truth was provided by histopathological analysis from pre-treatment stereotactic biopsy or at surgical resection. Our two-dimensional morphological analysis outperforms previous methods with high cross-validation accuracy of 97.9% and area under the receiver operating characteristic curve of 0.975 using a neural networks-based classifier.

A multilayer grow-or-go model for GBM: effects of invasive cells and anti-angiogenesis on growth

The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a new mathematical model that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model is able to replicate and justify the clinical observation of rebound growth when AA therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR) signal and the rebound tumor growth when AA is discontinued.

Computational modeling of 3D tumor growth and angiogenesis for chemotherapy evaluation

Solid tumors develop abnormally at spatial and temporal scales, giving rise to biophysical barriers that impact anti-tumor chemotherapy. This may increase the expenditure and time for conventional drug pharmacokinetic and pharmacodynamic studies. In order to facilitate drug discovery, we propose a mathematical model that couples three-dimensional tumor growth and angiogenesis to simulate tumor progression for chemotherapy evaluation. This application-oriented model incorporates complex dynamical processes including cell- and vascular-mediated interstitial pressure, mass transport, angiogenesis, cell proliferation, and vessel maturation to model tumor progression through multiple stages including tumor initiation, avascular growth, and transition from avascular to vascular growth. Compared to pure mechanistic models, the proposed empirical methods are not only easy to conduct but can provide realistic predictions and calculations. A series of computational simulations were conducted to demonstrate the advantages of the proposed comprehensive model. The computational simulation results suggest that solid tumor geometry is related to the interstitial pressure, such that tumors with high interstitial pressure are more likely to develop dendritic structures than those with low interstitial pressure.

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

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.

Capillary networks in tumor angiogenesis: from discrete endothelial cells to phase-field averaged descriptions via isogeometric analysis

Tumor angiogenesis, the growth of new capillaries from preexisting ones promoted by the starvation and hypoxia of cancerous cell, creates complex biological patterns. These patterns are captured by a hybrid model that involves high-order partial differential equations coupled with mobile, agent-based components. The continuous equations of the model rely on the phase-field method to describe the intricate interfaces between the vasculature and the host tissue. The discrete equations are posed on a cellular scale and treat tip endothelial cells as mobile agents. Here, we put the model into a coherent mathematical and algorithmic framework and introduce a numerical method based on isogeometric analysis that couples the discrete and continuous descriptions of the theory. Using our algorithms, we perform numerical simulations that show the development of the vasculature around a tumor. The new method permitted us to perform a parametric study of the model. Furthermore, we investigate different initial configurations to study the growth of the new capillaries. The simulations illustrate the accuracy and efficiency of our numerical method and provide insight into the dynamics of the governing equations as well as into the underlying physical phenomenon.

Predictive modeling of in vivo response to gemcitabine in pancreatic cancer

A clear contradiction exists between cytotoxic in-vitro studies demonstrating effectiveness of Gemcitabine to curtail pancreatic cancer and in-vivo studies failing to show Gemcitabine as an effective treatment. The outcome of chemotherapy in metastatic stages, where surgery is no longer viable, shows a 5-year survival <5%. It is apparent that in-vitro experiments, no matter how well designed, may fail to adequately represent the complex in-vivo microenvironmental and phenotypic characteristics of the cancer, including cell proliferation and apoptosis. We evaluate in-vitro cytotoxic data as an indicator of in-vivo treatment success using a mathematical model of tumor growth based on a dimensionless formulation describing tumor biology. Inputs to the model are obtained under optimal drug exposure conditions in-vitro. The model incorporates heterogeneous cell proliferation and death caused by spatial diffusion gradients of oxygen/nutrients due to inefficient vascularization and abundant stroma, and thus is able to simulate the effect of the microenvironment as a barrier to effective nutrient and drug delivery. Analysis of the mathematical model indicates the pancreatic tumors to be mostly resistant to Gemcitabine treatment in-vivo. The model results are confirmed with experiments in live mice, which indicate uninhibited tumor proliferation and metastasis with Gemcitabine treatment. By extracting mathematical model parameter values for proliferation and death from monolayer in-vitro cytotoxicity experiments with pancreatic cancer cells, and simulating the effects of spatial diffusion, we use the model to predict the drug response in-vivo, beyond what would have been expected from sole consideration of the cancer intrinsic resistance. We conclude that this integrated experimental/computational approach may enhance understanding of pancreatic cancer behavior and its response to various chemotherapies, and, further, that such an approach could predict resistance based on pharmacokinetic measurements with the goal to maximize effective treatment strategies.

IN SILICO MODELING OF TUMOR GROWTH

Tumor growth is strongly coupled with both residual stress generated during the growth process and also biochemical factors. Several models have already been proposed to capture tumor growth considering either diffusion of nutrients concentration or residual stresses inside a tumor. In this work, a new method was proposed to model the generated residual stress in a growing solid using the continuum framework. This method was coupled with energy metabolism to predict the behavior of a soft tissue tumor. Moreover, it was shown that the main reason of vascular collapse in the middle of a tumor or vascular reopening in the tumor core can be the residual stress, generated during the growth.

Modeling the mechanics of cancer: effect of changes in cellular and extra-cellular mechanical properties

Malignant transformation, though primarily driven by genetic mutations in cells, is also accompanied by specific changes in cellular and extra-cellular mechanical properties such as stiffness and adhesivity. As the transformed cells grow into tumors, they interact with their surroundings via physical contacts and the application of forces. These forces can lead to changes in the mechanical regulation of cell fate based on the mechanical properties of the cells and their surrounding environment. A comprehensive understanding of cancer progression requires the study of how specific changes in mechanical properties influences collective cell behavior during tumor growth and metastasis. Here we review some key results from computational models describing the effect of changes in cellular and extra-cellular mechanical properties and identify mechanistic pathways for cancer progression that can be targeted for the prediction, treatment, and prevention of cancer.

A hybrid model of the role of VEGF binding in endothelial cell migration and capillary formation

Vascular endothelial growth factor (VEGF) is the most studied family of soluble, secreted mediators of endothelial cell migration, survival, and proliferation. VEGF exerts its function by binding to specific tyrosine kinase receptors on the cell surface and transducing the effect through downstream signaling. In order to study the influence of VEGF binding on endothelial cell motion, we develop a hybrid model of VEGF-induced angiogenesis, based on the theory of reinforced random walks. The model includes the chemotactic response of endothelial cells to angiogenic factors bound to cell-surface receptors, rather than approximating this as a function of extracellular chemical concentrations. This allows us to capture biologically observed phenomena such as activation and polarization of endothelial cells in response to VEGF gradients across their lengths, as opposed to extracellular gradients throughout the tissue. We also propose a novel and more biologically reasonable functional form for the chemotactic sensitivity of endothelial cells, which is also governed by activated cell-surface receptors. This model is able to predict the threshold level of VEGF required to activate a cell to move in a directed fashion as well as an optimal VEGF concentration for motion. Model validation is achieved by comparison of simulation results directly with experimental data.

Cancer Stem Cells: A Minor Cancer Subpopulation that Redefines Global Cancer Features

In recent years cancer stem cells (CSCs) have been hypothesized to comprise only a minor subpopulation in solid tumors that drives tumor initiation, progression, and metastasis; the so-called “cancer stem cell hypothesis.” While a seemingly trivial statement about numbers, much is put at stake. If true, the conclusions of many studies of cancer cell populations could be challenged, as the bulk assay methods upon which they depend have, by, and large, taken for granted the notion that a “typical” cell of the population possesses the attributes of a cell capable of perpetuating the cancer, i.e., a CSC. In support of the CSC hypothesis, populations enriched for so-called “tumor-initiating” cells have demonstrated a corresponding increase in tumorigenicity as measured by dilution assay, although estimates have varied widely as to what the fractional contribution of tumor-initiating cells is in any given population. Some have taken this variability to suggest the CSC fraction may be nearly 100% after all, countering the CSC hypothesis, and that there are simply assay-dependent error rates in our ability to “reconfirm” CSC status at the cell level. To explore this controversy more quantitatively, we developed a simple cellular automaton model of CSC-driven tumor growth dynamics. Assuming CSC and non-stem cancer cells (CC) subpopulations coexist to some degree, we evaluated the impact of an environmentally dependent CSC symmetric division probability and a CC proliferation capacity on tumor progression and morphology. Our model predicts, as expected, that the frequency of CSC divisions that are symmetric highly influences the frequency of CSCs in the population, but goes on to predict the two frequencies can be widely divergent, and that spatial constraints will tend to increase the CSC fraction over time. Further, tumor progression times show a marked dependence on both the frequency of CSC divisions that are symmetric and on the proliferation capacities of CC. Together, these findings can explain, within the CSC hypothesis, the widely varying measures of stem cell fractions observed. In particular, although the CSC fraction is influenced by the (environmentally modifiable) CSC symmetric division probability, with the former converging to unity as the latter nears 100%, the CSC fraction becomes quite small even for symmetric division probabilities modestly lower than 100%. In the latter case, the tumor exhibits a clustered morphology and the CSC fraction steadily increases with time; more so on both counts when the death rate of CCs is higher. Such variations in CSC fraction and morphology are not only consistent with the CSC hypothesis, but lend support to it as one expected byproduct of the dynamical interactions that are predicted to take place among a relatively small CSC population, its CC counterpart, and the host compartment over time.

An integrated computational/experimental model of lymphoma growth

Non-Hodgkin's lymphoma is a disseminated, highly malignant cancer, with resistance to drug treatment based on molecular- and tissue-scale characteristics that are intricately linked. A critical element of molecular resistance has been traced to the loss of functionality in proteins such as the tumor suppressor p53. We investigate the tissue-scale physiologic effects of this loss by integrating in vivo and immunohistological data with computational modeling to study the spatiotemporal physical dynamics of lymphoma growth. We compare between drug-sensitive Eμ-myc Arf-/- and drug-resistant Eμ-myc p53-/- lymphoma cell tumors grown in live mice. Initial values for the model parameters are obtained in part by extracting values from the cellular-scale from whole-tumor histological staining of the tumor-infiltrated inguinal lymph node in vivo. We compare model-predicted tumor growth with that observed from intravital microscopy and macroscopic imaging in vivo, finding that the model is able to accurately predict lymphoma growth. A critical physical mechanism underlying drug-resistant phenotypes may be that the Eμ-myc p53-/- cells seem to pack more closely within the tumor than the Eμ-myc Arf-/- cells, thus possibly exacerbating diffusion gradients of oxygen, leading to cell quiescence and hence resistance to cell-cycle specific drugs. Tighter cell packing could also maintain steeper gradients of drug and lead to insufficient toxicity. The transport phenomena within the lymphoma may thus contribute in nontrivial, complex ways to the difference in drug sensitivity between Eμ-myc Arf-/- and Eμ-myc p53-/- tumors, beyond what might be solely expected from loss of functionality at the molecular scale. We conclude that computational modeling tightly integrated with experimental data gives insight into the dynamics of Non-Hodgkin's lymphoma and provides a platform to generate confirmable predictions of tumor growth.

Non-Hodgkin's lymphoma is a cancer that develops from white blood cells called lymphocytes in the immune system, whose role is to fight disease throughout the body. This cancer can spread throughout the whole body and be very lethal – in the US, one third of patients will die from this disease within five years of diagnosis. Chemotherapy is a usual treatment for lymphoma, but the cancer can become highly resistant to it. One reason is that a critical gene called p53 can become mutated and help the cancer to survive. In this work we investigate how cells with this mutation affect the cancer growth by performing experiments in mice and using a computer model. By inputting the model parameters based on data from the experiments, we are able to accurately predict the growth of the tumor as compared to tumor measurements in living mice. We conclude that computational modeling integrated with experimental data gives insight into the dynamics of Non-Hodgkin's lymphoma, and provides a platform to generate confirmable predictions of tumor growth.

A computational model for predicting nanoparticle accumulation in tumor vasculature

Vascular targeting of malignant tissues with systemically injected nanoparticles (NPs) holds promise in molecular imaging and anti-angiogenic therapies. Here, a computational model is presented to predict the development of tumor neovasculature over time and the specific, vascular accumulation of blood-borne NPs. A multidimensional tumor-growth model is integrated with a mesoscale formulation for the NP adhesion to blood vessel walls. The fraction of injected NPs depositing within the diseased vasculature and their spatial distribution is computed as a function of tumor stage, from 0 to day 24 post-tumor inception. As the malignant mass grows in size, average blood flow and shear rates increase within the tumor neovasculature, reaching values comparable with those measured in healthy, pre-existing vessels already at 10 days. The NP vascular affinity, interpreted as the likelihood for a blood-borne NP to firmly adhere to the vessel walls, is a fundamental parameter in this analysis and depends on NP size and ligand density, and vascular receptor expression. For high vascular affinities, NPs tend to accumulate mostly at the inlet tumor vessels leaving the inner and outer vasculature depleted of NPs. For low vascular affinities, NPs distribute quite uniformly intra-tumorally but exhibit low accumulation doses. It is shown that an optimal vascular affinity can be identified providing the proper balance between accumulation dose and uniform spatial distribution of the NPs. This balance depends on the stage of tumor development (vascularity and endothelial receptor expression) and the NP properties (size, ligand density and ligand-receptor molecular affinity). Also, it is demonstrated that for insufficiently developed vascular networks, NPs are transported preferentially through the healthy, pre-existing vessels, thus bypassing the tumor mass. The computational tool described here can effectively select an optimal NP formulation presenting high accumulation doses and uniform spatial intra-tumor distributions as a function of the development stage of the malignancy.

Hypoxic cell waves around necrotic cores in glioblastoma: a biomathematical model and its therapeutic implications

Glioblastoma is a rapidly evolving high-grade astrocytoma that is distinguished pathologically from lower grade gliomas by the presence of necrosis and microvascular hyperplasia. Necrotic areas are typically surrounded by hypercellular regions known as "pseudopalisades" originated by local tumor vessel occlusions that induce collective cellular migration events. This leads to the formation of waves of tumor cells actively migrating away from central hypoxia. We present a mathematical model that incorporates the interplay among two tumor cell phenotypes, a necrotic core and the oxygen distribution. Our simulations reveal the formation of a traveling wave of tumor cells that reproduces the observed histologic patterns of pseudopalisades. Additional simulations of the model equations show that preventing the collapse of tumor microvessels leads to slower glioma invasion, a fact that might be exploited for therapeutic purposes.

In silico modelling of tumour margin diffusion and infiltration: review of current status

As a result of advanced treatment techniques, requiring precise target definitions, a need for more accurate delineation of the Clinical Target Volume (CTV) has arisen. Mathematical modelling is found to be a powerful tool to provide fairly accurate predictions for the Microscopic Extension (ME) of a tumour to be incorporated in a CTV. In general terms, biomathematical models based on a sequence of observations or development of a hypothesis assume some links between biological mechanisms involved in cancer development and progression to provide quantitative or qualitative measures of tumour behaviour as well as tumour response to treatment. Generally, two approaches are taken: deterministic and stochastic modelling. In this paper, recent mathematical models, including deterministic and stochastic methods, are reviewed and critically compared. It is concluded that stochastic models are more promising to provide a realistic description of cancer tumour behaviour due to being intrinsically probabilistic as well as discrete, which enables incorporation of patient-specific biomedical data such as tumour heterogeneity and anatomical boundaries.

Multiscale cancer modeling

Simulating cancer behavior across multiple biological scales in space and time, i.e., multiscale cancer modeling, is increasingly being recognized as a powerful tool to refine hypotheses, focus experiments, and enable more accurate predictions. A growing number of examples illustrate the value of this approach in providing quantitative insights in the initiation, progression, and treatment of cancer. In this review, we introduce the most recent and important multiscale cancer modeling works that have successfully established a mechanistic link between different biological scales. Biophysical, biochemical, and biomechanical factors are considered in these models. We also discuss innovative, cutting-edge modeling methods that are moving predictive multiscale cancer modeling toward clinical application. Furthermore, because the development of multiscale cancer models requires a new level of collaboration among scientists from a variety of fields such as biology, medicine, physics, mathematics, engineering, and computer science, an innovative Web-based infrastructure is needed to support this growing community.

Integrated intravital microscopy and mathematical modeling to optimize nanotherapeutics delivery to tumors

Inefficient vascularization hinders the optimal transport of cell nutrients, oxygen, and drugs to cancer cells in solid tumors. Gradients of these substances maintain a heterogeneous cell-scale microenvironment through which drugs and their carriers must travel, significantly limiting optimal drug exposure. In this study, we integrate intravital microscopy with a mathematical model of cancer to evaluate the behavior of nanoparticle-based drug delivery systems designed to circumvent biophysical barriers. We simulate the effect of doxorubicin delivered via porous 1000 x 400 nm plateloid silicon particles to a solid tumor characterized by a realistic vasculature, and vary the parameters to determine how much drug per particle and how many particles need to be released within the vasculature in order to achieve remission of the tumor. We envision that this work will contribute to the development of quantitative measures of nanoparticle design and drug loading in order to optimize cancer treatment via nanotherapeutics.

Patient-calibrated agent-based modelling of ductal carcinoma in situ (DCIS): from microscopic measurements to macroscopic predictions of clinical progression

Ductal carcinoma in situ (DCIS)--a significant precursor to invasive breast cancer--is typically diagnosed as microcalcifications in mammograms. However, the effective use of mammograms and other patient data to plan treatment has been restricted by our limited understanding of DCIS growth and calcification. We develop a mechanistic, agent-based cell model and apply it to DCIS. Cell motion is determined by a balance of biomechanical forces. We use potential functions to model interactions with the basement membrane and amongst cells of unequal size and phenotype. Each cell's phenotype is determined by genomic/proteomic- and microenvironment-dependent stochastic processes. Detailed "sub-models" describe cell volume changes during proliferation and necrosis; we are the first to account for cell calcification. We introduce the first patient-specific calibration method to fully constrain the model based upon clinically-accessible histopathology data. After simulating 45 days of solid-type DCIS with comedonecrosis, the model predicts: necrotic cell lysis acts as a biomechanical stress relief and is responsible for the linear DCIS growth observed in mammography; the rate of DCIS advance varies with the duct radius; the tumour grows 7-10mm per year--consistent with mammographic data; and the mammographic and (post-operative) pathologic sizes are linearly correlated--in quantitative agreement with the clinical literature. Patient histopathology matches the predicted DCIS microstructure: an outer proliferative rim surrounds a stratified necrotic core with nuclear debris on its outer edge and calcification in the centre. This work illustrates that computational modelling can provide new insight on the biophysical underpinnings of cancer. It may 1-day be possible to augment a patient's mammography and other imaging with rigorously-calibrated models that help select optimal surgical margins based upon the patient's histopathologic data.

Numerical simulation of a thermodynamically consistent four-species tumor growth model

In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.

Multiparametric magnetic resonance imaging to differentiate high-grade gliomas and brain metastases

PURPOSE

To assess the performance of parameters used in conventional magnetic resonance imaging (MRI), perfusion-weighted MR imaging (PWI) and visual texture analysis, alone and in combination, to differentiate a single brain metastasis (MET) from glioblastoma multiforme (GBM).

PATIENTS AND METHODS

In a retrospective study of 50 patients (41 GBM and 14 MET) who underwent T2/FLAIR/T1(post-contrast) imaging and PWI, morphological (circularity, surface area), perfusion (rCBV in the ring-like tumor area, rCBV in the peritumoral area, percentage of signal intensity recovery at the end of first pass) and texture parameters in the peritumoral area were estimated. Statistical differences and performances were assessed using Wilcoxon's test and receiver operating characteristic curves, respectively. Multiparametric classification of tumors was performed using k-means clustering.

RESULTS

Significant statistical differences in circularity, surface area, rCBVs, percentage of signal intensity recovery and texture parameters (energy, entropy, homogeneity, correlation, inverse differential moment, sum average) were observed between MET and GBM (P<0.05). Moderate-to-good classification performances were found with these parameters. Clustering based on rCBV and texture parameters (contrast, sum average) differentiated MET from GBM with a sensitivity of 92% and a specificity of 71%.

CONCLUSION

Combining perfusion and visual texture parameters within a statistical classifier significantly improved the differentiation of a single brain MET and GBM.

An in vitro comparison of the effects of the iron-chelating agents, CP94 and dexrazoxane, on protoporphyrin IX accumulation for photodynamic therapy and/or fluorescence guided resection

Photodynamic therapy (PDT) utilizes the combined interaction of a photosensitizer, light and molecular oxygen to ablate tumor tissue. Maximizing the accumulation of the photosensitizer protoporphyrin IX (PpIX) within different cell types would be clinically useful. Dermatological PpIX-induced PDT regimes produce good clinical outcomes but this currently only applies when the lesion remains superficial. Also, as an adjuvant therapy for the treatment of primary brain tumors, fluorescence guided resection (FGR) and PDT can be used to highlight and destroy tumor cells unreachable by surgical resection. By employing iron chelators PpIX accumulation can be enhanced. Two iron-chelating agents, 1,2-diethyl-3-hydroxypyridin-4-one hydrochloride (CP94) and dexrazoxane, were individually combined with the porphyrin precursors aminolevulinic acid (ALA), methyl aminolevulinate (MAL) and hexyl aminolevulinate (HAL). Efficacies of the iron-chelating agents were compared by recording the PpIX fluorescence in human squamous epithelial carcinoma cells (A431) and human glioma cells (U-87 MG) every hour for up to 6 h. Coincubation of ALA/MAL/HAL with CP94 resulted in a greater accumulation of PpIX compared to that produced by coincubation of these congeners with dexrazoxane. Therefore the clinical employment of iron chelation, particularly with CP94 could potentially increase and/or accelerate the accumulation of ALA/MAL/HAL-induced PpIX for PDT or FGR.

Integrative physical oncology

Cancer is arguably the ultimate complex biological system. Solid tumors are microstructured soft matter that evolves as a consequence of spatio-temporal events at the intracellular (e.g., signaling pathways, macromolecular trafficking), intercellular (e.g., cell-cell adhesion/communication), and tissue (e.g., cell-extracellular matrix interactions, mechanical forces) scales. To gain insight, tumor and developmental biologists have gathered a wealth of molecular, cellular, and genetic data, including immunohistochemical measurements of cell type-specific division and death rates, lineage tracing, and gain-of-function/loss-of-function mutational analyses. These data are empirically extrapolated to a diagnosis/prognosis of tissue-scale behavior, e.g., for clinical decision. Integrative physical oncology (IPO) is the science that develops physically consistent mathematical approaches to address the significant challenge of bridging the nano (nm)-micro (µm) to macro (mm, cm) scales with respect to tumor development and progression. In the current literature, such approaches are referred to as multiscale modeling. In the present article, we attempt to assess recent modeling approaches on each separate scale and critically evaluate the current 'hybrid-multiscale' models used to investigate tumor growth in the context of brain and breast cancers. Finally, we provide our perspective on the further development and the impact of IPO.

Bright solitary waves in malignant gliomas

We put forward a nonlinear wave model describing the fundamental dynamical features of an aggressive type of brain tumors. Our model accounts for the invasion of normal tissue by a proliferating and propagating rim of active glioma cancer cells in the tumor boundary and the subsequent formation of a necrotic core. By resorting to numerical simulations, phase space analysis, and exact solutions we prove that bright solitary tumor waves develop in such systems. Possible implications of our model as a tool to extract relevant patient specific tumor parameters in combination with standard preoperative clinical imaging are also discussed.

Identification of intrinsic in vitro cellular mechanisms for glioma invasion

Invasion of malignant glioma is a highly complex phenomenon involving molecular and cellular processes at various spatio-temporal scales, whose precise interplay is still not fully understood. In order to identify the intrinsic cellular mechanisms of glioma invasion, we study an in vitro culture of glioma cells. By means of a computational approach, based on a cellular automaton model, we compare simulation results to the experimental data and deduce cellular mechanisms from microscopic and macroscopic observables (experimental data). For the first time, it is shown that the migration/proliferation dichotomy plays a central role in the invasion of glioma cells. Interestingly, we conclude that a diverging invasive zone is a consequence of this dichotomy. Additionally, we observe that radial persistence of glioma cells in the vicinity of dense areas accelerates the invasion process. We argue that this persistence results from a cell-cell repulsion mechanism. If glioma cell behavior is regulated through a migration/proliferation dichotomy and a self-repellent mechanism, our simulations faithfully reproduce all the experimental observations.

Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.

Multiscale modelling of vascular tumour growth in 3D: the roles of domain size and boundary conditions

We investigate a three-dimensional multiscale model of vascular tumour growth, which couples blood flow, angiogenesis, vascular remodelling, nutrient/growth factor transport, movement of, and interactions between, normal and tumour cells, and nutrient-dependent cell cycle dynamics within each cell. In particular, we determine how the domain size, aspect ratio and initial vascular network influence the tumour's growth dynamics and its long-time composition. We establish whether it is possible to extrapolate simulation results obtained for small domains to larger ones, by constructing a large simulation domain from a number of identical subdomains, each subsystem initially comprising two parallel parent vessels, with associated cells and diffusible substances. We find that the subsystem is not representative of the full domain and conclude that, for this initial vessel geometry, interactions between adjacent subsystems contribute to the overall growth dynamics. We then show that extrapolation of results from a small subdomain to a larger domain can only be made if the subdomain is sufficiently large and is initialised with a sufficiently complex vascular network. Motivated by these results, we perform simulations to investigate the tumour's response to therapy and show that the probability of tumour elimination in a larger domain can be extrapolated from simulation results on a smaller domain. Finally, we demonstrate how our model may be combined with experimental data, to predict the spatio-temporal evolution of a vascular tumour.

Physical oncology: a bench-to-bedside quantitative and predictive approach

Cancer models relating basic science to clinical care in oncology may fail to address the nuances of tumor behavior and therapy, as in the case, discussed herein, of the complex multiscale dynamics leading to the often-observed enhanced invasiveness, paradoxically induced by the very antiangiogenic therapy designed to destroy the tumor. Studies would benefit from approaches that quantitatively link the multiple physical and temporal scales from molecule to tissue in order to offer outcome predictions for individual patients. Physical oncology is an approach that applies fundamental principles from the physical and biological sciences to explain certain cancer behaviors as observable characteristics arising from the underlying physical and biochemical events. For example, the transport of oxygen molecules through tissue affects phenotypic characteristics such as cell proliferation, apoptosis, and adhesion, which in turn underlie the patient-scale tumor growth and invasiveness. Our review of physical oncology illustrates how tumor behavior and treatment response may be a quantifiable function of marginally stable molecular and/or cellular conditions modulated by inhomogeneity. By incorporating patient-specific genomic, proteomic, metabolomic, and cellular data into multiscale physical models, physical oncology could complement current clinical practice through enhanced understanding of cancer behavior, thus potentially improving patient survival.

An Adaptive Multigrid Algorithm for Simulating Solid Tumor Growth Using Mixture Models

In this paper we give the details of the numerical solution of a three-dimensional multispecies diffuse interface model of tumor growth, which was derived in (Wise et al., J. Theor. Biol. 253 (2008)) and used to study the development of glioma in (Frieboes et al., NeuroImage 37 (2007) and tumor invasion in (Bearer et al., Cancer Research, 69 (2009)) and (Frieboes et al., J. Theor. Biol. 264 (2010)). The model has a thermodynamic basis, is related to recently developed mixture models, and is capable of providing a detailed description of tumor progression. It utilizes a diffuse interface approach, whereby sharp tumor boundaries are replaced by narrow transition layers that arise due to differential adhesive forces among the cell-species. The model consists of fourth-order nonlinear advection-reaction-diffusion equations (of Cahn-Hilliard-type) for the cell-species coupled with reaction-diffusion equations for the substrate components. Numerical solution of the model is challenging because the equations are coupled, highly nonlinear, and numerically stiff. In this paper we describe a fully adaptive, nonlinear multigrid/finite difference method for efficiently solving the equations. We demonstrate the convergence of the algorithm and we present simulations of tumor growth in 2D and 3D that demonstrate the capabilities of the algorithm in accurately and efficiently simulating the progression of tumors with complex morphologies.

Discrimination between metastasis and glioblastoma multiforme based on morphometric analysis of MR images

BACKGROUND AND PURPOSE

Solitary MET and GBM are difficult to distinguish by using MR imaging. Differentiation is useful before any metastatic work-up or biopsy. Our hypothesis was that MET and GBM tumors differ in morphology. Shape analysis was proposed as an indicator for discriminating these 2 types of brain pathologies. The purpose of this study was to evaluate the accuracy of this approach in the discrimination of GBMs and brain METs.

MATERIALS AND METHODS

The dataset consisted of 33 brain MR imaging sets of untreated patients, of which 18 patients were diagnosed as having a GBM and 15 patients, as having solitary metastatic brain tumor. The MR imaging was segmented by using the K-means algorithm. The resulting set of classes (also called "clusters") represented the variety of tissues observed. A morphology-based approach allowed discrimination of the 2 types of tumors. This approach was validated by a leave-1-patient-out procedure.

RESULTS

A method was developed for the discrimination of GBMs and solitary METs. Two masses out of 33 were wrongly classified; the overall results were accurate in 93.9% of the observed cases.

CONCLUSIONS

A semiautomated method based on a morphologic analysis was developed. Its application was found to be useful in the discrimination of GBM from solitary MET.

Physical determinants of vascular network remodeling during tumor growth

The process in which a growing tumor transforms a hierarchically organized arterio-venous blood vessel network into a tumor specific vasculature is analyzed with a theoretical model. The physical determinants of this remodeling involve the morphological and hydrodynamic properties of the initial network, generation of new vessels (sprouting angiogenesis), vessel dilation (circumferential growth), vessel regression, tumor cell proliferation and death, and the interdependence of these processes via spatio-temporal changes of blood flow parameters, oxygen/nutrient supply and growth factor concentration fields. The emerging tumor vasculature is non-hierarchical, compartmentalized into well-characterized zones, displays a complex geometry with necrotic zones and "hot spots" of increased vascular density and blood flow of varying size, and transports drug injections efficiently. Implications for current theoretical views on tumor-induced angiogenesis are discussed.

Mathematical Oncology: How Are the Mathematical and Physical Sciences Contributing to the War on Breast Cancer?

Mathematical modeling has recently been added as a tool in the fight against cancer. The field of mathematical oncology has received great attention and increased enormously, but over-optimistic estimations about its ability have created unrealistic expectations. We present a critical appraisal of the current state of mathematical models of cancer. Although the field is still expanding and useful clinical applications may occur in the future, managing over-expectation requires the proposal of alternative directions for mathematical modeling. Here, we propose two main avenues for this modeling: 1) the identification of the elementary biophysical laws of cancer development, and 2) the development of a multiscale mathematical theory as the framework for models predictive of tumor growth. Finally, we suggest how these new directions could contribute to addressing the current challenges of understanding breast cancer growth and metastasis.

Dynamical model for assessment of anti-angiogenic therapy of cancer

Different experimental models have substantially established that the anti-angiogenic (AAG)group of drugs are able to control the growth of tumor mass by cutting down the nutritive supply to the cancer cells. The mechanism of action of this group of drugs acts on the cells of the vascular endothelium. Recently, different AAG drugs have been in clinical trials. Initial clinical trials showed that application of AAG drugs produced different sorts of toxicity in patients,so calibration of the doses and drug application schedules are very important at present.Hence, development of analytical models would definitely help in this respect, particularly atthe individual level. The analytical model presented here may help to make a judicious choice of drug doses and drug schedule to control the growth of the tumor system under the condition of malignancy.

Supratentorial WHO grade II glioma invasion: a morphologic study using sequential conventional MRI

The natural morphological growth and invasion of World Health Organization (WHO) grade II glioma has not been well documented in previous literature. This study retrospectively analysed the morphological invasive characteristics of adult. Data from 20 patients (15 men, 5 women; mean age, 38 years; age range, 22-64 years), who had supratentorial WHO grade II gliomas and consecutively underwent serial preoperative conventional MR examinations were retrospectively analysed, for change in location, and evidence of haemorrhage, enhancement, necrosis, peri-tumoural oedema and mass effect. Seven tumours, initially located in the grey matter (3, insula; 4, frontal cortices), expanded without definite invasion along fibres. Thirteen tumours, originated from the junction of grey and white matter, invaded in different orientations (4, contralateral invasion; 1, ipsilateral remote dissemination; 8, ipsilateral invasion via the surrounding fibres). The increased proportion of haemorrhage, enhancement, necrosis, peri-tumoural oedema and mass effect in the first and last pre-operative examinations were 14% (1/7), 29% (2/7), 43% (3/7), 43% (3/7) and 29% (2/7), respectively, for the 7 tumours initially located in grey matter, and 15% (2/13), 38% (5/13), 31% (4/13), 15% (2/13) and 38% (5/13) respectively for the 13 tumours initially located at the junction of grey and white matter. The growth of supratentorial WHO grade II glioma is a complicated process. The growth directionality may be determined by initial tumour location.

Predictions of tumour morphological stability and evaluation against experimental observations

The hallmark of malignant tumours is their spread into neighbouring tissue and metastasis to distant organs, which can lead to life threatening consequences. One of the defining characteristics of aggressive tumours is an unstable morphology, including the formation of invasive fingers and protrusions observed both in vitro and in vivo. In spite of extensive biological, clinical and modelling study and research at physical scales ranging from the molecular to the tissue, the driving dynamics of tumour invasiveness are not completely understood, partly because it is challenging to observe and study cancer as a multi-scale system. Mathematical modelling has been applied to provide further insights into these complex invasive and metastatic behaviours. Modelling a solid tumour as an incompressible fluid, we consider three possible constitutive relations to describe tumour growth, namely Darcy's law, Stokes' law and the combined Darcy-Stokes law. We study the tumour morphological stability described by each model and evaluate the consistency between theoretical model predictions and experimental data from in vitro three-dimensional multicellular tumour spheroids. The analysis reveals that the Stokes model is the most consistent with the experimental observations, and that it predicts our experimental tumour growth is marginally stable. We further show that it is feasible to extract parameter values from a limited set of data and create a self-consistent modelling framework that can be extended to the multi-scale study of cancer.

Front instabilities and invasiveness of simulated 3D avascular tumors

We use the Glazier-Graner-Hogeweg model to simulate three-dimensional (3D), single-phenotype, avascular tumors growing in an homogeneous tissue matrix (TM) supplying a single limiting nutrient. We study the effects of two parameters on tumor morphology: a diffusion-limitation parameter defined as the ratio of the tumor-substrate consumption rate to the substrate-transport rate, and the tumor-TM surface tension. This initial model omits necrosis and oxidative/hypoxic metabolism effects, which can further influence tumor morphology, but our simplified model still shows significant parameter dependencies. The diffusion-limitation parameter determines whether the growing solid tumor develops a smooth (noninvasive) or fingered (invasive) interface, as in our earlier two-dimensional (2D) simulations. The sensitivity of 3D tumor morphology to tumor-TM surface tension increases with the size of the diffusion-limitation parameter, as in 2D. The 3D results are unexpectedly close to those in 2D. Our results therefore may justify using simpler 2D simulations of tumor growth, instead of more realistic but more computationally expensive 3D simulations. While geometrical artifacts mean that 2D sections of connected 3D tumors may be disconnected, the morphologies of 3D simulated tumors nevertheless correlate with the morphologies of their 2D sections, especially for low-surface-tension tumors, allowing the use of 2D sections to partially reconstruct medically-important 3D-tumor structures.

Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.

A novel strategy for mechanism based computational drug discovery

Glioma, the common brain tumor, which arises from the glial cells, offers worse prognosis and therapy than any other tumors. Despite the genetic and pathological diversities of malignant gliomas, common signaling pathways that drive cellular proliferation, survival, invasion and angiogenesis have been identified. Very often, various tyrosine kinase receptors are inappropriately activated in human brain tumors and contribute to tumor malignancy. During such tumourous states where multiple pathways are involved, a few of them are responsbile for cell differentiation, proliferation and anti-apoptosis. Computational simulation studies of normal EGFR signaling in glioma together with the mutant EGFR mediated signaling and the MAPK signaling in glioma were carried out. There were no significant cross talks observed between the mutant EGFR and the MAPK pathways and thus from the simulation results, we propose a novel concept of 'multiple-targeting' that combines EGFR and Ras targeted therapy thereby providing a better therapeutic value against glioma. Diallyl Disulfide (DADS) that has been commonly used for Ras inhibition in glioma was taken for analyses and the effect of inhibiting the EGFR downstream signaling protein with this DADS was analyzed using the simulation and docking studies.

Applications of a new In vivo tumor spheroid based shell-less chorioallantoic membrane 3-D model in bioengineering research

The chicken chorioallantoic membrane (CAM) is a classical in vivo biological model in studies of angiogenesis. Combined with the right tumor system and experimental configuration this classical model can offer new approaches to investigating tumor processes. The increase in development of biotechnological devices for cancer diagnosis and treatment, calls for more sophisticated tumor models that can easily adapt to the technology, and provide a more accurate, stable and consistent platform for rapid quantitative and qualitative analysis. As we discuss a variety of applications of this novel in vivo tumor spheroid based shell-less CAM model in biomedical engineering research, we will show that it is extremely versatile and easily adaptable to an array of biomedical applications. The model is particularly useful in quantitative studies of the progression of avascular tumors into vascularized tumors in the CAM. Its environment is more stable, flat and has a large working area and wider field of view excellent for imaging and longitudinal studies. Finally, rapid data acquisition, screening and validation of biomedical devices and therapeutics are possible with the short experimental window.

Image guided personalization of reaction-diffusion type tumor growth models using modified anisotropic eikonal equations

Reaction-diffusion based tumor growth models have been widely used in the literature for modeling the growth of brain gliomas. Lately, recent models have started integrating medical images in their formulation. Including different tissue types, geometry of the brain and the directions of white matter fiber tracts improved the spatial accuracy of reaction-diffusion models. The adaptation of the general model to the specific patient cases on the other hand has not been studied thoroughly yet. In this paper, we address this adaptation. We propose a parameter estimation method for reaction-diffusion tumor growth models using time series of medical images. This method estimates the patient specific parameters of the model using the images of the patient taken at successive time instances. The proposed method formulates the evolution of the tumor delineation visible in the images based on the reaction-diffusion dynamics; therefore, it remains consistent with the information available. We perform thorough analysis of the method using synthetic tumors and show important couplings between parameters of the reaction-diffusion model. We show that several parameters can be uniquely identified in the case of fixing one parameter, namely the proliferation rate of tumor cells. Moreover, regardless of the value the proliferation rate is fixed to, the speed of growth of the tumor can be estimated in terms of the model parameters with accuracy. We also show that using the model-based speed, we can simulate the evolution of the tumor for the specific patient case. Finally, we apply our method to two real cases and show promising preliminary results.