A four-dimensional computer simulation model of the in vivo response to radiotherapy of glioblastoma multiforme: studies on the effect of clonogenic cell density

A Monte Carlo simulation-based decision support system for radiation oncologists in the treatment of glioblastoma multiforme

In the present research, we have developed a model-based crisp logic function statistical classifier decision support system supplemented with treatment planning systems for radiation oncologists in the treatment of glioblastoma multiforme (GBM). This system is based on Monte Carlo radiation transport simulation and it recreates visualization of treatment environments on mathematical anthropomorphic brain (MAB) phantoms. Energy deposition within tumour tissue and normal tissues are graded by quality audit factors which ensure planned dose delivery to tumour site thereby minimising damages to healthy tissues. The proposed novel methodology predicts tumour growth response to radiation therapy from a patient-specific medicine quality audit perspective. Validation of the study was achieved by recreating thirty-eight patient-specific mathematical anthropomorphic brain phantoms of treatment environments by taking into consideration density variation and composition of brain tissues. Dose computations accomplished through water phantom, tissue-equivalent head phantoms are neither cost-effective, nor patient-specific customized and is often less accurate. The above-highlighted drawbacks can be overcome by using open-source Electron Gamma Shower (EGSnrc) software and clinical case reports for MAB phantom synthesis which would result in accurate dosimetry with due consideration to the time factors. Considerable dose deviations occur at the tumour site for environments with intraventricular glioblastoma, haematoma, abscess, trapped air and cranial flaps leading to quality factors with a lower logic value of 0. Logic value of 1 depicts higher dose deposition within healthy tissues and also leptomeninges for majority of the environments which results in radiation-induced laceration.

Isotoxic dose escalated radiotherapy for glioblastoma based on diffusion-weighted MRI and tumor control probability-an in-silico study

OBJECTIVES

Glioblastoma (GBM) is the most common malignant primary brain tumor with local recurrence after radiotherapy (RT), the most common mode of failure. Standard RT practice applies the prescription dose uniformly across tumor volume disregarding radiological tumor heterogeneity. We present a novel strategy using diffusion-weighted (DW-) MRI to calculate the cellular density within the gross tumor volume (GTV) in order to facilitate dose escalation to a biological target volume (BTV) to improve tumor control probability (TCP).

METHODS

The pre-treatment apparent diffusion coefficient (ADC) maps derived from DW-MRI of ten GBM patients treated with radical chemoradiotherapy were used to calculate the local cellular density based on published data. Then, a TCP model was used to calculate TCP maps from the derived cell density values. The dose was escalated using a simultaneous integrated boost (SIB) to the BTV, defined as the voxels for which the expected pre-boost TCP was in the lowest quartile of the TCP range for each patient. The SIB dose was chosen so that the TCP in the BTV increased to match the average TCP of the whole tumor.

RESULTS

By applying a SIB of between 3.60 Gy and 16.80 Gy isotoxically to the BTV, the cohort's calculated TCP increased by a mean of 8.44% (ranging from 7.19 to 16.84%). The radiation dose to organ at risk is still under their tolerance.

CONCLUSIONS

Our findings indicate that TCPs of GBM patients could be increased by escalating radiation doses to intratumoral locations guided by the patient's biology (i.e., cellularity), moreover offering the possibility for personalized RT GBM treatments.

ADVANCES IN KNOWLEDGE

A personalized and voxel level SIB radiotherapy method for GBM is proposed using DW-MRI, which can increase the tumor control probability and maintain organ at risk dose constraints.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for three time and space-dependent factors that influence the efficiency of radiotherapy fractionation schemes—tumor cell repopulation, reoxygenation and redistribution of proliferative states. A special algorithm is developed, aimed at finding the fractionation schemes that provide increased tumor cure probability under the constraints of maximum normal tissue damage and maximum fractional dose. The optimization procedure is performed for varied radiosensitivity of tumor cells under the values of model parameters, corresponding to different degrees of tumor malignancy. The resulting optimized schemes consist of two stages. The first stages are aimed to increase the radiosensitivity of the tumor cells, remaining after their end, sparing the caused normal tissue damage. This allows to increase the doses during the second stages and thus take advantage of the obtained increased radiosensitivity. Such method leads to significant expansions in the curative ranges of the values of tumor radiosensitivity parameters. Overall, the results of this study represent the theoretical proof of concept that non-uniform radiotherapy fractionation schemes may be considerably more effective that uniform ones, due to the time and space-dependent effects.

Mathematical Modeling of the Effects of Tumor Heterogeneity on the Efficiency of Radiation Treatment Schedule

Radiotherapy uses high doses of energy to eradicate cancer cells and control tumors. Various treatment schedules have been developed and tried in clinical trials, yet significant obstacles remain to improving the radiotherapy fractionation. Genetic and non-genetic cellular diversity within tumors can lead to different radiosensitivity among cancer cells that can affect radiation treatment outcome. We propose a minimal mathematical model to study the effect of tumor heterogeneity and repair in different radiation treatment schedules. We perform stochastic and deterministic simulations to estimate model parameters using available experimental data. Our results suggest that gross tumor volume reduction is insufficient to control the disease if a fraction of radioresistant cells survives therapy. If cure cannot be achieved, protocols should balance volume reduction with minimal selection for radioresistant cells. We show that the most efficient treatment schedule is dependent on biology and model parameter values and, therefore, emphasize the need for careful tumor-specific model calibration before clinically actionable conclusions can be drawn.

Monte-Carlo model development for evaluation of current clinical target volume definition for heterogeneous and hypoxic glioblastoma

Clinical target volume (CTV) determination may be complex and subjective. In this work a microscopic-scale tumour model was developed to evaluate current CTV practices in glioblastoma multiforme (GBM) external radiotherapy. Previously, a Geant4 cell-based dosimetry model was developed to calculate the dose deposited in individual GBM cells. Microscopic extension probability (MEP) models were then developed using Matlab-2012a. The results of the cell-based dosimetry model and MEP models were combined to calculate survival fractions (SF) for CTV margins of 2.0 and 2.5 cm. In the current work, oxygenation and heterogeneous radiosensitivity profiles were incorporated into the GBM model. The genetic heterogeneity was modelled using a range of α/β values (linear-quadratic model parameters) associated with different GBM cell lines. These values were distributed among the cells randomly, taken from a Gaussian-weighted sample of α/β values. Cellular oxygen pressure was distributed randomly taken from a sample weighted to profiles obtained from literature. Three types of GBM models were analysed: homogeneous-normoxic, heterogeneous-normoxic, and heterogeneous-hypoxic. The SF in different regions of the tumour model and the effect of the CTV margin extension from 2.0-2.5 cm on SFs were investigated for three MEP models. The SF within the beam was increased by up to three and two orders of magnitude following incorporation of heterogeneous radiosensitivities and hypoxia, respectively, in the GBM model. However, the total SF was shown to be overdominated by the presence of tumour cells in the penumbra region and to a lesser extent by genetic heterogeneity and hypoxia. CTV extension by 0.5 cm reduced the SF by a maximum of 78.6  ±  3.3%, 78.5  ±  3.3%, and 77.7  ±  3.1% for homogeneous and heterogeneous-normoxic, and heterogeneous hypoxic GBMs, respectively. Monte-Carlo model was developed to quantitatively evaluate SF for genetically heterogeneous and hypoxic GBM with two CTV margins and three MEP distributions. The results suggest that photon therapy may not provide cure for hypoxic and genetically heterogeneous GBM. However, the extension of the CTV margin by 0.5 cm could be beneficial to delay the recurrence time for this tumour type due to significant increase in tumour cell irradiation.

A patient-specific computational model of hypoxia-modulated radiation resistance in glioblastoma using 18F-FMISO-PET

Glioblastoma multiforme (GBM) is a highly invasive primary brain tumour that has poor prognosis despite aggressive treatment. A hallmark of these tumours is diffuse invasion into the surrounding brain, necessitating a multi-modal treatment approach, including surgery, radiation and chemotherapy. We have previously demonstrated the ability of our model to predict radiographic response immediately following radiation therapy in individual GBM patients using a simplified geometry of the brain and theoretical radiation dose. Using only two pre-treatment magnetic resonance imaging scans, we calculate net rates of proliferation and invasion as well as radiation sensitivity for a patient's disease. Here, we present the application of our clinically targeted modelling approach to a single glioblastoma patient as a demonstration of our method. We apply our model in the full three-dimensional architecture of the brain to quantify the effects of regional resistance to radiation owing to hypoxia in vivo determined by [(18)F]-fluoromisonidazole positron emission tomography (FMISO-PET) and the patient-specific three-dimensional radiation treatment plan. Incorporation of hypoxia into our model with FMISO-PET increases the model-data agreement by an order of magnitude. This improvement was robust to our definition of hypoxia or the degree of radiation resistance quantified with the FMISO-PET image and our computational model, respectively. This work demonstrates a useful application of patient-specific modelling in personalized medicine and how mathematical modelling has the potential to unify multi-modality imaging and radiation treatment planning.

Mathematically modeling the biological properties of gliomas: A review

Although mathematical modeling is a mainstay for industrial and many scientific studies, such approaches have found little application in neurosurgery. However, the fusion of biological studies and applied mathematics is rapidly changing this environment, especially for cancer research. This review focuses on the exciting potential for mathematical models to provide new avenues for studying the growth of gliomas to practical use. In vitro studies are often used to simulate the effects of specific model parameters that would be difficult in a larger-scale model. With regard to glioma invasive properties, metabolic and vascular attributes can be modeled to gain insight into the infiltrative mechanisms that are attributable to the tumor's aggressive behavior. Morphologically, gliomas show different characteristics that may allow their growth stage and invasive properties to be predicted, and models continue to offer insight about how these attributes are manifested visually. Recent studies have attempted to predict the efficacy of certain treatment modalities and exactly how they should be administered relative to each other. Imaging is also a crucial component in simulating clinically relevant tumors and their influence on the surrounding anatomical structures in the brain.

Evaluation of current clinical target volume definitions for glioblastoma using cell-based dosimetry stochastic methods

OBJECTIVE

Determination of an optimal clinical target volume (CTV) is complex and remains uncertain. The aim of this study was to develop a glioblastoma multiforme (GBM) model to be used for evaluation of current CTV practices for external radiotherapy.

METHODS

The GBM model was structured as follows: (1) a Geant4 cellular model was developed to calculate the absorbed dose in individual cells represented by cubic voxels of 20 μm sides. The system was irradiated with opposing 6 MV X-ray beams. The beams encompassed planning target volumes corresponding to 2.0- and 2.5-cm CTV margins; (2) microscopic extension probability (MEP) models were developed using MATLAB(®) 2012a (MathWorks(®), Natick, MA), based on clinical studies reporting on GBM clonogenic spread; (3) the cellular dose distribution was convolved with the MEP models to evaluate cellular survival fractions (SFs) for both CTV margins.

RESULTS

A CTV margin of 2.5 cm, compared to a 2.0-cm CTV margin, resulted in a reduced total SF from 12.9% ± 0.9% to 3.6% ± 0.2%, 5.5% ± 0.4% to 1.2% ± 0.1% and 11.1% ± 0.7% to 3.0% ± 0.2% for circular, elliptical and irregular MEP distributions, respectively.

CONCLUSION

A Monte Carlo model was developed to quantitatively evaluate the impact of GBM CTV margins on total and penumbral SF. The results suggest that the reduction in total SF ranges from 3.5 to 5, when the CTV is extended by 0.5 cm.

ADVANCES IN KNOWLEDGE

The model provides a quantitative tool for evaluation of different CTV margins in terms of cell kill efficacy. Cellular platform of the tool allows future incorporation of cellular properties of GBM.

Optimization of radiation dosing schedules for proneural glioblastoma

Glioblastomas are the most aggressive primary brain tumor. Despite treatment with surgery, radiation and chemotherapy, these tumors remain uncurable and few significant increases in survival have been observed over the last half-century. We recently employed a combined theoretical and experimental approach to predict the effectiveness of radiation administration schedules, identifying two schedules that led to superior survival in a mouse model of the disease (Leder et al., Cell 156(3):603-616, 2014). Here we extended this approach to consider fractionated schedules to best minimize toxicity arising in early- and late-responding tissues. To this end, we decomposed the problem into two separate solvable optimization tasks: (i) optimization of the amount of radiation per dose, and (ii) optimization of the amount of time that passes between radiation doses. To ensure clinical applicability, we then considered the impact of clinical operating hours by incorporating time constraints consistent with operational schedules of the radiology clinic. We found that there was no significant loss incurred by restricting dosage to an 8:00 a.m. to 5:00 p.m. window. Our flexible approach is also applicable to other tumor types treated with radiotherapy.

A Proposed Paradigm Shift in Initializing Cancer Predictive Models with DCE-MRI Based PK Parameters: A Feasibility Study

Glioblastoma multiforme is the most aggressive type of glioma and the most common malignant primary intra-axial brain tumor. In an effort to predict the evolution of the disease and optimize therapeutical decisions, several models have been proposed for simulating the growth pattern of glioma. One of the latest models incorporates cell proliferation and invasion, angiogenic net rates, oxygen consumption, and vasculature. These factors, particularly oxygenation levels, are considered fundamental factors of tumor heterogeneity and compartmentalization. This paper focuses on the initialization of the cancer cell populations and vasculature based on imaging examinations of the patient and presents a feasibility study on vasculature prediction over time. To this end, pharmacokinetic parameters derived from dynamic contrast-enhanced magnetic resonance imaging using Toft’s model are used in order to feed the model. K^trans is used as a metric of the density of endothelial cells (vasculature); at the same time, it also helps to discriminate distinct image areas of interest, under a set of assumptions. Feasibility results of applying the model to a real clinical case are presented, including a study on the effect of certain parameters on the pattern of the simulated tumor.

Mathematical modeling of PDGF-driven glioblastoma reveals optimized radiation dosing schedules

Glioblastomas (GBMs) are the most common and malignant primary brain tumors and are aggressively treated with surgery, chemotherapy, and radiotherapy. Despite this treatment, recurrence is inevitable and survival has improved minimally over the last 50 years. Recent studies have suggested that GBMs exhibit both heterogeneity and instability of differentiation states and varying sensitivities of these states to radiation. Here, we employed an iterative combined theoretical and experimental strategy that takes into account tumor cellular heterogeneity and dynamically acquired radioresistance to predict the effectiveness of different radiation schedules. Using this model, we identified two delivery schedules predicted to significantly improve efficacy by taking advantage of the dynamic instability of radioresistance. These schedules led to superior survival in mice. Our interdisciplinary approach may also be applicable to other human cancer types treated with radiotherapy and, hence, may lay the foundation for significantly increasing the effectiveness of a mainstay of oncologic therapy. PAPERCLIP:

Estimating dose painting effects in radiotherapy: a mathematical model

Tumor heterogeneity is widely considered to be a determinant factor in tumor progression and in particular in its recurrence after therapy. Unfortunately, current medical techniques are unable to deduce clinically relevant information about tumor heterogeneity by means of non-invasive methods. As a consequence, when radiotherapy is used as a treatment of choice, radiation dosimetries are prescribed under the assumption that the malignancy targeted is of a homogeneous nature. In this work we discuss the effects of different radiation dose distributions on heterogeneous tumors by means of an individual cell-based model. To that end, a case is considered where two tumor cell phenotypes are present, which we assume to strongly differ in their respective cell cycle duration and radiosensitivity properties. We show herein that, as a result of such differences, the spatial distribution of the corresponding phenotypes, whence the resulting tumor heterogeneity can be predicted as growth proceeds. In particular, we show that if we start from a situation where a majority of ordinary cancer cells (CCs) and a minority of cancer stem cells (CSCs) are randomly distributed, and we assume that the length of CSC cycle is significantly longer than that of CCs, then CSCs become concentrated at an inner region as tumor grows. As a consequence we obtain that if CSCs are assumed to be more resistant to radiation than CCs, heterogeneous dosimetries can be selected to enhance tumor control by boosting radiation in the region occupied by the more radioresistant tumor cell phenotype. It is also shown that, when compared with homogeneous dose distributions as those being currently delivered in clinical practice, such heterogeneous radiation dosimetries fare always better than their homogeneous counterparts. Finally, limitations to our assumptions and their resulting clinical implications will be discussed.

Is it time for a new paradigm for systemic cancer treatment? Lessons from a century of cancer chemotherapy

U.S. SEER (Surveillance Epidemiology and End Results) data for age-adjusted mortality rates for all cancers combined for all races show only a modest overall 13% decline over the past 35 years. Moreover, the greatest contributor to cancer mortality is treatment-resistant metastatic disease. The accepted therapeutic paradigm for the past half-century for the treatment of advanced cancers has involved the use of systemic chemotherapy drugs cytotoxic for cycling cells (both normal and malignant) during DNA synthesis and/or mitosis. The failure of this therapeutic modality to achieve high-level, consistent rates of disease-free survival for some of the most common cancers, including tumors of the lung, colon breast, brain, melanoma, and others is the focus of this paper. A retrospective assessment of critical milestones in cancer chemotherapy indicates that most successful therapeutic regimens use cytotoxic cell cycle inhibitors in combined, maximum tolerated, dose-dense acute treatment regimens originally developed to treat acute lymphoblastic leukemia and some lymphomas. Early clinical successes in this area led to their wholesale application to the treatment of solid tumor malignancies that, unfortunately, has not produced consistent, long-term high cure rates for many common cancers. Important differences in therapeutic sensitivity of leukemias/lymphomas versus solid tumors can be explained by key biological differences that define the treatment-resistant solid tumor phenotype. A review of these clinical outcome data in the context of recent developments in our understanding of drug resistance mechanisms characteristic of solid tumors suggests the need for a new paradigm for the treatment of chemotherapy-resistant cancers. In contrast to reductionist approaches, the systemic approach targets both microenvironmental and systemic factors that drive and sustain tumor progression. These systemic factors include dysregulated inflammatory and oxidation pathways shown to be directly implicated in the development and maintenance of the cancer phenotype. The paradigm stresses the importance of a combined preventive/therapeutic approach involving adjuvant chemotherapies that incorporate anti-inflammatory and anti-oxidant therapeutics.

From patient-specific mathematical neuro-oncology to precision medicine

Gliomas are notoriously aggressive, malignant brain tumors that have variable response to treatment. These patients often have poor prognosis, informed primarily by histopathology. Mathematical neuro-oncology (MNO) is a young and burgeoning field that leverages mathematical models to predict and quantify response to therapies. These mathematical models can form the basis of modern “precision medicine” approaches to tailor therapy in a patient-specific manner. Patient-specific models (PSMs) can be used to overcome imaging limitations, improve prognostic predictions, stratify patients, and assess treatment response in silico. The information gleaned from such models can aid in the construction and efficacy of clinical trials and treatment protocols, accelerating the pace of clinical research in the war on cancer. This review focuses on the growing translation of PSM to clinical neuro-oncology. It will also provide a forward-looking view on a new era of patient-specific MNO.

Simulating radiotherapy effect in high-grade glioma by using diffusive modeling and brain atlases

Applying diffusive models for simulating the spatiotemporal change of concentration of tumour cells is a modern application of predictive oncology. Diffusive models are used for modelling glioblastoma, the most aggressive type of glioma. This paper presents the results of applying a linear quadratic model for simulating the effects of radiotherapy on an advanced diffusive glioma model. This diffusive model takes into consideration the heterogeneous velocity of glioma in gray and white matter and the anisotropic migration of tumor cells, which is facilitated along white fibers. This work uses normal brain atlases for extracting the proportions of white and gray matter and the diffusion tensors used for anisotropy. The paper also presents the results of applying this glioma model on real clinical datasets.

In silico modelling of treatment-induced tumour cell kill: developments and advances

Mathematical and stochastic computer (in silico) models of tumour growth and treatment response of the past and current eras are presented, outlining the aims of the models, model methodology, the key parameters used to describe the tumour system, and treatment modality applied, as well as reported outcomes from simulations. Fractionated radiotherapy, chemotherapy, and combined therapies are reviewed, providing a comprehensive overview of the modelling literature for current modellers and radiobiologists to ignite the interest of other computational scientists and health professionals of the ever evolving and clinically relevant field of tumour modelling.

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.

Monte Carlo role in radiobiological modelling of radiotherapy outcomes

Radiobiological models are essential components of modern radiotherapy. They are increasingly applied to optimize and evaluate the quality of different treatment planning modalities. They are frequently used in designing new radiotherapy clinical trials by estimating the expected therapeutic ratio of new protocols. In radiobiology, the therapeutic ratio is estimated from the expected gain in tumour control probability (TCP) to the risk of normal tissue complication probability (NTCP). However, estimates of TCP/NTCP are currently based on the deterministic and simplistic linear-quadratic formalism with limited prediction power when applied prospectively. Given the complex and stochastic nature of the physical, chemical and biological interactions associated with spatial and temporal radiation induced effects in living tissues, it is conjectured that methods based on Monte Carlo (MC) analysis may provide better estimates of TCP/NTCP for radiotherapy treatment planning and trial design. Indeed, over the past few decades, methods based on MC have demonstrated superior performance for accurate simulation of radiation transport, tumour growth and particle track structures; however, successful application of modelling radiobiological response and outcomes in radiotherapy is still hampered with several challenges. In this review, we provide an overview of some of the main techniques used in radiobiological modelling for radiotherapy, with focus on the MC role as a promising computational vehicle. We highlight the current challenges, issues and future potentials of the MC approach towards a comprehensive systems-based framework in radiobiological modelling for radiotherapy.

Monte Carlo radiotherapy simulations of accelerated repopulation and reoxygenation for hypoxic head and neck cancer

OBJECTIVE

A temporal Monte Carlo tumour growth and radiotherapy effect model (HYP-RT) simulating hypoxia in head and neck cancer has been developed and used to analyse parameters influencing cell kill during conventionally fractionated radiotherapy. The model was designed to simulate individual cell division up to 10(8) cells, while incorporating radiobiological effects, including accelerated repopulation and reoxygenation during treatment.

METHOD

Reoxygenation of hypoxic tumours has been modelled using randomised increments of oxygen to tumour cells after each treatment fraction. The process of accelerated repopulation has been modelled by increasing the symmetrical stem cell division probability. Both phenomena were onset immediately or after a number of weeks of simulated treatment.

RESULTS

The extra dose required to control (total cell kill) hypoxic vs oxic tumours was 15-25% (8-20 Gy for 5 × 2 Gy per week) depending on the timing of accelerated repopulation onset. Reoxygenation of hypoxic tumours resulted in resensitisation and reduction in total dose required by approximately 10%, depending on the time of onset. When modelled simultaneously, accelerated repopulation and reoxygenation affected cell kill in hypoxic tumours in a similar manner to when the phenomena were modelled individually; however, the degree was altered, with non-additive results. Simulation results were in good agreement with standard linear quadratic theory; however, differed for more complex comparisons where hypoxia, reoxygenation as well as accelerated repopulation effects were considered.

CONCLUSION

Simulations have quantitatively confirmed the need for patient individualisation in radiotherapy for hypoxic head and neck tumours, and have shown the benefits of modelling complex and dynamic processes using Monte Carlo methods.

3D modeling of effects of increased oxygenation and activity concentration in tumors treated with radionuclides and antiangiogenic drugs

PURPOSE

Formation of new blood vessels (angiogenesis) in response to hypoxia is a fundamental event in the process of tumor growth and metastatic dissemination. However, abnormalities in tumor neovasculature often induce increased interstitial pressure (IP) and further reduce oxygenation (pO2) of tumor cells. In radiotherapy, well-oxygenated tumors favor treatment. Antiangiogenic drugs may lower IP in the tumor, improving perfusion, pO2 and drug uptake, by reducing the number of malfunctioning vessels in the tissue. This study aims to create a model for quantifying the effects of altered pO2-distribution due to antiangiogenic treatment in combination with radionuclide therapy.

METHODS

Based on experimental data, describing the effects of antiangiogenic agents on oxygenation of GlioblastomaMultiforme (GBM), a single cell based 3D model, including 10(10) tumor cells, was developed, showing how radionuclide therapy response improves as tumor oxygenation approaches normal tissue levels. The nuclides studied were 90Y, 131I, 177Lu, and 211At. The absorbed dose levels required for a tumor control probability (TCP) of 0.990 are compared for three different log-normal pO2-distributions: micro1 = 2.483, sigma1 = 0.711; micro2 = 2.946, sigma2 = 0.689; micro3 = 3.689, and sigma3 = 0.330. The normal tissue absorbed doses will, in turn, depend on this. These distributions were chosen to represent the expected oxygen levels in an untreated hypoxic tumor, a hypoxic tumor treated with an anti-VEGF agent, and in normal, fully-oxygenated tissue, respectively. The former two are fitted to experimental data. The geometric oxygen distributions are simulated using two different patterns: one Monte Carlo based and one radially increasing, while keeping the log-normal volumetric distributions intact. Oxygen and activity are distributed, according to the same pattern.

RESULTS

As tumor pO2 approaches normal tissue levels, the therapeutic effect is improved so that the normal tissue absorbed doses can be decreased by more than 95%, while retaining TCP, in the most favorable scenario and by up to about 80% with oxygen levels previously achieved in vivo, when the least favourable oxygenation case is used as starting point. The major difference occurs in poorly oxygenated cells. This is also where the pO2-dependence of the oxygen enhancement ratio is maximal.

CONCLUSIONS

Improved tumor oxygenation together with increased radionuclide uptake show great potential for optimising treatment strategies, leaving room for successive treatments, or lowering absorbed dose to normal tissues, due to increased tumor response. Further studies of the concomitant use of antiangiogenic drugs and radionuclide therapy therefore appear merited.

High-grade glioma diffusive modeling using statistical tissue information and diffusion tensors extracted from atlases

Glioma, especially glioblastoma, is a leading cause of brain cancer fatality involving highly invasive and neoplastic growth. Diffusive models of glioma growth use variations of the diffusion-reaction equation in order to simulate the invasive patterns of glioma cells by approximating the spatiotemporal change of glioma cell concentration. The most advanced diffusive models take into consideration the heterogeneous velocity of glioma in gray and white matter, by using two different discrete diffusion coefficients in these areas. Moreover, by using diffusion tensor imaging (DTI), they simulate the anisotropic migration of glioma cells, which is facilitated along white fibers, assuming diffusion tensors with different diffusion coefficients along each candidate direction of growth. Our study extends this concept by fully exploiting the proportions of white and gray matter extracted by normal brain atlases, rather than discretizing diffusion coefficients. Moreover, the proportions of white and gray matter, as well as the diffusion tensors, are extracted by the respective atlases; thus, no DTI processing is needed. Finally, we applied this novel glioma growth model on real data and the results indicate that prognostication rates can be improved.

Computer simulation of tumour control probabilities after irradiation for varying intrinsic radio-sensitivity using a single cell based model

BACKGROUND

Currently, optimisation of the dose distribution and clinical acceptance are almost entirely based on the physical dose distribution and tumour control probability modelling is far from being routinely used as objective in treatment planning. For future individualised radiotherapeutic strategies, a reliable patient specific simulation model, taking into account customised tumour features, is needed to predict and improve treatment outcome.

MATERIALS AND METHODS

To approach these demands, a single cell and Monte-Carlo based model was developed, which enables three-dimensional tumour growth and radiation response simulation. Tumour cells were characterised by cell-associated features such as age, intrinsic radio-sensitivity, proliferation ability, and oxygenation status, while capillary cells were considered as sources of a radial-dependent oxygen profile. Response to radiation was simulated by the linear-quadratic model, taking into account the lower radio-sensitivity of poorly oxygenated tumour cells.

RESULTS

The present study shows the influence of the model components and demonstrates the impact of the intra- and inter-tumoural radio-sensitivity heterogeneity on the treatment response.

CONCLUSION

The simulation model adequately delineates the importance of the above described selected parameters on tumour control probability, providing an insight into the interplay of different physical and biological parameters, and its relevance for an individual tumour response.

Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach

Glioblastoma multiforme (GBM) is the most malignant form of primary brain tumors known as gliomas. They proliferate and invade extensively and yield short life expectancies despite aggressive treatment. Response to treatment is usually measured in terms of the survival of groups of patients treated similarly, but this statistical approach misses the subgroups that may have responded to or may have been injured by treatment. Such statistics offer scant reassurance to individual patients who have suffered through these treatments. Furthermore, current imaging-based treatment response metrics in individual patients ignore patient-specific differences in tumor growth kinetics, which have been shown to vary widely across patients even within the same histological diagnosis and, unfortunately, these metrics have shown only minimal success in predicting patient outcome. We consider nine newly diagnosed GBM patients receiving diagnostic biopsy followed by standard-of-care external beam radiation therapy (XRT). We present and apply a patient-specific, biologically based mathematical model for glioma growth that quantifies response to XRT in individual patients in vivo. The mathematical model uses net rates of proliferation and migration of malignant tumor cells to characterize the tumor's growth and invasion along with the linear-quadratic model for the response to radiation therapy. Using only routinely available pre-treatment MRIs to inform the patient-specific bio-mathematical model simulations, we find that radiation response in these patients, quantified by both clinical and model-generated measures, could have been predicted prior to treatment with high accuracy. Specifically, we find that the net proliferation rate is correlated with the radiation response parameter (r = 0.89, p = 0.0007), resulting in a predictive relationship that is tested with a leave-one-out cross-validation technique. This relationship predicts the tumor size post-therapy to within inter-observer tumor volume uncertainty. The results of this study suggest that a mathematical model can create a virtual in silico tumor with the same growth kinetics as a particular patient and can not only predict treatment response in individual patients in vivo but also provide a basis for evaluation of response in each patient to any given therapy.

Introduction of hypermatrix and operator notation into a discrete mathematics simulation model of malignant tumour response to therapeutic schemes in vivo. Some operator properties

The tremendous rate of accumulation of experimental and clinical knowledge pertaining to cancer dictates the development of a theoretical framework for the meaningful integration of such knowledge at all levels of biocomplexity. In this context our research group has developed and partly validated a number of spatiotemporal simulation models of in vivo tumour growth and in particular tumour response to several therapeutic schemes. Most of the modeling modules have been based on discrete mathematics and therefore have been formulated in terms of rather complex algorithms (e.g. in pseudocode and actual computer code). However, such lengthy algorithmic descriptions, although sufficient from the mathematical point of view, may render it difficult for an interested reader to readily identify the sequence of the very basic simulation operations that lie at the heart of the entire model. In order to both alleviate this problem and at the same time provide a bridge to symbolic mathematics, we propose the introduction of the notion of hypermatrix in conjunction with that of a discrete operator into the already developed models. Using a radiotherapy response simulation example we demonstrate how the entire model can be considered as the sequential application of a number of discrete operators to a hypermatrix corresponding to the dynamics of the anatomic area of interest. Subsequently, we investigate the operators’ commutativity and outline the “summarize and jump” strategy aiming at efficiently and realistically address multilevel biological problems such as cancer. In order to clarify the actual effect of the composite discrete operator we present further simulation results which are in agreement with the outcome of the clinical study RTOG 83–02, thus strengthening the reliability of the model developed.

Multi-scale, multi-resolution brain cancer modeling

In advancing discrete-based computational cancer models towards clinical applications, one faces the dilemma of how to deal with an ever growing amount of biomedical data that ought to be incorporated eventually in one form or another. Model scalability becomes of paramount interest. In an effort to start addressing this critical issue, here, we present a novel multi-scale and multi-resolution agent-based in silico glioma model. While 'multi-scale' refers to employing an epidermal growth factor receptor (EGFR)-driven molecular network to process cellular phenotypic decisions within the micro-macroscopic environment, 'multi-resolution' is achieved through algorithms that classify cells to either active or inactive spatial clusters, which determine the resolution they are simulated at. The aim is to assign computational resources where and when they matter most for maintaining or improving the predictive power of the algorithm, onto specific tumor areas and at particular times. Using a previously described 2D brain tumor model, we have developed four different computational methods for achieving the multi-resolution scheme, three of which are designed to dynamically train on the high-resolution simulation that serves as control. To quantify the algorithms' performance, we rank them by weighing the distinct computational time savings of the simulation runs versus the methods' ability to accurately reproduce the high-resolution results of the control. Finally, to demonstrate the flexibility of the underlying concept, we show the added value of combining the two highest-ranked methods. The main finding of this work is that by pursuing a multi-resolution approach, one can reduce the computation time of a discrete-based model substantially while still maintaining a comparably high predictive power. This hints at even more computational savings in the more realistic 3D setting over time, and thus appears to outline a possible path to achieve scalability for the all-important clinical translation.

In silico cancer modeling: is it ready for prime time?

At the dawn of the era of personalized, systems-driven medicine, computational or in silico modeling and the simulation of disease processes is becoming increasingly important for hypothesis generation and data integration in both experiments and clinics alike. Arguably, the use of these techniques is nowhere more visible than in oncology. To illustrate the field's vast potential, as well as its current limitations, we briefly review selected studies on modeling malignant brain tumors. Implications for clinical practice, and for clinical trial design and outcome prediction, are also discussed.

A mathematical model for brain tumor response to radiation therapy

Gliomas are highly invasive primary brain tumors, accounting for nearly 50% of all brain tumors (Alvord and Shaw in The pathology of the aging human nervous system. Lea & Febiger, Philadelphia, pp 210-281, 1991). Their aggressive growth leads to short life expectancies, as well as a fairly algorithmic approach to treatment: diagnostic magnetic resonance image (MRI) followed by biopsy or surgical resection with accompanying second MRI, external beam radiation therapy concurrent with and followed by chemotherapy, with MRIs conducted at various times during treatment as prescribed by the physician. Swanson et al. (Harpold et al. in J Neuropathol Exp Neurol 66:1-9, 2007) have shown that the defining and essential characteristics of gliomas in terms of net rates of proliferation (rho) and invasion (D) can be determined from serial MRIs of individual patients. We present an extension to Swanson's reaction-diffusion model to include the effects of radiation therapy using the classic linear-quadratic radiobiological model (Hall in Radiobiology for the radiologist. Lippincott, Philadelphia, pp 478-480, 1994) for radiation efficacy, along with an investigation of response to various therapy schedules and dose distributions on a virtual tumor (Swanson et al. in AACR annual meeting, Los Angeles, 2007).

An imaging-based tumour growth and treatment response model: investigating the effect of tumour oxygenation on radiation therapy response

A multiscale tumour simulation model employing cell-line-specific biological parameters and functional information derived from pre-therapy PET/CT imaging data was developed to investigate effects of different oxygenation levels on the response to radiation therapy. For each tumour voxel, stochastic simulations were performed to model cellular growth and therapeutic response. Model parameters were fitted to published preclinical experiments of head and neck squamous cell carcinoma (HNSCC). Using the obtained parameters, the model was applied to a human HNSCC case to investigate effects of different uniform and non-uniform oxygenation levels and results were compared for treatment efficacy. Simulations of the preclinical studies showed excellent agreement with published data and underlined the model's ability to quantitatively reproduce tumour behaviour within experimental uncertainties. When using a simplified transformation to derive non-uniform oxygenation levels from molecular imaging data, simulations of the clinical case showed heterogeneous tumour response and variability in radioresistance with decreasing oxygen levels. Once clinically validated, this model could be used to transform patient-specific data into voxel-based biological objectives for treatment planning and to investigate biologically optimized dose prescriptions.

Cell killing and resistance in pre-operative breast cancer chemotherapy

Background

Despite the recent development of technologies giving detailed images of tumours in vivo, direct or indirect ways to measure how many cells are actually killed by a treatment or are resistant to it are still beyond our reach.

Methods

We designed a simple model of tumour progression during treatment, based on descriptions of the key phenomena of proliferation, quiescence, cell killing and resistance, and giving as output the macroscopically measurable tumour volume and growth fraction. The model was applied to a database of the time course of volumes of breast cancer in patients undergoing pre-operative chemotherapy, for which the initial estimate of proliferating cells by the measure of the percentage of Ki67-positive cells was available.

Results

The analysis recognises different patterns of response to treatment. In one subgroup of patients the fitting implied drug resistance. In another subgroup there was a shift to higher sensitivity during the therapy. In the subgroup of patients where killing of cycling cells had the highest score, the drugs showed variable efficacy against quiescent cells.

Conclusion

The approach was feasible, providing items of information not otherwise available. Additional data, particularly sequential Ki67 measures, could be added to the system, potentially reducing uncertainty in estimates of parameter values.

Single-cell-based computer simulation of the oxygen-dependent tumour response to irradiation

Optimization of treatment plans in radiotherapy requires the knowledge of tumour control probability (TCP) and normal tissue complication probability (NTCP). Mathematical models may help to obtain quantitative estimates of TCP and NTCP. A single-cell-based computer simulation model is presented, which simulates tumour growth and radiation response on the basis of the response of the constituting cells. The model contains oxic, hypoxic and necrotic tumour cells as well as capillary cells which are considered as sources of a radial oxygen profile. Survival of tumour cells is calculated by the linear quadratic model including the modified response due to the local oxygen concentration. The model additionally includes cell proliferation, hypoxia-induced angiogenesis, apoptosis and resorption of inactivated tumour cells. By selecting different degrees of angiogenesis, the model allows the simulation of oxic as well as hypoxic tumours having distinctly different oxygen distributions. The simulation model showed that poorly oxygenated tumours exhibit an increased radiation tolerance. Inter-tumoural variation of radiosensitivity flattens the dose response curve. This effect is enhanced by proliferation between fractions. Intra-tumoural radiosensitivity variation does not play a significant role. The model may contribute to the mechanistic understanding of the influence of biological tumour parameters on TCP. It can in principle be validated in radiation experiments with experimental tumours.