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

Modelling and Bayesian adaptive prediction of individual patients' tumour volume change during radiotherapy

The aim of this study is to develop a mathematical modelling method that can predict individual patients’ response to radiotherapy, in terms of tumour volume change during the treatment. The main concept is to start from a population-average model, which is subsequently updated from an individual’s tumour volume measurement. The model becomes increasingly personalized and so too does the prediction it produces. This idea of adaptive prediction was realised by using a Bayesian approach for updating the model parameters. The feasibility of the developed method was demonstrated on the data from 25 non-small cell lung cancer patients treated with helical tomotherapy, during which tumour volume was measured from daily imaging as part of the image-guided radiotherapy. The method could provide useful information for adaptive treatment planning and dose scheduling based on the patient’s personalised response.

Image based modeling of tumor growth

Tumors are a main cause of morbidity and mortality worldwide. Despite the efforts of the clinical and research communities, little has been achieved in the past decades in terms of improving the treatment of aggressive tumors. Understanding the underlying mechanism of tumor growth and evaluating the effects of different therapies are valuable steps in predicting the survival time and improving the patients' quality of life. Several studies have been devoted to tumor growth modeling at different levels to improve the clinical outcome by predicting the results of specific treatments. Recent studies have proposed patient-specific models using clinical data usually obtained from clinical images and evaluating the effects of various therapies. The aim of this review is to highlight the imaging role in tumor growth modeling and provide a worthwhile reference for biomedical and mathematical researchers with respect to tumor modeling using the clinical data to develop personalized models of tumor growth and evaluating the effect of different therapies.

Modelling the radiotherapy effect in the reaction-diffusion equation

PURPOSE

In recent years, the reaction-diffusion (Fisher-Kolmogorov) equation has received much attention from the oncology research community due to its ability to describe the infiltrating nature of glioblastoma multiforme and its extraordinary resistance to any type of therapy. However, in a number of previous papers in the literature on applications of this equation, the term (R) expressing the 'External Radiotherapy effect' was incorrectly derived. In this note we derive an analytical expression for this term in the correct form to be included in the reaction-diffusion equation.

METHODS

The R term has been derived starting from the Linear-Quadratic theory of cell killing by ionizing radiation. The correct definition of R was adopted and the basic principles of differential calculus applied.

RESULTS

The compatibility of the R term derived here with the reaction-diffusion equation was demonstrated. Referring to a typical glioblastoma tumour, we have compared the results obtained using our expression for the R term with the 'incorrect' expression proposed by other authors.

Modeling the efficacy of the extent of surgical resection in the setting of radiation therapy for glioblastoma

Standard therapy for glioblastoma (GBM) includes maximal surgical resection and radiation therapy. While it is established that radiation therapy provides the greatest survival benefit of standard treatment modalities, the impact of the extent of surgical resection (EOR) on patient outcome remains highly controversial. While some studies describe no correlation between EOR and patient survival even up to total resection, others propose either qualitative (partial versus subtotal versus complete resection) or quantitative EOR thresholds, below which there is no correlation with survival. This work uses a mathematical model in the form of a reaction–diffusion partial differential equation to simulate tumor growth and treatment with radiation therapy and surgical resection based on tumor‐specific rates of diffusion and proliferation. Simulation of 36 tumors across a wide spectrum of diffusion and proliferation rates suggests that while partial or subtotal resections generally do not provide a survival advantage, complete resection significantly improves patient outcomes. Furthermore, our model predicts a tumor‐specific quantitative threshold below which EOR has no effect on patient survival and demonstrates that this threshold increases with tumor aggressiveness, particularly with the rate of proliferation. Thus, this model may serve as an aid for determining both when surgical resection is indicated as well as the surgical margins necessary to provide clinically significant improvements in patient survival. In addition, by assigning relative benefits to radiation and surgical resection based on tumor invasiveness and proliferation, this model confirms that (with the exception of the least aggressive tumors) the survival benefit of radiation therapy exceeds that of surgical resection.

Estimating subthreshold tumor on MRI using a 3D-DTI growth model for GBM: An adjunct to radiation therapy planning

Mathematical modeling and serial magnetic resonance imaging (MRI) used to calculate patient-specific rates of tumor diffusion, D, and proliferation, ρ, can be combined to simulate glioblastoma multiforme (GBM) growth. We showed that the proportion and distribution of tumor cells below the MRI threshold are determined by the D/ρ ratio of the tumor. As most radiation fields incorporate a 1‑3 cm margin to account for subthreshold tumor, accurate characterization of subthreshold tumor aids the design of optimal radiation fields. This study compared two models: a standard one‑dimensional (1D) isotropic model and a three‑dimensional (3D) anisotropic model using the advanced imaging method of diffusion tensor imaging (DTI) ‑ with regards to the D/ρ ratio's effect on the proportion and spatial extent of the subthreshold tumor. A validated reaction‑diffusion equation accounting for tumor diffusion and proliferation modeled tumor concentration in time and space. For the isotropic and anisotropic models, nine tumors with different D/ρ ratios were grown to a T1 radius of 1.5 cm. For each tumor, the percent and extent of tumor cells beyond the T2 radius were calculated. For both models, higher D/ρ ratios were correlated with a greater proportion and extent of subthreshold tumor. Anisotropic modeling demonstrated a higher proportion and extent of subthreshold tumor than predicted by the isotropic modeling. Because the quantity and distribution of subthreshold tumor depended on the D/ρ ratio, this ratio should influence radiation field demarcation. Furthermore, the use of DTI data to account for anisotropic tumor growth allows for more refined characterization of the subthreshold tumor based on the patient-specific D/ρ ratio.

In silico analysis suggests differential response to bevacizumab and radiation combination therapy in newly diagnosed glioblastoma

Recently, two phase III studies of bevacizumab, an anti-angiogenic, for newly diagnosed glioblastoma (GBM) patients were released. While they were unable to statistically significantly demonstrate that bevacizumab in combination with other therapies increases the overall survival of GBM patients, there remains a question of potential benefits for subpopulations of patients. We use a mathematical model of GBM growth to investigate differential benefits of combining surgical resection, radiation and bevacizumab across observed tumour growth kinetics. The differential hypoxic burden after gross total resection (GTR) was assessed along with the change in radiation cell kill from bevacizumab-induced tissue re-normalization when starting therapy for tumours at different diagnostic sizes. Depending on the tumour size at the time of treatment, our model predicted that GTR would remove a variable portion of the hypoxic burden ranging from 11% to 99.99%. Further, our model predicted that the combination of bevacizumab with radiation resulted in an additional cell kill ranging from 2.6×10⁷ to 1.1×10¹⁰ cells. By considering the outcomes given individual tumour kinetics, our results indicate that the subpopulation of patients who would receive the greatest benefit from bevacizumab and radiation combination therapy are those with large, aggressive tumours and who are not eligible for GTR.

A mathematical model of tumor growth and its response to single irradiation

BACKGROUND

Mathematical modeling of biological processes is widely used to enhance quantitative understanding of bio-medical phenomena. This quantitative knowledge can be applied in both clinical and experimental settings. Recently, many investigators began studying mathematical models of tumor response to radiation therapy. We developed a simple mathematical model to simulate the growth of tumor volume and its response to a single fraction of high dose irradiation. The modelling study may provide clinicians important insights on radiation therapy strategies through identification of biological factors significantly influencing the treatment effectiveness.

METHODS

We made several key assumptions of the model. Tumor volume is composed of proliferating (or dividing) cancer cells and non-dividing (or dead) cells. Tumor growth rate (or tumor volume doubling time) is proportional to the ratio of the volumes of tumor vasculature and the tumor. The vascular volume grows slower than the tumor by introducing the vascular growth retardation factor, θ. Upon irradiation, the proliferating cells gradually die over a fixed time period after irradiation. Dead cells are cleared away with cell clearance time. The model was applied to simulate pre-treatment growth and post-treatment radiation response of rat rhabdomyosarcoma tumors and metastatic brain tumors of five patients who were treated with Gamma Knife stereotactic radiosurgery (GKSRS).

RESULTS

By selecting appropriate model parameters, we showed the temporal variation of the tumors for both the rat experiment and the clinical GKSRS cases could be easily replicated by the simple model. Additionally, the application of our model to the GKSRS cases showed that the α-value, which is an indicator of radiation sensitivity in the LQ model, and the value of θ could be predictors of the post-treatment volume change.

CONCLUSIONS

The proposed model was successful in representing both the animal experimental data and the clinically observed tumor volume changes. We showed that the model can be used to find the potential biological parameters, which may be able to predict the treatment outcome. However, there is a large statistical uncertainty of the result due to the small sample size. Therefore, a future clinical study with a larger number of patients is needed to confirm the finding.

Spatial Metrics of Tumour Vascular Organisation Predict Radiation Efficacy in a Computational Model

Intratumoural heterogeneity is known to contribute to poor therapeutic response. Variations in oxygen tension in particular have been correlated with changes in radiation response in vitro and at the clinical scale with overall survival. Heterogeneity at the microscopic scale in tumour blood vessel architecture has been described, and is one source of the underlying variations in oxygen tension. We seek to determine whether histologic scale measures of the erratic distribution of blood vessels within a tumour can be used to predict differing radiation response. Using a two-dimensional hybrid cellular automaton model of tumour growth, we evaluate the effect of vessel distribution on cell survival outcomes of simulated radiation therapy. Using the standard equations for the oxygen enhancement ratio for cell survival probability under differing oxygen tensions, we calculate average radiation effect over a range of different vessel densities and organisations. We go on to quantify the vessel distribution heterogeneity and measure spatial organization using Ripley’s L function, a measure designed to detect deviations from complete spatial randomness. We find that under differing regimes of vessel density the correlation coefficient between the measure of spatial organization and radiation effect changes sign. This provides not only a useful way to understand the differences seen in radiation effect for tissues based on vessel architecture, but also an alternate explanation for the vessel normalization hypothesis.

In this paper we use a mathematical model, called a hybrid cellular automaton, to study the effect of different vessel distributions on radiation therapy outcomes at the cellular level. We show that the correlation between radiation outcome and spatial organization of vessels changes signs between relatively low and high vessel density. Specifically, that for relatively low vessel density, radiation efficacy is decreased when vessels are more homogeneously distributed, and the opposite is true, that radiation efficacy is improved, when vessel organisation is normalised in high densities. This result suggests an alteration to the vessel normalization hypothesis which states that normalisation of vascular beds should improve radio- and chemo-therapeutic response, but has failed to be validated in clinical studies. In this alteration, we show that Ripley’s L function allows discrimination between vascular architectures in different density regimes in which the standard hypothesis holds and does not hold. Further, we find that this information can be used to augment quantitative histologic analysis of tumours to aid radiation dose personalisation.

A mathematical approach to virus therapy of glioblastomas

BACKGROUND

It is widely believed that the treatment of glioblastomas (GBM) could benefit from oncolytic virus therapy. Clinical research has shown that Vesicular Stomatitis Virus (VSV) has strong oncolytic properties. In addition, mathematical models of virus treatment of tumors have been developed in recent years. Some experiments in vitro and in vivo have been done and shown promising results, but have been never compared quantitatively with mathematical models. We use in vitro data of this virus applied to glioblastoma.

RESULTS

We describe three increasingly realistic mathematical models for the VSV-GBM in vitro experiment with progressive incorporation of time-delay effects. For the virus dynamics, we obtain results consistent with the in vitro experimental speed data only when applying the more complex and comprehensive model, with time-delay effects both in the reactive and diffusive terms. The tumor speed is given by the minimum of a very simple function that nonetheless yields results within the experimental measured range.

CONCLUSIONS

We have improved a previous model with new ideas and carefully incorporated concepts from experimental results. We have shown that the delay time τ is the crucial parameter in this kind of models. We have demonstrated that our new model can satisfactorily predict the front speed for the lytic action of oncolytic VSV on glioblastoma observed in vitro. We provide a basis that can be applied in the near future to realistically simulate in vivo virus treatments of several cancers.

The mathematics of cancer: integrating quantitative models

Mathematical modelling approaches have become increasingly abundant in cancer research. The complexity of cancer is well suited to quantitative approaches as it provides challenges and opportunities for new developments. In turn, mathematical modelling contributes to cancer research by helping to elucidate mechanisms and by providing quantitative predictions that can be validated. The recent expansion of quantitative models addresses many questions regarding tumour initiation, progression and metastases as well as intra-tumour heterogeneity, treatment responses and resistance. Mathematical models can complement experimental and clinical studies, but also challenge current paradigms, redefine our understanding of mechanisms driving tumorigenesis and shape future research in cancer biology.

Predicting the Response of Breast Cancer to Neoadjuvant Therapy Using a Mechanically Coupled Reaction-Diffusion Model

Although there are considerable data on the use of mathematical modeling to describe tumor growth and response to therapy, previous approaches are often not of the form that can be easily applied to clinical data to generate testable predictions in individual patients. Thus, there is a clear need to develop and apply clinically relevant oncologic models that are amenable to available patient data and yet retain the most salient features of response prediction. In this study we show how a biomechanical model of tumor growth can be initialized and constrained by serial patient-specific magnetic resonance imaging data, obtained at two time points early in the course of therapy (before initiation and following one cycle of therapy), to predict the response for individual patients with breast cancer undergoing neoadjuvant therapy. Using our mechanics coupled modeling approach, we are able to predict, after the first cycle of therapy, breast cancer patients that would eventually achieve a complete pathologic response and those who would not, with receiver operating characteristic area under the curve (AUC) of 0.87, sensitivity of 92%, and specificity of 84%. Our approach significantly outperformed the AUCs achieved by standard (i.e., not mechanically coupled) reaction-diffusion predictive modeling (0.75), simple analysis of the tumor cellularity estimated from imaging data (0.73), and the Response Evaluation Criteria in Solid Tumors (0.71). Thus, we show the potential for mathematical model prediction for use as a prognostic indicator of response to therapy. The work indicates the considerable promise of image-driven biophysical modeling for predictive frameworks within therapeutic applications.

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.

A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation

BACKGROUND

Although altered protocols that challenge conventional radiation fractionation have been tested in prospective clinical trials, we still have limited understanding of how to select the most appropriate fractionation schedule for individual patients. Currently, the prescription of definitive radiotherapy is based on the primary site and stage, without regard to patient-specific tumor or host factors that may influence outcome. We hypothesize that the proportion of radiosensitive proliferating cells is dependent on the saturation of the tumor carrying capacity. This may serve as a prognostic factor for personalized radiotherapy (RT) fractionation.

METHODS

We introduce a proliferation saturation index (PSI), which is defined as the ratio of tumor volume to the host-influenced tumor carrying capacity. Carrying capacity is as a conceptual measure of the maximum volume that can be supported by the current tumor environment including oxygen and nutrient availability, immune surveillance and acidity. PSI is estimated from two temporally separated routine pre-radiotherapy computed tomography scans and a deterministic logistic tumor growth model. We introduce the patient-specific pre-treatment PSI into a model of tumor growth and radiotherapy response, and fit the model to retrospective data of four non-small cell lung cancer patients treated exclusively with standard fractionation. We then simulate both a clinical trial hyperfractionation protocol and daily fractionations, with equal biologically effective dose, to compare tumor volume reduction as a function of pretreatment PSI.

RESULTS

With tumor doubling time and radiosensitivity assumed constant across patients, a patient-specific pretreatment PSI is sufficient to fit individual patient response data (R(2) = 0.98). PSI varies greatly between patients (coefficient of variation >128 %) and correlates inversely with radiotherapy response. For this study, our simulations suggest that only patients with intermediate PSI (0.45-0.9) are likely to truly benefit from hyperfractionation. For up to 20 % uncertainties in tumor growth rate, radiosensitivity, and noise in radiological data, the absolute estimation error of pretreatment PSI is <10 % for more than 75 % of patients.

CONCLUSIONS

Routine radiological images can be used to calculate individual PSI, which may serve as a prognostic factor for radiation response. This provides a new paradigm and rationale to select personalized RT dose-fractionation.

Adaptive Mathematical Model of Tumor Response to Radiotherapy Based on CBCT Data

Mathematical modeling of tumor response to radiotherapy has the potential of enhancing the quality of the treatment plan, which can be even tailored on an individual basis. Lack of extensive in vivo validation has prevented, however, reliable clinical translation of modeling outcomes. Image-guided radiotherapy is a consolidated treatment modality based on computed tomographic (CT) imaging for tumor delineation and volumetric cone beam CT data for periodic checks during treatment. In this study, a macroscopic model of tumor growth and radiation response is proposed, being able to adapt along the treatment course as volumetric tumor data become available. Model parameter learning was based on cone beam CT images in 13 uterine cervical cancer patients, subdivided into three groups (G1, G2, G3) according to tumor type and treatment. Three group-specific parameter sets (PS1, PS2, and PS3) on one general parameter set (PSa) were applied. The corresponding average model fitting errors were 14%, 18%, 13%, and 21%, respectively. The model adaptation testing was performed using volume data of three patients, other than the ones involved in the parameter learning. The extrapolation performance of the general model was improved, while comparable prediction errors were found for the group-specific approach. This suggests that an online parameter tuning can overcome the limitations of a suboptimal patient stratification, which appeared otherwise a critical issue.

Radial expansion rates and tumor growth kinetics predict malignant transformation in contrast-enhancing low-grade diffuse astrocytoma

BACKGROUND

Contrast-enhancing low-grade diffuse astrocytomas are an understudied, aggressive subtype at increased risk because of few radiographic indications of malignant transformation. In the current study, we tested whether tumor growth kinetics could identify tumors that undergo malignant transformation to higher grades.

METHODS

Thirty patients with untreated diffuse astrocytomas (WHO II) that underwent tumor progression were enrolled. Contrast-enhancing and T2 hyperintense tumor regions were segmented and the radius of tumor at two time points leading to progression was estimated. Radial expansion rates were used to estimate proliferation and invasion rates using a biomathematical model.

RESULTS

Radial expansion rates for both contrast-enhancing (p = 0.0040) and T2 hyperintense regions (p = 0.0016) were significantly higher in WHO II-IV tumors compared with nontransformers. Similarly, model estimates showed a significantly higher proliferation (p = 0.0324) and invasion rate (p = 0.0050) in WHO II-IV tumors compared with nontransformers.

CONCLUSION

Tumor growth kinetics can identify contrast-enhancing diffuse astrocytomas undergoing malignant transformation.

Optimal radiation fractionation for low-grade gliomas: Insights from a mathematical model

We study optimal radiotherapy fractionations for low-grade glioma using mathematical models. Both space-independent and space-dependent models are studied. Two different optimization criteria have been developed, the first one accounting for the global effect of the tumor mass on the disease symptoms and the second one related to the delay of the malignant transformation of the tumor. The models are studied theoretically and numerically using the method of feasible directions. We have searched for optimal distributions of the daily doses dj in the standard protocol of 30 fractions using both models and the two different optimization criteria. The optimal results found in all cases are minor deviations from the standard protocol and provide only marginal potential gains. Thus, our results support the optimality of current radiation fractionations over the standard 6 week treatment period. This is also in agreement with the observation that minor variations of the fractionation have failed to provide measurable gains in survival or progression free survival, pointing out to a certain optimality of the current approach.

An inverse problem formulation for parameter estimation of a reaction-diffusion model of low grade gliomas

We present a numerical scheme for solving a parameter estimation problem for a model of low-grade glioma growth. Our goal is to estimate the spatial distribution of tumor concentration, as well as the magnitude of anisotropic tumor diffusion. We use a constrained optimization formulation with a reaction-diffusion model that results in a system of nonlinear partial differential equations. In our formulation, we estimate the parameters using partially observed, noisy tumor concentration data at two different time instances, along with white matter fiber directions derived from diffusion tensor imaging. The optimization problem is solved with a Gauss-Newton reduced space algorithm. We present the formulation and outline the numerical algorithms for solving the resulting equations. We test the method using a synthetic dataset and compute the reconstruction error for different noise levels and detection thresholds for monofocal and multifocal test cases.

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.

Joint tumor growth prediction and tumor segmentation on therapeutic follow-up PET images

Tumor response to treatment varies among patients. Patient-specific prediction of tumor evolution based on medical images during the treatment can help to build and adapt patient's treatment planning in a non-invasive way. Personalized tumor growth modeling allows patient-specific prediction by estimating model parameters based on individual's images. The model parameters are often estimated by optimizing a cost function constructed based on the tumor delineations. In this paper, we propose a joint framework for tumor growth prediction and tumor segmentation in the context of patient's therapeutic follow ups. Throughout the treatment, a series of sequential positron emission tomography (PET) images are acquired for tumor response monitoring. We propose to take into account the predicted information, which is used in combination with the random walks (RW) algorithm, to develop an automatic tumor segmentation method on PET images. Moreover, we propose an iterative scheme of RW, making the segmentation more performant. Furthermore, the obtained segmentation is applied to the process of model parameter estimation so as to get the model based prediction of tumor evolution. We evaluate our methods on 7 lung tumor patients, totaling 29 PET exams, under radiotherapy by comparing the obtained tumor prediction and tumor segmentation with manual tumor delineation by expert. Our system produces promising results when compared to the state-of-the-art methods.

Patient-specific characterization of the invasiveness and proliferation of low-grade gliomas using serial MR imaging and a mathematical model of tumor growth

Low-grade gliomas (LGGs) represent a significant proportion of hemispheric gliomas in adults. Although less aggressive than glioblastomas (GBMs), they have a broad range of biologic behavior, and often a limited prognosis. The aim of the present study was to explore LGG growth kinetics through a combination of routine MRI imaging and a novel adaptation of a mathematical tumor model. MRI imaging in 14 retrospectively identified grade II LGGs that showed some tumor enhancement was used to assess tumor radii at two separate time-points. This information was combined with a reaction-diffusion partial-differential equation model of tumor growth to calculate diffusion (D) and proliferation (ρ) coefficients for each tumor, representing measures of tumor invasiveness and cellular multiplication, respectively. The results were compared to previously published data on GBMs. The average value of D was 0.034 mm(2)/day and ρ was 0.0056/day. Grade II LGGs had a broad range of D and ρ. On average, the proliferation coefficient ρ was significantly lower than previously published values for GBM, by about an order of magnitude. The diffusion coefficient, modeling invasiveness, however, was only slightly lower but without statistical significance. It was possible to calculate detailed growth kinetic parameters for some LGGs, potentially providing a new way to assess tumor aggressiveness and possibly gauge prognosis. Even within a single-grade (WHO II), LGGs were found to have broad range of D and ρ, possibly correlating to their variable biologic behavior. Overall, the model parameters suggest that LGG is less aggressive than GBM based primarily on a lower index of tumor proliferation rather than on lesser invasiveness.

Mathematical modelling of tumour volume dynamics in response to stereotactic ablative radiotherapy for non-small cell lung cancer

This paper reports a modelling study of tumour volume dynamics in response to stereotactic ablative radiotherapy (SABR). The main objective was to develop a model that is adequate to describe tumour volume change measured during SABR, and at the same time is not excessively complex as lacking support from clinical data. To this end, various modelling options were explored, and a rigorous statistical method, the Akaike information criterion, was used to help determine a trade-off between model accuracy and complexity. The models were calibrated to the data from 11 non-small cell lung cancer patients treated with SABR. The results showed that it is feasible to model the tumour volume dynamics during SABR, opening up the potential for using such models in a clinical environment in the future.

Patient-specific mathematical neuro-oncology: using a simple proliferation and invasion tumor model to inform clinical practice

Glioblastoma multiforme (GBM) is the most common malignant primary brain tumor associated with a poor median survival of 15–18 months, yet there is wide heterogeneity across and within patients. This heterogeneity has been the source of significant clinical challenges facing patients with GBM and has hampered the drive toward more precision or personalized medicine approaches to treating these challenging tumors. Over the last two decades, the field of Mathematical Neuro-oncology has grown out of desire to use (often patient-specific) mathematical modeling to better treat GBMs. Here, we will focus on a series of clinically relevant results using patient-specific mathematical modeling. The core model at the center of these results incorporates two hallmark features of GBM, proliferation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\rho )$$\end{document}(ρ) and invasion (D), as key parameters. Based on routinely obtained magnetic resonance images, each patient’s tumor can be characterized using these two parameters. The Proliferation-Invasion (PI) model uses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}ρ and D to create patient-specific growth predictions. The PI model, its predictions, and parameters have been used in a number of ways to derive biological insight. Beyond predicting growth, the PI model has been utilized to identify patients who benefit from different surgery strategies, to prognosticate response to radiation therapy, to develop a treatment response metric, and to connect clinical imaging features and genetic information. Demonstration of the PI model’s clinical relevance supports the growing role for it and other mathematical models in routine clinical practice.

Kinetic Models for Predicting Cervical Cancer Response to Radiation Therapy on Individual Basis Using Tumor Regression Measured In Vivo With Volumetric Imaging

This article describes a macroscopic mathematical modeling approach to capture the interplay between solid tumor evolution and cell damage during radiotherapy. Volume regression profiles of 15 patients with uterine cervical cancer were reconstructed from serial cone-beam computed tomography data sets, acquired for image-guided radiotherapy, and used for model parameter learning by means of a genetic-based optimization. Patients, diagnosed with either squamous cell carcinoma or adenocarcinoma, underwent different treatment modalities (image-guided radiotherapy and image-guided chemo-radiotherapy). The mean volume at the beginning of radiotherapy and the end of radiotherapy was on average 23.7 cm(3) (range: 12.7-44.4 cm(3)) and 8.6 cm(3) (range: 3.6-17.1 cm(3)), respectively. Two different tumor dynamics were taken into account in the model: the viable (active) and the necrotic cancer cells. However, according to the results of a preliminary volume regression analysis, we assumed a short dead cell resolving time and the model was simplified to the active tumor volume. Model learning was performed both on the complete patient cohort (cohort-based model learning) and on each single patient (patient-specific model learning). The fitting results (mean error: ∼ 16% and ∼ 6% for the cohort-based model and patient-specific model, respectively) highlighted the model ability to quantitatively reproduce tumor regression. Volume prediction errors of about 18% on average were obtained using cohort-based model computed on all but 1 patient at a time (leave-one-out technique). Finally, a sensitivity analysis was performed and the data uncertainty effects evaluated by simulating an average volume perturbation of about 1.5 cm(3) obtaining an error increase within 0.2%. In conclusion, we showed that simple time-continuous models can represent tumor regression curves both on a patient cohort and patient-specific basis; this discloses the opportunity in the future to exploit such models to predict how changes in the treatment schedule (number of fractions, doses, intervals among fractions) might affect the tumor regression on an individual basis.

Cellular and molecular portrait of eleven human glioblastoma cell lines under photon and carbon ion irradiation

This study aimed to examine the cellular and molecular long-term responses of glioblastomas to radiotherapy and hadrontherapy in order to better understand the biological effects of carbon beams in cancer treatment. Eleven human glioblastoma cell lines, displaying gradual radiosensitivity, were irradiated with photons or carbon ions. Independently of p53 or O(6)-methylguanine-DNA methyltransferase(1) status, all cell lines responded to irradiation by a G2/M phase arrest followed by the appearance of mitotic catastrophe, which was concluded by a ceramide-dependent-apoptotic cell death. Statistical analysis demonstrated that: (i) the SF2(2) and the D10(3) values for photon are correlated with that obtained in response to carbon ions; (ii) regardless of the p53, MGMT status, and radiosensitivity, the release of ceramide is associated with the induction of late apoptosis; and (iii) the appearance of polyploid cells after photon irradiation could predict the Relative Biological Efficiency(4) to carbon ions. This large collection of data should increase our knowledge in glioblastoma radiobiology in order to better understand, and to later individualize, appropriate radiotherapy treatment for patients who are good candidates.

Modeling the Interplay Between Tumor Volume Regression and Oxygenation in Uterine Cervical Cancer During Radiotherapy Treatment

This paper describes a patient-specific mathematical model to predict the evolution of uterine cervical tumors at a macroscopic scale, during fractionated external radiotherapy. The model provides estimates of tumor regrowth and dead-cell reabsorption, incorporating the interplay between tumor regression rate and radiosensitivity, as a function of the tumor oxygenation level. Model parameters were estimated by minimizing the difference between predicted and measured tumor volumes, these latter being obtained from a set of 154 serial cone-beam computed tomography scans acquired on 16 patients along the course of the therapy. The model stratified patients according to two different estimated dynamics of dead-cell removal and to the predicted initial value of the tumor oxygenation. The comparison with a simpler model demonstrated an improvement in fitting properties of this approach (fitting error average value <5%, p < 0.01), especially in case of tumor late responses, which can hardly be handled by models entailing a constant radiosensitivity, failing to model changes from initial severe hypoxia to aerobic conditions during the treatment course. The model predictive capabilities suggest the need of clustering patients accounting for cancer cell line, tumor staging, as well as microenvironment conditions (e.g., oxygenation level).

Invasion and proliferation kinetics in enhancing gliomas predict IDH1 mutation status

BACKGROUND

Glioblastomas with a specific mutation in the isocitrate dehydrogenase 1 (IDH1) gene have a better prognosis than gliomas with wild-type IDH1.

METHODS

Here we compare the IDH1 mutational status in 172 contrast-enhancing glioma patients with the invasion profile generated by a patient-specific mathematical model we developed based on MR imaging.

RESULTS

We show that IDH1-mutated contrast-enhancing gliomas were relatively more invasive than wild-type IDH1 for all 172 contrast-enhancing gliomas as well as the subset of 158 histologically confirmed glioblastomas. The appearance of this relatively increased, model-predicted invasive profile appears to be determined more by a lower model-predicted net proliferation rate rather than an increased model-predicted dispersal rate of the glioma cells. Receiver operator curve analysis of the model-predicted MRI-based invasion profile revealed an area under the curve of 0.91, indicative of a predictive relationship. The robustness of this relationship was tested by cross-validation analysis of the invasion profile as a predictive metric for IDH1 status.

CONCLUSIONS

The strong correlation between IDH1 mutation status and the MRI-based invasion profile suggests that use of our tumor growth model may lead to noninvasive clinical detection of IDH1 mutation status and thus lead to better treatment planning, particularly prior to surgical resection, for contrast-enhancing gliomas.

Mathematical and computational modeling in biology at multiple scales

A variety of topics are reviewed in the area of mathematical and computational modeling in biology, covering the range of scales from populations of organisms to electrons in atoms. The use of maximum entropy as an inference tool in the fields of biology and drug discovery is discussed. Mathematical and computational methods and models in the areas of epidemiology, cell physiology and cancer are surveyed. The technique of molecular dynamics is covered, with special attention to force fields for protein simulations and methods for the calculation of solvation free energies. The utility of quantum mechanical methods in biophysical and biochemical modeling is explored. The field of computational enzymology is examined.

A computational model incorporating neural stem cell dynamics reproduces glioma incidence across the lifespan in the human population

Glioma is the most common form of primary brain tumor. Demographically, the risk of occurrence increases until old age. Here we present a novel computational model to reproduce the probability of glioma incidence across the lifespan. Previous mathematical models explaining glioma incidence are framed in a rather abstract way, and do not directly relate to empirical findings. To decrease this gap between theory and experimental observations, we incorporate recent data on cellular and molecular factors underlying gliomagenesis. Since evidence implicates the adult neural stem cell as the likely cell-of-origin of glioma, we have incorporated empirically-determined estimates of neural stem cell number, cell division rate, mutation rate and oncogenic potential into our model. We demonstrate that our model yields results which match actual demographic data in the human population. In particular, this model accounts for the observed peak incidence of glioma at approximately 80 years of age, without the need to assert differential susceptibility throughout the population. Overall, our model supports the hypothesis that glioma is caused by randomly-occurring oncogenic mutations within the neural stem cell population. Based on this model, we assess the influence of the (experimentally indicated) decrease in the number of neural stem cells and increase of cell division rate during aging. Our model provides multiple testable predictions, and suggests that different temporal sequences of oncogenic mutations can lead to tumorigenesis. Finally, we conclude that four or five oncogenic mutations are sufficient for the formation of glioma.

A novel bicompartmental mathematical model of glioblastoma multiforme

Glioblastoma multiforme (GBM) is the most common primary CNS neoplasm, and continues to have a dismal prognosis. A widely-used approach to the mathematical modeling of GBM involves utilizing a reaction-diffusion model of cell density as a function of space and time, which accounts for both the infiltrative nature of the tumor using a diffusion term, and the proliferation of tumor cells using a proliferation term. The current paper extends the standard models by incorporating an advection term to account for the so-called 'cell streaming' which is often seen with GBM, where some of the tumor cells seem to stream widely along the white matter pathways. The current paper introduces a bicompartmental GBM model in the form of coupled partial differential equations with a component of dispersive cells. The parameters needed for this model are explored. It is shown that this model can account for the rapid distant dispersal of GBM cells in the CNS, as well as such phenomena as multifocal gliomas with tumor foci distant from the core tumor site. The model suggests a higher percentage of tumor cells below the threshold of MRI images in comparison to the standard model. By incorporating an advection component, the proposed model is able to account for phenomena such as multicentric gliomas and rapid distant dispersion of a small fraction of tumor cells throughout the CNS, features important to the prognosis of GBM, but not easily accounted for by current models.

Patient-specific metrics of invasiveness reveal significant prognostic benefit of resection in a predictable subset of gliomas

Object

Malignant gliomas are incurable, primary brain neoplasms noted for their potential to extensively invade brain parenchyma. Current methods of clinical imaging do not elucidate the full extent of brain invasion, making it difficult to predict which, if any, patients are likely to benefit from gross total resection. Our goal was to apply a mathematical modeling approach to estimate the overall tumor invasiveness on a patient-by-patient basis and determine whether gross total resection would improve survival in patients with relatively less invasive gliomas.

Methods

In 243 patients presenting with contrast-enhancing gliomas, estimates of the relative invasiveness of each patient's tumor, in terms of the ratio of net proliferation rate of the glioma cells to their net dispersal rate, were derived by applying a patient-specific mathematical model to routine pretreatment MR imaging. The effect of varying degrees of extent of resection on overall survival was assessed for cohorts of patients grouped by tumor invasiveness.

Results

We demonstrate that patients with more diffuse tumors showed no survival benefit (P = 0.532) from gross total resection over subtotal/biopsy, while those with nodular (less diffuse) tumors showed a significant benefit (P = 0.00142) with a striking median survival benefit of over eight months compared to sub-totally resected tumors in the same cohort (an 80% improvement in survival time for GTR only seen for nodular tumors).

Conclusions

These results suggest that our patient-specific, model-based estimates of tumor invasiveness have clinical utility in surgical decision making. Quantification of relative invasiveness assessed from routinely obtained pre-operative imaging provides a practical predictor of the benefit of gross total resection.

Identifying predictors of early growth response and adverse radiation effects of vestibular schwannomas to radiosurgery

PURPOSE

To determine whether pre-treatment growth rate of vestibular schwannomas (VS) predict response to radiosurgery.

METHODS

A retrospective review of a prospectively maintained database of all VS patients treated with 12Gy prescription dose between September 2005 and June 2011 at our institution using the Leksell Model 4C Gamma Knife Unit was conducted. Patients who had a minimum of 12-months clinical and radiological assessment before and after radiosurgery were included in this study. Tumor growth rates were calculated using specific growth rate (SGR). Tumor volumes were measured on FIESTA-MRI scans using ITK-SNAP v2.2.

RESULTS

Following radiosurgery, twenty-seven (42.9%) patients showed a significant decrease in volume after one year, twenty-nine (46.0%) stabilized, and seven (11.1%) continued to grow. There was no correlation between VS pre-treatment SGRs with post-treatment SGRs (p = 0.34), and incidence of adverse radiation effects (ARE). The reduction in tumors' SGRs after radiosurgery was proportional to pre-treatment SGRs, although this correlation was not statistically significant (p = 0.19). Analysis of risk factors revealed a positive correlation between post-treatment SGRs and incidence of non-auditory complications, most of which were attributed to ARE (p = 0.047).

CONCLUSION

Pre-treatment growth rate of VS does not predict tumor response to radiosurgery or incidence of ARE. VS with higher SGRs post-radiosurgery are more likely to experience ARE.

Glucose-lactate metabolic cooperation in cancer: insights from a spatial mathematical model and implications for targeted therapy

A recent study has hypothesised a glucose-lactate metabolic symbiosis between adjacent hypoxic and oxygenated regions of a developing tumour, and proposed a treatment strategy to target this symbiosis. However, in vivo experimental support remains inconclusive. Here we develop a minimal spatial mathematical model of glucose-lactate metabolism to examine, in principle, whether metabolic symbiosis is plausible in human tumours, and to assess the potential impact of inhibiting it. We find that symbiosis is a robust feature of our model system-although on the length scale at which oxygen supply is diffusion-limited, its occurrence requires very high cellular metabolic activity-and that necrosis in the tumour core is reduced in the presence of symbiosis. Upon simulating therapeutic inhibition of lactate uptake, we predict that targeted treatment increases the extent of tissue oxygenation without increasing core necrosis. The oxygenation effect is correlated strongly with the extent of wild-type hypoxia and only weakly with wild-type symbiotic behaviour, and therefore may be promising for radiosensitisation of hypoxic, lactate-consuming tumours even if they do not exhibit a spatially well-defined symbiosis. Finally, we conduct in vitro experiments on the U87 glioblastoma cell line to facilitate preliminary speculation as to where highly malignant tumours might fall in our parameter space, and find that these experiments suggest a weakly symbiotic regime for U87 cells, thus raising the new question of what relationship might exist between symbiosis and tumour malignancy.

Computer implementation of a new therapeutic model for GBM tumor

Modeling the tumor behavior in the host organ as function of time and radiation dose has been a major study in the previous decades. Here the effort in estimation of cancerous and normal cell proliferation and growth in glioblastoma multiform (GBM) tumor is presented. This paper introduces a new mathematical model in the form of differential equation of tumor growth. The model contains dose delivery amount in the treatment scheme as an input term. It also can be utilized to optimize the treatment process in order to increase the patient survival period. Gene expression programming (GEP) as a new concept is used for estimating this model. The LQ model has also been applied to GEP as an initial value, causing acceleration and improvement of the algorithm estimation. The model shows the number of the tumor and normal brain cells during the treatment process using the status of normal and cancerous cells in the initiation of treatment, the timing and amount of dose delivery to the patient, and a coefficient that describes the brain condition. A critical level is defined for normal cell when the patient's death occurs. In the end the model has been verified by clinical data obtained from previous accepted formulae and some of our experimental resources. The proposed model helps to predict tumor growth during treatment process in which further treatment processes can be controlled.

Improving treatment strategies for patients with metastatic castrate resistant prostate cancer through personalized computational modeling

Metastatic castrate resistant prostate cancer (mCRPC) is responsible for the majority of prostate cancer deaths with the median survival after diagnosis being 2 years. The metastatic lesions often arise in the skeleton, and current treatment options are primarily palliative. Using guidelines set forth by the National Comprehensive Cancer Network (NCCN), the medical oncologist has a number of choices available to treat the metastases. However, the sequence of those treatments is largely dependent on the patient history, treatment response and preferences. We posit that the utilization of personalized computational models and treatment optimization algorithms based on patient specific parameters could significantly enhance the oncologist's ability to choose an optimized sequence of available therapies to maximize overall survival. In this perspective, we used an integrated team approach involving clinicians, researchers, and mathematicians, to generate an example of how computational models and genetic algorithms can be utilized to predict the response of heterogeneous mCRPCs in bone to varying sequences of standard and targeted therapies. The refinement and evolution of these powerful models will be critical for extending the overall survival of men diagnosed with mCRPC.

Gene therapy enhances chemotherapy tolerance and efficacy in glioblastoma patients

BACKGROUND

Temozolomide (TMZ) is one of the most potent chemotherapy agents for the treatment of glioblastoma. Unfortunately, almost half of glioblastoma tumors are TMZ resistant due to overexpression of methylguanine methyltransferase (MGMT(hi)). Coadministration of O6-benzylguanine (O6BG) can restore TMZ sensitivity, but causes off-target myelosuppression. Here, we conducted a prospective clinical trial to test whether gene therapy to confer O6BG resistance in hematopoietic stem cells (HSCs) improves chemotherapy tolerance and outcome.

METHODS

We enrolled 7 newly diagnosed glioblastoma patients with MGMT(hi) tumors. Patients received autologous gene-modified HSCs following single-agent carmustine administration. After hematopoietic recovery, patients underwent O6BG/TMZ chemotherapy in 28-day cycles. Serial blood samples and tumor images were collected throughout the study. Chemotherapy tolerance was determined by the observed myelosuppression and recovery following each cycle. Patient-specific biomathematical modeling of tumor growth was performed. Progression-free survival (PFS) and overall survival (OS) were also evaluated.

RESULTS

Gene therapy permitted a significant increase in the mean number of tolerated O6BG/TMZ cycles (4.4 cycles per patient, P < 0.05) compared with historical controls without gene therapy (n = 7 patients, 1.7 cycles per patient). One patient tolerated an unprecedented 9 cycles and demonstrated long-term PFS without additional therapy. Overall, we observed a median PFS of 9 (range 3.5-57+) months and OS of 20 (range 13-57+) months. Furthermore, biomathematical modeling revealed markedly delayed tumor growth at lower cumulative TMZ doses in study patients compared with patients that received standard TMZ regimens without O6BG.

CONCLUSION

These data support further development of chemoprotective gene therapy in combination with O6BG and TMZ for the treatment of glioblastoma and potentially other tumors with overexpression of MGMT.

TRIAL REGISTRATION

Clinicaltrials.gov NCT00669669.

FUNDING

R01CA114218, R01AI080326, R01HL098489, P30DK056465, K01DK076973, R01HL074162, R01CA164371, R01NS060752, U54CA143970.

Oedema-based model for diffuse low-grade gliomas: application to clinical cases under radiotherapy

OBJECTIVES

Diffuse low-grade gliomas are characterized by slow growth. Despite appropriate treatment, they change inexorably into more aggressive forms, jeopardizing the patient's life. Optimizing treatments, for example with the use of mathematical modelling, could help to prevent tumour regrowth and anaplastic transformation. Here, we present a model of the effect of radiotherapy on such tumours. Our objective is to explain observed delay of tumour regrowth following radiotherapy and to predict its duration.

MATERIALS AND METHODS

We have used a migration-proliferation model complemented by an equation describing appearance and draining of oedema. The model has been applied to clinical data of tumour radius over time, for a population of 28 patients.

RESULTS

We were able to show that draining of oedema accounts for regrowth delay after radiotherapy and have been able to fit the clinical data in a robust way. The model predicts strong correlation between high proliferation coefficient and low progression-free gain of lifetime, due to radiotherapy among the patients, in agreement with clinical studies. We argue that, with reasonable assumptions, it is possible to predict (precision ~20%) regrowth delay after radiotherapy and the gain of lifetime due to radiotherapy.

CONCLUSIONS

Our oedema-based model provides an early estimation of individual duration of tumour response to radiotherapy and thus, opens the door to the possibility of personalized medicine.

Prediction of lung tumor evolution during radiotherapy in individual patients with PET

We propose a patient-specific model based on partial differential equation to predict the evolution of lung tumors during radiotherapy. The evolution of tumor cell density is formulated by three terms: 1) advection describing the advective flux transport of tumor cells, 2) proliferation representing the tumor cell proliferation modeled as Gompertz differential equation, and 3) treatment quantifying the radiotherapeutic efficacy from linear quadratic formulation. We consider that tumor cell density variation can be derived from positron emission tomography images, the novel idea is to model the advection term by calculating 3D optical flow field from sequential images. To estimate patient-specific parameters, we propose an optimization between the predicted and observed images, under a global constraint that the tumor volume decreases exponentially as radiation dose increases. A thresholding on the predicted tumor cell densities is then used to define tumor contours, tumor volumes and maximum standardized uptake values (SUVmax). Results obtained on seven patients show a satisfying agreement between the predicted tumor contours and those drawn by an expert.

Selecting radiotherapy dose distributions by means of constrained optimization problems

The main steps in planning radiotherapy consist in selecting for any patient diagnosed with a solid tumor (i) a prescribed radiation dose on the tumor, (ii) bounds on the radiation side effects on nearby organs at risk and (iii) a fractionation scheme specifying the number and frequency of therapeutic sessions during treatment. The goal of any radiotherapy treatment is to deliver on the tumor a radiation dose as close as possible to that selected in (i), while at the same time conforming to the constraints prescribed in (ii). To this day, considerable uncertainties remain concerning the best manner in which such issues should be addressed. In particular, the choice of a prescription radiation dose is mostly based on clinical experience accumulated on the particular type of tumor considered, without any direct reference to quantitative radiobiological assessment. Interestingly, mathematical models for the effect of radiation on biological matter have existed for quite some time, and are widely acknowledged by clinicians. However, the difficulty to obtain accurate in vivo measurements of the radiobiological parameters involved has severely restricted their direct application in current clinical practice.In this work, we first propose a mathematical model to select radiation dose distributions as solutions (minimizers) of suitable variational problems, under the assumption that key radiobiological parameters for tumors and organs at risk involved are known. Second, by analyzing the dependence of such solutions on the parameters involved, we then discuss the manner in which the use of those minimizers can improve current decision-making processes to select clinical dosimetries when (as is generally the case) only partial information on model radiosensitivity parameters is available. A comparison of the proposed radiation dose distributions with those actually delivered in a number of clinical cases strongly suggests that solutions of our mathematical model can be instrumental in deriving good quality tests to select radiotherapy treatment plans in rather general situations.

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.

Radiotherapy planning for glioblastoma based on a tumor growth model: improving target volume delineation

Glioblastoma differ from many other tumors in the sense that they grow infiltratively into the brain tissue instead of forming a solid tumor mass with a defined boundary. Only the part of the tumor with high tumor cell density can be localized through imaging directly. In contrast, brain tissue infiltrated by tumor cells at low density appears normal on current imaging modalities. In current clinical practice, a uniform margin, typically two centimeters, is applied to account for microscopic spread of disease that is not directly assessable through imaging. The current treatment planning procedure can potentially be improved by accounting for the anisotropy of tumor growth, which arises from different factors: anatomical barriers such as the falx cerebri represent boundaries for migrating tumor cells. In addition, tumor cells primarily spread in white matter and infiltrate gray matter at lower rate. We investigate the use of a phenomenological tumor growth model for treatment planning. The model is based on the Fisher-Kolmogorov equation, which formalizes these growth characteristics and estimates the spatial distribution of tumor cells in normal appearing regions of the brain. The target volume for radiotherapy planning can be defined as an isoline of the simulated tumor cell density. This paper analyzes the model with respect to implications for target volume definition and identifies its most critical components. A retrospective study involving ten glioblastoma patients treated at our institution has been performed. To illustrate the main findings of the study, a detailed case study is presented for a glioblastoma located close to the falx. In this situation, the falx represents a boundary for migrating tumor cells, whereas the corpus callosum provides a route for the tumor to spread to the contralateral hemisphere. We further discuss the sensitivity of the model with respect to the input parameters. Correct segmentation of the brain appears to be the most crucial model input. We conclude that the tumor growth model provides a method to account for anisotropic growth patterns of glioma, and may therefore provide a tool to make target delineation more objective and automated.

Radiotherapy planning for glioblastoma based on a tumor growth model: implications for spatial dose redistribution

Gliomas differ from many other tumors as they grow infiltratively into the brain parenchyma rather than forming a solid tumor mass with a well-defined boundary. Tumor cells can be found several centimeters away from the central tumor mass that is visible using current imaging techniques. The infiltrative growth characteristics of gliomas question the concept of a radiotherapy target volume that is irradiated to a homogeneous dose-the standard in current clinical practice. We discuss the use of the Fisher-Kolmogorov glioma growth model in radiotherapy treatment planning. The phenomenological tumor growth model assumes that tumor cells proliferate locally and migrate into neighboring brain tissue, which is mathematically described via a partial differential equation for the spatio-temporal evolution of the tumor cell density. In this model, the tumor cell density drops approximately exponentially with distance from the visible gross tumor volume, which is quantified by the infiltration length, a parameter describing the distance at which the tumor cell density drops by a factor of e. This paper discusses the implications for the prescribed dose distribution in the periphery of the tumor. In the context of the exponential cell kill model, an exponential fall-off of the cell density suggests a linear fall-off of the prescription dose with distance. We introduce the dose fall-off rate, which quantifies the steepness of the prescription dose fall-off in units of Gy mm(-1). It is shown that the dose fall-off rate is given by the inverse of the product of radiosensitivity and infiltration length. For an infiltration length of 3 mm and a surviving fraction of 50% at 2 Gy, this suggests a dose fall-off of approximately 1 Gy mm(-1). The concept is illustrated for two glioblastoma patients by optimizing intensity-modulated radiotherapy plans. The dose fall-off rate concept reflects the idea that infiltrating gliomas lack a defined boundary and are characterized by a continuous fall-off of the density of infiltrating tumor cells. The approach can potentially be used to individualize the prescribed dose distribution if better methods to estimate radiosensitivity and infiltration length on a patient by patient basis become available.

Current advances in mathematical modeling of anti-cancer drug penetration into tumor tissues

Delivery of anti-cancer drugs to tumor tissues, including their interstitial transport and cellular uptake, is a complex process involving various biochemical, mechanical, and biophysical factors. Mathematical modeling provides a means through which to understand this complexity better, as well as to examine interactions between contributing components in a systematic way via computational simulations and quantitative analyses. In this review, we present the current state of mathematical modeling approaches that address phenomena related to drug delivery. We describe how various types of models were used to predict spatio-temporal distributions of drugs within the tumor tissue, to simulate different ways to overcome barriers to drug transport, or to optimize treatment schedules. Finally, we discuss how integration of mathematical modeling with experimental or clinical data can provide better tools to understand the drug delivery process, in particular to examine the specific tissue- or compound-related factors that limit drug penetration through tumors. Such tools will be important in designing new chemotherapy targets and optimal treatment strategies, as well as in developing non-invasive diagnosis to monitor treatment response and detect tumor recurrence.

Toward patient-specific, biologically optimized radiation therapy plans for the treatment of glioblastoma

PURPOSE

To demonstrate a method of generating patient-specific, biologically-guided radiotherapy dose plans and compare them to the standard-of-care protocol.

METHODS AND MATERIALS

We integrated a patient-specific biomathematical model of glioma proliferation, invasion and radiotherapy with a multiobjective evolutionary algorithm for intensity-modulated radiation therapy optimization to construct individualized, biologically-guided plans for 11 glioblastoma patients. Patient-individualized, spherically-symmetric simulations of the standard-of-care and optimized plans were compared in terms of several biological metrics.

RESULTS

The integrated model generated spatially non-uniform doses that, when compared to the standard-of-care protocol, resulted in a 67% to 93% decrease in equivalent uniform dose to normal tissue, while the therapeutic ratio, the ratio of tumor equivalent uniform dose to that of normal tissue, increased between 50% to 265%. Applying a novel metric of treatment response (Days Gained) to the patient-individualized simulation results predicted that the optimized plans would have a significant impact on delaying tumor progression, with increases from 21% to 105% for 9 of 11 patients.

CONCLUSIONS

Patient-individualized simulations using the combination of a biomathematical model with an optimization algorithm for radiation therapy generated biologically-guided doses that decreased normal tissue EUD and increased therapeutic ratio with the potential to improve survival outcomes for treatment of glioblastoma.

A mechanically coupled reaction-diffusion model for predicting the response of breast tumors to neoadjuvant chemotherapy

There is currently a paucity of reliable techniques for predicting the response of breast tumors to neoadjuvant chemotherapy. The standard approach is to monitor gross changes in tumor size as measured by physical exam and/or conventional imaging, but these methods generally do not show whether a tumor is responding until the patient has received many treatment cycles. One promising approach to address this clinical need is to integrate quantitative in vivo imaging data into biomathematical models of tumor growth in order to predict eventual response based on early measurements during therapy. In this work, we illustrate a novel biomechanical mathematical modeling approach in which contrast enhanced and diffusion weighted magnetic resonance imaging data acquired before and after the first cycle of neoadjuvant therapy are used to calibrate a patient-specific response model which subsequently is used to predict patient outcome at the conclusion of therapy. We present a modification of the reaction-diffusion tumor growth model whereby mechanical coupling to the surrounding tissue stiffness is incorporated via restricted cell diffusion. We use simulations and experimental data to illustrate how incorporating tissue mechanical properties leads to qualitatively and quantitatively different tumor growth patterns than when such properties are ignored. We apply the approach to patient data in a preliminary dataset of eight patients exhibiting a varying degree of responsiveness to neoadjuvant therapy, and we show that the mechanically coupled reaction-diffusion tumor growth model, when projected forward, more accurately predicts residual tumor burden at the conclusion of therapy than the non-mechanically coupled model. The mechanically coupled model predictions exhibit a significant correlation with data observations (PCC = 0.84, p < 0.01), and show a statistically significant >4 fold reduction in model/data error (p = 0.02) as compared to the non-mechanically coupled model.

Computational medicine: translating models to clinical care

Because of the inherent complexity of coupled nonlinear biological systems, the development of computational models is necessary for achieving a quantitative understanding of their structure and function in health and disease. Statistical learning is applied to high-dimensional biomolecular data to create models that describe relationships between molecules and networks. Multiscale modeling links networks to cells, organs, and organ systems. Computational approaches are used to characterize anatomic shape and its variations in health and disease. In each case, the purposes of modeling are to capture all that we know about disease and to develop improved therapies tailored to the needs of individuals. We discuss advances in computational medicine, with specific examples in the fields of cancer, diabetes, cardiology, and neurology. Advances in translating these computational methods to the clinic are described, as well as challenges in applying models for improving patient health.

Modeling Tumor-Associated Edema in Gliomas during Anti-Angiogenic Therapy and Its Impact on Imageable Tumor

Glioblastoma, the most aggressive form of primary brain tumor, is predominantly assessed with gadolinium-enhanced T1-weighted (T1Gd) and T2-weighted magnetic resonance imaging (MRI). Pixel intensity enhancement on the T1Gd image is understood to correspond to the gadolinium contrast agent leaking from the tumor-induced neovasculature, while hyperintensity on the T2/FLAIR images corresponds with edema and infiltrated tumor cells. None of these modalities directly show tumor cells; rather, they capture abnormalities in the microenvironment caused by the presence of tumor cells. Thus, assessing disease response after treatments impacting the microenvironment remains challenging through the obscuring lens of MR imaging. Anti-angiogenic therapies have been used in the treatment of gliomas with spurious results ranging from no apparent response to significant imaging improvement with the potential for extremely diffuse patterns of tumor recurrence on imaging and autopsy. Anti-angiogenic treatment normalizes the vasculature, effectively decreasing vessel permeability and thus reducing tumor-induced edema, drastically altering T2-weighted MRI. We extend a previously developed mathematical model of glioma growth to explicitly incorporate edema formation allowing us to directly characterize and potentially predict the effects of anti-angiogenics on imageable tumor growth. A comparison of simulated glioma growth and imaging enhancement with and without bevacizumab supports the current understanding that anti-angiogenic treatment can serve as a surrogate for steroids and the clinically driven hypothesis that anti-angiogenic treatment may not have any significant effect on the growth dynamics of the overall tumor cell populations. However, the simulations do illustrate a potentially large impact on the level of edematous extracellular fluid, and thus on what would be imageable on T2/FLAIR MR. Additionally, by evaluating virtual tumors with varying growth kinetics, we see tumors with lower proliferation rates will have the most reduction in swelling from such treatments.

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.