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

Mathematical modeling of brain metastases growth and response to therapies: A review

Brain metastases (BMs) are the most common intracranial tumor type and a significant health concern, affecting approximately 10% to 30% of all oncological patients. Although significant progress is being made, many aspects of the metastatic process to the brain and the growth of the resulting lesions are still not well understood. There is a need for an improved understanding of the growth dynamics and the response to treatment of these tumors. Mathematical models have been proven valuable for drawing inferences and making predictions in different fields of cancer research, but few mathematical works have considered BMs. This comprehensive review aims to establish a unified platform and contribute to fostering emerging efforts dedicated to enhancing our mathematical understanding of this intricate and challenging disease. We focus on the progress made in the initial stages of mathematical modeling research regarding BMs and the significant insights gained from such studies. We also explore the vital role of mathematical modeling in predicting treatment outcomes and enhancing the quality of clinical decision-making for patients facing BMs.

Tumor growth for remodeling process: A 2D approach

This paper aims to present a comprehensive framework for coupling tumor-bone remodeling processes in a 2-dimensional geometry. This is achieved by introducing a bio-inspired damage that represents the growing tumor, which subsequently affects the main populations involved in the remodeling process, namely, osteoclasts, osteoblasts, and bone tissue. The model is constructed using a set of differential equations based on the Komarova's and Ayati's models, modified to incorporate the bio-inspired damage that may result in tumor mass formation. Three distinct models were developed. The first two models are based on the Komarova's governing equations, with one demonstrating an osteolytic behavior and the second one an osteoblastic model. The third model is a variation of Ayati's model, where the bio-inspired damage is induced through the paracrine and autocrine parameters, exhibiting an osteolytic behavior. The obtained results are consistent with existing literature, leading us to believe that our in-silico experiments will serve as a cornerstone for paving the way towards targeted interventions and personalized treatment strategies, ultimately improving the quality of life for those affected by these conditions.

In situtumor model for longitudinal in silico imaging trials

Objective.In this article, we introduce a computational model for simulating the growth of breast cancer lesions accounting for the stiffness of surrounding anatomical structures.Approach.In our model, ligaments are classified as the most rigid structures while the softer parts of the breast are occupied by fat and glandular tissues As a result of these variations in tissue elasticity, the rapidly proliferating tumor cells are met with differential resistance. It is found that these cells are likely to circumvent stiffer terrains such as ligaments, instead electing to proliferate preferentially within the more yielding confines of the breast's soft topography. By manipulating the interstitial tumor pressure in direct proportion to the elastic constants of the tissues surrounding the tumor, this model thus creates the potential for realizing a database of unique lesion morphology sculpted by the distinctive topography of each local anatomical infrastructure. We modeled the growth of simulated lesions within volumes extracted from fatty breast models, developed by Graffet alwith a resolution of 50μm generated with the open-source and readily available Virtual Imaging Clinical Trials for Regulatory Evaluation (VICTRE) imaging pipeline. To visualize and validate the realism of the lesion models, we leveraged the imaging component of the VICTRE pipeline, which replicates the siemens mammomat inspiration mammography system in a digital format. This system was instrumental in generating digital mammogram (DM) images for each breast model containing the simulated lesions.Results.By utilizing the DM images, we were able to effectively illustrate the imaging characteristics of the lesions as they integrated with the anatomical backgrounds. Our research also involved a reader study that compared 25 simulated DM regions of interest (ROIs) with inserted lesions from our models with DM ROIs from the DDSM dataset containing real manifestations of breast cancer. In general the simulation time for the lesions was approximately 2.5 hours, but it varied depending on the lesion's local environment.Significance.The lesion growth model will facilitate and enhance longitudinal in silico trials investigating the progression of breast cancer.

A Mathematical Model of Stroma-Supported Allometric Tumor Growth

Mounting empirical research suggests that the stroma, or interface between healthy and cancerous tissue, is a critical determinate of cancer invasion. At the same time, a cancer cell's location and potential to proliferate can influence its sensitivity to cancer treatments. In this paper, we use ordinary differential equations to develop spatially structured models for solid tumors wherein the growth of tumor components is coordinated. The model tumors feature two components, a proliferating peripheral growth region, which potentially includes a mix of cancerous and noncancerous stroma cells, and a solid tumor core. Mathematical and numerical analysis are used to investigate how coordinated expansion of the tumor growth region and core can influence overall growth dynamics in a variety of tumor types. Model assumptions, which are motivated by empirical and in silico solid tumor research, are evaluated through comparison to tumor volume data and existing models of tumor growth.

Mechanotransduction in tumor dynamics modeling

Mechanotherapy is a groundbreaking approach to impact carcinogenesis. Cells sense and respond to mechanical stimuli, translating them into biochemical signals in a process known as mechanotransduction. The impact of stress on tumor growth has been studied in the last three decades, and many papers highlight the role of mechanics as a critical self-inducer of tumor fate at the in vitro and in vivo biological levels. Meanwhile, mathematical models attempt to determine laws to reproduce tumor dynamics. This review discusses biological mechanotransduction mechanisms and mathematical-biomechanical models together. The aim is to provide a common framework for the different approaches that have emerged in the literature from the perspective of tumor avascularity and to provide insight into emerging mechanotherapies that have attracted interest in recent years.

Model-informed experimental design recommendations for distinguishing intrinsic and acquired targeted therapeutic resistance in head and neck cancer

The promise of precision medicine has been limited by the pervasive resistance to many targeted therapies for cancer. Inferring the timing (i.e., pre-existing or acquired) and mechanism (i.e., drug-induced) of such resistance is crucial for designing effective new therapeutics. This paper studies cetuximab resistance in head and neck squamous cell carcinoma (HNSCC) using tumor volume data obtained from patient-derived tumor xenografts. We ask if resistance mechanisms can be determined from this data alone, and if not, what data would be needed to deduce the underlying mode(s) of resistance. To answer these questions, we propose a family of mathematical models, with each member of the family assuming a different timing and mechanism of resistance. We present a method for fitting these models to individual volumetric data, and utilize model selection and parameter sensitivity analyses to ask: which member(s) of the family of models best describes HNSCC response to cetuximab, and what does that tell us about the timing and mechanisms driving resistance? We find that along with time-course volumetric data to a single dose of cetuximab, the initial resistance fraction and, in some instances, dose escalation volumetric data are required to distinguish among the family of models and thereby infer the mechanisms of resistance. These findings can inform future experimental design so that we can best leverage the synergy of wet laboratory experimentation and mathematical modeling in the study of novel targeted cancer therapeutics.

Dynamic characterization of breast cancer response to neoadjuvant therapy using biophysical metrics of spatial proliferation

Current tools to assess breast cancer response to neoadjuvant chemotherapy cannot reliably predict disease eradication, which if possible, could allow early cessation of therapy. In this work, we assessed the ability of an image data-driven mathematical modeling approach for dynamic characterization of breast cancer response to neoadjuvant therapy. We retrospectively analyzed patients enrolled in the I-SPY 2 TRIAL at the Atrium Health Wake Forest Baptist Comprehensive Cancer Center. Patients enrolled on the study received four MR imaging examinations during neoadjuvant therapy with acquisitions at baseline (T₀), 3-weeks/early-treatment (T₁), 12-weeks/mid-treatment (T₂), and completion of therapy prior to surgery (T₃). We use a biophysical mathematical model of tumor growth to generate spatial estimates of tumor proliferation to characterize the dynamics of treatment response. Using histogram summary metrics to quantify estimated tumor proliferation maps, we found strong correlation of mathematical model-estimated tumor proliferation with residual cancer burden, with Pearson correlation coefficients ranging from 0.88 and 0.97 between T₀ and T₂, representing a significant improvement from conventional assessment methods of change in mean apparent diffusion coefficient and functional tumor volume. This data shows the significant promise of imaging-based biophysical mathematical modeling methods for dynamic characterization of patient-specific response to neoadjuvant therapy with correlation to residual disease outcomes.

Integrating mechanism-based modeling with biomedical imaging to build practical digital twins for clinical oncology

Digital twins employ mathematical and computational models to virtually represent a physical object (e.g., planes and human organs), predict the behavior of the object, and enable decision-making to optimize the future behavior of the object. While digital twins have been widely used in engineering for decades, their applications to oncology are only just emerging. Due to advances in experimental techniques quantitatively characterizing cancer, as well as advances in the mathematical and computational sciences, the notion of building and applying digital twins to understand tumor dynamics and personalize the care of cancer patients has been increasingly appreciated. In this review, we present the opportunities and challenges of applying digital twins in clinical oncology, with a particular focus on integrating medical imaging with mechanism-based, tissue-scale mathematical modeling. Specifically, we first introduce the general digital twin framework and then illustrate existing applications of image-guided digital twins in healthcare. Next, we detail both the imaging and modeling techniques that provide practical opportunities to build patient-specific digital twins for oncology. We then describe the current challenges and limitations in developing image-guided, mechanism-based digital twins for oncology along with potential solutions. We conclude by outlining five fundamental questions that can serve as a roadmap when designing and building a practical digital twin for oncology and attempt to provide answers for a specific application to brain cancer. We hope that this contribution provides motivation for the imaging science, oncology, and computational communities to develop practical digital twin technologies to improve the care of patients battling cancer.

The biophysics of cancer: emerging insights from micro- and nanoscale tools

Cancer is a complex and dynamic disease that is aberrant both biologically and physically. There is growing appreciation that physical abnormalities with both cancer cells and their microenvironment that span multiple length scales are important drivers for cancer growth and metastasis. The scope of this review is to highlight the key advancements in micro- and nano-scale tools for delineating the cause and consequences of the aberrant physical properties of tumors. We focus our review on three important physical aspects of cancer: 1) solid mechanical properties, 2) fluid mechanical properties, and 3) mechanical alterations to cancer cells. Beyond posing physical barriers to the delivery of cancer therapeutics, these properties are also known to influence numerous biological processes, including cancer cell invasion and migration leading to metastasis, and response and resistance to therapy. We comment on how micro- and nanoscale tools have transformed our fundamental understanding of the physical dynamics of cancer progression and their potential for bridging towards future applications at the interface of oncology and physical sciences.

Quantitative magnetic resonance imaging and tumor forecasting of breast cancer patients in the community setting

This protocol describes a complete data acquisition, analysis and computational forecasting pipeline for employing quantitative MRI data to predict the response of locally advanced breast cancer to neoadjuvant therapy in a community-based care setting. The methodology has previously been successfully applied to a heterogeneous patient population. The protocol details how to acquire the necessary images followed by registration, segmentation, quantitative perfusion and diffusion analysis, model calibration, and prediction. The data collection portion of the protocol requires ~25 min of scanning, postprocessing requires 2-3 h, and the model calibration and prediction components require ~10 h per patient depending on tumor size. The response of individual breast cancer patients to neoadjuvant therapy is forecast by application of a biophysical, reaction-diffusion mathematical model to these data. Successful application of the protocol results in coregistered MRI data from at least two scan visits that quantifies an individual tumor's size, cellularity and vascular properties. This enables a spatially resolved prediction of how a particular patient's tumor will respond to therapy. Expertise in image acquisition and analysis, as well as the numerical solution of partial differential equations, is required to carry out this protocol.

Clinical assessment of a biophysical model for distinguishing tumor progression from radiation necrosis

PURPOSE

The efficacy of an imaging-driven mechanistic biophysical model of tumor growth for distinguishing radiation necrosis from tumor progression in patients with enhancing lesions following stereotactic radiosurgery (SRS) for brain metastasis is validated.

METHODS

We retrospectively assessed the model using 73 patients with 78 lesions and histologically confirmed radiation necrosis or tumor progression. Postcontrast T1-weighted MRI images were used to extract parameters for a mechanistic reaction-diffusion logistic growth model mechanically coupled to the surrounding tissue. The resulting model was then used to estimate mechanical stress fields, which were then compared with edema visualized on FLAIR imaging using DICE similarity coefficients. DICE, model, and standard radiographic morphometric analysis parameters were evaluated using a receiver operating characteristic (ROC) curve for prediction of radiation necrosis or tumor progression. Multivariate logistic regression models were then constructed using mechanistic model parameters or advanced radiomic features. An independent validation was performed to evaluate predictive performance.

RESULTS

Tumor cell proliferation rate resulted in ROC AUC = 0.86, 95% CI: 0.76-0.95, P < 0.0001, 74% sensitivity and 95% specificity) and DICE similarity coefficient associated with high stresses demonstrated an ROC AUC = 0.93, 95% CI: 0.86-0.99, P < 0.0001, 81% sensitivity and 95% specificity. In a multivariate logistic regression model using an independent validation dataset, mechanistic modeling parameters had an ROC AUC of 0.95, with 94% sensitivity and 96% specificity.

CONCLUSIONS

Imaging-driven biophysical modeling of tumor growth represents a novel method for accurately predicting clinically significant tumor behavior.

Biologically-Based Mathematical Modeling of Tumor Vasculature and Angiogenesis via Time-Resolved Imaging Data

Tumor-associated vasculature is responsible for the delivery of nutrients, removal of waste, and allowing growth beyond 2-3 mm3. Additionally, the vascular network, which is changing in both space and time, fundamentally influences tumor response to both systemic and radiation therapy. Thus, a robust understanding of vascular dynamics is necessary to accurately predict tumor growth, as well as establish optimal treatment protocols to achieve optimal tumor control. Such a goal requires the intimate integration of both theory and experiment. Quantitative and time-resolved imaging methods have emerged as technologies able to visualize and characterize tumor vascular properties before and during therapy at the tissue and cell scale. Parallel to, but separate from those developments, mathematical modeling techniques have been developed to enable in silico investigations into theoretical tumor and vascular dynamics. In particular, recent efforts have sought to integrate both theory and experiment to enable data-driven mathematical modeling. Such mathematical models are calibrated by data obtained from individual tumor-vascular systems to predict future vascular growth, delivery of systemic agents, and response to radiotherapy. In this review, we discuss experimental techniques for visualizing and quantifying vascular dynamics including magnetic resonance imaging, microfluidic devices, and confocal microscopy. We then focus on the integration of these experimental measures with biologically based mathematical models to generate testable predictions.

Modeling of Glioma Growth with Mass Effect by Longitudinal Magnetic Resonance Imaging

It is well-known that expanding glioblastomas typically induce significant deformations of the surrounding parenchyma (i.e., the so-called ?mass effect?). In this study, we evaluate the performance of three mathematical models of tumor growth: 1) a reaction-diffusion-advection model which accounts for mass effect (RDAM), 2) a reaction-diffusion model with mass effect that is consistent only in the case of small deformations (RDM), and 3) a reaction-diffusion model that does not include the mass effect (RD). The models were calibrated with magnetic resonance imaging (MRI) data obtained during tumor development in a murine model of glioma (n = 9). We obtained T₂-weighted and contrast-enhanced T₁-weighted MRI at 6 time points over 10 days to determine the spatiotemporal variation in the mass effect and tumor concentration, respectively. We calibrated the three models using data 1) at the first four, 2) only at the first and fourth, and 3) only at the third and fourth time points. Each of these calibrations were run forward in time to predict the volume fraction of tumor cells at the conclusion of the experiment. The diffusion coefficient for the RDAM model (median of 10.65 ? 10^-3 mm²d^-1) is significantly less than those for the RD and RDM models (17.46 ? 10^-3 mm²d^-1 and 19.38 ? 10^-3 mm²d^-1, respectively). The tumor concentrations for the RD, RDM, and RDAM models have medians of 40.2%, 32.1%, and 44.7%, respectively, for the calibration using data from the first four time points. The RDM model most accurately predicts tumor growth, while the RDAM model presents the least variation in its estimates of the diffusion coefficient and proliferation rate. This study demonstrates that the mathematical models capture both tumor development and mass effect observed in experiments.

Evaluating patient-specific neoadjuvant regimens for breast cancer via a mathematical model constrained by quantitative magnetic resonance imaging data

The ability to accurately predict response and then rigorously optimize a therapeutic regimen on a patient-specific basis, would transform oncology. Toward this end, we have developed an experimental-mathematical framework that integrates quantitative magnetic resonance imaging (MRI) data into a biophysical model to predict patient-specific treatment response of locally advanced breast cancer to neoadjuvant therapy. Diffusion-weighted and dynamic contrast-enhanced MRI data is collected prior to therapy, after 1 cycle of therapy, and at the completion of the first therapeutic regimen. The model is initialized and calibrated with the first 2 patient-specific MRI data sets to predict response at the third, which is then compared to patient outcomes (N = 18). The model's predictions for total cellularity, total volume, and the longest axis at the completion of the regimen are significant within expected measurement precision (P< 0.05) and strongly correlated with measured response (P < 0.01). Further, we use the model to investigate, in silico, a range of (practical) alternative treatment plans to achieve the greatest possible tumor control for each individual in a subgroup of patients (N = 13). The model identifies alternative dosing strategies predicted to achieve greater tumor control compared to the standard of care for 12 of 13 patients (P < 0.01). In summary, a predictive, mechanism-based mathematical model has demonstrated the ability to identify alternative treatment regimens that are forecasted to outperform the therapeutic regimens the patients clinically. This has important implications for clinical trial design with the opportunity to alter oncology care in the future.

Towards integration of 64Cu-DOTA-trastuzumab PET-CT and MRI with mathematical modeling to predict response to neoadjuvant therapy in HER2 + breast cancer

While targeted therapies exist for human epidermal growth factor receptor 2 positive (HER2 +) breast cancer, HER2 + patients do not always respond to therapy. We present the results of utilizing a biophysical mathematical model to predict tumor response for two HER2 + breast cancer patients treated with the same therapeutic regimen but who achieved different treatment outcomes. Quantitative data from magnetic resonance imaging (MRI) and 64Cu-DOTA-trastuzumab positron emission tomography (PET) are used to estimate tumor density, perfusion, and distribution of HER2-targeted antibodies for each individual patient. MRI and PET data are collected prior to therapy, and follow-up MRI scans are acquired at a midpoint in therapy. Given these data types, we align the data sets to a common image space to enable model calibration. Once the model is parameterized with these data, we forecast treatment response with and without HER2-targeted therapy. By incorporating targeted therapy into the model, the resulting predictions are able to distinguish between the two different patient responses, increasing the difference in tumor volume change between the two patients by > 40%. This work provides a proof-of-concept strategy for processing and integrating PET and MRI modalities into a predictive, clinical-mathematical framework to provide patient-specific predictions of HER2 + treatment response.

Generating patient-specific virtual tumor populations with reaction-diffusion models and molecular imaging data

The use of mathematical tumor growth models coupled to noisy imaging data has been suggested as a possible component in the push towards precision medicine. We discuss the generation of population and patient-specific virtual populations in this context, providing in silico experiments to demonstrate how intra- and inter-patient heterogeneity can be estimated by applying rigorous statistical procedures to noisy molecular imaging data, and how the noise properties of such data can be analyzed to estimate uncertainties in predicted patient outcomes.

Characterization of multicellular breast tumor spheroids using image data-driven biophysical mathematical modeling

Multicellular tumor spheroid (MCTS) systems provide an in vitro cell culture model system which mimics many of the complexities of an in vivo solid tumor and tumor microenvironment, and are often used to study cancer cell growth and drug efficacy. Here, we present a coupled experimental-computational framework to estimate phenotypic growth and biophysical tumor microenvironment properties. This novel framework utilizes standard microscopy imaging of MCTS systems to drive a biophysical mathematical model of MCTS growth and mechanical interactions. By extending our previous in vivo mechanically-coupled reaction–diffusion modeling framework we developed a microscopy image processing framework capable of mechanistic characterization of MCTS systems. Using MDA-MB-231 breast cancer MCTS, we estimated biophysical parameters of cellular diffusion, rate of cellular proliferation, and cellular tractions forces. We found significant differences in these model-based biophysical parameters throughout the treatment time course between untreated and treated MCTS systems, whereas traditional size-based morphometric parameters were inconclusive. The proposed experimental-computational framework estimates mechanistic MCTS growth and invasion parameters with significant potential to assist in better and more precise assessment of in vitro drug efficacy through the development of computational analysis methodologies for three-dimensional cell culture systems to improve the development and evaluation of antineoplastic drugs.

Mechanism-Based Modeling of Tumor Growth and Treatment Response Constrained by Multiparametric Imaging Data

Multiparametric imaging is a critical tool in the noninvasive study and assessment of cancer. Imaging methods have evolved over the past several decades to provide quantitative measures of tumor and healthy tissue characteristics related to, for example, cell number, blood volume fraction, blood flow, hypoxia, and metabolism. Mechanistic models of tumor growth also have matured to a point where the incorporation of patient-specific measures could provide clinically relevant predictions of tumor growth and response. In this review, we identify and discuss approaches that use multiparametric imaging data, including diffusion-weighted magnetic resonance imaging, dynamic contrast-enhanced magnetic resonance imaging, diffusion tensor imaging, contrast-enhanced computed tomography, [¹⁸F]fluorodeoxyglucose positron emission tomography, and [¹⁸F]fluoromisonidazole positron emission tomography to initialize and calibrate mechanistic models of tumor growth and response. We focus the discussion on brain and breast cancers; however, we also identify three emerging areas of application in kidney, pancreatic, and lung cancers. We conclude with a discussion of the future directions for incorporating multiparametric imaging data and mechanistic modeling into clinical decision making for patients with cancer.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

Optimal Control Theory for Personalized Therapeutic Regimens in Oncology: Background, History, Challenges, and Opportunities

Optimal control theory is branch of mathematics that aims to optimize a solution to a dynamical system. While the concept of using optimal control theory to improve treatment regimens in oncology is not novel, many of the early applications of this mathematical technique were not designed to work with routinely available data or produce results that can eventually be translated to the clinical setting. The purpose of this review is to discuss clinically relevant considerations for formulating and solving optimal control problems for treating cancer patients. Our review focuses on two of the most widely used cancer treatments, radiation therapy and systemic therapy, as they naturally lend themselves to optimal control theory as a means to personalize therapeutic plans in a rigorous fashion. To provide context for optimal control theory to address either of these two modalities, we first discuss the major limitations and difficulties oncologists face when considering alternate regimens for their patients. We then provide a brief introduction to optimal control theory before formulating the optimal control problem in the context of radiation and systemic therapy. We also summarize examples from the literature that illustrate these concepts. Finally, we present both challenges and opportunities for dramatically improving patient outcomes via the integration of clinically relevant, patient-specific, mathematical models and optimal control theory.

A Review of Mathematical Models for Tumor Dynamics and Treatment Resistance Evolution of Solid Tumors

Increasing knowledge of intertumor heterogeneity, intratumor heterogeneity, and cancer evolution has improved the understanding of anticancer treatment resistance. A better characterization of cancer evolution and subsequent use of this knowledge for personalized treatment would increase the chance to overcome cancer treatment resistance. Model‐based approaches may help achieve this goal. In this review, we comprehensively summarized mathematical models of tumor dynamics for solid tumors and of drug resistance evolution. Models displayed by ordinary differential equations, algebraic equations, and partial differential equations for characterizing tumor burden dynamics are introduced and discussed. As for tumor resistance evolution, stochastic and deterministic models are introduced and discussed. The results may facilitate a novel model‐based analysis on anticancer treatment response and the occurrence of resistance, which incorporates both tumor dynamics and resistance evolution. The opportunities of a model‐based approach as discussed in this review can be of great benefit for future optimizing and personalizing anticancer treatment.

Biophysical model-based parameters to classify tumor recurrence from radiation-induced necrosis for brain metastases

PURPOSE

Stereotactic radiosurgery (SRS) is used for local control treatment of patients with intracranial metastases. As a result of SRS, some patients develop radiation-induced necrosis. Radiographically, radiation-induced necrosis can appear similar to tumor recurrence in magnetic resonance (MR) T₁ -weighted contrast-enhanced imaging, T₂ -weighted MR imaging, and Fluid-Attenuated Inversion Recovery (FLAIR) MR imaging. Radiographic ambiguities often necessitate invasive brain biopsies to determine lesion etiology or cause delayed subsequent therapy initiation. We use a biomechanically coupled tumor growth model to estimate patient-specific model parameters and model-derived measures to noninvasively classify etiology of enhancing lesions in this patient population.

METHODS

In this initial, preliminary retrospective study, we evaluated five patients with tumor recurrence and five with radiation-induced necrosis. Longitudinal patient-specific MR imaging data were used in conjunction with the model to parameterize tumor cell proliferation rate and tumor cell diffusion coefficient, and Dice correlation coefficients were used to quantify degree of correlation between model-estimated mechanical stress fields and edema visualized from MR imaging.

RESULTS

Results found four statistically relevant parameters which can differentiate tumor recurrence and radiation-induced necrosis.

CONCLUSIONS

This preliminary investigation suggests potential of this framework to noninvasively determine the etiology of enhancing lesions in patients who previously underwent SRS for intracranial metastases.

An adjoint-based method for a linear mechanically-coupled tumor model: Application to estimate the spatial variation of murine glioma growth based on diffusion weighted magnetic resonance imaging

We present an efficient numerical method to quantify the spatial variation of glioma growth based on subject-specific medical images using a mechanically-coupled tumor model. The method is illustrated in a murine model of glioma in which we consider the tumor as a growing elastic mass that continuously deforms the surrounding healthy-appearing brain tissue. As an inverse parameter identification problem, we quantify the volumetric growth of glioma and the growth component of deformation by fitting the model predicted cell density to the cell density estimated using the diffusion-weighted magnetic resonance imaging (DW-MRI) data. Numerically, we developed an adjoint-based approach to solve the optimization problem. Results on a set of experimentally measured, in vivo rat glioma data indicate good agreement between the fitted and measured tumor area and suggest a wide variation of in-plane glioma growth with the growth-induced Jacobian ranging from 1.0 to 6.0.

Mathematical models of tumor cell proliferation: A review of the literature

INTRODUCTION

A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Incorporating drug delivery into an imaging-driven, mechanics-coupled reaction diffusion model for predicting the response of breast cancer to neoadjuvant chemotherapy: theory and preliminary clinical results

Clinical methods for assessing tumor response to therapy are largely rudimentary, monitoring only temporal changes in tumor size. Our goal is to predict the response of breast tumors to therapy using a mathematical model that utilizes magnetic resonance imaging (MRI) data obtained non-invasively from individual patients. We extended a previously established, mechanically coupled, reaction-diffusion model for predicting tumor response initialized with patient-specific diffusion weighted MRI (DW-MRI) data by including the effects of chemotherapy drug delivery, which is estimated using dynamic contrast-enhanced (DCE-) MRI data. The extended, drug incorporated, model is initialized using patient-specific DW-MRI and DCE-MRI data. Data sets from five breast cancer patients were used-obtained before, after one cycle, and at mid-point of neoadjuvant chemotherapy. The DCE-MRI data was used to estimate spatiotemporal variations in tumor perfusion with the extended Kety-Tofts model. The physiological parameters derived from DCE-MRI were used to model changes in delivery of therapy drugs within the tumor for incorporation in the extended model. We simulated the original model and the extended model in both 2D and 3D and compare the results for this five-patient cohort. Preliminary results show reductions in the error of model predicted tumor cellularity and size compared to the experimentally-measured results for the third MRI scan when therapy was incorporated. Comparing the two models for agreement between the predicted total cellularity and the calculated total cellularity (from the DW-MRI data) reveals an increased concordance correlation coefficient from 0.81 to 0.98 for the 2D analysis and 0.85 to 0.99 for the 3D analysis (p  <  0.01 for each) when the extended model was used in place of the original model. This study demonstrates the plausibility of using DCE-MRI data as a means to estimate drug delivery on a patient-specific basis in predictive models and represents a step toward the goal of achieving individualized prediction of tumor response to therapy.

Biophysical Modeling of In Vivo Glioma Response After Whole-Brain Radiation Therapy in a Murine Model of Brain Cancer

PURPOSE

To develop and investigate a set of biophysical models based on a mechanically coupled reaction-diffusion model of the spatiotemporal evolution of tumor growth after radiation therapy.

METHODS AND MATERIALS

Post-radiation therapy response is modeled using a cell death model (Md), a reduced proliferation rate model (Mp), and cell death and reduced proliferation model (Mdp). To evaluate each model, rats (n = 12) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging (MRI) and contrast-enhanced MRI at 7 time points over 2 weeks. Rats received either 20 or 40 Gy between the third and fourth imaging time point. Diffusion-weighted MRI was used to estimate tumor cell number within enhancing regions in contrast-enhanced MRI data. Each model was fit to the spatiotemporal evolution of tumor cell number from time point 1 to time point 5 to estimate model parameters. The estimated model parameters were then used to predict tumor growth at the final 2 imaging time points. The model prediction was evaluated by calculating the error in tumor volume estimates, average surface distance, and voxel-based cell number.

RESULTS

For both the rats treated with either 20 or 40 Gy, significantly lower error in tumor volume, average surface distance, and voxel-based cell number was observed for the Mdp and Mp models compared with the Md model. The Mdp model fit, however, had significantly lower sum squared error compared with the Mp and Md models.

CONCLUSIONS

The results of this study indicate that for both doses, the Mp and Mdp models result in accurate predictions of tumor growth, whereas the Md model poorly describes response to radiation therapy.

Mechanically Coupled Reaction-Diffusion Model to Predict Glioma Growth: Methodological Details

Biophysical models designed to predict the growth and response of tumors to treatment have the potential to become a valuable tool for clinicians in care of cancer patients. Specifically, individualized tumor forecasts could be used to predict response or resistance early in the course of treatment, thereby providing an opportunity for treatment selection or adaption. This chapter discusses an experimental and modeling framework in which noninvasive imaging data is used to initialize and parameterize a subject-specific model of tumor growth. This modeling approach is applied to an analysis of murine models of glioma growth.

Tumor Cell Load and Heterogeneity Estimation From Diffusion-Weighted MRI Calibrated With Histological Data: an Example From Lung Cancer

Diffusion-weighted magnetic resonance imaging (DWI) is a key non-invasive imaging technique for cancer diagnosis and tumor treatment assessment, reflecting Brownian movement of water molecules in tissues. Since densely packed cells restrict molecule mobility, tumor tissues produce usually higher signal (a.k.a. less attenuated signal) on isotropic maps compared with normal tissues. However, no general quantitative relation between DWI data and the cell density has been established. In order to link low-resolution clinical cross-sectional data with high-resolution histological information, we developed an image processing and analysis chain, which was used to study the correlation between the diffusion coefficient (D value) estimated from DWI and tumor cellularity from serial histological slides of a resected non-small cell lung cancer tumor. Color deconvolution followed by cell nuclei segmentation was performed on digitized histological images to determine local and cell-type specific 2d (two-dimensional) densities. From these, the 3d cell density was inferred by a model-based sampling technique, which is necessary for the calculation of local and global 3d tumor cell count. Next, DWI sequence information was overlaid with high-resolution CT data and the resected histology using prominent anatomical hallmarks for co-registration of histology tissue blocks and non-invasive imaging modalities' data. The integration of cell numbers information and DWI data derived from different tumor areas revealed a clear negative correlation between cell density and D value. Importantly, spatial tumor cell density can be calculated based on DWI data. In summary, our results demonstrate that tumor cell count and heterogeneity can be predicted from DWI data, which may open new opportunities for personalized diagnosis and therapy optimization.

Selection and Validation of Predictive Models of Radiation Effects on Tumor Growth Based on Noninvasive Imaging Data

The use of mathematical and computational models for reliable predictions of tumor growth and decline in living organisms is one of the foremost challenges in modern predictive science, as it must cope with uncertainties in observational data, model selection, model parameters, and model inadequacy, all for very complex physical and biological systems. In this paper, large classes of parametric models of tumor growth in vascular tissue are discussed including models for radiation therapy. Observational data is obtained from MRI of a murine model of glioma and observed over a period of about three weeks, with X-ray radiation administered 14.5 days into the experimental program. Parametric models of tumor proliferation and decline are presented based on the balance laws of continuum mixture theory, particularly mass balance, and from accepted biological hypotheses on tumor growth. Among these are new model classes that include characterizations of effects of radiation and simple models of mechanical deformation of tumors. The Occam Plausibility Algorithm (OPAL) is implemented to provide a Bayesian statistical calibration of the model classes, 39 models in all, as well as the determination of the most plausible models in these classes relative to the observational data, and to assess model inadequacy through statistical validation processes. Discussions of the numerical analysis of finite element approximations of the system of stochastic, nonlinear partial differential equations characterizing the model classes, as well as the sampling algorithms for Monte Carlo and Markov chain Monte Carlo (MCMC) methods employed in solving the forward stochastic problem, and in computing posterior distributions of parameters and model plausibilities are provided. The results of the analyses described suggest that the general framework developed can provide a useful approach for predicting tumor growth and the effects of radiation.

A mechanically coupled reaction-diffusion model that incorporates intra-tumoural heterogeneity to predict in vivo glioma growth

While gliomas have been extensively modelled with a reaction-diffusion (RD) type equation it is most likely an oversimplification. In this study, three mathematical models of glioma growth are developed and systematically investigated to establish a framework for accurate prediction of changes in tumour volume as well as intra-tumoural heterogeneity. Tumour cell movement was described by coupling movement to tissue stress, leading to a mechanically coupled (MC) RD model. Intra-tumour heterogeneity was described by including a voxel-specific carrying capacity (CC) to the RD model. The MC and CC models were also combined in a third model. To evaluate these models, rats (n = 14) with C6 gliomas were imaged with diffusion-weighted magnetic resonance imaging over 10 days to estimate tumour cellularity. Model parameters were estimated from the first three imaging time points and then used to predict tumour growth at the remaining time points which were then directly compared to experimental data. The results in this work demonstrate that mechanical-biological effects are a necessary component of brain tissue tumour modelling efforts. The results are suggestive that a variable tissue carrying capacity is a needed model component to capture tumour heterogeneity. Lastly, the results advocate the need for additional effort towards capturing tumour-to-tissue infiltration.

The Attenuation Distribution Across the Long Axis of Breast Cancer Liver Metastases at CT: A Quantitative Biomarker for Predicting Overall Survival

OBJECTIVE

The objective of our study was to compare attenuation distribution across the long-axis (ADLA) measurements, Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1, and Choi criteria for predicting overall survival (OS) in patients with metastatic breast cancer treated with bevacizumab.

MATERIALS AND METHODS

We obtained HIPAA-compliant data from a prospective, multisite, phase 3 trial of bevacizumab for the treatment of metastatic breast cancer. For patients with one or more liver metastases measuring 15 mm or larger at baseline, we evaluated up to two target liver lesions using RECIST, Choi criteria, and ADLA measurements, with the latter defined as the SD of the CT attenuation values of each pixel along the tumor long-axis diameter. The optimal percentage change threshold for defining an ADLA response was computed by cross-validation analysis in a Cox model. The log-rank test was applied to evaluate RECIST, Choi criteria, and ADLA for discriminating patients with superior OS. The predictive accuracies of all three techniques were compared using Brier scores and areas under the ROC curve (AUC). All analyses were performed separately using best overall response (BOR) and response at the first follow-up time point (FU1).

RESULTS

One hundred sixty-four patients met the inclusion criteria. A 25% decrease in the ADLA measurement from baseline was the optimal ADLA response threshold for BOR and FU1. RECIST, Choi criteria, and ADLA successfully identified patients with superior OS when using BOR (RECIST, p = 0.02; Choi and ADLA, p < 0.001), but only Choi criteria and ADLA measurements were successful when using FU1 (RECIST, p = 0.43; Choi and ADLA, p < 0.001). In a direct comparison, ADLA measurements outperformed both RECIST and Choi criteria using BOR (95% CI for Brier score differences, ADLA-RECIST [-0.58 to -0.08] and ADLA-Choi [-0.55 to -0.06]; 95% CI for AUC differences, ADLA-RECIST [0.16-0.33] and ADLA-Choi [0.17-0.36]) as well as using FU1 (95% CI for Brier score differences, ADLA-RECIST [-0.77 to -0.08] and ADLA-Choi [-0.58 to -0.03]; 95% CI for AUC differences, ADLA-RECIST [0.22-0.39] and ADLA-Choi [0.01-0.22]).

CONCLUSION

ADLA measurements may be a useful noninvasive indicator of cancer treatment response. Because ADLA measurements may be extracted relatively easily using existing radiologist workflows, further investigation of the ADLA technique is warranted.

Three-dimensional Image-based Mechanical Modeling for Predicting the Response of Breast Cancer to Neoadjuvant Therapy

The use of quantitative medical imaging data to initialize and constrain mechanistic mathematical models of tumor growth has demonstrated a compelling strategy for predicting therapeutic response. More specifically, we have demonstrated a data-driven framework for prediction of residual tumor burden following neoadjuvant therapy in breast cancer that uses a biophysical mathematical model combining reaction-diffusion growth/therapy dynamics and biomechanical effects driven by early time point imaging data. Whereas early work had been based on a limited dimensionality reduction (two-dimensional planar modeling analysis) to simplify the numerical implementation, in this work, we extend our framework to a fully volumetric, three-dimensional biophysical mathematical modeling approach in which parameter estimates are generated by an inverse problem based on the adjoint state method for numerical efficiency. In an in silico performance study, we show accurate parameter estimation with error less than 3% as compared to ground truth. We apply the approach to patient data from a patient with pathological complete response and a patient with residual tumor burden and demonstrate technical feasibility and predictive potential with direct comparisons between imaging data observation and model predictions of tumor cellularity and volume. Comparisons to our previous two-dimensional modeling framework reflect enhanced model prediction of residual tumor burden through the inclusion of additional imaging slices of patient-specific data.

Selection, calibration, and validation of models of tumor growth

This paper presents general approaches for addressing some of the most important issues in predictive computational oncology concerned with developing classes of predictive models of tumor growth. First, the process of developing mathematical models of vascular tumors evolving in the complex, heterogeneous, macroenvironment of living tissue; second, the selection of the most plausible models among these classes, given relevant observational data; third, the statistical calibration and validation of models in these classes, and finally, the prediction of key Quantities of Interest (QOIs) relevant to patient survival and the effect of various therapies. The most challenging aspects of this endeavor is that all of these issues often involve confounding uncertainties: in observational data, in model parameters, in model selection, and in the features targeted in the prediction. Our approach can be referred to as "model agnostic" in that no single model is advocated; rather, a general approach that explores powerful mixture-theory representations of tissue behavior while accounting for a range of relevant biological factors is presented, which leads to many potentially predictive models. Then representative classes are identified which provide a starting point for the implementation of OPAL, the Occam Plausibility Algorithm (OPAL) which enables the modeler to select the most plausible models (for given data) and to determine if the model is a valid tool for predicting tumor growth and morphology (in vivo). All of these approaches account for uncertainties in the model, the observational data, the model parameters, and the target QOI. We demonstrate these processes by comparing a list of models for tumor growth, including reaction-diffusion models, phase-fields models, and models with and without mechanical deformation effects, for glioma growth measured in murine experiments. Examples are provided that exhibit quite acceptable predictions of tumor growth in laboratory animals while demonstrating successful implementations of OPAL.

MR Imaging Biomarkers in Oncology Clinical Trials

The authors discuss eight areas of quantitative MR imaging that are currently used (RECIST, DCE-MR imaging, DSC-MR imaging, diffusion MR imaging) in clinical trials or emerging (CEST, elastography, hyperpolarized MR imaging, multiparameter MR imaging) as promising techniques in diagnosing cancer and assessing or predicting response of cancer to therapy. Illustrative applications of the techniques in the clinical setting are summarized before describing the current limitations of the methods.

Multi-scale Modeling in Clinical Oncology: Opportunities and Barriers to Success

Hierarchical processes spanning several orders of magnitude of both space and time underlie nearly all cancers. Multi-scale statistical, mathematical, and computational modeling methods are central to designing, implementing and assessing treatment strategies that account for these hierarchies. The basic science underlying these modeling efforts is maturing into a new discipline that is close to influencing and facilitating clinical successes. The purpose of this review is to capture the state-of-the-art as well as the key barriers to success for multi-scale modeling in clinical oncology. We begin with a summary of the long-envisioned promise of multi-scale modeling in clinical oncology, including the synthesis of disparate data types into models that reveal underlying mechanisms and allow for experimental testing of hypotheses. We then evaluate the mathematical techniques employed most widely and present several examples illustrating their application as well as the current gap between pre-clinical and clinical applications. We conclude with a discussion of what we view to be the key challenges and opportunities for multi-scale modeling in clinical oncology.

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill

It has been hypothesized that continuously releasing drug molecules into the tumor over an extended period of time may significantly improve the chemotherapeutic efficacy by overcoming physical transport limitations of conventional bolus drug treatment. In this paper, we present a generalized space- and time-dependent mathematical model of drug transport and drug-cell interactions to quantitatively formulate this hypothesis. Model parameters describe: perfusion and tissue architecture (blood volume fraction and blood vessel radius); diffusion penetration distance of drug (i.e., a function of tissue compactness and drug uptake rates by tumor cells); and cell death rates (as function of history of drug uptake). We performed preliminary testing and validation of the mathematical model using in vivo experiments with different drug delivery methods on a breast cancer mouse model. Experimental data demonstrated a 3-fold increase in response using nano-vectored drug vs. free drug delivery, in excellent quantitative agreement with the model predictions. Our model results implicate that therapeutically targeting blood volume fraction, e.g., through vascular normalization, would achieve a better outcome due to enhanced drug delivery.

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.

Predicting in vivo glioma growth with the reaction diffusion equation constrained by quantitative magnetic resonance imaging data

Reaction-diffusion models have been widely used to model glioma growth. However, it has not been shown how accurately this model can predict future tumor status using model parameters (i.e., tumor cell diffusion and proliferation) estimated from quantitative in vivo imaging data. To this end, we used in silico studies to develop the methods needed to accurately estimate tumor specific reaction-diffusion model parameters, and then tested the accuracy with which these parameters can predict future growth. The analogous study was then performed in a murine model of glioma growth. The parameter estimation approach was tested using an in silico tumor 'grown' for ten days as dictated by the reaction-diffusion equation. Parameters were estimated from early time points and used to predict subsequent growth. Prediction accuracy was assessed at global (total volume and Dice value) and local (concordance correlation coefficient, CCC) levels. Guided by the in silico study, rats (n = 9) with C6 gliomas, imaged with diffusion weighted magnetic resonance imaging, were used to evaluate the model's accuracy for predicting in vivo tumor growth. The in silico study resulted in low global (tumor volume error <8.8%, Dice >0.92) and local (CCC values >0.80) level errors for predictions up to six days into the future. The in vivo study showed higher global (tumor volume error >11.7%, Dice <0.81) and higher local (CCC <0.33) level errors over the same time period. The in silico study shows that model parameters can be accurately estimated and used to accurately predict future tumor growth at both the global and local scale. However, the poor predictive accuracy in the experimental study suggests the reaction-diffusion equation is an incomplete description of in vivo C6 glioma biology and may require further modeling of intra-tumor interactions including segmentation of (for example) proliferative and necrotic regions.

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.

Multi-site clinical evaluation of DW-MRI as a treatment response metric for breast cancer patients undergoing neoadjuvant chemotherapy

PURPOSE

To evaluate diffusion weighted MRI (DW-MR) as a response metric for assessment of neoadjuvant chemotherapy (NAC) in patients with primary breast cancer using prospective multi-center trials which provided MR scans along with clinical outcome information.

MATERIALS AND METHODS

A total of 39 patients with locally advanced breast cancer accrued from three different prospective clinical trials underwent DW-MR examination prior to and at 3-7 days (Hull University), 8-11 days (University of Michigan) and 35 days (NeoCOMICE) post-treatment initiation. Thirteen patients, 12 of which participated in treatment response study, from UM underwent short interval (<1hr) MRI examinations, referred to as "test-retest" for examination of repeatability. To further evaluate stability in ADC measurements, a thermally controlled diffusion phantom was used to assess repeatability of diffusion measurements. MRI sequences included contrast-enhanced T1-weighted, when appropriate, and DW images acquired at b-values of 0 and 800 s/mm2. Histogram analysis and a voxel-based analytical technique, the Parametric Response Map (PRM), were used to derive diffusion response metrics for assessment of treatment response prediction.

RESULTS

Mean tumor apparent diffusion coefficient (ADC) values generated from patient test-retest examinations were found to be very reproducible (|ΔADC|<0.1x10-3mm2/s). This data was used to calculate the 95% CI from the linear fit of tumor voxel ADC pairs of co-registered examinations (±0.45x10-3mm2/s) for PRM analysis of treatment response. Receiver operating characteristic analysis identified the PRM metric to be predictive of outcome at the 8-11 (AUC = 0.964, p = 0.01) and 35 day (AUC = 0.770, p = 0.05) time points (p<.05) while whole-tumor ADC changes where significant at the later 35 day time interval (AUC = 0.825, p = 0.02).

CONCLUSION

This study demonstrates the feasibility of performing a prospective analysis of DW-MRI as a predictive biomarker of NAC in breast cancer patients. In addition, we provide experimental evidence supporting the use of sensitive analytical tools, such as PRM, for evaluating ADC measurements.

Multimodality functional imaging in radiation therapy planning: relationships between dynamic contrast-enhanced MRI, diffusion-weighted MRI, and 18F-FDG PET

OBJECTIVES

Biologically guided radiotherapy needs an understanding of how different functional imaging techniques interact and link together. We analyse three functional imaging techniques that can be useful tools for achieving this objective.

MATERIALS AND METHODS

The three different imaging modalities from one selected patient are ADC maps, DCE-MRI, and 18F-FDG PET/CT, because they are widely used and give a great amount of complementary information. We show the relationship between these three datasets and evaluate them as markers for tumour response or hypoxia marker. Thus, vascularization measured using DCE-MRI parameters can determine tumour hypoxia, and ADC maps can be used for evaluating tumour response.

RESULTS

ADC and DCE-MRI include information from 18F-FDG, as glucose metabolism is associated with hypoxia and tumour cell density, although 18F-FDG includes more information about the malignancy of the tumour. The main disadvantage of ADC maps is the distortion, and we used only low distorted regions, and extracellular volume calculated from DCE-MRI can be considered equivalent to ADC in well-vascularized areas.

CONCLUSION

A dataset for achieving the biologically guided radiotherapy must include a tumour density study and a hypoxia marker. This information can be achieved using only MRI data or only PET/CT studies or mixing both datasets.

Toward a science of tumor forecasting for clinical oncology

We propose that the quantitative cancer biology community makes a concerted effort to apply lessons from weather forecasting to develop an analogous methodology for predicting and evaluating tumor growth and treatment response. Currently, the time course of tumor response is not predicted; instead, response is only assessed post hoc by physical examination or imaging methods. This fundamental practice within clinical oncology limits optimization of a treatment regimen for an individual patient, as well as to determine in real time whether the choice was in fact appropriate. This is especially frustrating at a time when a panoply of molecularly targeted therapies is available, and precision genetic or proteomic analyses of tumors are an established reality. By learning from the methods of weather and climate modeling, we submit that the forecasting power of biophysical and biomathematical modeling can be harnessed to hasten the arrival of a field of predictive oncology. With a successful methodology toward tumor forecasting, it should be possible to integrate large tumor-specific datasets of varied types and effectively defeat one cancer patient at a time.

Deformable registration for quantifying longitudinal tumor changes during neoadjuvant chemotherapy

PURPOSE

To evaluate DRAMMS, an attribute-based deformable registration algorithm, compared to other intensity-based algorithms, for longitudinal breast MRI registration, and to show its applicability in quantifying tumor changes over the course of neoadjuvant chemotherapy.

METHODS

Breast magnetic resonance images from 14 women undergoing neoadjuvant chemotherapy were analyzed. The accuracy of DRAMMS versus five intensity-based deformable registration methods was evaluated based on 2,380 landmarks independently annotated by two experts, for the entire image volume, different image subregions, and patient subgroups. The registration method with the smallest landmark error was used to quantify tumor changes, by calculating the Jacobian determinant maps of the registration deformation.

RESULTS

DRAMMS had the smallest landmark errors (6.05 ± 4.86 mm), followed by the intensity-based methods CC-FFD (8.07 ± 3.86 mm), NMI-FFD (8.21 ± 3.81 mm), SSD-FFD (9.46 ± 4.55 mm), Demons (10.76 ± 6.01 mm), and Diffeomorphic Demons (10.82 ± 6.11 mm). Results show that registration accuracy also depends on tumor versus normal tissue regions and different patient subgroups.

CONCLUSIONS

The DRAMMS deformable registration method, driven by attribute-matching and mutual-saliency, can register longitudinal breast magnetic resonance images with a higher accuracy than several intensity-matching methods included in this article. As such, it could be valuable for more accurately quantifying heterogeneous tumor changes as a marker of response to treatment.

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.