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

Abstract: 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.

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

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.

Mathematical modelling of trastuzumab-induced immune response in an in vivo murine model of HER2+ breast cancer

Abstract: The goal of this study is to develop an integrated, mathematical–experimental approach for understanding the interactions between the immune system and the effects of trastuzumab on breast cancer that overexpresses the human epidermal growth factor receptor 2 (HER2+). A system of coupled, ordinary differential equations was constructed to describe the temporal changes in tumour growth, along with intratumoural changes in the immune response, vascularity, necrosis and hypoxia. The mathematical model is calibrated with serially acquired experimental data of tumour volume, vascularity, necrosis and hypoxia obtained from either imaging or histology from a murine model of HER2+ breast cancer. Sensitivity analysis shows that model components are sensitive for 12 of 13 parameters, but accounting for uncertainty in the parameter values, model simulations still agree with the experimental data. Given theinitial conditions, the mathematical model predicts an increase in the immune infiltrates over time in the treated animals. Immunofluorescent staining results are presented that validate this prediction by showing an increased co-staining of CD11c and F4/80 (proteins expressed by dendritic cells and/or macrophages) in the total tissue for the treated tumours compared to the controls (equation M9). We posit that the proposed mathematical–experimental approach can be used to elucidate driving interactions between the trastuzumab-induced responses in the tumour and the immune system that drive the stabilization of vasculature while simultaneously decreasing tumour growth—conclusions revealed by the mathematical model that were not deducible from the experimental data alone.

Magnetization Transfer MRI of Breast Cancer in the Community Setting: Reproducibility and Preliminary Results in Neoadjuvant Therapy

Abstract: Repeatability and reproducibility of magnetization transfer magnetic resonance imaging of the breast, and the ability of this technique to assess the response of locally advanced breast cancer to neoadjuvant therapy (NAT), are determined. Reproducibility scans at 3 different 3 T scanners, including 2 scanners in community imaging centers, found a 16.3% difference (n = 3) in magnetization transfer ratio (MTR) in healthy breast fibroglandular tissue. Repeatability scans (n = 10) found a difference of ∼8.1% in the MTR measurement of fibroglandular tissue between the 2 measurements. Thus, MTR is repeatable and reproducible in the breast and can be integrated into community imaging clinics. Serial magnetization transfer magnetic resonance imaging performed at longitudinal time points during NAT indicated no significant change in average tumoral MTR during treatment. However, histogram analysis indicated an increase in the dispersion of MTR values of the tumor during NAT, as quantified by higher standard deviation (P = .005), higher full width at half maximum (P = .02), and lower kurtosis (P = .02). Patients’ stratification into those with pathological complete response (pCR; n = 6) at the conclusion of NAT and those with residual disease (n = 9) showed wider distribution of tumor MTR values in patients who achieved pCR after 2-4 cycles of NAT, as quantified by higher standard deviation (P = .02), higher full width at half maximum (P = .03), and lower kurtosis (P = .03). Thus, MTR can be used as an imaging metric to assess response to breast NAT.

Abstract 924: A mathematical-experimental approach for predicting host responses in a preclinical model for trastuzumab-treated HER2+ breast cancer

Introduction: Trastuzumab, a targeted therapy for human epidermal growth factor receptor 2 (HER2) positive breast cancer, induces cell cycle arrest and inhibits HER2 expression. Trastuzumab has been shown to improve vascular delivery of subsequent cytotoxic therapies, but the mechanism by which it regulates tumor-associated angiogenesis is not well characterized. Therefore, we developed an integrated, mathematical-experimental approach to systematically investigate the interactions of tumor-growth characteristics—vasculature, immune response, hypoxia, and necrosis—and to evaluate their effects on tumor response to treatment in a murine model of HER2+ breast cancer.

Shocks and rarefactions arise in a two-phase model with logistic growth

Multi-phase or mixture models are often used to describe the dynamics of complex fluids. In this work, we use a general transformation to reduce the two-phase system of one spatial and time variable to a system of a single variable. Then we assess the behavior of solutions for the inviscid two-phase model with logistic growth. The growth rate widely impacts the behavior of the solution, producing either shocks or rarefactions. Increasing growth increases the frequency and spread of these waves, and eliminating growth reduces solutions to continuous traveling waves. This analysis generalizes a class of asymptotic/linear results for gel swelling as well as showing the extraordinary richness in the molding framework.

Particular Solutions for a Two-Phase Model with a Sharp Interface

Abstract: Two-phase models can be used to describe the dynamics of mixed materials and can be applied to many physical and biological phenomena. For example, these types of models have been used to describe the dynamics of cancer, biofilms, cytoplasm, and hydrogels. Frequently the physical domain separates into a region of mixed material immersed in a region of pure fluid solvent. Previous works have found a perturbation solution to capture the front velocity at the initial time of contact between the polymer network and pure solvent, then approximated the solution to the sharp-interface at other points in time. The primary purpose of this work is to use a symmetry transformation to capture an exact solution to this two-phase problem with asharp-interface. This solution is useful for a variety of reasons. First, the exact solution replicates the numeric results, but it also captures the dynamics of the volume profile at the boundary between phases for arbitrary time scales. Also, the solution accounts for dispersion of the network further away from the boundary. Further, our findings suggest that an infinite number of exact solutions of various classes exist for the two-phase system, which may give further insights into the behaviors of the general two-phase model.

Nonclassical symmetries of a class of Burgers’ systems

Abstract: The nonclassical symmetries of a class of Burgers’ systems are considered. This study was initialized by Cherniha and Serov with a restriction on the form of the nonclassical symmetry operator. In this paper we remove this restriction and solve the determining equations to show that (1) a new form of a Burgers’ system exists that admits a nonclassical symmetry and (2) a Burgers’ system exists that is linearizable.