The Cahn-Hilliard equation involves fourth-order spatial derivatives. Finite element solutions are not common because primal variational formulations of fourthorder operators are well defined and integrable only if the finite element basis functions are piecewise smooth and globally C 1 -continuous. There are a very limited number of two-dimensional finite elements possessing C 1 -continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two-and three-dimensional geometric flexibility, compact support, and, most importantly, the possibility of C 1 and higher-order continuity. A NURBSbased variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology. There are a very limited number of two-dimensional finite elements possessing C1-continuity applicable to complex geometries, but none in three-dimensions. We propose Isogeometric Analysis as a technology that possesses a unique combination of attributes for complex problems involving higher-order differential operators, namely, higher-order accuracy, robustness, two-and three-dimensional geometric exibility, compact support, and, most importantly, the possibility of C1 and higher-order continuity. A NURBS-based variational formulation for the Cahn-Hilliard equation was tested on two-and three-dimensional problems. We present steady state solutions in two-dimensions and, for the first time, in three-dimensions. To achieve these results an adaptive time-stepping method is introduced. We also present a technique for desensitizing calculations to dependence on mesh refinement. This enables the calculation of topologically correct solutions on coarse meshes, opening the way to practical engineering applications of phase-field methodology.

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We present an introduction to Isogeometric Analysis, a new methodology for solving partial differential equations (PDEs) based on a synthesis of Computer Aided Design (CAD) and Finite Element Analysis (FEA) technologies. A prime motivation for the development of Isogeometric Analysis is to simplify the process of building detailed analysis models for complex engineering systems from CAD representations, a major bottleneck in the overall engineering process. However, we also show that Isogeometric Analysis is a powerful methodology for providing more accurate solutions of PDEs, and we summarize recently obtained mathematical results and describe open problems.

Active surveillance (AS) is a suitable management option for many newly-diagnosed prostate cancer (PCa) cases, which usually exhibit low to intermediate clinical risk. In AS, patients are closely monitored via multiparametric magnetic resonance imaging (mpMRI), prostate-specific antigen tests, and biopsies until these reveal an increase in PCa risk that warrants treatment. Thus, AS can contribute to reduce the current overtreatment of PCa and avoid treatment side-effects that may adversely impact patients’ lives without necessarily improving their longevity. However, current AS protocols rely on assessing PCa according to population-based studies, which complicates the design of personalized monitoring plans and the early detection of tumor progression. To address these issues, we propose to predict PCa growth using patient-specific forecasts based on an mpMRI-informed mechanistic model. Here, we present a preliminary study in a cohort of seven PCa patients who enrolled in AS and had three mpMRI scans over a period of 1.4 to 4.9 years. Our mechanistic model describes PCa growth in terms of tumor cell density dynamics, consisting of mobility and proliferation modeled via a diffusion process and a logistic law, respectively. We segment each patient’s prostate geometry on T2-weighted images and estimate tumor cell density from standard apparent diffusion coefficient (ADC) maps. To facilitate PCa modeling, we co-register each patient’s mpMRI datasets with a biomechanical elastic method that aligns the prostate segmentations across the imaging datasets. The model is initialized with the data from the first mpMRI scan. Model calibration relies on a nonlinear least-squares algorithm that minimizes the mismatch between ADC-measured and model-predicted tumor cell density at the second mpMRI date. For validation, we initialize the patient-specific calibrated model with the second mpMRI dataset and forecast PCa growth at the third mpMRI date, which we compare to the corresponding ADC-estimated tumor cell density. Model calibration resulted in a concordance correlation coefficient (CCC) > 0.99 for both tumor volume and global tumor cell count across the cohort. For each patient, the spatial fit of tumor cell density at the second scan date rendered a median Dice score and CCC of 0.79 and 0.52, respectively. Model validation resulted in CCCs of 0.88 and 0.81 for tumor volume and global tumor cell count across the cohort. Comparing the patient-specific ADC-estimated and model forecasts of tumor cell density at the third scan date, we obtained a median Dice score and CCC of 0.65 and 0.46, respectively. Thus, while improvement and further testing in larger cohorts are still needed, these results do suggest that our computational forecasting approach is a promising technology to predict the spatiotemporal growth of PCa during AS on a patient-specific basis. Citation Format: Guillermo Lorenzo, Jon S. Heiselman, Michael A. Liss, Michael I. Miga, Hector Gomez, Thomas E. Yankeelov, Thomas J. Hughes, Alessandro Reali. Patient-specific forecasting of prostate cancer growth during active surveillance using an imaging-informed mechanistic model [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 5064.

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