500 examples and problems of applied differential equations

This book highlights an unprecedented number of real-life applications of differential equations together with the underlying theory and techniques. The problems and examples presented here touch on key topics in the discipline, including first order (linear and nonlinear) differential equations, second (and higher) order differential equations, first order differential systems, the Runge–Kutta method, and nonlinear boundary value problems. Applications include growth of bacterial colonies, commodity prices, suspension bridges, spreading rumors, modeling the shape of a tsunami, planetary motion, quantum mechanics, circulation of blood in blood vessels, price-demand-supply relations, predator-prey relations, and many more. Upper undergraduate and graduate students in Mathematics, Physics and Engineering will find this volume particularly useful, both for independent study and as supplementary reading. While many problems can be solved at the undergraduate level, a number of challenging real-life applications have also been included as a way to motivate further research in this vast and fascinating field

Multiple Aneurysms Anatomy Challenge 2018 (MATCH)Phase I: Segmentation

Purpose: Advanced morphology analysis and image-based hemodynamic simulations are increasingly used to assess the rupture risk of intracranial aneurysms (IAs). However, the accuracy of those results strongly depends on the quality of the vessel wall segmentation. Methods: To evaluate state-of-the-art segmentation approaches, the Multiple Aneurysms AnaTomy CHallenge (MATCH) was announced. Participants carried out segmentation in three anonymized 3D DSA datasets (left and right anterior, posterior circulation) of a patient harboring five IAs. Qualitative and quantitative inter-group comparisons were carried out with respect to aneurysm volumes and ostia. Further, over- and undersegmentation were evaluated based on highly resolved 2D images. Finally, clinically relevant morphological parameters were calculated. Results: Based on the contributions of 26 participating groups, the findings reveal that no consensus regarding segmentation software or underlying algorithms exists. Qualitative similarity of the aneurysm representations was obtained. However, inter-group differences occurred regarding the luminal surface quality, number of vessel branches considered, aneurysm volumes (up to 20%) and ostium surface areas (up to 30%). Further, a systematic oversegmentation of the 3D surfaces was observed with a difference of approximately 10% to the highly resolved 2D reference image. Particularly, the neck of the ruptured aneurysm was overrepresented by all groups except for one. Finally, morphology parameters (e.g., undulation and non-sphericity) varied up to 25%. Conclusions: MATCH provides an overview of segmentation methodologies for IAs and highlights the variability of surface reconstruction. Further, the study emphasizes the need for careful processing of initial segmentation results for a realistic assessment of clinically relevant morphological parameters

The Computational Fluid Dynamics Rupture Challenge 2013 - Phase II: Variability of Hemodynamic Simulations in Two Intracranial Aneurysms

With the increased availability of computational resources, the past decade has seen a rise in the use of computational fluid dynamics (CFD) for medical applications. There has been an increase in the application of CFD to attempt to predict the rupture of intracranial aneurysms, however, while many hemodynamic parameters can be obtained from these computations, to date, no consistent methodology for the prediction of the rupture has been identified. One particular challenge to CFD is that many factors contribute to its accuracy; the mesh resolution and spatial/temporal discretization can alone contribute to a variation in accuracy. This failure to identify the importance of these factors and identify a methodology for the prediction of ruptures has limited the acceptance of CFD among physicians for rupture prediction. The International CFD Rupture Challenge 2013 seeks to comment on the sensitivity of these various CFD assumptions to predict the rupture by undertaking a comparison of the rupture and blood-flow predictions from a wide range of independent participants utilizing a range of CFD approaches. Twenty-six groups from 15 countries took part in the challenge. Participants were provided with surface models of two intracranial aneurysms and asked to carry out the corresponding hemodynamics simulations, free to choose their own mesh, solver, and temporal discretization. They were requested to submit velocity and pressure predictions along the centerline and on specified planes. The first phase of the challenge, described in a separate paper, was aimed at predicting which of the two aneurysms had previously ruptured and where the rupture site was located. The second phase, described in this paper, aims to assess the variability of the solutions and the sensitivity to the modeling assumptions. Participants were free to choose boundary conditions in the first phase, whereas they were prescribed in the second phase but all other CFD modeling parameters were not prescribed. In order to compare the computational results of one representative group with experimental results, steady-flow measurements using particle image velocimetry (PIV) were carried out in a silicone model of one of the provided aneurysms. Approximately 80% of the participating groups generated similar results. Both velocity and pressure computations were in good agreement with each other for cycle-averaged and peaksystolic predictions. Most apparent outliers (results that stand out of the collective) were observed to have underestimated velocity levels compared to the majority of solutions, but nevertheless identified comparable flow structures. In only two cases, the results deviate by over 35% from the mean solution of all the participants. Results of steady CFD simulations of the representative group and PIV experiments were in good agreement. The study demonstrated that while a range of numerical schemes, mesh resolution, and solvers was used, similar flow predictions were observed in the majority of cases. To further validate the computational results, it is suggested that time-dependent measurements should be conducted in the future. However, it is recognized that this study does not include the biological aspects of the aneurysm, which needs to be considered to be able to more precisely identify the specific rupture risk of an intracranial aneurysm

Variability of computational fluid dynamics solutions for pressure and flow in a giant aneurysm: the ASME 2012 Summer Bioengineering Conference CFD Challenge.

Stimulated by a recent controversy regarding pressure drops predicted in a giant aneurysm with a proximal stenosis, the present study sought to assess variability in the prediction of pressures and flow by a wide variety of research groups. In phase I, lumen geometry, flow rates, and fluid properties were specified, leaving each research group to choose their solver, discretization, and solution strategies. Variability was assessed by having each group interpolate their results onto a standardized mesh and centerline. For phase II, a physical model of the geometry was constructed, from which pressure and flow rates were measured. Groups repeated their simulations using a geometry reconstructed from a micro-computed tomography (CT) scan of the physical model with the measured flow rates and fluid properties. Phase I results from 25 groups demonstrated remarkable consistency in the pressure patterns, with the majority predicting peak systolic pressure drops within 8% of each other. Aneurysm sac flow patterns were more variable with only a few groups reporting peak systolic flow instabilities owing to their use of high temporal resolutions. Variability for phase II was comparable, and the median predicted pressure drops were within a few millimeters of mercury of the measured values but only after accounting for submillimeter errors in the reconstruction of the life-sized flow model from micro-CT. In summary, pressure can be predicted with consistency by CFD across a wide range of solvers and solution strategies, but this may not hold true for specific flow patterns or derived quantities. Future challenges are needed and should focus on hemodynamic quantities thought to be of clinical interest.</p

The Computational Fluid Dynamics Rupture Challenge 2013—Phase II: variability of hemodynamic simulations in two intracranial aneurysms

With the increased availability of computational resources, the past decade has seen a rise in the use of computational fluid dynamics (CFD) for medical applications. There has been an increase in the application of CFD to attempt to predict the rupture of intracranial aneurysms, however, while many hemodynamic parameters can be obtained from these computations, to date, no consistent methodology for the prediction of the rupture has been identified. One particular challenge to CFD is that many factors contribute to its accuracy; the mesh resolution and spatial/temporal discretization can alone contribute to a variation in accuracy. This failure to identify the importance of these factors and identify a methodology for the prediction of ruptures has limited the acceptance of CFD among physicians for rupture prediction. The International CFD Rupture Challenge 2013 seeks to comment on the sensitivity of these various CFD assumptions to predict the rupture by undertaking a comparison of the rupture and blood-flow predictions from a wide range of independent participants utilizing a range of CFD approaches. Twenty-six groups from 15 countries took part in the challenge. Participants were provided with surface models of two intracranial aneurysms and asked to carry out the corresponding hemodynamics simulations, free to choose their own mesh, solver, and temporal discretization. They were requested to submit velocity and pressure predictions along the centerline and on specified planes. The first phase of the challenge, described in a separate paper, was aimed at predicting which of the two aneurysms had previously ruptured and where the rupture site was located. The second phase, described in this paper, aims to assess the variability of the solutions and the sensitivity to the modeling assumptions. Participants were free to choose boundary conditions in the first phase, whereas they were prescribed in the second phase but all other CFD modeling parameters were not prescribed. In order to compare the computational results of one representative group with experimental results, steady-flow measurements using particle image velocimetry (PIV) were carried out in a silicone model of one of the provided aneurysms. Approximately 80% of the participating groups generated similar results. Both velocity and pressure computations were in good agreement with each other for cycle-averaged and peaksystolic predictions. Most apparent "outliers" (results that stand out of the collective) were observed to have underestimated velocity levels compared to the majority of solutions, but nevertheless identified comparable flow structures. In only two cases, the results deviate by over 35% from the mean solution of all the participants. Results of steady CFD simulations of the representative group and PIV experiments were in good agreement. The study demonstrated that while a range of numerical schemes, mesh resolution, and solvers was used, similar flow predictions were observed in the majority of cases. To further validate the computational results, it is suggested that time-dependent measurements should be conducted in the future. However, it is recognized that this study does not include the biological aspects of the aneurysm, which needs to be considered to be able to more precisely identify the specific rupture risk of an intracranial aneurysm

Multiple Aneurysms AnaTomy CHallenge 2018 (MATCH): Phase I: Segmentation

Purpose-Advanced morphology analysis and image-based hemodynamic simulations are increasingly used to assess the rupture risk of intracranial aneurysms (IAs). However, the accuracy of those results strongly depends on the quality of the vessel wall segmentation. Methods-To evaluate state-of-the-art segmentation approaches, the Multiple Aneurysms AnaTomy CHallenge (MATCH) was announced. Participants carried out segmentation in three anonymized 3D DSA datasets (left and right anterior, posterior circulation) of a patient harboring five IAs. Qualitative and quantitative inter-group comparisons were carried out with respect to aneurysm volumes and ostia. Further, over- and undersegmentation were evaluated based on highly resolved 2D images. Finally, clinically relevant morphological parameters were calculated. Results-Based on the contributions of 26 participating groups, the findings reveal that no consensus regarding segmentation software or underlying algorithms exists. Qualitative similarity of the aneurysm representations was obtained. However, inter-group differences occurred regarding the luminal surface quality, number of vessel branches considered, aneurysm volumes (up to 20%) and ostium surface areas (up to 30%). Further, a systematic oversegmentation of the 3D surfaces was observed with a difference of approximately 10% to the highly resolved 2D reference image. Particularly, the neck of the ruptured aneurysm was overrepresented by all groups except for one. Finally, morphology parameters (e.g., undulation and non-sphericity) varied up to 25%. Conclusions-MATCH provides an overview of segmentation methodologies for IAs and highlights the variability of surface reconstruction. Further, the study emphasizes the need for careful processing of initial segmentation results for a realistic assessment of clinically relevant morphological parameters

Variability of computational fluid dynamics solutions for pressure and flow in a giant aneurysm: the ASME 2012 Summer Bioengineering Conference CFD Challenge

Stimulated by a recent controversy regarding pressure drops predicted in a giant aneurysm with a proximal stenosis, the present study sought to assess variability in the prediction of pressures and flow by a wide variety of research groups. In phase I, lumen geometry, flow rates, and fluid properties were specified, leaving each research group to choose their solver, discretization, and solution strategies. Variability was assessed by having each group interpolate their results onto a standardized mesh and centerline. For phase II, a physical model of the geometry was constructed, from which pressure and flow rates were measured. Groups repeated their simulations using a geometry reconstructed from a micro-computed tomography (CT) scan of the physical model with the measured flow rates and fluid properties. Phase I results from 25 groups demonstrated remarkable consistency in the pressure patterns, with the majority predicting peak systolic pressure drops within 8% of each other. Aneurysm sac flow patterns were more variable with only a few groups reporting peak systolic flow instabilities owing to their use of high temporal resolutions. Variability for phase II was comparable, and the median predicted pressure drops were within a few millimeters of mercury of the measured values but only after accounting for submillimeter errors in the reconstruction of the life-sized flow model from micro-CT. In summary, pressure can be predicted with consistency by CFD across a wide range of solvers and solution strategies, but this may not hold true for specific flow patterns or derived quantities. Future challenges are needed and should focus on hemodynamic quantities thought to be of clinical interest