On Finite-Difference Solutions of the Heat Equation in Spherical Coordinates

Thibault, Jules; Bergeron, Simon; Bonin, Hugues W.

doi:10.1080/10407788708913597

Cited by 18 publications

(13 citation statements)

References 5 publications

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“…To ensure comparability to the simulated standard-of-care doses, optimized doses were converted to spherical symmetry before assessing the decision criteria and simulation. Simulations were run in accordance with the methodology presented in [7] , except MATLAB’s pdepe function was replaced by in-house implementation of Crank-Nicolson numerical scheme [20] . Space is discretized in spherical symmetry where each spatial location x i in x indicates a point from the center of the tumor along a radius out to a maximum of 8 cm in intervals of.08 mm.…”

Section: Methodsmentioning

confidence: 99%

Toward Patient-Specific, Biologically Optimized Radiation Therapy Plans for the Treatment of Glioblastoma

et al. 2013

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PurposeTo demonstrate a method of generating patient-specific, biologically-guided radiotherapy dose plans and compare them to the standard-of-care protocol.Methods and MaterialsWe integrated a patient-specific biomathematical model of glioma proliferation, invasion and radiotherapy with a multiobjective evolutionary algorithm for intensity-modulated radiation therapy optimization to construct individualized, biologically-guided plans for 11 glioblastoma patients. Patient-individualized, spherically-symmetric simulations of the standard-of-care and optimized plans were compared in terms of several biological metrics.ResultsThe integrated model generated spatially non-uniform doses that, when compared to the standard-of-care protocol, resulted in a 67% to 93% decrease in equivalent uniform dose to normal tissue, while the therapeutic ratio, the ratio of tumor equivalent uniform dose to that of normal tissue, increased between 50% to 265%. Applying a novel metric of treatment response (Days Gained) to the patient-individualized simulation results predicted that the optimized plans would have a significant impact on delaying tumor progression, with increases from 21% to 105% for 9 of 11 patients.ConclusionsPatient-individualized simulations using the combination of a biomathematical model with an optimization algorithm for radiation therapy generated biologically-guided doses that decreased normal tissue EUD and increased therapeutic ratio with the potential to improve survival outcomes for treatment of glioblastoma.

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Section: Methodsmentioning

confidence: 99%

Toward Patient-Specific, Biologically Optimized Radiation Therapy Plans for the Treatment of Glioblastoma

et al. 2013

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“…Differential equations were solved numerically. Second order partial differential equations, the equations representing heat conduction and oxygen diffusion within the pellets, were solved by using an implicit finite-difference method [31,32]. The solution of first order ordinary differential equations was obtained by using Adams Moulton's method which uses the first five values of the dependent variable to calculate the next value [33].…”

Section: Results and Discussion (16)mentioning

confidence: 99%

Modelling of a Pilot Scale Iron Ore Pellet Induration Furnace

Küçükada¹,

Thibault²,

Hodouin³

et al. 1994

Canadian Metallurgical Quarterly

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In this work, a mathematical model of the pilot scale iron ore pellet induration furnace is presented. The validity of the model has been verified by performing separate experiments of subprocesses: drying, firing and cooling. The experiments were performed in such a way that the phenomena of heat transfer, drying and coke combustion could be studied separately. To do this, the green pellets were first dried, then heated to elevated temperatures at which the coke combustion commences and the pellet strength begins to increase. Finally, the pellets were cooled by passing ambient air across the bed.The pellet induration model is based on the equations describing the phenomena of heat and mass transfer, gas flow, drying and combustion of coke. The experiments and the pellet induration cycle were simulated by solving the differential equations. The pellet bed was divided into differential height elements and the phenomenological equations were solved by numerical methods. This type of model is very useful in many applications such as designing of new plants, evaluating the effectiveness of different process parameters and training the operators and engineers. In addition, it can also serve as a module for on-line control of the moving grate furnace operations. ) pellet or gas humidity (kg m-3 pellet or gas) axial distance or bed height within the pot-grate (m).

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“…No-flux boundary conditions were applied for all species on both the ventral and dorsal boundaries ( Figure 1I). To avoid the singularity that happens on the top points where the elevation angle θ is equal to zero, we used a hollow-shaft approximation method presented by Thibault et al (Thibault et al 1987 in this study. The growth of the domain is achieved by adding finite-difference layers at the marginal region on elevation direction.…”

Section: Measurement Of Source Distributionsmentioning

confidence: 99%

Evaluation of BMP-mediated patterning in a 3D mathematical model of the zebrafish blastula embryo

Wang

Mullins

et al. 2019

Preprint

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Bone Morphogenetic Proteins (BMPs) play an important role in dorsal-ventral (DV) patterning of the early zebrafish embryo. BMP signaling is regulated by a network of extracellular and intracellular factors that impact the range and signaling of BMP ligands. Recent advances in understanding the mechanism of pattern formation support a source-sink mechanism, however it is not clear how the source-sink mechanism shapes patterns in 3D, nor how sensitive the pattern is to biophysical rates and boundary conditions along both the anteroposterior (AP) and DV axes of the embryo. We propose a new three-dimensional growing Partial Differential Equation (PDE)-based model to simulate the BMP patterning process during the blastula stage. This model provides a starting point to elucidate how different mechanisms and components work together in 3D to create and maintain the BMP gradient in the embryo. We also show how the 3D model fits the BMP signaling gradient data at multiple time points along both axes. Furthermore, sensitivity analysis of the model suggests that the spatiotemporal patterns of Chordin and BMP ligand gene expression are dominant drivers of shape in 3D and more work is needed to quantify the spatiotemporal profiles of gene and protein expression to further refine the models.

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On Finite-Difference Solutions of the Heat Equation in Spherical Coordinates

Cited by 18 publications

References 5 publications

Toward Patient-Specific, Biologically Optimized Radiation Therapy Plans for the Treatment of Glioblastoma

Toward Patient-Specific, Biologically Optimized Radiation Therapy Plans for the Treatment of Glioblastoma

Modelling of a Pilot Scale Iron Ore Pellet Induration Furnace

Evaluation of BMP-mediated patterning in a 3D mathematical model of the zebrafish blastula embryo

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