Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

Raghavan, Raghu; Brady, Martin

doi:10.1002/eng2.12108

Cited by 4 publications

(5 citation statements)

References 15 publications

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“…These are quite substantial flow speeds. In fact, a simple geometric calculation, as well as direct measurements [ 24 , 36 ], shows that flow speeds due to the infusions studied in this paper as close as 1 cm from the catheter are less than micron/second. One might expect such flow speeds to seriously affect the fluid and particle distributions, and that any simulation which neglected these would be more substantially in disagreement with data than ours seem to be.…”

Section: Discussionmentioning

confidence: 99%

“…It would be more robust to measure the flux at the start of the infusions for distribution estimates. We have developed and tested these ideas in animal models: the experiments have been reported in [ 36 , 49 , 50 ]; and the theory behind it as well as some of the results have appeared in [ 24 ]. The protocol for the studies reported here did not allow for application of these ideas, as discussed in Section 2.2.2 , nor were the methods sufficiently developed at the time of these trials for implementation.…”

Section: Discussionmentioning

confidence: 99%

“…We therefore developed and implemented a new method to compute solutions to the equations for fluid flow with the boundary valued being a specified flux (related, via K , to the gradient of the pressure projected along a direction transverse to the boundary). This method is fully described in [ 24 ], where the “transverse” direction just mentioned is carefully defined. For isotropic K , this is the direction normal to the boundary.…”

Section: Methodsmentioning

confidence: 99%

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Determinants of Intraparenchymal Infusion Distributions: Modeling and Analyses of Human Glioblastoma Trials

Brady¹,

Raghavan²,

Sampson

2020

Pharmaceutics

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Intra-parenchymal injection and delivery of therapeutic agents have been used in clinical trials for brain cancer and other neurodegenerative diseases. The complexity of transport pathways in tissue makes it difficult to envision therapeutic agent distribution from clinical MR images. Computer-assisted planning has been proposed to mitigate risk for inadequate delivery through quantitative understanding of infusion characteristics. We present results from human studies and simulations of intratumoral infusions of immunotoxins in glioblastoma patients. Gd-DTPA and 124I-labeled human serum albumin (124I-HSA) were co-infused with the therapeutic, and their distributions measured in MRI and PET. Simulations were created by modeling tissue fluid mechanics and physiology and suggested that reduced distribution of tracer molecules within tumor is primarily related to elevated loss rates computed from DCE. PET-tracer on the other hand shows that the larger albumin molecule had longer but heterogeneous residence times within the tumor. We found over two orders of magnitude variation in distribution volumes for the same infusion volumes, with relative error ~20%, allowing understanding of even anomalous infusions. Modeling and measurement revealed that key determinants of flow include infusion-induced expansion and loss through compromised BBB. Opportunities are described to improve computer-assisted CED through iterative feedback between simulations and imaging.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

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Determinants of Intraparenchymal Infusion Distributions: Modeling and Analyses of Human Glioblastoma Trials

Brady¹,

Raghavan²,

Sampson

2020

Pharmaceutics

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“…In this section, we summarize the formulas which serve as the basis for simulating random walks of fixed step length and random times, in contrast to the conventional Gaussian process that simulates the walk in a fixed time with random step lengths. The detailed justification and derivation of the formulas quoted below are available in Reference 8, while applications to Dirichlet problems are discussed in Reference 2, and to Neumann problems in Reference 11. We quote only as much of the formulas as we need to proceed with the discussion of the problem of computing the inverse distribution for generating samples.…”

Section: Hitting Time Distributions: Summary Of Previous Resultsmentioning

confidence: 99%

Inverse functions for Monte Carlo simulations with applications to hitting time distributions

Ben‐David

Raghavan²

2020

Engineering Reports

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Random sampling is a ubiquitous tool in simulations and modeling in a variety of applications. There are efficient algorithms for several known distributions, but in general, one must resort to computing or approximating the inverse of the distributions to generate random samples. In certain physical and biomedical applications with which we have been particularly concerned, it has proven to be more efficient to provide random times for a walk of a fixed length, rather than the conventional random step lengths in a given time step for the walker. For these, the hitting‐time distributions must be sampled, but the problem is that the distributions contain complicated expressions for which no efficient method exists to compute an inverse. In this article, we explore a well‐known probability (the F‐ratio distribution)—whose inverse is efficiently computable—as an alternative to generating look‐up tables and interpolations to obtain the required stochastic time samples. We transform the two‐parameter F‐distribution into a four‐parameter distribution and we match the first four moments of the distribution to the moments of the exact hitting‐time distribution. We find that this simple procedure approximates the hitting‐time distribution well and produces very small errors. Future Monte Carlo simulations in a number of fields of applications may benefit from our method.

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“…(In the Appendix below, we explain only the solution to Dirichlet problems. The Neumann problems require more preparation to explain, and our approach to these is available in Reference .) Particles were started at (25,0,0), about halfway between the source and the outer boundary.…”

Section: Methodsmentioning

confidence: 99%

Hitting time distributions for efficient simulations of drift‐diffusion processes

Raghavan¹

2020

Engineering Reports

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Numerical solutions to partial differential equations in anisotropic, heterogeneous media obtained by their probabilistic representations are useful for a number of purposes, including our own interests in biomedical simulations. These solutions are obtained by a walk with both random and deterministic components hitting a boundary. Hitting time distributions are required to efficiently simulate such processes. The distributions for hitting time T and place XT on a surface for a particle undergoing both diffusion and drift are presented here, generalizing the previous work. Importantly, the distributions directly obtained are, on the whole, not useful for numerical work, being expressed as very poorly and nonuniformly convergent series. Resummed series with rapid convergence for probabilities over a useful range of the hitting time to the boundary are here obtained in different ways. A numerical inverse is constructed to implement the random walk. We have used these hitting time distributions, combined with efficient inverse function construction, to obtain speed‐up of simulations of fluid and particle transport by permitting the steps allowed for a given accuracy to be as large as possible.

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Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

Cited by 4 publications

References 15 publications

Determinants of Intraparenchymal Infusion Distributions: Modeling and Analyses of Human Glioblastoma Trials

Determinants of Intraparenchymal Infusion Distributions: Modeling and Analyses of Human Glioblastoma Trials

Inverse functions for Monte Carlo simulations with applications to hitting time distributions

Hitting time distributions for efficient simulations of drift‐diffusion processes

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