Convection Diffusion Reaction Model Research Articles

Spatiotemporal modeling of radiopharmaceutical transport in solid tumors: Application to 177Lu-PSMA therapy of prostate cancer

Background and objective177Lu-labeled prostate-specific membrane antigen (PSMA) radiopharmaceutical therapy (RPT) represents a pivotal advancement in addressing prostate cancer. However, existing therapies, while promising, remain incompletely understood and optimized. Computational models offer potential insights into RPTs, aiding in clinical drug delivery enhancement. In this study, we investigate the impact of various physiological parameters on the delivery of 177Lu-PSMA-617 RPT using the convection-diffusion-reaction (CDR) model. MethodsOur investigation encompasses tumor geometry and surrounding tissue, characterized by well-defined boundaries and initial conditions. Utilizing the finite element method, we solve governing equations across a range of parameters: dissociation constant KD (1, 0.1, 0.01 [nM]), internalization rate (0.01–0.0001 [min−1]), diverse tumor shapes, and variable necrotic zone sizes. This model can provide an accurate analysis of radiopharmaceutical delivery from the injection site to the tumor cell, including drug transport in the vascular, interstitial, and intracellular spaces, and considering important parameters (e.g., drug extravasation from microvessels or to lymphatic vessels, the extracellular matrix, receptors, and intracellular space). ResultsOur findings reveal significant enhancements in tumor-absorbed doses as KD decreases. This outcome can be attributed to the higher affinity of radiopharmaceuticals for PSMA receptors as KD diminishes, facilitating a more efficient binding and retention of the therapeutic agent within the tumor microenvironment. Additionally, tumor-absorbed doses for KD ∼ 1 [nM] show an upward trend with higher internalization rates. This observation can be rationalized by considering that a greater internalization rate would result in a higher proportion of radiopharmaceuticals being taken up by tumor cells after binding to receptors on the cell surface. Notably, tumor shape and necrotic zone size exhibit limited influence on tumor absorbed dose. ConclusionsThe present study employs the CDR model to explore the role of physiological parameters in shaping 177Lu-PSMA-617 RPT delivery. These findings provide insights for improving prostate cancer therapy by understanding radiopharmaceutical transport dynamics. This computational approach contributes to advancing our understanding of radiopharmaceutical delivery mechanisms and has implications for enhancing treatment efficacy.

Support vector regression for the temperature-stimulated drug release

This paper presents a new coupled nonlinear convection–diffusion–reaction model for the temperature-stimulated drug release and a numerical algorithm based on a machine learning technique. By incorporating the fractional derivatives into the model, we take advantage of anomalous diffusion in biological tissues. In the proposed technique a least-squares support vector regression is developed for the numerical simulation of the problem. The Bernstein polynomials are used as the kernel function and some training points are considered for the residual system of the coupled problem in an inverse process. Some numerical experiments are carried out to support the method and its spectral convergence.

Exploring a Convection–Diffusion–Reaction Model of the Propagation of Forest Fires: Computation of Risk Maps for Heterogeneous Environments

The propagation of a forest fire can be described by a convection–diffusion–reaction problem in two spatial dimensions, where the unknowns are the local temperature and the portion of fuel consumed as functions of spatial position and time. This model can be solved numerically in an efficient way by a linearly implicit–explicit (IMEX) method to discretize the convection and nonlinear diffusion terms combined with a Strang-type operator splitting to handle the reaction term. This method is applied to several variants of the model with variable, nonlinear diffusion functions, where it turns out that increasing diffusivity (with respect to a given base case) significantly enlarges the portion of fuel burnt within a given time while choosing an equivalent constant diffusivity or a degenerate one produces comparable results for that quantity. In addition, the effect of spatial heterogeneity as described by a variable topography is studied. The variability of topography influences the local velocity and direction of wind. It is demonstrated how this variability affects the direction and speed of propagation of the wildfire and the location and size of the area of fuel consumed. The possibility to solve the base model efficiently is utilized for the computation of so-called risk maps. Here the risk associated with a given position in a sub-area of the computational domain is quantified by the rapidity of consumption of a given amount of fuel by a fire starting in that position. As a result, we obtain that, in comparison with the planar case and under the same wind conditions, the model predicts a higher risk for those areas where both the variability of topography (as expressed by the gradient of its height function) and the wind velocity are influential. In general, numerical simulations show that in all cases the risk map with for a non-planar topography includes areas with a reduced risk as well as such with an enhanced risk as compared to the planar case.

Drug delivery mathematical modeling for pressure controlled bioreactor

We analyze a mathematical model for drug delivery monitoring subject to flow-through bioreactor. We use a convection–diffusion–reaction model combined with PID control. Enzyme-containing bioreactor converts a prodrug into an active drug. This approach connects two aspects of drug delivery, mechanical pumps and prodrugs. Drug delivery monitoring is performed by adjusting the prodrug inflow pressure. We model several dynamic treatment regimes. A combined algorithm of treatment regimes can be used for personalized treatment. A control-aided system allows us to reduce the impact of pumping fluctuations, diffusion slowdown, and drug production rate reduction.

A robust numerical scheme for the simulation of nonlinear convection–diffusion–reaction equation

In this article, the polynomial based differential quadrature method is used to develop a numerical scheme for solving convection–diffusion–reaction model with a general reaction term. The numerical scheme is found applicable to a wide set of problems including diffusion, convection–diffusion and diffusion–reaction. The validation of the method is performed through simulations of Newell-Whitehead, Burger-Fisher, and Burger-Huxley equations. The proposed numerical method is found better than many numerical methods presented in the literature. Vector form of numerical scheme is presented to implement algorithm efficiently. The compatibility, error and computational complexity of the numerical scheme are studied in detail.

Personalized image-based tumor growth prediction in a convection-diffusion-reaction model.

Inter-individual heterogeneity of tumors leads to non-effectiveness of unique therapy plans. This issue has caused a growing interest in the field of personalized medicine and its application in tumor growth evaluation. Accordingly, in this paper, a framework of personalized medicine is presented for growth prediction of brain glioma tumors. A convection-diffusion-reaction model is used as the patient-specific tumor growth model which is associated with multimodal magnetic resonance images (MRIs). Two parameters of intracellular area fraction (ICAF) and metabolic rate have been used to incorporate the physiological data obtained from medical images into the model. The framework is tested on the data of two cases of glioma tumors to document the approach; parameter estimation is made using particle swarm optimization (PSO) and genetic algorithm (GA) and the model is evaluated by comparing the predicted tumors with the observed tumors in terms of root mean square error of the ICAF maps (IRMSE), relative area difference (RAD) and Dice's coefficient (DC). Results show the differences of IRMSE, RAD and DC in 4.1 ∓ 1.15%, 0.099 ∓ 0.041 and 85.5 ∓ 7.5%, respectively. Survival times are estimated by assuming the tumor radius of 35mm as the fatal burden. Results confirm that less-diffusive tumors lead to higher survival times. The represented framework makes it possible to personally predict the growth behavior of glioma tumors only based on patients' routine MRIs and provides a basis for modeling the personalized therapy and walking in the path of personalized medicine.

Evaluation of o-xylene and other volatile organic compounds removal using a xylene-acclimated biotrickling filter

In this study, performance evaluation for the gas-phase o-xylene removal using a xylene-acclimated biotrickling filter (BTF) was conducted. Substrate interactions during aerobic biodegradation of three poorly soluble compounds, both individually and in paired mixtures (namely, o-xylene and ethyl acetate, o-xylene and dichloromethane, which are common solvents used by pharmaceutical industry), were also investigated. Experimental results indicate that a maximum elimination capacity of 99.3 g·m−3·h−1 (70% removal) was obtained at an o-xylene loading rate of 143.0 g·m−3·h−1, while the top packing layer (one-third height of the three packing layers) only contributed about 13% to the total elimination capacity. Kinetic constants for o-xylene biodegradation and the pattern of o-xylene removal performance along the height of the BTF were obtained through the modified Michaelis–Menten kinetics and convection–diffusion reaction model, respectively. A reduction of removal efficiency in o-xylene (83.2–74.5% removal at a loading rate of 40.3 g·m−3·h−1 for the total volatile organic compound (VOC) loading rate of 79 g·m−3·h−1) in the presence of ethyl acetate (100% removal) was observed, while enhanced o-xylene removal efficiency (71.6–78.6% removal at a loading rate of 45.1 g·m−3·h−1 for the total VOC loading rate of 90 g·m−3·h−1) was achieved in the presence of dichloromethane (35.6% removal). This work shows that a BTF with xylene-acclimated microbial consortia has the ability to remove several poorly soluble compounds, which would advance the knowledge on the treatment of pharmaceutical VOC emissions.

Coupled particulate and continuum model for nanoparticle targeted delivery

Prediction of nanoparticle (NP) distribution in a vasculature involves transport phenomena at various scales and is crucial for the evaluation of NP delivery efficiency. A combined particulate and continuum model is developed to model NP transport and delivery processes. In the particulate model ligand–receptor binding kinetics is coupled with Brownian dynamics to study NP binding on a microscale. An analytical formula is derived to link molecular level binding parameters to particulate level adhesion and detachment rates. The obtained NP adhesion rates are then coupled with a convection–diffusion–reaction model to study NP transport and delivery at macroscale. The binding results of the continuum model agree well with those from the particulate model. The effects of shear rate, particle size and vascular geometry on NP adhesion are investigated. Attachment rates predicted by the analytical formula also agree reasonably well with the experimental data reported in literature. The developed coupled model that links ligand–receptor binding dynamics to NP adhesion rate along with macroscale transport and delivery processes may serve as a faster evaluation and prediction tool to determine NP distribution in complex vascular networks.

Adaptive finite element simulation of stack pollutant emissions over complex terrains

A three-dimensional finite element model for the pollutant dispersion is presented. In these environmental processes over a complex terrain, a mesh generator capable of adapting itself to the topographic characteristics is essential. The first stage of the model consists on the construction of an adaptive tetrahedral mesh of a rectangular region bounded in its lower part by the terrain and in its upper part by a horizontal plane. Once the mesh is constructed, an adaptive local refinement of tetrahedra is used in order to capture the plume rise. Wind measurements are used to compute an interpolated wind field, that is modified by using a mass-consistent model and perturbing its vertical component to introduce the plume rise effect. Then, we use an Eulerian convection–diffusion–reaction model to simulate the pollutant dispersion. In this work, the transport of pollutants is considered and dry deposition is formulated as a boundary condition. The discretization of the stack geometry allows to define the emissions as boundary conditions. The proposed model uses an adaptive finite element space discretization, a Crank-Nicolson time scheme, and a splitting operator. This approach has been applied in La Palma island. Finally, numerical results and conclusions are presented.

Development of a convection–diffusion–reaction model for solving Maxwell's equations in frequency domain

SUMMARYWe propose in this study a numerically accurate and computationally efficient convection–diffusion–reaction finite difference scheme to discretize the full‐vector and semi‐vector optical waveguide equations. The scheme formulated in a grid stencil of five nodal points for solving the three‐dimensional waveguide equations employs the locally analytic solution. In this three‐dimensional study, calculations were carried out for the investigation of wave propagation in diffused channel, rectangular and rib types of optical waveguide. Copyright © 2011 John Wiley & Sons, Ltd.

A generalized 1-dimensional particle transport method for convection diffusion reaction model

This paper is devoted to the development of numerical method to deal with convection diffusion problem with reaction term (CDR) and convection diffusion dominated problem (stiff chemical reaction). The technique is based on the unifying Eulerian–Lagrangian schemes (particle transport method) under the framework of operator splitting method. In the computational domain, the particle set is assigned to solve the convection reaction subproblem along the characteristic curves which are formed by convective velocity. At each time step, convection, diffusion and reaction terms are solved separately by assuming that each phenomenon occurs separately in a sequential fashion. Moreover, adaptivities and projection techniques are used to add particles in the regions of high gradients, discontinuities and transfer a solution from particle set onto grid point respectively. The numerical results show that the particle transport method has improved the solutions of CDR problems. Nevertheless, the method is time consuming when compared with other classical techniques e.g., method of lines.

Linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in homogeneous autocatalytic reactors

Abstract In this paper, we perform linear stability analysis of high- and low-dimensional models for describing mixing-limited pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors. We consider three different models of varying dimensionality—the 3D convection-diffusion-reaction (CDR) model is the high dimensional one, and the Liapunov–Schmidt reduction based spatially averaged two-dimensional CDR model and its regularized form are the two low-dimensional ones. For each of these three models, steady state bifurcation diagrams that show the presence of multiple steady states were obtained and the stability of these multiple steady states to transverse perturbations was analyzed using linear stability analysis. Parametric analysis of the steady state bifurcation diagrams shows that for sufficiently large values of transverse Peclet number p, mixing-limited patterns may emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Comparison of the bifurcation diagrams and the stability boundaries of the two low-dimensional models with that of the 3D CDR model reveals that the regularized form of the low-dimensional model has higher accuracy and a larger region of validity than the averaged form and is therefore recommended over the latter.

Dynamic Simulation of Mixing-Limited Pattern Formation in Homogeneous Autocatalytic Reactions

Interaction between transport and reaction generates a variety of complex spatio-temporal patterns in chemical reactors. These patterned states, which are typically initiated by autocatalytic effects and sustained by differences in diffusion/local mixing rates, often cause undesired effects in the reactor. In this work, we analyze the dynamic evolution of mixing-limited spatial pattern formation in fast, homogeneous autocatalytic reactions occurring in isothermal tubular reactors using two-dimensional (2-D) convection-diffusion-reaction (CDR) models that are obtained through rigorous spatial averaging of the three-dimensional (3-D) CDR model using Liapunov-Schmidt technique of bifurcation theory. We use the spatially-averaged 2-D CDR model (and its "regularized" form) to perform steady-state bifurcation analysis that captures the region of multiple solutions, and we analyze the stability of these multiple steady states to transverse perturbations using linear stability analysis. Parametric analyses of the steady-state bifurcation diagrams and stability boundaries show that when transverse mixing is significantly slower than the rate of autocatalytic reaction, mixing-limited patterns emerge from the unstable middle branch that connects the ignition and extinction points of an S-shaped bifurcation curve. Our dynamic simulations show the emergence of three different types of spatial patterns namely, Band, Anti-phase and Target, depending on the nature of transverse perturbation. The temporal evolution of these patterns consists of rapid intensification of the concentration-segregation process (especially when transverse mixing is much slower than reaction) followed by slow diffusion-mediated return to symmetry that occurs at time scales much larger than the reactor residence time. Our parametric analysis of the dynamics reveals that while larger Péclet numbers (both axial and transverse) increase the stability and decay time of the patterned states, larger Damköhler numbers lead to faster ignition resulting in the opposite effect.

An effective explicit pressure gradient scheme implemented in the two-level non-staggered grids for incompressible Navier–Stokes equations

In this paper, an improved two-level method is presented for effectively solving the incompressible Navier–Stokes equations. This proposed method solves a smaller system of nonlinear Navier–Stokes equations on the coarse mesh and needs to solve the Oseen-type linearized equations of motion only once on the fine mesh level. Within the proposed two-level framework, a prolongation operator, which is required to linearize the convective terms at the fine mesh level using the convergent Navier–Stokes solutions computed at the coarse mesh level, is rigorously derived to increase the prediction accuracy. This indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic. Solution convergence can therefore be accelerated. For the sake of numerical accuracy, momentum equations are discretized by employing the general solution for the two-dimensional convection–diffusion–reaction model equation. The convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms. The converged solution is, thus, very high in accuracy as well as in yielding a quadratic spatial rate of convergence. For the sake of programming simplicity and computational efficiency, pressure gradient terms are rigorously discretized within the explicit framework in the non-staggered grid system. The proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme, which accommodates the dispersion-relation-preserving property, have been both rigorously justified from the predicted Navier–Stokes solutions.

Analytical study of effects of finger-growth velocity on reaction characteristics of reactive miscible viscous fingering by using a convection–diffusion–reaction model

Reactive miscible viscous fingering occurs when a reactive and miscible less-viscous liquid displaces a more-viscous liquid. Effects of bulk finger-growth velocity on reaction characteristics of miscible viscous fingering with a chemical reaction were studied analytically by using a convection–diffusion–reaction model. The model assumes the existence of a distinct interface between both liquids, assumes the existence of a two-dimensional, steady, stagnated flow field in the less-viscous liquid, and assumes an infinite chemical reaction rate. The model was then used to determine the reaction characteristics, such as the location of the reaction surface and the profile of the product, as functions of the velocity and initial reactant concentrations. The results reveal that the effects of the velocity on the reaction characteristics can be divided into low-, intermediate-, and high-velocity regions. In the low-velocity region, the reaction characteristics strongly depend on the reactant concentrations. In the intermediate-velocity region, the dependence of the reaction characteristics on the reactant concentrations decreases with increasing velocity. In the high-velocity region, the reaction characteristics are nearly independent of the reactant concentrations. Experiments confirm the existence of these three velocity regions predicted by the model.

First-principles modeling, scaling laws and design of structured photocatalytic oxidation reactors for air purification

First-principles, predictive engineering models provide a sound theoretical basis for quantifying the inherent light energy utilization capabilities and performance limitations of candidate commercial photocatalytic oxidation reactor configurations. In particular, these models provide insight into the similarities and differences between photoreactors based on structured honeycombed monoliths, and those based on reticulated foams or other random catalyst supports.For honeycombed monoliths, a deterministic first-principles radiation field model provides the channel wall light intensity profile down the length of a single channel in the monolith. A three-dimensional developing flow convection–diffusion reaction model employing this radiation field submodel predicts the velocity and concentration fields. The model shows that light intensity gradients in a monolith of typical dimensions are severe, that only a fraction of the monolith can be effectively photo-activated, and as a consequence process performance is largely controlled by light distribution. For a given light source and photocatalyst combination, reactor performance scales according to the aspect ratio of the channeled monolith, the Reynolds number, and the Dahmköhler number.For randomly structured monoliths, the radiation field must be determined by probablistic methods. Monte Carlo simulations show that the radiation field in such random porous structure scales according to the pore size distribution and the void fraction, and the photocatalyst film thickness. Reactor performance scales by the radiation field, the Peclet number, the Stanton number, and the Dahmköhler number. The complex interrelationship between the random structure of the monolith and the resulting radiation field and mass transfer behavior makes scaling of these reactor types particularly difficult.