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Model development and numerical simulation

Overview

A theoretical model for determination of liver tissue temperature during RFA has been developed with a novel approach to the standard bio-heat equation. The model comprises both a tissue and a blood sub-volume, with separate, coupled differential equations. Simulations have shown the importance of considering the two sub-volumes separately, even when the blood sub-volume is a small fraction of the total volume. Analytical solutions have been obtained for simple cases and related to existing models.

Further refinement of the bio-heat model, based on experimental findings has been done including detailed analysis of the treatment of individual vessels based on their thermal significance. Also refinement of the cell death model based on the cell experiments conducted by MUG and the power distribution model complemented the current model's update. Preliminary work on patient-specific model parameters and multi-scale cellular modelling is ongoing.

The modified bio-heat equation has been solved using the stabilised finite element method on an unstructured tetrahedral grid. The volume mesh was created from a smoothed and reduced version of the segmented surface mesh boundary. In order to provide a more detailed vessel structure near the target ablation zone, we now limit the volume mesh to a predefined region surrounding the ablation zone. This produces a more accurate liver geometry (with less elements) and an improved solution time compared to the models shown during the first review. The simulation tool has been optimised both in terms of performance (solver time) and accuracy in relation to the pre-clinical animal experiments. The numerical speedups include implementation of the parallelisation procedures for the modified bio-heat equations and the 3-state cell death model. We have also simplified the heat generation model from the needle and tailored the blood flow model for the reduced mesh geometry. Improvements in solver time and model accuracy are ongoing.

Details:

Theoretical model of RFA processes

The fundamental basis of our bio-heat model is based on the coupling between tissue and blood temperatures and use of a porous medium model to describe the heat sink effects of small vessels, whilst explicitly modelling those vessels that are known to be thermally significant. Through use of a theoretical analysis of the coupled equations, we have developed a new non-dimensional number that provides a measure of the thermal significance of individual vessels.

One addition to the model has been a more detailed simulation of the power distributed from the RITA probe. To avoid solving the full electrical field (which is computationally expensive), we have developed a novel way to characterise the power distribution as a series of weighted point sources, based on theoretical analysis of the electric field. This has now been implemented and increases the quality of fit to the data considerably, giving an accurate, yet simple, representation of the power delivery (manuscript in preparation). Our cell death model has been further refined and reduced to a three-state model, which has been fitted to the experimental data provided by partners at MUG. This model is the simplest characterisation possible and the first such validated model in the literature, providing a balance between two-state models (simple, yet incapable of mimicking cell recovery) and infinite-state models (accurate, yet computationally impractical).

Two new strands to our modelling work are the incorporation of patient-specific model parameters and development of a multi-scale approach. For the former, we have been able to propose changes dependent upon the cirrhosis level, based on the Child-Pugh classification and data from the literature. As simulations are performed, we will be able to determine the importance of this parameter on the treatment planning. We are also developing a cellular model of the response to heating, by exploring the effects of temperature variations on membrane potential and ionic currents as well as changes in cell volume.

The modelling approach that we have developed is the first known attempt to perform a practical implementation of such a detailed model for ablation in the liver: as we continue to cycle the experimental/mathematical loop we will be able further to refine and to improve our model.

Numerical modelling

We have continued to develop the simulation tool both in terms of performance (solver time) and accuracy in relation to the pre-clinical animal experiments. The numerical speedups include implementation of the parallelisation procedures for the modified bio-heat equations and the 3-state cell death model. Part of the optimisation also includes adapting the code to the specific needs of the pig experiment model and the RFA procedures used.

We have looked at different ways to model the heat generation from the needle and the blood flow in the liver tissue. We no longer explicitly calculate the joule heating from the electric current distribution, instead we define heat generation point sources at certain locations on the needle geometry. These point sources are derived from the joule heating distribution which in turn is calculated from the gradient of the static current distribution around the very thin and long needle type geometry. The source point heat distribution is used in all models and does not require the use of the static current solver for each analysis.

To model the blood flow correctly in the bio-heat equation, we've implemented both a source/sink point and an explicit vessel wall boundary condition. This allows us to either assume the blood enters and leaves the domain at the segmented vessel ends (point sources) or model a porosity on the vessel walls to allow for a more averaged flow distribution. In both cases the boundary conditions are adjusted to obtain the correct bulk liver blood mass flow taken from literature.

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