A point source model to represent heat distribution without calculating the Joule heat during radiofrequency ablation

Abstract

Numerous liver cancer oncologists suggest bridging therapies to limit cancer growth until donors are available. Interventional radiology including radiofrequency ablation (RFA) is one such bridging therapy. This locoregional therapy aims to produce an optimal amount of heat to kill cancer cells, where the heat is produced by a radiofrequency (RF) needle. Less experienced Interventional Radiologists (IRs) require a software-assisted smart solution to predict the optimal heat distribution as both overkilling and untreated cancer cells are problematic treatments. Therefore, two of the big three partial differential equations, 1) heat equation (Pennes, Journal of Applied Physiology, 1948, 1, 93–122) to predict the heat distribution and 2) Laplace equation (Prakash, Open Biomed. Eng. J., 2010, 4, 27–38) for electric potential along with different cell death models (O’Neill et al., Ann. Biomed. Eng., 2011, 39, 570–579) are widely used in the last three decades. However, solving two differential equations and a cell death model is computationally expensive when the number of finite compact coverings of a liver topological structure increases in millions. Since the heat source from the Joule losses Q r = σ|∇V|2 is obtained from Laplace equation σΔV = 0, it is called the Joule heat model. The traditional Joule heat model can be replaced by a point source model to obtain the heat source term. The idea behind this model is to solve σΔV = δ 0 where δ 0 is a Dirac-delta function. Therefore, using the fundamental solution of the Laplace equation (Evans, Partial Differential Equations, 2010) we represent the solution of the Joule heat model using an alternative model called the point source model which is given by the Gaussian distribution. Q r x = ∑ x i ∈ Ω 1 K ∑ i c i e − | x − x i | 2 2 σ 2 where K and c i are obtained by using needle parameters. This model is employed in one of our software solutions called RFA Guardian (Voglreiter et al., Sci. Rep., 2018, 8, 787) which predicted the treatment outcome very well for more than 100 patients.