Mathematical modeling of cryogenic processes in biotissues and optimization of the cryosurgery operations N. A. Kudryashov, K. E. Shilnikov National Research Nuclear University “MEPhI” Department of Applied Mathematics VIII-th conference Working group on mathematical models and numerical methods in biomathematics Moscow, October 31 – November 3, 2016

Introduction 130% growth of prostate cancer incidence during the decade Wide range of applicability of the cryosurgery Difficulties of cryosurgery planning on the real tumors of an arbitrary shape Finding of the ways to improve the effectiveness and safety of carrying out the cryosurgery

Aims and problems solved The development of effective numerical algorithms and program complexes for the planning and optimization of the cryosurgery operations Problems solved: Multiscale and macroscopically averaged models for bioheat transfer processes taking into account the peculiarities of cell-scale cryogenic processes Numerical algorithms and computerized tools for three dimensional macroscopic modeling of cryotip-based and cutaneous cryosurgery Numerical algorithms for cryosurgery optimization based on the direct operation modeling

Introduction. Cell scale. Cryogenic processes in biotissues Intracellular crystallization Cell dehydratation Extracellular crystallization Influence of cooling rate on the cell processes: Low cooling rate: extracellular crystallization is dominant (cryoconservation) High cooling rate: intracellular crystallization is dominant (cryusurgery) ►►Dependence of freezing interval on the cooling rate

Cryogenic processes in biotissues Dehydratation Dehydratation as the mechanism of osmotic regulation Mazur (1965) Cell membrane permeability Levin et. al. (1976)

Cryogenic processes in biotissues. Crystallization Intracellular crystallization Toner et al. (1990) Extracellular crystallization Pazhayannur et. al. (1996)

Differential model for cell scale crystallization Fraction of unfrozen water Cell phase distribution: + initial conditions

Dependence of phase distribution on the cooling rate. Numerical results Cooling rate 80K/min Cooling rate 5K/min

Multiscale model for cryogenic bioheat transfer + initial and boundary conditions

Mathematical modeling of the cryosurgery with the spherical cryotip. Governing equations

Mathematical modeling of the cryosurgery with the spherical cryotip. Initial and boundary conditions Conditions on the cryotip’s working surface : i.e. Consistence condition

Mathematical modeling of the cryosurgery with the spherical cryotip Numerical algorithm Explicit finite volume approximation for energy conservation law Semi-implicit Euler-Cauchy scheme for the dynamic system on phase distribution parameters in each grid cell

Numerical results Macroscopic mesh: Microscopic time discetization:

Method of averaged phase distribution Averaging parameters: 1. - characteristic temperature of full freezing 2. ,

Mathematical modeling of the cryosurgery with the spherical cryotip Macroscopic statement + initial and boundary conditions

Comparison of numerical results obtained by both models proposed with the experimental data Time dependence of freezing front propagation depth Experiment: Junkun et. al. (1999)

Three dimensional numerical modeling of the cryosurgery Main assumptions Tumor tissue sizes are much less then the organ’s sizes Computation area allows the construction of indexed mesh Cryotips’ sizes are much less compared to the characteristic computation area sizes Phase averaging parameters are close to apriori estimations

Three dimensional numerical modeling of the cryosurgery Problem statement

Three dimensional numerical modeling of the cryosurgery Numerical algorithm (1) Shilnikov E.V. (2014)

Three dimensional numerical modeling of the cryosurgery Numerical algorithm (2) Flux relaxation method for the improving the scheme stability conditions: Chetverushkin B.N., Shilnikov E.V. (2010)

Three dimensional numerical modeling of the cryosurgery Grid convergence Step, м Healthy tissue destructed, м^3 Tumor tissue undestructed, м^3 0.00167 5.44e-7 3.41e-6 0.00125 5.27e-7 3.28e-6 0.001 5.17e-7 3.35e-6 0.00083 5.21e-7 3.25e-6 0.00071 5.24e-7 3.3e-6 0.00063 5.15e-7 3.31e-6 0.0005 5.2e-7 3.29e-6 grid cells grid cells grid cells

Three dimensional numerical modeling of the cryosurgery Numerical results(1) Geometry of the problem Resulting critical iso-surfaces localization relatively to the tumor tissue Propagation of critical iso-surfaces t=30 s t=90 s t=180 s t=285 s

Three dimensional numerical modeling of the cryosurgery Numerical results(2) t=15 sec t=90 sec t=150 sec t=300 sec t=450 sec t=600 sec

Pareto optimization of the cryosurgery operations Objective variables: Problem statemanent: Objective functions: volume of injured healthy tissue (VH) and volume of undestructed tumor tissue(VT) Quasi-gradient scheme:

Optimization of probe based cryosurgery Compromise optimum. Numerical results. t=15 sec t=90 sec t=150 sec t=300 sec t=450 sec t=600 sec Increasing of tumor tissue destruction effectiveness by 23% 3-fold decrease of healthy tissue injury

Optimization of the cutaneous cryosurgery Suppressing heaters method(1) Boundary conditions: Free propagation of the tissue necrosis front

Optimization of the cutaneous cryosurgery Suppressing heaters method(2) Resulting tissue necrosis front localization Healthy tissue injury decreased by 30% against the increasing of duration of the operation by 25%

Conclusion The multiscale and phase averaged macroscopic models for the cryogenic bioheat transfer are proposed as the tools for predicting the results of cryosurgery with respect to peculiarities of phase change processes in biological solutes. The quasi-gradient method is adopted for the optimization of the cryosurgery operations Numerical experiments on the model problems showed that the developed planning tools can cause a valuable increasing of effectiveness and safety of cryosurgery operations performing

Publications N.A. Kudryashov, K.E. Shilnikov. Numerical modeling and optimization of the cryosurgery operations // Journal of Computational and Applied Mathematics. 2015, No. 290, pp. 259–267. N.A. Kudryashov, K.E. Shilnikov. Numerical simulation of cryosurgeries and optimization of probe placement // Computational Mathematics and Mathematical Physics. Volume 55, Issue 9, pp. 1579–1589. N.A. Kudryashov, K.E. Shilnikov. Numerical Simulation of the Effect of Localization of the Front of Cellular Necrosis during Cutaneous Cryosurgery // Mathematical Models and Computer Simulations. 2016, Vol. 8, No. 6, pp. 680–688. N.A. Kudryashov, K.E. Shilnikov. Nonlinear bioheat transfer models and multi-objective numerical optimization of the cryosurgery operations // Proceedings of 13-th International Conference of Numerical Analysis and Applied Mathematics, Greece, 2015. AIP Conf. Proc. 1738, 230009 (2016); http://dx.doi.org/10.1063/1.4952018

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