1. 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

2. 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

3. 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

4. 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

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

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

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

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

9. Multiscale model for cryogenic bioheat transfer
+ initial and boundary conditions

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

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

12. 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

13. Numerical results
Macroscopic mesh:
Microscopic time discetization:

14. Method of averaged phase distribution
Averaging parameters:
characteristic temperature of full freezing ,

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

16. 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)

17. 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

18. Three dimensional numerical modeling of the cryosurgery
Problem statement

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

20. 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)

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

22. 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

23. 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

24. 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:

25. 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

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

27. 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%

28. 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

29. 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, AIP Conf. Proc. 1738, (2016);

30. Thank you for your attention