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3D numerical simulation of real-time temperature field in a hyperthermia cancer treatment using OcTree meshes

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Abstract

This paper presents a computational strategy for real-time monitoring of hyperthermia treatment of cancer using nanoparticles. In this strategy, a numerical scheme based on the finite volume method is used for the simulation of heat diffusion in a synthetic biological system. This synthetic model is discretized with OcTree meshes, which can perform local refinement, allowing a good representation of complex geometries, besides having advantages on the use of computational resources. For real-time estimation of temperature in the three-dimensional model, the capture of the temperature field on the external surface of the model is simulated. With the data from these simulations, inverse problems are proposed and solved using the L-BFGS method. The damage caused by a temperature increase in healthy and diseased tissues is estimated, thus simulating a criterion for treatment safety. Numerical examples are analyzed with tumors at different depths, illustrating some particularities of this approach.

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Abbreviations

A :

Empirical constant used in damage simulation, \(\hbox {s}^{-1}\)

c :

Specific heat, J/\(\hbox {Kg}^{\circ }\mathrm{C}\)

\(c_b\) :

Specific heat of the blood, J/\(\hbox {Kg}^{\circ }\mathrm{C}\)

E :

Activation energy for the reaction, J/mol

\(F_j\) :

Face of a control volume

h :

Grid spacing, m

\(n_f\) :

Number of faces

\(n_v\) :

Number of volumes

\(Q_m\) :

Metabolic heat generation, W/\(\hbox {m}^3\)

\(Q_r\) :

Internal heat energy emerged from the nanoparticles, W/\(\hbox {m}^3\)

R :

Universal gas constant, J/\(\hbox {mol}^{\circ }\mathrm{C}\)

\({\mathbf {R}}\) :

Real numbers set

\(S_i\) :

Control volume surface

\(V_i\) :

Control volume

\(W_b\) :

Blood perfusion rate, Kg/s/\(\hbox {m}^3\)

\(W_b^*\) :

Weighted average of the nonlinear perfusion, Kg/s/\(\hbox {m}^3\)

\(\kappa \) :

Thermal conductivity, W/\(\hbox {m}^{\circ }\mathrm{C}\)

\(\rho \) :

Density, Kg/\(\hbox {m}^3\)

\(\phi _a\) :

Arterial temperature, \(^{\circ }\mathrm{C}\)

\(\phi \) :

Tissue temperature, \(^{\circ }\mathrm{C}\)

\(\Delta t\) :

Time increment, s

\(\Omega \) :

Tissue damage

\(\chi \) :

Concentration of nanoparticles in a biological environment

\(\mathbf {U_o}\) :

Temperature values observed on the skin surface

\(\mathbf {U_s}\) :

Temperature values obtained with the computational simulation

\({\mathbf {F}}\) :

Residue function

References

  1. Minkowycz W, Sparrow EM, Abraham JP (2012) Nanoparticle heat transfer and fluid flow. CRC Press, Boca Raton

    Google Scholar 

  2. Huang S, Hainfeld J (2013) Intravenous magnetic nanoparticle cancer hyperthermia. Int J Nanomed 2521–2532

  3. Hervault A, Thanh NTK (2014) Magnetic nanoparticle-based therapeutic agents for thermo-chemotherapy treatment of cancer. Nanoscale 6(20):11553–11573. https://doi.org/10.1039/c4nr03482a

    Article  Google Scholar 

  4. Jha S, Sharma PK, Malviya R (2016) Hyperthermia: role and risk factor for cancer treatment. Achiev Life Sci. https://doi.org/10.1016/j.als.2016.11.004

    Article  Google Scholar 

  5. Zienkiewicz OC, Morgan K (1983) Finite element and approximation. John Wiley Edition

  6. Barth TJ (1992) Aspects of unstructured grids and finite-volume solvers for the Euler and Navier–Stokes equations, agard repo Edition

  7. Popinet S (2003) Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J Comput Phys 190(2):572–600

    Article  MathSciNet  Google Scholar 

  8. Haber E, Heldmann S (2007) An octree multigrid method for quasi-static Maxwell’s equations with highly discontinuous coefficients. J Comput Phys 223(2):783–796

    Article  MathSciNet  Google Scholar 

  9. Horesh L, Haber E (2011) A second order discretization of maxwell’s equations in the quasi-static regime on octree grids. SIAM J Sci Comput 33(5):2805–2822

    Article  MathSciNet  Google Scholar 

  10. Valente AG, Di Bartolo L, Mansur WJ (2015) 3D seismic modeling using staggered-grid OcTree mesh. In: SEG Technical Program Expanded Abstracts 2015, pp 3754–3758

  11. Valente A, Loureiro F, Di Bartolo L, Mansur WJ (2018) Computer simulation of hyperthermia with nanoparticles using an OcTree finite volume technique. Int Commun Heat Mass Transf 91:248–255. https://doi.org/10.1016/j.icheatmasstransfer.2017.12.021

    Article  Google Scholar 

  12. Liu DC, Nocedal J (1989) On the limited memory BFGS method for large scale optimization. Math Program 45:503–528

    Article  MathSciNet  Google Scholar 

  13. Fletcher R (1980) Practical methods of optimization—unconstrained op. Wiley, USA

    MATH  Google Scholar 

  14. Haber E (2005) Quasi-newton methods for large-scale electromagnetic inverse problems. Inverse Probl 21(1):305

    Article  MathSciNet  Google Scholar 

  15. de Souza RVM, Peters FC, Mansur WJ, dos Santos Salles H, Bautista KDT, Santos LA (2019) Gii regularization technique for seismic data inversion. J Appl Geophys 160:229–235. https://doi.org/10.1016/j.jappgeo.2018.11.020

    Article  Google Scholar 

  16. Steinberg I, Tamir G, Gannot I (2018) A reconstruction method for the estimation of temperatures of multiple sources applied for nanoparticle-mediated hyperthermia. Molecules. https://doi.org/10.3390/molecules23030670

    Article  Google Scholar 

  17. Chen G, Stang J, Haynes M, Leuthardt E, Moghaddam M (2018) Real-time three-dimensional microwave monitoring of interstitial thermal therapy. IEEE Trans Biomed Eng 65(3):528–538. https://doi.org/10.1109/TBME.2017.2702182

    Article  Google Scholar 

  18. Mello FM, Rodrigues HF, Silveira-Lacerda E, Bakuzis AF, Zufelato N, Capistrano G (2017) Precise determination of the heat delivery during in vivo magnetic nanoparticle hyperthermia with infrared thermography. Phys Med Biol 62(10):4062–4082. https://doi.org/10.1088/1361-6560/aa6793

    Article  Google Scholar 

  19. Harris GR, Herman BA, Myers MR (2011) A comparison of the thermal-dose equation and the intensity-time product, itm, for predicting tissue damage thresholds. Ultrasound Med Biol 1:1. https://doi.org/10.1016/j.ultrasmedbio.2011.01.005

    Article  Google Scholar 

  20. Minkowycz W (2009) Advances in numerical heat transfer, vol III. CRC Press, Boca Raton

    Book  Google Scholar 

  21. Song C W, Lokshina A, Rhee J G, Patten M, Levitt S H (1984) Implication of blood flow in hyperthermic treatment of tumors. IEEE Trans Biomed Eng BME–31(1):9–16. https://doi.org/10.1109/TBME.1984.325364

    Article  Google Scholar 

  22. Lang J, Erdmann B, Seebass M (1999) Impact of nonlinear heat transfer on temperature control in regional hyperthermia. IEEE Trans Biomed Eng 46(9):1129–1138

    Article  Google Scholar 

  23. Loureiro F, Mansur W, Wrobel L, Silva J (2014) The explicit green’s approach with stability enhancement for solving the bioheat transfer equation. Int J Heat Mass Transf 76:393–404

    Article  Google Scholar 

  24. Reis RF, Loureiro FS, Lobosco M (2016) 3D numerical simulations on GPUs of hyperthermia with nanoparticles by a nonlinear bioheat model. J Comput Appl Math 295:35–47

    Article  MathSciNet  Google Scholar 

  25. Rossmann C, Haemmerich D (2014) Review of temperature dependence of thermal properties, dielectric properties, and perfusion of biological tissues at hyperthermic and ablation temperatures. Crit Rev Biomed Eng 42(6)

  26. Valvano J, Cochran J, Diller K (1985) Thermal conductivity and diffusivity of biomaterials measured with self-heated thermistors. Int J Thermophys 6(3):301–311

    Article  Google Scholar 

  27. Diller KR (1992) Modeling of bioheat transfer processes at high and low temperatures. Adv Heat Transf 22:157–357

    Article  Google Scholar 

  28. Welch A (1985) Laser irradiation of tissue. Heat Transf Med Biol 2:135–179

    Article  Google Scholar 

  29. Dormy E, Tarantola A (1995) Numerical simulation of elastic wave propagation using a finite volume method. J Geophys Res Solid Earth 1978–2012 100(B2):2123–2133

    Google Scholar 

  30. Reddy JN (2014) An introduction to nonlinear finite element analysis: with applications to heat transfer, fluid mechanics, and solid mechanics, OUP Oxford

  31. Van der Vorst HA (2003) Iterative Krylov methods for large linear systems, vol 13. Cambridge University Press, Cambridge

    Book  Google Scholar 

  32. Habashy TM, Abubakar A (2004) A general framework for constraint minimization for the inversion of electromagnetic measurements. Progr Electromagn Res 46:265–312

    Article  Google Scholar 

  33. Bell N, Garland M (2012) Cusp: generic parallel algorithms for sparse matrix and graph computations, version 0.3.0 http://cusp-library.googlecode.com

  34. Johannsen M, Gneveckow U, Eckelt L, Feussner A, Waldöfner N, Scholz R, Deger S, Wust P, Loening S, Jordan A (2005) Clinical hyperthermia of prostate cancer using magnetic nanoparticles: presentation of a new interstitial technique. Int J Hypertherm 21(7):637–647

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge the research support from CAPES (Brazilian Federal Agency for Support and Evaluation of Graduate Education), from CNPq (Brazilian Council for Scientific and Technological Development) with the Grant Nos.: 423794/2016-7, 306.933/2014-4, 423114/2018-2, 306191/2018-0 and 151474/2018-4, and from FAPERJ (Research Foundation of the State of Rio de Janeiro) with the Grant Nos.: 203.021/2017, 203.234/2016, and 210.210/2016. We also thank Petrobras for its support. IN MEMORIAM: Franciane \({\mathrm{Peters}}^\dagger \) (May 22, 1985–July 15, 2020). The authors dedicate this paper to the extraordinary colleague and friend Franciane Peters, co-author of this work. In a very precocious way, dear “Fran” left and left us her example of commitment, competence, and especially how human we should be.

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Authors and Affiliations

  1. COPPE/Department of Civil Engineering/LAMEMO, Federal University of Rio de Janeiro, Av. Athos da Silveira Ramos, 149, Rio de Janeiro, Brazil

    André Valente, Franciane Conceição Peters, Raphael Vieira Menezes de Souza & Webe João Mansur

  2. Federal Institute of Rio de Janeiro, Av. República do Paraguai, 120, Duque de Caxias, Brazil

    André Valente

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  1. André Valente
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Correspondence to André Valente.

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Valente, A., Peters, F.C., de Souza, R.V.M. et al. 3D numerical simulation of real-time temperature field in a hyperthermia cancer treatment using OcTree meshes. J Braz. Soc. Mech. Sci. Eng. 43, 15 (2021). https://doi.org/10.1007/s40430-020-02760-1

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  • DOI: https://doi.org/10.1007/s40430-020-02760-1

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