Sensitivity Analysis of Temperature Field and Parameter Identification in Burned and Healthy Skin Tissue

  • E. MajchrzakEmail author
  • M. Paruch
  • M. Dziewoński
  • S. Freus
  • K. Freus
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 49)


In the chapter, problems connected with the numerical modeling of bioheat transfer processes are presented. In particular the non-homogeneous system of a burn wound and healthy tissue is considered. The heat exchange between sub-domains and environment is described by a system of partial differential equations (the Pennes equations) supplemented by adequate boundary conditions. The first goal of the research is the estimation of the changes of temperature fields due to perturbations in thermal parameters using the direct method of sensitivity analysis. Both the basic problem and additional ones concerning the sensitivity with respect to selected parameters are solved using the boundary element method. The second goal is the problem of burn wound shape identification. The additional information necessary to solve such a task results from the knowledge of temperature distribution on the external surface of skin tissue. At the stage of solving the inverse problem, a gradient method has been used. In the final part of the chapter the results of computations are shown.


Bioheat transfer Sensitivity analysis Boundary element method Burn wounds Parameter identification Gradient method 



The article and research are financed within the project N R13 0124 10 sponsored by Polish National Centre for Research and Development.


  1. 1.
    Romero Mendez, R., Jimenez-Lozano, J.N., Sen, M., Gonzalez, F.J.: Analytical solution of a Pennes equation for burn-depth determination from infrared thermographs. Math. Med. Biol. 27, 21–38 (2010). doi: 10.1093/imammb/dqp010 Google Scholar
  2. 2.
    Srinivas, S.M., de Boer, J.F., et al.: Determination of burn depth by polarization-sensitive optical coherence tomography, J. Biomed. Opt. 9(1), 207–212 (2004)Google Scholar
  3. 3.
    Riordan, C.L., et al.: Noncontact laser doppler imaging in burn depth analysis of the extremities. J. Burn Care Rehabil. 177–186 (2003)Google Scholar
  4. 4.
    Rumiński, J., Kaczmarek, M., Renkielska, A., Nowakowski, A.: Thermal parametric imaging in the evaluation of skin burn depth. IEEE Trans. Biomed. Eng. 54(2), 303–312 (2007)CrossRefGoogle Scholar
  5. 5.
    Pennes, H.H.: Analysis of tissue and arterial blood temperatures in the resting human forearm. J. Appl. Physiol. 1, 93–122 (1948)Google Scholar
  6. 6.
    Majchrzak, E., Dziatkiewicz, G., Paruch, M.: The modelling of heating a tissue subjected to external electromagnetic field. Acta Bioeng. Biomech. 10(2), 29–37 (2008)Google Scholar
  7. 7.
    Majchrzak, E., Paruch, M.: Identification of electromagnetic field parameters assuring the cancer destruction during hyperthermia treatment. Inverse Probl. Sci. Eng. 19(1), 45–58 (2011)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kleiber, M.: Parameter Sensitivity. Wiley, Chichester (1997)zbMATHGoogle Scholar
  9. 9.
    Dems, K., Rousselet, B.: Sensitivity analysis for transient heat conduction in a solid body. Struct. Optim. 17, 36–45 (1999)Google Scholar
  10. 10.
    Jasiński, M.: Sensitivity analysis of transient bioheat transfer with perfusion rate dependent on tissue injury. Comput. Assist. Mech. Eng. Sci. 16, 267–277 (2009)MathSciNetGoogle Scholar
  11. 11.
    Jasiński, M.: Investigation of tissue thermal damage process with application of direct sensitivity method. Mol. Cell. Biomech 10(3), 201–232 (2013)Google Scholar
  12. 12.
    Brebbia, C.A., Dominguez, J.: Boundary Elements, An Introductory Course, CMP, McGraw-Hill Book Company, London (1992)Google Scholar
  13. 13.
    Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques. Springer, Berlin (1984)CrossRefzbMATHGoogle Scholar
  14. 14.
    Nowak, A.J.: Chapter 3: Solving linear heat conduction problems by the multiple reciprocity boundary element method. In: Wrobel, L.C., Brebbia, C.A. (eds.) Boundary Element Methods In Heat Transfer, pp. 63–132. Computational Mechanics Publications, Southampton, Boston (1992)Google Scholar
  15. 15.
    Paruch, M., Majchrzak, E.: Identification of tumor region parameters using evolutionary algorithm and multiple reciprocity boundary element method. Eng. Appl. Artif. Intell. 20, 647–655 (2007)CrossRefGoogle Scholar
  16. 16.
    Kurpisz, K., Nowak, A.J.: Inverse Thermal Problems, pp. 259–298. Computational Mechanics Publications, Southampton-Boston (1995)zbMATHGoogle Scholar
  17. 17.
    Burczyński, T.: Sensitivity analysis, optimization and inverse problems. In: Beskos, D., Maier, G. (eds.) Boundary Element Advances in Solid Mechanics, pp. 245–307. Springer, New York (2003)CrossRefGoogle Scholar
  18. 18.
    Tortorelli, D.A., Zixian, W.: A systematic approach to shape sensitivity analysis. Int. J. Solids Struct. 1181–1212 (1993)Google Scholar
  19. 19.
    Sokołowski, J., Zolesio, J.-P.: Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Series in Computational Mathematics, Springer (1992)Google Scholar
  20. 20.
    Mochnacki, B., Majchrzak, E.: Application of the shape sensitivity analysis in numerical modelling of solidification process. In: THERMEC’2006, Pts 1–5 Book Series: Materials Science Forum, vol. 539–543, pp. 2524–2529 (2006)Google Scholar
  21. 21.
    Ciesielski, M., Dziewonski, M., Freus, S.:Scanning method of temperature distribution of human body by device registering encircling images. In: Desing and Computation of Modern Engineering Materials. Advanced Structured Materials, vol. 54, pp. 97–106 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • E. Majchrzak
    • 1
    Email author
  • M. Paruch
    • 1
  • M. Dziewoński
    • 1
  • S. Freus
    • 2
  • K. Freus
    • 2
  1. 1.Institute of Computational Mechanics and Engineering, Silesian University of TechnologyGliwicePoland
  2. 2.Institute of Mathematics, Czestochowa University of TechnologyCzęstochowaPoland

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