Thermal wave model of bioheat transfer with modified Riemann–Liouville fractional derivative

In this paper a new fractional thermal wave model of the bioheat transfer (FTWMBT) caused by spatial heating is built using Taylor’s series expansion of modified Riemann–Liouville fractional derivatives. A one-dimensional analytical solution of the FTWMBT in a finite medium is obtained. The FTWMBT in the case (α = 1) interpolates the standard thermal wave model of bioheat transfer and the well-known Pennes’ bioheat equation (τ = 0). Finally, numerical results are presented graphically for various values of different parameters. This study demonstrates that fractional models can provide a unified approach to examine the heat transfer in biological tissue.

[1]  T. Lu,et al.  Non-Fourier analysis of skin biothermomechanics , 2008 .

[2]  Guy Jumarie,et al.  Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions , 2009, Appl. Math. Lett..

[3]  A. Barabasi,et al.  Lung tissue viscoelasticity: a mathematical framework and its molecular basis. , 1994, Journal of applied physiology.

[4]  Lisa X. Xu,et al.  New thermal wave aspects on burn evaluation of skin subjected to instantaneous heating , 1999, IEEE Transactions on Biomedical Engineering.

[5]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[6]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[7]  ON FRACTIONAL RELAXATION , 2003 .

[8]  I. Podlubny.  Fractional differential equations , 1998 .

[9]  K Darvish,et al.  Frequency dependence of complex moduli of brain tissue using a fractional Zener model , 2005, Physics in medicine and biology.

[10]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[11]  Rudolf Hilfer,et al.  Experimental evidence for fractional time evolution in glass forming materials , 2002 .

[12]  Jing Liu,et al.  Preliminary survey on the mechanisms of the wave-like behaviors of heat transfer in living tissues , 2000 .

[13]  Kai-Nan An,et al.  Dynamic mechanical properties of agarose gels modeled by a fractional derivative model. , 2004, Journal of biomechanical engineering.

[14]  Praveen Kumar Gupta,et al.  Solution of fractional bioheat equations by finite difference method and HPM , 2011, Math. Comput. Model..

[15]  Wenchang Tan,et al.  An anomalous subdiffusion model for calcium spark in cardiac myocytes , 2007 .

[16]  Y. Povstenko.  Fractional Cattaneo-Type Equations and Generalized Thermoelasticity , 2011 .

[17]  P. Maass,et al.  Sparsity regularization for parameter identification problems , 2012 .

[18]  Karen Willcox,et al.  Kinetics and kinematics for translational motions in microgravity during parabolic flight. , 2009, Aviation, space, and environmental medicine.

[19]  Ralf Metzler,et al.  Fractional dynamics : recent advances , 2011 .

[20]  Xiaoyun Jiang,et al.  The fractional finite Hankel transform and its applications in fractal space , 2009 .

[21]  Trifce Sandev,et al.  Generalized Langevin equation with a three parameter Mittag-Leffler noise , 2011 .

[22]  Marina V. Shitikova,et al.  Analysis of Dynamic Behaviour of Viscoelastic Rods Whose Rheological Models Contain Fractional Derivatives of Two Different Orders , 2001 .

[23]  Ralf Metzler,et al.  Velocity and displacement correlation functions for fractional generalized Langevin equations , 2012 .

[24]  Mingyu Xu,et al.  The time fractional heat conduction equation in the general orthogonal curvilinear coordinate and the cylindrical coordinate systems , 2010 .

[25]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[26]  B Suki,et al.  Effects of collagenase and elastase on the mechanical properties of lung tissue strips. , 2000, Journal of applied physiology.

[27]  Guy Jumarie,et al.  From Lagrangian mechanics fractal in space to space fractal Schrödinger’s equation via fractional Taylor’s series , 2009 .

[28]  C. Fetecau,et al.  EXACT SOLUTIONS FOR THE FLOW OF A VISCOELASTIC FLUID INDUCED BY A CIRCULAR CYLINDER SUBJECT TO A TIME DEPENDENT SHEAR STRESS , 2010 .

[29]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[30]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[31]  B. Suki,et al.  Dynamic properties of lung parenchyma: mechanical contributions of fiber network and interstitial cells. , 1997, Journal of applied physiology.

[32]  Ralf Metzler,et al.  Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative , 2011, Journal of Physics A: Mathematical and Theoretical.

[33]  Damian Craiem,et al.  Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics , 2010, Physical biology.