PAMM · Proc. Appl. Math. Mech. 17, 213 – 214 (2017) / DOI 10.1002/pamm.201710076 Heat transfer in multi-phase porous media with application to cancer detection Angela Niedermeyer 1, ∗ , Yousef Heider 1 , Marcus Stoffel 1 , and Bernd Markert 1 1 Templergraben 64, 52062 Aachen Heat transport in biological tissues and its modelling with different approaches, such as Pennes’s bioheat equation and porous media theories, is investigated in this study. Breast cancer detection is envisaged as an application. The suitability of the widely-used Pennes model for breast cancer detection by means of infrared imaging is investigated through experimental and numerical examples. c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction and heat transport models for biological tissues A currently investigated approach for breast cancer detection is the analysis of skin temperature, which can be measurably influenced by additional heat generation in the presence of a cancer. Due to the fact that heat transfer in biological tissues, especially breast tissue, is complex and cancerous tumours are often located far from the surface, the exact determination of parameters, such as location and size, from infrared thermographic (IRT) images is challenging. Therefore, computer aided analysis based on a thermo-mechanical model is envisaged. Pennes’s widely-used single-phase model [1] has been one of the earliest models for energy transport in living tissue and is given as follows: ρ s c ps ∂T s ∂t = ∇· (k s ∇T s )+ q m + c pb ω b (T a − T s ). (1) Herein, ρ, c p , T , t, k, q m , ω are the density, specific heat, temperature, time, thermal conductivity, metabolic heat production and mean blood perfusion rate, respectively. s, a indicate tissue and arterial parameters. Porous media (PM) models are based on a macroscopic description of tissue as a two-phase material with solid phase (ϕ s ) and fluid phase (blood, ϕ b ), statistically distributed over a representative volume element. An energy balance equation for each constituent can be defined (a. o. [2]): (1 − ε)(ρc p ) s ∂ 〈T 〉 s ∂t = ∇· (k a s · ∇〈T 〉 s )+ h bs ( 〈T 〉 b −〈T 〉 s ) + q m (1 − ε) for Ω s , (2) ε (ρc p ) b  ∂ 〈T 〉 b ∂t + v b · ∇〈T 〉 b  = ∇· ( k a b · ∇〈T 〉 b ) + h bs ( 〈T 〉 s −〈T 〉 b ) for Ω b , (3) where ε, 〈T 〉, k b ,h bs , v b are the porosity, local averaged temperature, effective thermal conductivity tensor, interstitial con- vective heat transfer coefficient and blood velocity, respectively. Although it is argued that PM models are very adequate for modelling of biological tissues [3] and more suited than Pennes’s bioheat equation for the modelling of heat transfer in biological tissues (a. o. [2,4]), the latter has been used till today in many research works due to its simplicity. 2 Simplified model of the female breast A simplified model of the female breast is used to study the applicability of Pennes’s bioheat equation (1) to breast cancer detection. Stationary conditions are assumed in this case. The breast is considered a hemisphere consisting of glandular tissue and an outer layer of fatty tissue. The tumour is modelled as a sphere inside the glandular tissue. Physical tissue parameters are taken from literature [5, 6]. The flat surface is set to a constant core body temperature of T cb = 37 ◦ C and the cancerous tumour to a constant temperature of T c = 42 ◦ C. The boundary condition at the skin surface was modelled using a combined heat transfer coefficient accounting for convection, evaporation and radiation (k s ∂T ∂n = −h (T − T ∞ ), h = 13.5 W m 2 K7 ). Figs. 1 a), b) show heat transfer excluding effects of perfusion. In Figs. 1 c), d), it is shown how the isotropic perfusion term in Pennes’s bioheat equation results in a rather even temperature profile, where the cancerous heat source has only a minute effect on the surface temperature distribution. For breast cancer detection requiring exact local temperatures, Pennes’s model might not be the preferred option due to its assumption that blood perfusion is isotropic and proportional to the difference of the temperature at a given location and a uniform arterial temperature. To analyse the applicability to tissues with large vessels, further experiments and simulations were conducted. 3 Experiments and simulation with a blood vessel model IRT images were taken of the surface of a polyoxymethylene rectangular cuboid (0.14 m x 0.16 m x 0.06 m) incorporating flow channels (d f =0.01 m, T f = 38 ◦ C) and a small cylindrical heat source (d c =0.004 m, l c =0.01 m, T c = 43 ◦ C) representing ∗ Corresponding author: e-mail angela.niedermeyer@iam.rwth-aachen.de, phone +49 241 80 90037, fax +49 241 80 92231 c  2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim