## 6 Mammary Tissue Modeling
### Inverse Modeling  
Inverse thermal modeling techniques can be applied to estimate the exact location and size of a breast lesion. Attempts to evaluate the blood perfusion rate of this tumor are also discussed by different authors. The problem of the identification of both thermophysical (thermal conductivity, perfusion coefficient, and metabolic heat generation) and geometrical parameters (location and size) of a tumor lesion was investigated by [PERSON] and [PERSON] [84]. These authors addressed both 2D and 3D problems using an evolutionary algorithm coupled with the multiple reciprocity boundary element method. In the second and more interesting problem, the breast was modeled as one half of the ellipsoid, including a tumor lesion with a similar geometry. Real values of unknown parameters were obtained, but the algorithm implemented by the authors was time-consuming, especially in the case of a 3D problem. [PERSON] and [PERSON] [85] applied the curve fitting method to simultaneously estimate the size, radial location, and angular location (described by two angular coordinates) of a tumor present in a 3D human breast. In particular, a simplified hemispherical model of the breast based on the Pennes bio-heat equation was used to compute the skin surface temperature profile. The suggested procedure did not require solving the governing equation, and the estimation was very fast. The authors reported smaller estimation errors when a tumor of larger size located close to the skin surface was considered. The same inverse procedure was previously tested by the same authors, considering a simplified 2D geometry [86]. More accurate finite element 3D breast models were investigated in the inverse analysis performed by [PERSON] and [PERSON] [87]. In particular, these authors suggested both an undeformed model and a dropped (ptosis) breast model; the former concerns a hemispherical model consisting of three layers (lobule, fat, and skin layers) at the base of which a muscular layer is placed, while the latter is partitioned into three parts, namely a lower hemispherical part, a cylindrical middle segment, and an upper cylindrical wedge region showing a 45\({}^{\circ}\) angle with the chest. In both models, a spherical cancerous lesion is submerged entirely in the lobular region. Simulated steady-state temperature distributions computed through the described models were then used in a customized genetic algorithm to retrieve metabolic heat generation and lesion location. The authors reported a very good accuracy of the estimated parameters, even when a noisy level of 10% is considered. A surrogate 3D breast geometry was developed by [PERSON] et al. [88] using the breast profile curve obtained from IR images. Moreover, an external prosthesis was used to dimension the thoracic wall in order to adjust the total volume of the model. Additionally, a cancerous lesion was accurately inserted in the model using the information provided by an ultrasound examination. Once the bio-heat transfer problem was defined, the authors applied the sequential quadratic programming method to estimate the thermal conductivities of glandular and nodule tissues. The authors claimed that their new geometry allowedthe tumor thermal conductivity to be estimated compared to that defined by other authors in a similar work [89]. A 2D hemispherical model of an axisymmetric tumorous breast was applied by [PERSON] and [PERSON] [90] in an inverse analysis to simultaneously estimate the size, location, and blood perfusion of the lesion. In particular, the considered hemispherical model involves six tissue layers (namely epidermis, papillary dermis, reticular dermis, fat, gland, and muscle layers) and a spherical tumor located at the axis of the hemisphere. It is worth noting that the thickness of the gland and the muscle layers are not uniform in the breast model (as shown in Figure 8b), whereas the others are uniform.