## 6 Mammary Tissue Modeling
### Inverse Modeling
The Pennes bio-heat equation was solved in such a domain to generate both the steady-state and transient surface temperature distributions recordable by IR imaging. In particular, the cooling phase was modeled by imposing a constant-temperature boundary condition. Then, the Levenberg-Marquardt algorithm was iteratively used to estimate the tumor features. The authors performed several numerical tests by varying the depth, radii, and blood perfusion of the tumor. They concluded that the simultaneous estimation of blood perfusion, size, and location of the tumor is possible only when transient temperature data are considered. [PERSON] et al. [18] estimated the location of the geometric center of the three major types of mammary lesions using an anatomical 2D breast model without the need for prior knowledge of the tumor's thermophysical properties. The authors simulated the steady-state temperature distribution of the breast surface by adding random errors to the computed temperatures. Also, using four constraint functions, the search area was defined, and within it, the presence of a circular tumor localized at every 1 mm was hypothesized. For each possible tumor position, an auxiliary temperature profile was computed by setting all the tumor thermal properties equal to one. Then the correlation between the simulated temperature distribution and auxiliary temperature profiles was found. The estimated geometric center of the lesion was defined as that that yielded the highest correlation. The authors reported a maximum error for lesion center estimation of 0.32 cm when an invasive lobular carcinoma is located 5 cm underneath the skin surface. The simulated annealing technique coupled with the dual reciprocity boundary element method was applied by [PERSON] et al. [91] to simultaneously estimate the geometrical (size) and thermophysical (blood perfusion rate and metabolic heat generation) parameters of a breast tumor. The inverse procedure was tested using input temperature data simulated through a bio-heat transfer problem concerning a simplified 2D breast model in which a lesion with different sizes and locations was embedded. The authors underlined that their method does not require the computation of sensitivities or search directions. To retrieve the radius and location of a spherical tumor in a human breast, [PERSON] and [PERSON] [92] applied artificial neural networks in conjunction with the surface temperature distributions computed from a simplified hemispherical finite element breast model. In particular, the studied model considers the breast as a gland with uniform properties. Moreover, to solve the inverse problem, these authors developed a multilayer feed-forward artificial neural network with back propagation learning algorithms using the MATLAB neural network toolbox. This algorithm was trained using the surface temperatures obtained by varying the tumor parameters in the forward problem. The authors reported accuracies of 90% and 95% in the estimation of the position and radius of the tumor, respectively, when no random noise in the temperature data was considered. A combination of artificial neural networks and genetic algorithms was used by [PERSON] and [PERSON] [93] to prove the possibility of estimating the depth, diameter, and heat generation rates of a breast lesion from simulated surface temperature distributions. In detail, the artificial neural network was trained to provide the temperature distributions of the breast surface as a function of the tumor features, while the genetic algorithm was involved in the estimation procedure. A finite element 2D hemispherical model of the breast described in [94] and consisting of a subcutaneous fat layer, a gland core, a non-uniform muscle layer, and the thoracic wall was used to generate the simulated temperature data. The authors reported good estimation accuracy with low random noise levels. [PERSON] et al. [95] applied a trained dynamic neural network model to simultaneously estimate the depth, size, and metabolic heat generation rate of a breast lesion.