Reconstruction of the Stress-free Hyperelastic Parameters of Breast Tissue: Machine-Learning Based Inverse Problem Technique


  • Xi Feng Western University
  • Abbas Samani


Breast Tissue, Stress-free State, Hyperelastic Parameters, Inverse Problem, Neural Network



A key to simulate the breast deformation using the finite element method (FEM) is the accurate mechanical properties of its tissue. The breast tissue is known to undergo large deformation under loading pertaining to medical intervention. As such, its properties are best described by hyperelastic parameters. Breast tissues exhibit significant deformation even under sole gravity loading experienced by the tissue in the preloading phase of traditional mechanical testing techniques. These techniques ignore this initial stressed state, leading to inaccurate estimates of the tissue hyperelastic parameters, hence impacting estimated breast deformation. In the context of computer assisted medical diagnosis and intervention such inaccuracies may translate into inaccurate diagnosis or ineffective intervention. To address this issue, a robust method is necessary to estimate the tissue stress-free hyperelastic parameters using their counterpart obtained under gravity loading.

  1. Methods

We propose a machine-learning based inverse-problem solution to convert hyperelastic parameters of the breast obtained from conventional mechanical testing [1,2] to their stress-free counterparts. In this study we investigated this conversion for the Yeoh and 1st order Ogden models. For this purpose, each hyperelastic parameter reported in the literature is scaled down incrementally down to 50% of its value. To construct a data space of the stress-free parameters, various combinations of the scaled down parameters are formed after checking the Drucker Stability condition, leading to over 800 points. For each point in this space, a uniaxial test is simulated using FEM where the gravity preloading is included, to obtain simulated stress-strain data. This data is fitted in accordance with the hyperelastic model to estimate the corresponding hyper-elastic parameters under gravity preloading conditions. This will lead to two data spaces of hyperelastic parameters, one in the stress-free and the other in the gravity preloading states. To map the latter space to the former, we construct a neural network (NN) that contains three layers with 80 hidden neurons in each layer. Once this NN is trained, it can be used to convert any breast tissue hypereleastic parameter set obtained in traditional mechanical testing to its stress-free state counterpart.

  • Results

We calculated the distance between predicted and true unloaded hyperelastic parameter points, and used the r2 parameter to measure the accuracy of results obtained from the NN. For the two independent models: Yeoh and 1st Ogden, the best predicted accuracy was obtained at 0.91 and 0.86, respectively.

  1. Conclusion

The proposed method is capable of predicting the stress-free hyperelastic of the breast tissue using their loaded counterpart with high accuracy. While only two hyperelastic models were investigated, the method can be adapted with other models and to other types of tissues (e.g brain and liver).


  1. Samani A, Plewes D. A method to measure the hyperelastic parameters of ex vivo breast tissue samples. Phys Med Biol. 2004 Sep 21;49(18):4395-405.
  2. Dempsey SCH, O'Hagan JJ, Samani A. Measurement of the hyperelastic properties of 72 normal homogeneous and heterogeneous ex vivo breast tissue samples. J Mech Behav Biomed Mater. 2021 Dec;124:104794.




How to Cite

X. Feng and A. Samani, “Reconstruction of the Stress-free Hyperelastic Parameters of Breast Tissue: Machine-Learning Based Inverse Problem Technique”, CMBES Proc., vol. 45, May 2023.