## 2 Materials and Methods
### Evaluation of the Image Data
#### 2.4.4 Elastic Properties  
It is not possible to determine the elastic modulus \(E(\mathbf{x})\) directly from the displacement field or the strain image, so additional information is necessary to get quantitative results. The three-dimensional deformation of the tissue was measured in a two-dimensional plane. Therefore, we also reduced the model to a two-dimensional problem by making assumptions. We used the plane-stress approximation, which assumes that all out of plane stresses are zero [45]. The reconstruction of \(E(\mathbf{x})\) from a displacement field is an inverse problem. In our algorithm, we discretized the input to linear triangular elements. The linearity was only valid for small strains. [PERSON] et al. [46] showed on ex vivo porcine VFs that the transition point from linearity to non-linearity was approximately at a strain value of 0.122. Consequently, we only evaluated measurements with lower strain values.  
The quality of the estimation depends on the magnitude of the displacement field, due to the smaller signal-to-noise ratio [47]. To obtain greater displacements in the registration process, every hundredth frame was used. We solved this by using OpenQSEI, which is an open source FEM solver of elastic problems based on MATLAB [48].  
The computation of \(E(\mathbf{x})\) is an inverse problem in the form of:  
\[\mathbf{u}(\mathbf{x},t)=U(E(\mathbf{x}))+e \tag{10}\]  
in which \(U(E(\mathbf{x}))\) is the forward solution, \(e\) the assumed noise, and \(\mathbf{u}(\mathbf{x},t)\) the measured displacement field. The assumed noise \(e\) was considered Gaussian-distributed and summed up all noise that occurred in the measurement process so far. The solution of Equation (10) can be found by solving:  
\[l=\operatorname*{arg\,min}_{E_{L}\leq E\leq E_{U}}\{\|(L_{e}(\mathbf{u}(\mathbf{x},t) )-U(E(\mathbf{x})))\|^{2}+p_{E}(E)\} \tag{11}\]  
The observation noise covariance matrix is expressed by \(L_{e}^{T}L_{e}=C_{e}^{-1}\); furthermore, \(E_{L}\) and \(E_{U}\) define the lower and upper limit of \(E\). The solution can be plagued by modeling errors, noise, and outlier data [47; 49; 50]. Therefore and because the inverse problem can be ill-posed, the computation needs a regularization term \(p_{E}(E)\). The choice of \(p_{E}(E)\) has a major impact on the result. We decided to use a Tikhonov uninformative regularization, because we expected no sharp features and a non-homogeneous background [47]. The Tikhonov regularization parameter \(\alpha\) was chosen according to the noise level. If \(\alpha\) was large, the regularization tampered with the solution. Consequently, if \(\alpha\) was too small, the noise would destroy the solution. The right choice of \(\alpha\) needs to account for those influences [51] (p. 123).  
The choice of the boundary conditions is another crucial part of the FEM, and it has to be oriented on the physical settings (Figure 1a). In our case, the displacements at the tissue-transducer interface were close to zero (Figure 4). In the FEM, all displacements at this boundary were considered to be zero. The second boundary condition, which was used in the model, was a virtual force \(F_{v}\) compressing the tissue (Figure 4). The force was seen as virtual, because in the reference system of the transducer, and consequently of the image, the tissue was compressed towards the transducer (in the positive y-direction). The real force, applied by the transducer, was compressing the tissue in the negative y-direction. Using those two boundary conditions, OpenQSEI solved the inverse problem (Equation (11)) with an iterative constrained Gauss-Newton optimization algorithm [48].