# Quantitative Optical Coherence Elastography for Robust Stiffness Assessment
## 4 Robust Stiffness Characterization Based on qOCE Measurement  
The elastic modulus of a material (\(E\)) is defined as the ratio between the stress (\(\sigma\)) and the strain (\(\epsilon\)): \(E=\sigma/\epsilon\). However, direct measurement of \(E\) is challenging because the stress and the strain are defined in the 3D space and have spatial variation. In a conventional indentation test for the measurement of stiffness, a material sample is compressed by an indenter. The reaction force (\(F\)) and indenter displacement (\(h\)) are measured. The elastic modulus is extracted using the \(F\)-\(h\) relationship based on simplified material models. With the assumption that an infinitely thick sample is compressed by a flat cylindrical indentor, the \(F\)-\(h\) relationship can be expressed as Equation (1) where \(R\) indicates the radius of the indentor and \(\
u\) indicates the Poisson's ratio [18]. However, the assumption of an infinitely thick sample is often not realistic. Therefore, the accuracy of stiffness assessment based on Equation (1) largely depends on the boundary conditions. In a modified model, a constant coefficient (\(\kappa\)) was introduced to take the indentor geometry (\(R\)) and sample thickness (\(T\)) into consideration (\(\kappa=\kappa(R/T)\)) (Equation (2)) [19]. However, the spatial variation of mechanical properties is often unknown during in situ tissue characterization:  
\[F=\frac{2 RhE}{1-\
u^{2}} \tag{1}\]  
\[F=\kappa\left(\frac{R}{T}\right)\frac{2 RhE}{1-\
u^{2}} \tag{2}\]  
To better utilize data obtained from qOCE measurement for robust stiffness assessment, we consider the analytical solution shown in Equation (3) for the axial displacement \(d(z)\) within an isotropic, linearly elastic sample indented by a flat cylindrical punch (qOCE probe) [20]. Here, \(h\) indicates the displacement of the indenter, \(z\) indicates the axial (depth) coordinate in the 3D space, Im indicates to take the imaginary part of a complex number and \(i\) is the imaginary unit. Assuming an incompressible sample (\(\
u=0.5\)), we take the ratio between \(d(z)\) (Equation (3)) and \(F\) (Equation (1)), and define the result as \(m_{qOCE}(z)\) that is inversely proportional to the elastic modulus of the material (Equation (4)). Notably, \(M_{R}\) on the right-hand side of Equation (4) has an analytical expression shown in Equation (5). Therefore, the elastic modulus (\(E\)) can be extracted by fitting the linear model shown in Equation (6) where \(m_{qOCE}\) is a vector obtained using qOCE data: \(m_{qOCE}=d(z)/F_{qOCE}\), and vector \(M_{R}=M_{R}(z)\) derives from the above analytical solution (Equation (5)). The extraction of sample elastic modulus using qOCE data is illustrated in Figure 2:  
\[d(z)|_{x=0,y=0}=\frac{h}{\pi(1-v)}\text{Im}\bigg{[}2(1-v)\log(2z+2 iR)-\frac{z }{z+iR}\bigg{]}, \tag{3}\]  
\[m_{qOCE}(z)=\frac{d(z)}{F}=\frac{1}{E}M_{R}(z), \tag{4}\]  
\[M_{R}(z)=\frac{3}{4\pi R}\text{Im}\bigg{[}2(1-v)\log(2z+2 iR)-\frac{z}{z+iR} \bigg{]}, \tag{5}\]\[m_{\text{qOCE}}=\frac{1}{E}M_{R} \tag{6}\]  
The above method (Figure 2) allows more robust assessment of sample stiffness. First, our method quantifies both geometric deformation and reaction force, which is essential for the measurement of elastic modulus (Equations (1) and (3)). In comparison, OCE that only measures sample deformation lacks the capability to quantify material properties. Secondly, a conventional indentation test measures the reaction force and the indenter displacement \(h\) that is significantly affected by the boundary condition of the measurement. Our method, on the other hand, uses local displacement (\(d(z)\)) of the sample rather than the global deformation. It is expected to result in improved robustness because \(d(z)\) in close proximity to the probe tip is largely determined by the probe geometry and the stress established at the contact plane. Other geometric factors, such as the sample thickness, play a much less significant role in determining the local displacement field immediately under the probe tip.