# Quantitative Optical Coherence Elastography for Robust Stiffness Assessment
## 5 Results
### Quantification of Elastic Modulus Using qOCE Data  
We performed qOCE measurement on a cylindrical PDMS phantom made with a 10:1 base-to-agent ratio (\(E\) = 2.6 MPa, 6 mm in thickness and 25 mm in diameter). We translated the probe at a speed of 0.1 mm/s and acquired an OCT signal at a 50 kHz Ascan rate. By translating the probe with different displacements, different reaction forces were obtained (Figure 5a) from the in-line fiber-optic force sensor. The displacement fields (\(d(z)\)) established within the sample were also acquired (Figure 5b). By normalizing \(d(z)\) with the corresponding reaction force \(F\), \(m_{qOCE}(z)\) was obtained for difference indenter displacements (Figure 5c). As indicated by Equations (4)-(6), \(m_{qOCE}(z)\) remains the same for the same sample when the indenter is translated with different displacements. This can be observed in Figure 5c, where results obtained with different probe displacement are similar. Furthermore, with the expression of \(M_{R}(z)\) shown in Equation (5) and the known elastic modulus of the material, we were able to synthesize a \(m_{qOCE}(z)\) curve in Figure 5c (black, solid curve). It can be observed that the analytical solution (black curve) provides satisfactory approximation to the experimental data in a limited depth range (\(z<0.6\) mm). As the depth increases, \(m_{qOCE}(z)\) extracted from qOCE data deviates from the analytical solution more significantly. This is because Doppler based displacement tracking becomes less accurate with signal attenuation as depth. Moreover, the actual displacement at a larger depth becomes more dependent on the measurement geometry. Therefore, the analytical solution (the  
Figure 4: Magnitude optical coherence tomography (OCT) signal (black curve to the right axis), and depth resolved displacements extracted through Doppler analysis (red and blue curves) to the left axis. Figure 3: Force measurement of the qOCE instrument compared with the readings from a commercial force gauge. black curve in Figure 5c) is valid within a limited depth range and it is crucial to choose an appropriate depth range of qOCE data for the assessment of sample stiffness. Figure 5: (**a**) reaction force corresponding to different probe displacement; (**b**) depth resolved sample displacement corresponding to different probe displacement; (**c**) \(m_{qOCE}\)(z) obtained from experimental data and obtained analytically. Using experimentally acquired \(m_{qOCE}(z)\) shown in Figure 5c, and \(M_{R}(z)\) obtained analytically (Equation (5)), we performed linear fitting of Equation (6). The resultant elastic moduli are shown in Figure 6a where the error bars indicate 95% confidential interval of the fitting. The known stiffness of PDMS fabricated with a 10:1 base-to-agent ratio is also shown in Figure 6a as the red line, which is consistent with the results obtained by fitting qOCE data. For comparison, we also extracted the elastic modulus using the apparent stress (\(\sigma_{\rm a}=F/A\) where \(F\) is the force reading from the in-line force sensor and \(A\) is the cross-sectional area of the qOCE probe) and the apparent strain (\(\epsilon_{\rm a}=h/T\) where \(h\) is the known probe displacement): \(\sigma_{\rm a}=E\epsilon_{\rm a}\). By fitting this linear relationship using \(\sigma_{\rm a}\) and \(\epsilon_{\rm a}\) at for different probe displacement (Figure 6b), we obtained an overestimated elastic modulus of 10.4 MPa. In addition, with the \(F\)-\(h\) data (Figure 6c), we also extracted the elastic modulus using Equation (1), which led to an underestimated elastic modulus of 2 MPa. The analysis based on apparent stress and apparent strain results in overestimation of stiffness because of the underlying assumption that the strain is uniformly distributed within the entire thickness of the sample. In fact, the deformation of the sample under compression limited to the volume in close proximity to the probe.