## 4 Discussion  
We examine the quantitative impact of analyzing experimental DCS measurements made at short SDS using Equation (2), for different phantoms (with varying optical transport and flow properties) and use MC simulations to directly compare theoretical calculations across range of SDS. Simulations showed expected behavior, where long SDS DCS data well matched and short SDS data being poorly matched for direct, forward comparisons to calculations from Equation (2) Figure 2. However, DCS data obtained at short SDS were well fit using theory at the cost of errors in extracted flow coefficients Figures 3 and 4. Relative changes in perfusion could still be quantified exactly using theory, even for SDS \(<\) 0.1 cm Figures 5 and 6. as long as the optical properties of the media considered were not changing. MC simulations were shown to be accurate for calibrating and correcting theoretically retrieved coefficients, given the SDS and optical properties of the medium Figures 7 and 8. Equation (2) could also directly be used to examine changes in flow in phantoms but required that flow dynamics include both Brownian and Random flow components Figure 9. As shown in Figure 10a,b, extracted both the diffusive flow and random flow coefficients extracted from theoretical analyses of experimental data (at SDS of 0.15 cm) increased with experimentally increasing flow rates. However, the Brownian flow component \(D_{B}\) changed proportionally with flow rate only when the flow channel was submerged deeper below the surface (estimated to be more than 500 \(\upmu\)m here). While, for shallower channel depths the flow speed \(V\) linearly changed with experimentally controlled flow speed. Relative changes in extracted flow coefficients in phantom experiments when flow was actively pumped showed that the ratios of theoretically extracted flow speeds were almost exactly as expected experimentally, but only when the channel depth was shallowest Figure 10d. These results taken together with relative flow changes extracted in MC simulations for semi-infinite media Figure 8, indicate that theoretical analysis of DCS measurements using Equation (2) is robust and can be used to quantitatively assess changes of relative flow within turbid media, even at short SDS as long as the optical properties and SDS do not change. These conclusions are in accordance with previous findings regarding the accuracy absolute flow rates vs. relative flow at longer SDS [13; 16; 53]. Our results support findings from recent studies, where Equation (2) was successfully used to examine relative changes in DCS flow coefficients from both simulations and in vivo studies in animal models, at short SDS [27; 28; 30; 31]. However, our findings also indicate that quantification of relative perfusion changes reported by these studies would be estimated more accurately, if the relative changes in perfusion at the short SDS channels were computed with MC-derived correction factors with corresponding changes in media optical properties they measured. We were also able to predict experimental measurements with MC simulations across all phantoms used Figures 11 and 12. These data showed excellent fits in all phantoms (\(\chi^{2}<\) 0.03). Given that inputs to tissue models and experiments were tightly coupled and constrained, these data indicate good fidelity of the experimental phantom system used here. Although MC simulations for active flow phantoms matched changes in relative flow rates used experimentally, the absolute value of flow rates (listed in Table 2) were nearly 3.7 times lower than those expected--i.e., the average flow speed in the channel experimentally (given a tube of radius 0.08 cm, and flow rate \(F_{1}=1.56\) mL/h) would be \(V_{1}=2.1\times 10^{-2}\) cm/s but the MC inputs for flow speed of in layer 2 was \(V_{1}=5.7\times 10^{-3}\) cm/s (data in Figure 12b).