## 3 Results
To that end, Equation (3) shows how the surface area-to-total volume ratio 'S' of a fiber mesh scaffold is related to its porosity and fiber diameter (see Supplementary Materials for derivation):  
\[S_{Fiber}=\frac{4(1-\varepsilon)}{D_{Fiber}} \tag{3}\]  
where D\({}_{Fiber}\) is the fiber diameter and \(\varepsilon\) is the scaffold porosity. Similarly, Equation (4) shows how the surface area-to-total volume ratio 'S' of a foam scaffold is related to its porosity and salt grain diameter (see Supplementary Materials for derivation):  
\[S_{Foam}=\frac{6\varepsilon}{D_{SalfGnin}} \tag{4}\]  
where D\({}_{SalfGnin}\) is the diameter of the salt grains used for leaching, and \(\varepsilon\) is the scaffold porosity. Therefore, for a known scaffold porosity and diameter combination, one can relate these geometric parameters to the surface area-to-total volume ratio of the scaffold (and thus, to the results of this manuscript). It is also important to note that while S\({}_{Fiber}\) goes down with an increasing porosity \(\varepsilon\), S\({}_{Foam}\) displays the opposite behavior, as can be seen from the Equations (3) and (4). Interestingly, the survival distances in the Y and Z directions (i.e., perpendicular to the flow) go down with the increasing flow rate (see Figure 4). The reason behind this is likely due to the particles becoming entrained by the carrier fluid. Consequently, they move less in the Y and Z directions per time step, relative to their displacement in the X. As a result, there is a smaller probability of collisions with the walls in the former directions. However, since both the Y and Z survival distances are an order of magnitude smaller than those in the X direction, the total survival distance is dominated by the trend displayed by the latter. Figure 3: Survival distance in the stream-wise direction as a function of perfusion flow rate and scaffold architecture. Data is plotted for the limiting case of the fully reactive Lagrangian Scalar Tracking (rLST) particles. Another quantity of interest is the survival time of the O\({}_{2}\) molecules in the scaffolds. Figure 5 plots this quantity as a function of the flow rate and the specific surface area of the scaffolds for the limiting case of fully reactive rLST particles. Conversely to the survival distance in Figure 3, the survival time in Figure 5 decreases with the flow rate, though the effect of the specific surface area remains the same, and the survival time varies inversely-proportionally to it. Combining the results from both of the figures, it becomes apparent that fully reactive molecules get carried to a farther distance by a higher flow rate. However, they take a shorter time to get consumed by the cells on the surface of the scaffolds. This is especially true for the scaffold geometries with a higher exposed surface area. Figure 4: Survival distances in the Y & Z directions, as a function of the scaffold geometry and perfusion flow rate. Data is plotted for the limiting case of the fully reactive rLST particles. Both scaffolds types are chosen to have a similar specific surface area. Figure 5: Survival time in the stream-wise direction as a function of perfusion flow rate and scaffold architecture. Data is plotted for the limiting case of the fully reactive rLST particles. Ultimately, it is of interest to measure the effective first order reaction rate constant \(\mathrm{k_{eff}}\), with which the O\({}_{2}\) molecules get consumed in the scaffolds, after accounting for the mass transfer limitations. To that end, Figure 6 is a plot of \(\mathrm{k_{eff}}\) as a function of the flow rate and specific surface area of the scaffold for the limiting case of an infinitely fast surface consumption of O\({}_{2}\). It reaffirms the previously observed trends, which show that the O\({}_{2}\) is consumed faster in the scaffold structures with the higher surface area-to-solid volume ratios.