# Method for Measuring Absolute Optical Properties of Turbid Samples in a Standard Cuvette
## 4 Discussion  
The method presented appears to be feasible in measuring absolute \(\mu_{a}\) and \(\mu_{s}^{\prime}\) in a standard cuvette (\(45\,\mathrm{mm}\times 10\,\mathrm{mm}\times 10\,\mathrm{mm}\)). This is significant given that typical/traditional measurements of \(\mu_{a}\) and \(\mu_{s}^{\prime}\) with diffuse optical methods require large sample volumes (on the order of liters) and careful instrumental calibration. In this case, small samples volumes may be used (on the order of \(1\,\mathrm{mL}\) to \(10\,\mathrm{mL}\)) without the need for calibration of optode coupling (as described in Section 3.1.2). To summarize, we started the development of this measurement method by choosing which data we intended to collect from the cuvette, namely \(\ln|\mathrm{DR}\{\widetilde{T}\}|\) and \(\angle\mathrm{DR}\{\widetilde{T}\}\), and determining how these data vary in respect to the desired recovered properties, namely \(\mu_{a}\) and \(\mu_{s}^{\prime}\) (Figure 3). This leads to the development of a fit for \(\mu_{a}\) and \(\mu_{s}^{\prime}\) and a careful examination of \(\chi^{2}\) space (Section 3.2). From this examination, one major result was discovered, this being that \(\mu_{a}\) is less determined (has a broad local minimum area in \(\chi^{2}\) space) compared to \(\mu_{s}^{\prime}\). This is the first potential limitation of this method since often \(\mu_{a}\) is in-fact the targeted property of interest while \(\mu_{s}^{\prime}\) may be considered a confound. Despite this it appears that this weakness mainly occurs when \(\mu_{s}^{\prime}\) is small (\(<\)\(1\,\mathrm{mm}^{-1}\); Figure 6) telling us that the method has its main strength when the sample is highly scattering. Given that most commercial spectrometers designed for cuvette measurement require a non-scattering sample. We also simulated two types of confounds that may lead to incorrect recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\). First, we investigated how instrumental noise would propagate through the measurement to the recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\) (Section 3.3.1). Here we confirmed what was expected when the \(\chi^{2}\) space was examined, specifically that \(\mu_{a}\) has a higher relative error compared to \(\mu_{s}^{\prime}\) (Figure 8 and Table 1). However, this error becomes comparable when \(\mu_{s}^{\prime}\) is high. For example, with \(\sigma_{|\widetilde{T}|}/|\widetilde{T}|=0.01\) and \(\sigma_{\widetilde{T}}=1.7\,\mathrm{mrad}=0.1^{\circ}\):  
* If \(\mu_{a}=0.005\,\mathrm{mm}^{-1}\) & \(\mu_{s}^{\prime}=0.5\,\mathrm{mm}^{-1}\) then \(\mu_{a}\) has an error of \(20\,\%\) and \(\mu_{s}^{\prime}\) of \(2\,\%\). * If \(\mu_{a}=0.020\,\mathrm{mm}^{-1}\) & \(\mu_{s}^{\prime}=2.0\,\mathrm{mm}^{-1}\) then \(\mu_{a}\) has an error of \(1\,\%\) and \(\mu_{s}^{\prime}\) of \(0.9\,\%\). Therefore, we see that this method really shines when the sample is very diffuse, which is another way of saying highly scattering. Investigation of the effect of incorrectly assumed boundary conditions on the fit results was also done (Section 3.3.2). Many different boundary conditions have been extensively studied and modeled in the past [34, 35, 36, 37, 38], however just because conditions can be modeled does not mean that there is knowledge of them in practice which can be corrected for. Given that a diffusion theory model was used, we varied boundary conditions in terms of \(n_{i}\) and \(n_{o}\). Again, we found that \(\mu_{a}\) is more likely to be incorrectly recovered compared to \(\mu_{s}^{\prime}\). However, further we found that \(n_{i}\) had the largest effect on the recovered \(\mu_{a}\) and \(\mu_{s}^{\prime}\) (Figure 9). This in principle is a short-coming of the method since one could argue that if the method were implemented, \(n_{o}\) could be controlled through instrument design but \(n_{i}\) would be unknown. However, Figure 9 shows that the effect of \(n_{i}\) is suppressed when \(n_{o}\) is large, suggesting a relationship to \(n_{r}\).