## 3 Design Optimization
### Performance Metrics & Design Criteria
#### 3.3.7 Definition of Liquid Volumes  
In centrifugally automated bioanalytical protocols, the precision \(\Delta U_{0,i}\) of loaded liquid volumes \(U_{0,i}\) by structures \(\Gamma_{i}\) directly enters mixing ratios of sample and reagents, and thus assay quantitation as well as valving performance in terms of critical spin rates \(\Omega\) (5), mostly through the radial product \(\overline{\tau}\Delta\tau\) representing the spatial distribution \(\Lambda=\Lambda(U_{0})\). Uncertainties \(\Delta U_{0}\) in \(U_{0}\) are generated during introduction of the liquid into \(\Gamma\) through pipetting, and subsequent serial transfer through LUOs and their valves before arriving in a final, end-point detection chamber.  
Figure 6(a) reveals the significant growth of \(\Delta\Omega\) with the spread \(\Delta U_{0}\) (with otherwise default tolerances \(\{\Delta\gamma_{k}\}\), see Appendix A.1); the structure \(\Gamma\) in Figure 6(b) is optimized to keep the standard deviation \(\Delta\Omega\) at bay for (rather large) \(\Delta U_{0}=10\)\(\mu\)L.  
On the outlet side, valves retain a certain fraction of the introduced liquid volume \(U_{0}\). While reproducible losses \(U_{\text{loss}}\) may readily be compensated via the layout, according to (5) and (6), they tend to raise space requirements; their impact can be mitigated through minimizing the following metric:  
\[\overline{U}_{\text{loss}}=\frac{U_{\text{loss}}}{U_{0}}\approx\frac{d_{0} \cdot w_{0}\cdot(R-r_{0})+U_{\text{iso}}+d\cdot w\cdot Z}{U_{0}} \tag{14}\]  
which assumes that the entire liquid filling \(\Gamma\) below its crest point \(R_{\text{crest}}\) is not transferred upon actuation in purely overflow driven CP-DF valving (see Figure 1 for parameter definitions).  
Assuming that deviations from this model mainly result from the pulley effects of the liquid plug of volume \(d\cdot w\cdot Z\) residing in the inbound segment (Figure 1), the metric.  
\[\overline{\Delta U}_{\text{loss}}=\frac{\Delta U_{\text{loss}}}{U_{\text{ loss}}}\approx\frac{d\cdot w\cdot Z}{d_{0}\cdot w_{0}\cdot(R-r_{0})+U_{\text{iso}}+d \cdot w\cdot Z} \tag{15}\]  
accounts for its statistical \(\Delta U_{\text{loss}}\) with respect to the total residual volume \(U_{\text{loss}}\). The value of \(\overline{\Delta U}_{\text{loss}}\) (15) should ideally vanish to minimize the band width \(\Delta\Omega\) (7) and improve the related bioanalytical precision, which is the case for merely overflow-driven liquid  
Figure 7: (a) Spread \(\Delta\Omega\) as a function of the precision of the loaded volume \(\Delta U_{0}\) with default other values \(\{\Delta\gamma_{k}\}\), with the dotted lines representing the values used and obtained for the default parameters (see Appendix A.1). (b) Structure \(\Gamma\) (excerpt) optimized for minimum standard deviation \(\Delta\Omega\) caused by a comparatively poor volume precision \(\Delta U_{0}\).  
transfer to the outer DF chamber. Note that an experimental characterization or a refined, computational fluid dynamic (CFD) simulation of the course of valving is required to obtain a more accurate estimate of \(U_{\text{loss}}\) and \(\Delta U_{\text{loss}}\).