A Closed-Form Outlet Identity and Open-Source Reference Implementation for Controlled Incremental Filtration

Abstract

Controlled Incremental Filtration (CIF) is a microfluidic approach for size-based particle separation: a center channel feeds the sample, and two flanking side channels collect filtrate as a controlled fraction f_gap of the center-channel flow is diverted laterally through gaps; the channel geometry varies along the device to control f_gap, whose value sets the particle size cutoff. CIF has been demonstrated for bead enrichment (Gifford 2014), leukoreduction of platelet-rich plasma (Xia 2016, 2022), leukapheresis (Lezzar 2022), and platelet size sorting (Dinh 2024). CIF design has two key parameters: the size cutoff (set by f_gap) and the volumetric filtrate fraction F. The latter was previously the numerical endpoint of an iterative gap-by-gap recursion, with no closed-form prediction; here we address that gap.

Under pressure equilibration (PE) and the boundary condition Q_s(0) = 0 shared across the CIF lineage, F is given exactly by F = 2·R_c / (R_s + 2·R_c) in outlet geometry alone, equivalent to the textbook current-divider rule for the CIF outlet (Oh et al. 2012). We derive this identity, verify it across five reference devices in three architectural regimes, and release an open-source reference implementation with a 237-validator, 19-stamp reproducibility suite.

To validate, we simulated the Dinh 2024 V1 and V3 outlets in three-dimensional Stokes flow (COMSOL Multiphysics 6.3). The closed-form predicts F_closed = 0.5592 (V1) and 0.7189 (V3), about 10% above the published 0.505 ± 0.033 and 0.649 ± 0.036; adding 3-D physics closes the residual to within 1σ (F_COMSOL = 0.5070 and 0.6281; σ-distances +0.06 and −0.58). The multi-variant ratio ln(1 − F_V3) / ln(1 − F_V1) confirms the agreement: 1.549 (closed-form), 1.489 ± 0.203 (paper), 1.398 (COMSOL). The 10% residual is substantially reduced by 3-D outlet bifurcation, with the asymmetry between V1 and V3 (α_3D = 0.811 vs. 0.660) providing a geometry-dependent signature. Two alternative mechanisms are smaller in magnitude (channel-depth tolerance shifts F by less than 0.05σ; PE holds with at least a 48× safety margin across all five reference devices), and the named subcategories of w_s-specific fabrication offset do not reproduce the observed asymmetry: isotropic PDMS shrinkage leaves F essentially invariant at fixed channel depth (Lee and Lee 2008), and aspect-ratio-dependent shrinkage predicts the opposite V1 > V3 ordering. The implementation is released under an MIT license at https://github.com/Austin-Routt/cif-reference-implementation, lowering the barrier for researchers seeking to apply, extend, audit, or build upon the Gifford-variant CIF design framework.