Topic I — A Microfluidic Biosensor

channel flow · analyte transport · Langmuir binding on a sensor patch · tier: standard

Motivation

A point-of-care sensor detects a pathogen or biomarker when target molecules carried by the flow bind to receptors on a small sensor patch in a microchannel. How fast the signal rises — and therefore the detection time and limit — depends on whether binding is limited by the reaction or by the transport of molecules to the surface, which depletes a boundary layer above the patch. A model answers design questions directly: flow rate, channel height, patch length.

Related fields: microfluidic point-of-care biosensors, biochips, interface engineering. Course weeks: 11 (Navier–Stokes/Stokes, mixed spaces), 12 (advection– diffusion with reactions, SUPG).

Model

2D channel of height H (or axisymmetric/3D as a stretch), sensor patch of length L_s on the bottom wall. Stokes or Navier–Stokes flow (low Reynolds number), analyte concentration c in the liquid, bound surface concentration B on the patch: \partial_tc+\mathbf u\cdot\nabla c=D\Delta c,\qquad \partial_tB=k_{\rm on}c\,(b_m-B)-k_{\rm off}B\ \text{ on the patch},\qquad -D\partial_nc=-\partial_tB\ \text{ on the patch}, c=c_0 at the inlet, natural outflow, no flux on the other walls. The surface ODE lives on the boundary: a field on the patch facets (a submesh or a boundary-only DG0 space) or one unknown per patch dof.

Design outputs: B(t) averaged over the patch; the time to reach 50 % of the equilibrium signal against the flow rate; the Damköhler number at which transport starts to dominate.

Suggested parameters

H = 10–100 µm, L_s = 10–500 µm, mean velocity 0.1–10 mm/s, D=10^{-11} m²/s (protein), k_{\rm on}=10^6 M⁻¹s⁻¹, k_{\rm off}=10^{-3} s⁻¹, b_m=10^{-11} mol/m², c_0 = 1 pM – 1 nM. Péclet numbers \mathrm{Pe}_H=\bar uH/D up to 10⁵ — SUPG and a graded mesh at the patch.

Required tasks

  1. Flow field (check against the parabolic profile).
  2. Transport without binding (steady); then with fast irreversible capture (perfect sink, c=0 on the patch): the transport-limited flux.
  3. The full coupled problem in time; B(t) for several flow rates and c_0.
  4. A map of the regimes in the Damköhler number k_{\rm on}b_mH/D (or its patch-length version) and the Péclet number.

Verification specific to this topic

  • Reaction-limited limit (Da ≪ 1, fast transport, c\approx c_0 at the surface): B(t)=\frac{k_{\rm on}c_0b_m}{k_{\rm on}c_0+k_{\rm off}}\big(1-e^{-(k_{\rm on}c_0+k_{\rm off})t}\big) — exact; your model must approach it as k_{\rm on}\to0.
  • Transport-limited flux (perfect sink): for a long thin depletion layer the flux follows the Lévêque scaling, \propto(\text{shear rate})^{1/3} — check the exponent by varying the flow rate (Squires, Messinger & Manalis 2008).
  • Mass balance: molecules captured = inflow − outflow − change in the channel, at every time.
  • MMS for the advection–diffusion with the Robin-type surface coupling.

Stretch goals

  1. Two analytes competing for the same receptors.
  2. A 3D channel with a finite-width patch.
  3. Electrokinetic or acoustic enhancement modelled as a body force in the flow.

Milestones

M1 flow · M2 perfect-sink transport and the 1/3 exponent · M3 reaction-limited limit · M4 coupled time-dependent model · M5 regime map.

Pitfalls

  • The concentration boundary layer at the patch is thin at high Péclet numbers: under-resolving it overestimates the flux. Refine until the captured amount stops changing.
  • Units: k_{\rm on} in M⁻¹s⁻¹ and b_m in mol/m² — convert once and print the dimensionless groups.
  • Oscillations from the abrupt patch edges (the flux is singular there): SUPG and refinement; report patch averages, not edge values.

References

  • T. M. Squires, R. J. Messinger, S. R. Manalis, “Making it stick: convection, reaction and diffusion in surface-based biosensors”, Nat. Biotechnol. 26 (2008) 417–426.
  • D. G. Myszka et al., “Extending the range of rate constants available from BIACORE: interpreting mass transport-influenced binding data”, Biophys. J. 75 (1998) 583–594.