MiroFlow: Microfluidic Flow & Bio-Particle Analysis

• MiroFlow is a suite of experimental techniques using coherent laser Doppler imaging and microfluidic designs to quantify microflow velocities and bio-particle dynamics.
• It employs heterodyne digital holography and hyperbolic channel configurations to achieve high spatial resolution and sensitivity in mapping microscale flows.
• Applications include studying DNA coil-stretch transitions, actin filament dynamics, and protein aggregate mechanics under controlled microenvironments.

MiroFlow denotes a suite of experimental techniques and device designs for quantifying microscale fluid flows and the mechanical dynamics of suspended particles via advanced optical and microfluidic strategies. In the laser Doppler imaging implementation, MiroFlow deploys wide-field, frequency-tunable heterodyne digital holography to produce quantitative, spatially resolved maps of microflow velocities using the Doppler signature of tracer particles (Gross et al., 2013). In microfluidic mechanical characterisation, MiroFlow refers to an optimised hyperbolic channel combined with real-time particle tracking, enabling measurements of the dynamical response of bio-particles—such as DNA, actin, and protein aggregates—under precisely controlled straining flow (Liu et al., 2020). Both implementations target high sensitivity and spatial resolution without mechanical scanning, facilitating the study of complex microenvironments in life and physical sciences.

1. Physical and Optical Principles

The MiroFlow technique for microflow velocimetry is predicated on coherent laser illumination of tracer particle suspensions, where each scattering event introduces a Doppler frequency shift in the scattered optical field, directly encoding the instantaneous velocity vector of the particle. The frequency shift arises from the momentum transfer $q = k_s - k_i$ between incident ($k_i$) and scattered ($k_s$) photon wavevectors, resulting in a phase modulation of the scattered field proportional to $q \cdot v(t)$. Consequently, the optical frequency is locally shifted by $\omega_D(t) = q \cdot v(t)$. Heterodyne detection, realized by mixing the scattered field with a frequency-shifted reference (local oscillator), transforms Doppler frequency information into pixel-resolved interferometric beat signals. By systematically tuning the LO shift, MiroFlow recovers the Doppler spectrum $S(x, y; \Delta\omega)$ across the entire detector field, providing direct access to the spectral distribution of particle velocities in the flow (Gross et al., 2013).

In microfluidic implementations, MiroFlow employs 3D planar contraction–expansion channels designed for idealized extensional flows—namely, geometries approximating hyperbolic streamlines (walls obeying $xy = \text{constant}$). The design delivers spatial profiles that generate constant, homogeneous strain-rate regions, critical for evaluating conformation dynamics and mechanical properties of single biomolecules and aggregates (Liu et al., 2020).

2. Mathematical Framework and Analytical Characterisation

For laser Doppler imaging, the Doppler angular-frequency shift for a moving particle is given by

$\omega_D = (k_s - k_i) \cdot v,$

where $k_i$ and $k_s$ are the incident and scattered wave-vectors in air, with magnitude $2\pi / \lambda_0$. In the configuration where the incidence angle to the detector plane is $\alpha$, and the detection is perpendicular to the flow plane, the mean Doppler shift is

$$\langle\omega_D\rangle = -\frac{2\pi}{\lambda_0} \sin \alpha \langle v \rangle.$$

The mean frequency shift per pixel, $\langle\Delta f\rangle$, is computed as the first-order moment of the measured Doppler spectrum:

$$\langle\Delta f\rangle = \frac{1}{2\pi} \cdot \frac{\sum_{\Delta\omega} \Delta\omega S(\Delta\omega)}{\sum S(\Delta\omega)}.$$

This yields pixel-resolved directional velocity maps.

For microfluidic straining flows, the governing equations are the incompressible Navier–Stokes, reduced in the low-Reynolds regime to Stokes flow, with the strain rate along the channel centreline expressed as

$$\dot\varepsilon(x) = \frac{\partial u_x}{\partial x}\Big|_{y=0,\,z=H/2},$$

with $u_x$ computed numerically via finite differences.

A parametric target velocity profile $\tilde u(\tilde x)$ is defined along the channel, constructed with plateau, linear, and quadratic transition regions. Optimisation minimizes the integral deviation between simulated and target velocity:

$$J = \int_{-\tilde l_h}^{\tilde l_h} [u_{\text{num}}(x) - u_{\text{target}}(x)]^2\,dx.$$

For mechanical characterisation, the Weissenberg number is defined as

$Wi = \lambda\,\dot\varepsilon,$

where $\lambda$ is the relaxation time of the object, and the Hencky strain is

$$\varepsilon_H = \int \dot\varepsilon\,dt \approx \ln(CR)\quad (\text{for } AR \ll 1).$$

3. Experimental Apparatus and Data Acquisition

Laser Doppler Imaging Configuration

The system employs a single-mode CW diode laser ($\lambda_0 = 658\,\text{nm}$, $P = 80$\,mW), split into object (illumination) and reference (LO) arms via a prism. Two AOMs in the LO path enable a tunable frequency shift $\Delta\omega$, facilitating spectral scanning. Off-axis holography (using a $10$\,mm lens in the LO arm to create a $1^\circ$ tilt) supports lensless Fourier holography at the camera—a $1280 \times 1024$ PixelFly CCD (pixel size $6.7\,\mu$m; frame rate $4$\,Hz; exposure $\tau_E=125$\,ms) at $50$\,cm working distance. For each LO detuning, $n=8$ phase-stepped images are acquired for demodulation and Fourier reconstruction (Gross et al., 2013).

Microfluidic Device and Tracking

Channels are fabricated in PDMS by soft lithography, with measured parameters $H=100\pm1\,\mu$m, $w_c=100\pm5\,\mu$m, $w_u=800\pm10\,\mu$m, $\tilde l_v=1$, $\tilde l_h=1.5$. The flow is controlled by a syringe pump ($Q=3.1-5.5$\,nL/s yields $U_u\approx 100\,\mu$m/s), and upstream three-inlet configurations focus sample particles near the centreline. Particle-tracking velocimetry (PTV) is deployed with $1\,\mu$m fluorescent beads (imaged at $50-125$\,Hz). Simultaneous imaging is performed on a Zeiss Observer A1 inverted microscope with a 63$\,\times$ oil immersion objective and Hamamatsu ORCA-Flash 4.0LT camera. Motorised stage tracking synchronizes image acquisition and particle position, maintaining sub-field-of-view drift ($<20\,\mu$m) and negligible motion blur ($<1\,\mu$m exposure at $50$\,ms) (Liu et al., 2020).

Parameter	Laser Doppler Imaging (Gross et al., 2013)	Microfluidics/Tracking (Liu et al., 2020)
Spatial resolution	$6.7\,\mu$m pixel (diffraction-limited); sub-$\mu$m with optics	$1-3\,\mu$m (dependent on optics & tracking)
Velocity range	$\pm 100\,\mu$m/s (at $\alpha=45^\circ$)	$10\,\mu$m/s to $1$\,mm/s (by $Q$)
Strain rate	N/A (measures flow velocities)	$0.3 - 5$\,s$^{-1}$ (strain region)
Field of view	$8.6\,\text{mm} \times 6.8\,\text{mm}$	$\sim 200\,\mu$m, stage tracked

4. Signal Processing, Performance, and Quantitative Limits

The recorded interferograms are processed by phase-shifting demodulation algorithms to extract the complex field $H_O(x, y; \Delta\omega)$. Inverse Fourier transformations reconstruct $E_O$ in the spatial domain, from which the spectral power $S(x, y; \Delta\omega) = A |E_O|^2$ is computed. Shot-noise is quantified from object-free regions, and the signal-to-noise ratio is mapped as $$S_\text{dB}(x, y; \Delta\omega) = 10 \log_{10} \left[S / N\right]$$.

Full-range LO detuning provides access to the local Doppler spectra. The zeroth-order spectral moment yields maps of scatterer concentration, while the first moment produces directional velocity maps.

Demonstrated quantitative limits include: spatial resolution of $6.7\,\mu$m (diffraction-limited), field-of-view $8.6 \times 6.8$\,mm$^2$, velocity sensitivity down to $10\,\mu$m/s, with $>30$\,dB dynamic range in dark regions, and frequency resolution as low as $2$\,mHz (Brownian motion typically imposes broader spectral widths in low-velocity regimes) (Gross et al., 2013).

5. Device Optimisation and Flow Homogeneity

For microfluidic implementations, the device geometry is parameterised with NURBS, and an iterative numerical optimisation minimizes the deviation between computed and target centreline velocity profiles under low-Reynolds number Stokes flow. Typical values: contraction ratio $CR=8$, aspect ratio $AR=8$, strain region length $\tilde l_h=1.5$ (giving homogeneous plateau $L_h=\tilde l_h w_u$), and transition zone length $\tilde l_v=1$. Relative inhomogeneity $\delta$ is maintained at $<10\%$ across the straining plateau.

Comparative simulations show that abrupt or 45$^\circ$-tapered geometries yield sharp, inhomogeneous peaks in $\dot\varepsilon$, while the optimised channel sustains a flat strain-rate plateau. Validation by PTV confirms agreement of measured and computed centreline velocities to within a few percent (Liu et al., 2020).

6. Biological Applications and Case Studies

MiroFlow’s optimised microfluidic platforms support direct observation of the mechanical response of biomolecules and supramolecular aggregates under controlled elongational and compressional forces.

• Actin Filaments: Persistence length $\ell_p \approx 17\,\mu$m, contour length $L_c = 5 - 50\,\mu$m. In extension, Brownian undulations are suppressed; in compression, viscous buckling and 3D coiling are observed. End-to-end distance $L_{ee}(x)$ and transverse fluctuation $a_\perp(x)$ are quantified.
• DNA Molecules: T4 GT7, $L_c \approx 72\,\mu$m, $\ell_p = 57$\,nm; relaxation time $\lambda \approx 4.3$\,s. MiroFlow accesses coil–stretch transitions at $Wi = \lambda\dot\varepsilon > 1$, revealing diverse stretching dynamics, molecular individualism, and breakage events.
• Protein Aggregates: IgG4, size $50-100\,\mu$m, modulus $E\approx 0.1$\,MPa. Under flow, no deformation is detected; instead, 3D rotational reconfigurations (orientation angle $\theta(x)$, tumbling, flips) are analysed as aggregates progress through channel constrictions.

The Hencky strain experienced is geometry-determined ($\varepsilon_H \approx \ln CR$ for $AR \ll 1$), independent of flow-rate. Operating at low $Re \sim 10^{-3}-10^{-2}$ ensures creeping-flow conditions. The system can probe $Wi\sim 1-10$, covering the threshold for polymer coil–stretch transitions (Liu et al., 2020).

7. Representative Imaging and Dynamic Mapping

Laser Doppler MiroFlow produces frequency-resolved maps: $S_\text{dB}(x, y)$ at various detuning frequencies $\Delta f$ highlight channels where local flow matches the LO frequency shift, confirming spatial velocity inhomogeneities. Summed spectra reveal scatterer accumulations (e.g., near dead-ends), while the first spectral moment $⟨\Delta f⟩(x,y)$ provides detailed maps of velocity fields. Spectral widths in low-velocity ROIs are dominated by Brownian broadening ($\sim 31$\,Hz rms).

For microfluidic mechanical studies, detailed mechanical trajectories—including end-to-end distances for actin or DNA, and orientation for aggregates—are extracted as a function of longitudinal position. The system achieves consistent synchrony between motorised tracking and object motion, with particle displacement maintained within $<20\,\mu$m of the optical axis.

A plausible implication is that the MiroFlow approach, by integrating spectral and real-space mapping with tunable microfluidic landscapes, enables systematic, multimodal characterisation of both flows and soft matter deformations with high temporal and spatial precision.

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References:

• (Gross et al., 2013) Atlan, M., Gross, M., & Leng, J., "Laser Doppler Imaging of Microflow", 2013.
• (Liu et al., 2020) Liu, A., et al., "Optimised hyperbolic microchannels for the mechanical characterisation of bio-particles", 2020.