Figure 1.
Schematic of the LSR optical setup
[12]–[14], [16]. Light from a randomly polarized He-Ne laser (632 nm, 30 mW) is coupled into a single mode fiber (SMF600). The beam is polarized, collimated, and focused (focal length 25 cm, 50 µm spot size) at the sample surface. A beam-splitter is used to ensure speckle patterns are acquired at 180° back-scattering geometry. The cross-polarized component of back-scattered light is focused at the CMOS sensor of a high-speed camera (PL-761, Pixelink, Ontario, Canada), equipped with a focusing lens system (MLH-10×, Computar, Commack, NY). The acquired speckle frame series are transferred to a high-speed computer for further processing.
Figure 2.
Detailed flow chart of the compensation algorithm.
Block 1: Speckle acquisition and g2(t) calculation: Speckle frame series are acquired with sufficient frame rate, ROI, and pixel to speckle size ratio. Speckle intensity temporal autocorrelation curves, g2(t), are evaluated for phantom and tissue samples using sufficient temporal and spatial averaging. Block 2: Measurement of optical properties: The radial remittance profile is evaluated from temporally averaged speckle intensities and is converted to the photon flux, ψ(ρ). Optical properties of the sample (μa and μ′s ) are derived experimentally by fitting the photon flux profile to the model obtained from steady-state diffusion theory. Block 3: PSCT-MCRT for simulating g2(t)-MSD expression: Experimentally evaluated optical properties, LSR configuration, and sample geometry are used in the PSCT-MCRT simulation to derive an expression for g2(t) as a function of MSD. Block 4: Evaluating MSD and |G*(ω)|: Following the measurement of MSD using the modified expression, logarithmic slope of MSD, α(t) = ∂ log <Δr2(t)>/∂ log t, is calculated and replaced in the simplified GSER to evaluate the viscoelastic modulus [18]–[20], [22], [23], [25], [26], [36]–[41]. Here KB is the Boltzman constant (1.38×10−23), T is temperature (degrees kelvin), a is the scattering particle size, <Δr2(1/ω)> corresponds to <Δr2(t)>, evaluated at t = 1/ω, ω = 1/t is the frequency, and Γ represents the gamma function.
Figure 3.
LSR of aqueous glycerol mixtures of different viscosities.
Speckle intensity temporal autocorrelation curves, g2exp(t), for aqueous glycerol mixtures (100% G, 90%G-10%W, 80%G-20%W, 70%G-30%W, and 60%G-40%W) with 0.1% volume fraction TiO2 scattering particles. It is observed that for higher viscosity liquids speckle intensity temporal autocorrelation decays slower.
Figure 4.
LSR of 90% glycerol-10% water mixtures with varying scattering concentrations.
Speckle intensity temporal autocorrelation curves, g2exp(t), for aqueous glycerol mixtures of 90%G-10%W and various concentrations of TiO2 scattering particles (0.04%–2%, corresponding to μ′s : 1.3–84.8 mm−1, N = 18), along with theoretical DLS and DWS curves (dotted lines). By changing the scattering concentration g2exp(t) curves sweep the transition area between the two theoretical limits. This data demonstrates the dependence of g2exp(t) on optical scattering in samples with identical mechanical properties.
Figure 5.
MSD of scattering particles, derived using DWS expression, and the corresponding magnitude viscoelastic modulus |G*(ω)| curves for 90% glycerol-10% water mixtures.
In panel (a), MSD is extracted from g2exp(t) assuming the validity of Diffusion approximation. Considerable variability is observed between MSD curves associated with different scattering concentrations. In panel (b) Generalized Stokes'-Einstein Relation is used to calculate |G*(ω)| from MSD obtained from Diffusion approximation. The curves fail to match the results of conventional rheometry and are biased by the corresponding scattering concentrations. Moreover, significant variation is observed between the evaluated modulus of sample with different scattering concentrations.
Figure 6.
Radial photon flux profile of 90% glycerol-10% water mixtures with varying scattering concentrations and the corresponding theoretical and experimental estimation of the reduced scattering coefficient, μ′s.
Panel (a) shows the photon flux profile of the glycerol suspensions. It is observed that for samples of higher scattering particles' concentration, the net backscattered signal level increases. At the same time, the curves decay faster as a function of radial distance. Transport albedo (μ′s/(μ′s+μa)) and effective attenuation coefficient (√μa (μ′s+μa)) are derived by fitting the photon flux to theoretical models of the steady-state diffusion theory to further extract μ′s and μa [51]. In panel (b) Mie theory estimates of μ′s are shown, which are derived based on TiO2 particle size, source wavelength, and the ratio of refractive indices of TiO2 particles and glycerol solutions(refractive index mismatch). A close correspondence is observed between experimental and theoretical measurements of the μ′s (R = 0.96, P<0.0001), especially at lower scattering concentrations. For higher scattering concentrations, potential sedimentation of scattering particles, and particle interactions lead to distortion of photon flux curves and saturation of evaluated parameters.
Figure 7.
Compensated MSD of scattering particles and the corresponding magnitude viscoelastic modulus, |G*(ω)|, of 90% glycerol-10% water-TiO2 suspensions.
Panel (a) depicts the corrected MSD curves, deduced from g2exp(t) curves of Fig. 4 using eqn. (6). The modified expression of eqn. (6) resulted from PSCT-MCRT simulation of photon propagation and correlation transfer in LSR experimental setup considering the exact sample geometry and optical properties. Compared to Fig. 5(a), variability of MSD curves is significantly reduced, especially at intermediate times. Residual small deviations, still observed at very early or long times, are most likely due to electronic noise and speckle blurring, respectively. In panel (b) Generalized Stokes'-Einstein Relation is used to calculate |G*(ω)| from corrected MSD. It is observed that the variability between measured |G*(ω)| for different concentrations is considerably reduced, compared to Fig. 5(b). Moreover, a high correspondence is observed between LSR results for |G*(ω)| and mechanical rheometry.
Figure 8.
Speckle intensity temporal autocorrelation curves of synovial fluid and vitreous humor.
Panel (a) depicts the measured g2(t) curves of synovial fluid samples mixed in with TiO2 particles of different concentrations (0.08%, 0.1%, 0.15%, 0.19%, corresponding to μ′s :4.0, 5.1, 7.6, and 10.1 mm−1, respectively), and panel (b) displays the curves corresponding to vitreous humor samples mixed in with TiO2 particles (0.08%, 0.1%, 0.15, corresponding to μ′s :4.0, 5.1, and 7.6 mm−1, respectively). It is observed that early decay accelerates by increasing the scattering coefficient. At longer times, there is an artificial increase of the curve plateau level due to blurring of rapidly fluctuating speckle patterns and insufficient camera frame rate.
Figure 9.
LSR results of |G*(ω)| for synovial fluid and vitreous humor measured with and without optical scattering correction.
The red diamonds are the average |G*(ω)| moduli, of synovial fluid (panel (a)) and vitreous humor (panel (b)) samples of Fig. 8, obtained from LSR by using the DWS expression (eqn. (3)). The red error bars correspond to standard error values. The purple squares represent the average |G*(ω)| moduli, obtained from the corrected MSD values using eqn. (6), and the purple error bars correspond to the standard errors. Also depicted in this figure are the |G*(ω)| values for the samples measured using a conventional rheometer (black solid line, round markers). While LSR results compensated for optical scattering show close correspondence with rheology measurements, the DWS-based approach results in an offset of about one decade relative to conventional testing results.