3.1.

No-Flow Conditions

Here, we briefly summarize some background theoretical results from Refs. 10, 12, and 35 needed to calculate the viscosity of a stagnant fluid from coherent light scattering measurements. This is a well-established problem in DLS theory;¹⁰ the more viscous the fluid, the slower are the speeds of the Brownian particles suspended in it, resulting in smaller Doppler shifts in the scattered radiation and a narrower power spectrum. Under stagnant conditions, DLS theory states that the power spectrum can be described by a Lorentzian function¹⁰

Thus, given the radius of the spherical Brownian scatterers and knowing the experimental temperature, the viscosity of the suspending fluid can be determined by fitting a Lorentzian to the experimentally measured power spectrum, determining its HWHM and solving Eq. (2).

3.2.

Flowing Conditions

In the case of Brownian particles suspended in a flowing fluid, the mathematical model for the power spectrum must take into account two sources of spectral broadening: the first is due to the random Doppler shifts caused by the Brownian motion as discussed above and the second is due to the dynamic speckle pattern on the detector that results from the flow.⁴⁴ Recently, both of these effects have been taken into account under OCT conditions.^12,11,35 In particular, Eq. (18) from Ref. 12 provides a mathematical expression that models the power spectrum of the scattered optical field using a Voigt function, which is the convolution in frequency space of a Lorentzian Eq. (1) (Brownian Doppler broadening) with a Gaussian¹² that may be Doppler shifted $G(f-f_D)=\exp \{-[\pi {\tau }_t{(f-f_D)}^2]\}$ (translational speckle fluctuation broadening), where $f_D$ is the Doppler shift due to flow, $f_D=\frac{2v_z}{\lambda }$, $\lambda =\frac{{\lambda }_0}{n}$, and $v_z$ is the flow velocity component in the axial direction ($v_z=0$ for the special case of perpendicular flow). The Voigt function, $W(f-f_D)$, can be expressed analytically in terms of the complex error function erf¹²

The challenges in fitting the Voigt function [Eq. (3)] to data using a least squares approach are well-known in the literature.^45–48 One problem is that Eq. (3) contains a product of an exponential and complementary error function of the same argument, $\frac{1-i2\pi {\tau }_b(f-f_D)}{2\frac{{\tau }_b}{{\tau }_t}}$; as the argument grows (with increasing frequency in the fitting region), the exponential rapidly increases, while the complementary error function rapidly decreases, causing stability problems in the fitting procedure for large $f$ [$f>500\mathrm{Hz}$ was problematic for fitting Eq. (3) in our data]. To fit the full spectra, we used the MATLAB^® algorithm from Ref. 47, which is accurate to within ${10}^{-13}$.

This summarizes the main theoretical predictions from Refs. 10, 12, and 35 needed to calculate the viscosity of a fluid from coherent light scattering measurements. In Sec. 4 below, the viscosities of fluid samples are determined by fitting Eq. (1) (no-flow case) and Eq. (3) (flow cases) to our experimental spectral data and then using Eq. (2) to calculate the viscosity of the fluid samples. We note that the ${\tau }_b$ in Eq. (3) is identical to the ${\tau }_b$ in Eq. (1); this is because Eq. (3) was derived via a convolution in frequency space of Eq. (1) and a Gaussian function, as described above.

4.1.

No-Flow Case

To determine the sample viscosity for the no-flow case, we fit Eq. (1) to the experimentally obtained power spectra with ${\tau }_b$ as the only fitting parameter and then use Eq. (2) to compute the viscosity. Figure 2(a) shows the experimentally obtained power spectra for a single depth in the sample, with Eq. (1) fitted (solid lines) for each glucose concentration. As seen, the fit is good throughout the studied concentration range. The derived results from the experimental fits of Fig. 2, as well as from additional glucose concentrations, are shown in Table 1 (top rows). Our determined viscosity value for pure water is $(0.904\pm 0.008)\mathrm{mPa}·\mathrm{s}$, which is within $\sim 1\%$ of the accepted literature value at 23.8°C of ($(0.914\pm 0.0005)\mathrm{mPa}·\mathrm{s}$⁴⁹ (see Table 1), indicating high accuracy for our methodology in turbid water suspensions under no-flow conditions. Assessing the accuracy of the glucose results is more challenging, owing to the incomplete (and often conflicting, see discussion in Sec. 1) set of literature values for comparison. We thus used an interpolation method (explained in more detail below) to generate “literature values” at 23.8°C from the existing 20°C and 30°C literature data (Refs. 34 and 50, respectively). The resulting values are shown in the bottom row of Table 1. Comparing the top and bottom row entries across the measured glucose concentration range, we note an accuracy of $\sim \pm 1\%$ in the no-flow colloidal glucose suspensions, except for the 1600 mM value which differs by $\sim 2.5\%$.

Fig. 2

(a) Power spectra of backscattered radiation from Brownian particles (polystyrene microspheres) suspended in stagnant aqueous glucose solutions in the 0 to 1600 mM range. Symbols are experimentally obtained power spectrum values and lines are theoretical fits of Eq. (1) with ${\tau }_b$ as the fitting parameter. (b) ${\mathit{HWHM}}_{\mathit{b}}$ of the power spectra in (a), as a function of depth in the capillary. Symbols are the ${\mathit{HWHM}}_{\mathit{b}}$ values from the theoretical fits of Eq. (1) and lines are the average fitted values for each concentration.

Table 1

Brownian contributions to measured spectral widths HWHMb and corresponding viscosity and diffusivity values [via Eq. (2)] in stagnant and flowing suspensions of microspheres in water with varying amounts of dissolved glucose. All numbers following the ± symbols are the numerical values of the combined standard uncertainty.30

In the no-flow case, the spectrum broadening is caused solely by the axial component of the random Doppler shifts produced by the jittering Brownian particles. Therefore, we would expect the ${\mathit{HWHM}}_{\mathit{b}}$ of the measured Lorentzian power spectra to be independent of depth into the sample. Indeed, aside from some statistical variation, this independence of depth is shown in Fig. 2(b), which plots the ${\mathit{HWHM}}_{\mathit{b}}$ values of the fitted Lorentzian as a function of depth across the capillary, with respect to the center of the capillary. Since the spectral width does not depend on the $z$-coordinate, the measured viscosity [calculated via Eq. (2)] also exhibits no depth dependence, as expected (mean values across depths are presented in Table 1).

4.2.

Perpendicular Geometry Flow Case

For these experiments, the direction of flow was set perpendicular to the OCT sample arm beam axis, by minimizing the Doppler shift in the power spectrum (i.e., having the peak centered at $f_D\to 0$). The viscosity of each sample was determined by fitting an algorithm approximation^45,47 of Eq. (3) to the measured power spectra and then using Eq. (2) to compute the viscosity for each depth. Figure 3(a) shows the experimentally obtained power spectra from the OCT voxel located at the center of the glass capillary, for different glucose concentrations. The lines in Fig. 3(a) are the fits of the algorithm approximation of Eq. (3) to the spectra, with both ${\tau }_t$ and ${\tau }_b$ as fitting parameters. As seen, Eq. (3) describes the data well across all measured glucose concentrations. The middle rows in Table 1 show the resulting numbers derived from these fits, averaged across all depths. Comparing to the top-row (no-flow) results, we note the consistency of our methodology in determining viscosity values under these two different experimental conditions. Comparing to the literature values bottom row, we note the technique’s accuracy for glucose viscosity determination under the perpendicular-flow conditions.

Fig. 3

(a) Analogous to Fig. 2(a) but now with flow perpendicular to the OCT sample arm beam. Symbols are experimentally obtained power spectrum values and lines are the fits of the algorithm approximation of Eq. (3) to the spectra with both ${\tau }_t$ and ${\tau }_b$ as fitting parameters. (b) Brownian motion contribution (${\mathit{HWHM}}_{\mathit{b}}$) to power spectra in (a), as a function of depth across the capillary. Symbols are the ${\mathit{HWHM}}_{\mathit{b}}$ calculated via Eq. (2), using ${\tau }_b$ values from the fits of Eq. (3) and lines are the average ${\mathit{HWHM}}_{\mathit{b}}$ for each glucose concentration.

Figure 3(b) plots the Brownian motion contribution (${\mathit{HWHM}}_{\mathit{b}}$) to the width of the fitted Voigt as a function of depth in the capillary; as expected, viscosity was independent of depth. Further, calculation of the flow speed $v$ via Eq. (4) using the fitted ${\tau }_t$ values as a function of depth showed good agreement with the theoretically expected parabolic flow profile (with $<5\%$ RMS deviation from the expected flow profile over all depths, for each glucose concentration—data not shown). The good fits in the presence of flow, the depth independence of the results, and the close agreement with literature values all validate our recently proposed mathematical OCT DLS model.^12,35

4.3.

Doppler Angle Flow Case

Having examined the special cases of no-flow and perpendicular-flow situations, we now move on to the most general case, with flow at an arbitrary (non-perpendicular) angle. Here, we must take into account the Doppler shifted peak centered at $f_D$ due to the axial flow component. In OCT, we measure both the amplitude and phase of the scattered electric field. However, in our data analysis, we are using only the real part of the electric field sampled in time (purely, real function of time), the Fourier transform of which is Hermitian, resulting in a power spectrum that is symmetric about the $f=0$ baseband. Therefore, when performing the fitting, we need to include the peak shifted to the negative frequencies, and the fitting function in Eq. (3) becomes

We now determine the viscosity of each phantom, starting by fitting Eq. (5) (via the algorithm approximation)^45,47 to the spectrum data. The Doppler angle $\theta$ chosen for the experiments was $81°$ in air. Figure 4 plots the experimentally obtained power spectrum from the voxel at the center of the glass capillary with the 0 mM data in Fig. 4(a) and the 100, 800, and 1600 mM data in Fig. 4(b). The lines in Fig. 4 show the fits of Eq. (5) with $f_D$, ${\tau }_t$, and ${\tau }_b$ as the fitting parameters. As seen, the fit is good for all glucose concentrations. The lower rows in Table 1 show the resulting derived values. Comparing to the top-row (no-flow) results, we again note the consistency of our methodology in determining viscosity values under these different experimental conditions. Comparing to the literature values in the bottom row, we note the technique’s accuracy for glucose viscosity determination under the Doppler angle-flow conditions. Specifically, for the case of pure water, the well-established literature value of $0.914\mathrm{mPa}·\mathrm{s}$ differs by $\sim 1.5\%$ from our flow measurements. This experimental consistency and good accuracy further validate the mathematical OCT DLS model originally developed in Refs. 12 and 35.

Fig. 4

Power spectra of backscattered radiation from flowing Brownian particles (polystyrene microspheres) in the aqueous glucose solutions, with a Doppler angle $\theta =81°$ in air. Data were collected from the center of capillary voxel, where the flow speed was $v=1.94\mathrm{mm}/\mathrm{s}$ and the velocity gradients were minimal. Symbols are experimentally obtained power spectrum data, and the fits of the algorithm approximation to Eq. (5) are shown as solid lines (where $f_D$, ${\tau }_t$, and ${\tau }_b$ are the fitting parameters). (a) 0 mM glucose, (b) 100, 800, and 1600 mM glucose.

Assuming a parabolic flow profile, we calculated $v=1.94\mathrm{mm}/\mathrm{s}$ at the center of the capillary, and the expected Doppler peak position for 0 mM glucose from the Doppler shift formula is $f_D=\frac{2v_{z}n}{{\lambda }_0}=518\mathrm{Hz}$. This is in very close agreement with the best-fit peak position as shown in Fig. 4(a). Further, note that the positions of the Doppler peaks in Fig. 4(b) are not identical for the various glucose concentrations, for two reasons. First, the position of the Doppler peak depends on the glucose concentration through the refractive index, both directly in the formula $f_D=\frac{2v_{z}n}{{\lambda }_0}$, as well as indirectly through $v_z$ (which depends on the Doppler angle in air and the refractive index of the suspending fluid). Second, the flow speeds in each of the experiments were likely not exactly the same: slight average flow speed variations do not significantly impact the measured width of the peak, and thus do not impact the measured viscosity values, yet the position of the peak in the frequency domain is very sensitive to these slight flow variations. Overall though, increasing glucose concentration narrows the spectral width of the peak, as expected from increasing viscosity which dampens the Brownian motion of the scatterers; the fact that we can measure and quantify this effect even under angled-flow conditions is encouraging.

We also point out that for the angled flow case, even slight velocity gradients within the OCT measurement voxel will cause artifacts.¹¹ In fact, the only depth in the capillary where it was possible to accurately measure the viscosity was at its center position, where the velocity gradient of the parabolic flow profile was close to zero. For all other depths, the velocity gradient is enough to cause additional spectral broadening and thus yield inaccurate results. Within $40\mu \mathrm{m}$ from the center of the capillary, the velocity gradient causes a change in the Voigt spectral width (departure from theory) of up to 7%.

4.4.

Measurement Uncertainty Analysis

Here, we briefly explain our uncertainty calculations. We can solve Eq. (2) for viscosity $\eta =\frac{4k_{B}T{n}^2}{3{\lambda }_o^{2}R·{\mathit{HWHM}}_{\mathit{b}}}$ and diffusivity $D=\frac{{\lambda }_o^2·{\mathit{HWHM}}_{\mathit{b}}}{8\pi n^2}$; aside from the constant $k_B$, each variable in the RHS of these equations has an associated “standard uncertainty, $u$” (the standard deviation of the assumed probability distribution of the variable, either measured directly, provided by the manufacturer, or estimated). GUM details how to mathematically combine these individual standard uncertainties to produce the combined standard uncertainty, $u_c$, on viscosity, $u_c(\eta )$, and on diffusivity, $u_c(D)$. By applying the GUM procedure, we assume that these individual components of uncertainty are statistically independent.

Table 1 presents a results summary. It lists the ${\mathit{HWHM}}_{\mathit{b}}$ values and corresponding diffusivity and viscosity values for all three experimental conditions. For the first ${\mathit{HWHM}}_{\mathit{b}}$ row (no-flow case), the numbers following the ± symbols are the numerical values of the standard uncertainty, $u({\mathit{HWHM}}_{\mathit{b}})$, calculated directly from the confidence interval (from the MATLAB^® fitting procedure) on the fitted parameter ${\tau }_b$; thus, $u({\mathit{HWHM}}_{\mathit{b}})$ is a measure of the quality of the fit of the Lorentzian to the data. The MATLAB^® fitting procedure assumes that the fitted parameter ${\tau }_b$ is a Gaussian random variable.

For the second and third ${\mathit{HWHM}}_{\mathit{b}}$ rows (flow cases), the numbers following the ± symbols are the numerical values of the standard uncertainty $u({\mathit{HWHM}}_b)$, which we assume has two contributions: $u{({\mathit{HWHM}}_{\mathit{b}})}_{\mathit{\text{flow}}}\equiv \sqrt{{(randomcomponent)}^2+{(biasedcomponent)}^2}$. The first is the (random) fitting uncertainty on ${\mathit{HWHM}}_{\mathit{b}}$ calculated directly from the confidence interval (from the MATLAB^® fitting procedure) on the fitted parameter ${\tau }_b$. The second is a non-random component (bias component), which we estimate as the difference between the measured ${\mathit{HWHM}}_{\mathit{b}}$ values and the no-flow ${\mathit{HWHM}}_{\mathit{b}}$ values: $|{\eta }_{\mathrm{perp}\text{-}\text{flow}}-{\eta }_{\text{no-flow}}|$ for the perpendicular-flow values and $|{\eta }_{\text{angled-flow}}-{\eta }_{\text{no-flow}}|$ for the angled flow values.

For the no-flow and perpendicular-flow cases, the listed ${\mathit{HWHM}}_{\mathit{b}}$ values are averages taken over depths within $\sim 40\mu \mathrm{m}$ from the capillary center. For the $\theta =81°$ experiment, only the data from the center of capillary location are listed. To calculate the combined standard uncertainties for diffusivity $u_c(D)$ and viscosity $u_c(\eta )$, for all three experimental conditions, the following values and their associated standard uncertainties were used: $T=(23.8\pm 0.2)°\mathrm{C}$, ${\lambda }_0=(1313.1\pm 0.07)\mathrm{nm}$,³⁹ $n=1.320-1.355\pm 0.001$,^40,41 and $R=(107.55\pm 0.75)\mathrm{nm}$,⁴² along with the $u({\mathit{HWHM}}_{\mathit{b}})$ values described above. Although the following uncertainties are not used in our uncertainty calculations, we list them for completeness: $w_0=(11.5\pm 0.2)\mu \mathrm{m}$,⁵¹ $\theta =(81\pm 1)°$, and $v=0-1.94\mathrm{mm}/\mathrm{s}$ with a relative standard uncertainty of 1%.⁵²

Despite the close agreement in the derived viscosity between the no-flow, perpendicular-flow, and Doppler angle flow cases, the no-flow data have a much smaller $u_c(\eta )$. In fact, the (random) component of uncertainty for the ${\mathit{HWHM}}_{\mathit{b}}$ due to the spectral fitting was very similar for all three experiments—the no-flow data fits [Lorentzian, Eq. (1)] and the flow data fits [Voigt, algorithm approximation for Eq. (3)]. The larger uncertainty values under flowing conditions arise from the bias component (discussed above), which is likely caused by some departure of the beam from a Gaussian profile due to spatial noise, and some fluctuations in the flow velocity. In the no-flow case, the main factor limiting the accuracy of viscosity measurements is the uncertainty on the particle radius. All the other components contributing to the combined uncertainty ($T$, $n$, ${\lambda }_0$, and ${\mathit{HWHM}}_{\mathit{b}}$) are considerably smaller.

As an aside, we note that to produce the averaged spectrum with less fluctuations, it is important to have enough scatterers in the scattering volume, $\sim 100$ or more [we used $\sim 5000\mathrm{per}\text{voxel}$ to get high signal to noise (S/N)]. With smaller numbers, the scattering becomes non-Gaussian,⁵³ and the noise fluctuations in the measured signal (and resulting power spectrum) become considerably higher, which will lead to a larger contribution from the (random) fitting uncertainty.

4.5.

Comparison with Literature

We begin this section with a brief comparison of our uncertainty values with those published in a previous OCT study,¹³ where the diffusivity of spherical Brownian particles in water was measured. First, under no-flow conditions and using a single parameter fit of ${\tau }_b$, the standard uncertainty on the measurement of diffusivity in Ref. 13 is $\sim 30\times$ higher than the calculated standard uncertainty in this paper (see Table 1). Second, under perpendicular-flow conditions and using a two-parameter fit of ${\tau }_b$ and ${\tau }_t$, the standard uncertainty on the measurement of diffusivity in Ref. 13 is $\sim 3\times$ higher than the calculated standard uncertainty in this paper.

Next, we compare our measured viscosity values with literature, and thus establish the technique’s accuracy. Since there are no reliable data in the literature for the viscosity of aqueous glucose solutions at our experimental temperature of 23.8°C, we used data sources at 20°C (Ref. 34) and at 30°C (Ref. 50), along with the following (slightly modified) empirical equation from Ref. 54 and also used in Ref. 32

Fig. 5

Graphical summary of all viscosity data and comparison with literature. No-flow (circles), perpendicular-flow (crosses), and Doppler angle flow values (triangles) are the experimentally measured viscosity results summarized in Table 1. Theoretical curves: (1) resulting fit of Eq. (6) to viscosity data from Ref. 34 (top dashed curve, 20°C), (2) resulting fit of Eq. (6) to viscosity data from Ref. 50 (bottom dashed-dot curve, 30°C), (3) Eq. (6) with the parameter value $c$ calculated by linearly interpolating the fitted $c$ values from the 20°C and 30°C fits to 23.8°C and with ${\eta }_o$ set to the viscosity of water at 23.8°C (solid curve). The inset shows a zoomed-in view in the lower glucose concentration range of potential biomedical relevance.

We now discuss briefly blood viscosity and follow this with a discussion of the potential of our methodology for blood glucometry measurements. The biomedically relevant range of blood viscosity is from 3 to $50\mathrm{mPa}·\mathrm{s}$.⁵⁵ The largest modifier of blood viscosity is blood shear rate, which becomes important in small capillaries under systolic conditions. Models that take into account the impact of shear rate on viscosity are well-developed;⁵⁵ this will be considered in our future theoretical model developments. Other important factors affecting blood viscosity are hematocrit and erythrocyte aggregation (rouleaux formation).^18,19 These depend on glucose only slightly and can be measured independently.⁵⁶ Further, in diabetic patients, it has been demonstrated that an increase in blood glucose concentration from 5 to 9 mM yields a $\sim 4\%$ increase in blood plasma viscosity and a $\sim 3\%$ increase in whole blood viscosity;⁵⁶ note for context that the relevant noninvasive physiological blood glucometry range is $\sim 1$ to 50 mM.⁵⁷ In summary, glucose concentration is not the largest modifier of blood viscosity, although it does contribute. It seems possible that the effects of other relevant blood viscosity-modifying parameters can be accounted for, and thus glucose content may be derivable from measurements of blood viscosity.

To assess the potential of our methodology for blood glucometry measurements, we use the experimental parameters from our earlier publication (Ref. 9, rat blood) and from a more recent study (Ref. 28, mouse blood); both of these report correlation times of OCT backscattered radiation as a function of glucose concentration. We estimate the uncertainty of glucose concentration determination via $u(x)=\frac{\partial x}{\partial {\tau }_b}·u({\tau }_b)$, where $x$ is the glucose concentration in units of molality, $u({\tau }_b)$ is the uncertainty on the correlation times given in the papers, and the derivative can be estimated from the data in the papers. We then compare our estimates with the reported values of $u(x)=\pm 0.8\mathrm{mM}$⁹ and $u(x)=\pm 5\mathrm{mM}$.²⁸

Our uncertainty for the spectral width measurements no-flow conditions is 30 mHz (see Table 1); we assume the same for the case of measuring the power spectrum from blood under no-flow conditions.†† In Ref. 9 (rat blood), the correlation time at $x=20\mathrm{mM}$ is 9.52 ms. We can calculate the expected uncertainty in correlation time measurement if our power spectrum technique was used: $u({\tau }_b)=2\pi {({\tau }_b)}^{2}u({\mathit{HWHM}}_{\mathit{b}})=2\pi {(9.52\mathrm{ms})}^2·0.03\mathrm{Hz}=0.02\mathrm{ms}$. The corresponding expected uncertainty in glucose concentration measurement at $x=20\mathrm{mM}$ is thus $u(x)=\frac{\partial x}{\partial {\tau }_b}·u({\tau }_b)=8.2\frac{\mathrm{mM}}{\mathrm{ms}}·0.02\mathrm{ms}=0.18\mathrm{mM}$, where we have used the table in Ref. 9 to estimate the derivative $\frac{\partial x}{\partial {\tau }_b}$. Comparing this with the initial reported precision of $\pm 0.8\mathrm{mM}$,⁹ our estimated $\sim 4\times$ improvement in $u(x)$ for rat blood is encouraging.

In Ref. 28 (mouse blood diluted with phosphate buffered saline), the correlation time at $x=20\mathrm{mM}$ is 7.1 ms (Fig. 5 of Ref. 28). Using this, we again calculate the expected uncertainty in correlation time measurement if our power spectrum measurement technique was used: $u({\tau }_b)=2\pi {({\tau }_b)}^{2}u({\mathit{HWHM}}_{\mathit{b}})=2\pi {(7.1\mathrm{ms})}^2·0.03\mathrm{Hz}=0.01\mathrm{ms}$. The corresponding expected uncertainty in glucose concentration measurement at $x=20\mathrm{mM}$ is $u(x)=\frac{\partial x}{\partial {\tau }_b}·u({\tau }_b)=2.0\frac{\mathrm{mM}}{\mathrm{ms}}·0.01\mathrm{ms}=0.02\mathrm{mM}$, where we have used the data in Fig. 5 of Ref. 28 to estimate the derivative $\frac{\partial x}{\partial {\tau }_b}$. This estimated $\sim 250\times$ improvement in $u(x)$ for (diluted) mouse blood is also very encouraging.

Recapping, we have made OCT measurements of the power spectrum of backscattered radiation from aqueous glucose phantoms containing small microspheres in dilute concentration. This satisfies the conditions of our recently developed theory (dilute suspension noninteracting particles and single scattering regime).^12,35 We are currently refining our theory to describe more realistic tissue situations that include varying scatterer shape and concentration, inter-particle interaction, and multiple scattering effects; this is quite challenging and will be reported in separate forthcoming publications when successful.