2.1.

Definition of Average Vessel Diameter

Throughout this paper, the volume-weighted average vessel diameter is considered, rather than the number-weighted average. Hence, if we have four vessels with a diameter of $10\mu \mathrm{m}$ and one with the diameter of $20\mu \mathrm{m}$ within a certain volume, all with the same length $l$, the average vessel diameter of those five vessels is $15\mu \mathrm{m}$ rather than $12\mu \mathrm{m}$. Although not explicitly defined, the same interpretation of the average vessel diameter is done in previous papers on the topic, for example, Svaasand et al.,⁸ van Veen et al.,⁹ and Fredriksson et al.¹⁰

2.2.

Modeling Doppler Spectra

Laser Doppler power spectra were calculated from tissue models, as previously described in Fredriksson et al.,¹² outlined in the flowchart in Fig. 1. The calculations were based on three-layered models of skin tissue with one epidermis layer of varying thickness ($t_{\mathrm{epi}}$), one upper dermis layer with a fixed thickness of 0.2 mm, and one lower dermis with an infinite thickness. The reduced scattering coefficient (${\mu }_{\mathrm{s}}^{\prime }$) was the same for all three layers. The epidermis layer contained a varying volume fraction of melanin ($f_{\mathrm{mel}}$) as absorber and no blood, whereas the dermis layers contained blood of varying volume fraction ($f_{\text{blood},1}$ and $f_{\text{blood},2}$), hemoglobin oxygen saturation ($S_{{\mathrm{O}}_2}$), flow speed distribution ($P_v$), and average vessel diameter ($D$). The light absorption in the epidermis layer (${\mu }_{\mathrm{a},0}$) was given by the melanin, whereas the absorption in the dermis layers (${\mu }_{\mathrm{a},1}$ and ${\mu }_{\mathrm{a},2}$) was given by the volume fraction of blood and its oxygen saturation at the given wavelength.

Fig. 1

Flowchart for calculating a Doppler power spectrum from the three-layered model with a given set of parameters.

The three-layered model was Monte Carlo simulated for various epidermis thicknesses and reduced scattering coefficients, where the pathlengths for all detected photons were stored and analyzed as pathlength distributions for each layer. For the three-layered model, six pathlength distributions were created, one describing the pathlength for light that had only been propagated in the first layer, one distribution over pathlengths in the first layer for light that had been propagating in the two first layers, and so on. The Monte Carlo simulations in Ref. 12 were based on a setup with optical fibers, but the same principle was used in this study for a camera setup with even illumination.

The effect of absorption, blood flow speed distribution, and average vessel diameter was added in a fast postprocessing step. In short, the intensity from each pathlength was reduced by applying Beer–Lambert’s law with the given absorption in each layer. Single shifted optical Doppler spectra for each speed $v$ in the speed distribution were calculated, as described in Fredriksson and Larsson.¹⁷ These spectra were then successively cross-correlated taking into account different degrees of multiple Doppler shifts; i.e., the optical Doppler spectrum for light Doppler shifted exactly two times was calculated as the autocorrelation of the single-shifted spectrum, the spectrum for light shifted exactly three times was calculated as the two-times shifted spectrum cross-correlated with the single-shifted spectrum, and so on. The optical Doppler spectrum for non-Doppler shifted light was given by the Dirac delta function (i.e., all energy located at frequency 0).

The resulting multiple Doppler shifted spectra were calculated using the shift distributions, i.e., the distributions over the number of Doppler shifts the light undergoes for each pathlength in the pathlength distribution. First, a distribution over the number of Doppler shifts the light undergoes when crossing a single vessel is calculated. That distribution can be approximated by a Poisson distribution with the expectation value approximated by the average vessel diameter $D$ times the scattering coefficient of blood ${\mu }_{\mathrm{s},\text{blood}}=222{\mathrm{mm}}^{-1}$:

That distribution is used for calculating an optical Doppler spectrum for each speed in the speed distribution representing the spectra for passing through a single vessel. In reality, some light will cross the vessel in the periphery of the vessel, i.e., with a photon path shorter than the diameter, whereas some light will cross the vessel with a rather flat angle, i.e., with a path longer than the diameter. However, simulations have shown that a Poisson distribution with the expectation value given in Eq. (1) is a good approximation.

A second distribution was then calculated describing the number of vessels the light passes. For a given pathlength $d$, the number of vessels that the light passes through can be described by a Poisson distribution with the expectation value:

For light that had been propagating in both blood layers, the resulting optical Doppler spectrum was calculated as the cross-correlation of the spectra from the first and second blood layers. The final optical Doppler spectrum was then calculated by weighting the optical Doppler spectra from the light that had been propagated in only the first layer, the first and second layer, and all three layers, depending on the intensity. The Doppler power spectrum $P(f)$ was finally calculated as the autocorrelation of the optical Doppler spectrum. A detailed mathematical description of the calculations of the Doppler power spectrum is found in Fredriksson et al.¹²

At first, a standard setup of the three-layered model was used. This standard model had an epidermal thickness of $75\mu \mathrm{m}$, a reduced scattering coefficient of $3.0{\mathrm{mm}}^{-1}$ (anisotropy factor $g=0.8$), and absorption coefficients of 0.15, 0.0043, and $0.0043{\mathrm{mm}}^{-1}$ for the epidermis layer and the two blood layers, respectively. The distribution of flow speeds had an average speed of $1\mathrm{mm}/\mathrm{s}$ and was decreasing with speed. The vessel diameter parameter $D$ was varied between 5 and $100\mu \mathrm{m}$.

In order to evaluate the vessel packaging effect in the more general case, 2000 three-layered models with random parameters $t_{\mathrm{epi}}$, ${\mu }_{\mathrm{s}}^{\prime }$, $f_{\mathrm{mel}}$, $f_{\text{blood},1}$, $f_{\text{blood},2}$, $S_{{\mathrm{O}}_2}$, and $P_v$ were generated. The distribution of the parameters in these 2000 models is summarized in Table 1.

Table 1

Average, standard deviation, and median values of the model parameters in the 2000 random models. Row four is the average blood fraction in the two layers, whereas row five denotes the relationship between the blood fraction in the two layers. The two last rows give the average speed of the speed distribution Pv and the width of the speed distribution in relation to the average speed, respectively.

2.3.

Validation Simulations of Discrete Vessel Models

Two validation models were created and Monte Carlo simulated. Both models consisted of a homogeneous semi-infinite scattering layer with a refractive index of 1.4 and a reduced scattering coefficient ${\mu }_{\mathrm{s}}^{\prime }$ of $2.0{\mathrm{mm}}^{-1}$ containing various amounts of vessels (cylinder shaped) oriented parallel to the surface with random directions in that plane. The vessels all had a length of 20 mm and were randomly but homogeneously (on a large scale) distributed within a radius of 6 mm from the central axis of the model, at a random depth between 0 and 5 mm. The light source had a diameter of 1.0 mm and illumined the model perpendicularly. All photons that were backscattered from the model within a radius of 3 mm and an angle of 0.3 rad were detected. The medium outside the modeled tissue had a refractive index of 1.0. Hence, a part of the light was refracted in the surface and not detected. The blood in the vessels had a scattering coefficient ${\mu }_{\mathrm{s}}$ of $222{\mathrm{mm}}^{-1}$ and a Gegenbauer kernel scattering phase function²¹ with parameters ${\alpha }_{\mathrm{Gk}}=1.0$ and $g_{\mathrm{Gk}}=0.984$, resulting in an anisotropy factor $g$ of 0.991 and a reduced scattering coefficient ${\mu }_{\mathrm{s}}^{\prime }$ of $2.0{\mathrm{mm}}^{-1}$. The absorption coefficient ${\mu }_{\mathrm{a}}$ of the blood was set to $0.5{\mathrm{mm}}^{-1}$. These optical properties are valid at 780 nm for 50% oxygenated blood with a hematocrit of 43% and a mean cell hemoglobin concentration of $345\mathrm{g}/\mathrm{l}$ RBC.^12,15

The first model contained 200 vessels all with a diameter of $50\mu \mathrm{m}$. All vessels had a parabolic flow profile with average velocity of $1\mathrm{mm}/\mathrm{s}$ ($2\mathrm{mm}/\mathrm{s}$ in the axial center of the vessel, 0 at the periphery). The blood vessels occupied 0.61% of the sampling volume in the model.

The second model contained 2000 vessels with vessel diameters having a volume weighted gamma distribution. The average velocity in the vessels was proportional to the diameter squared so that, for example, a vessel with a diameter of $50\mu \mathrm{m}$ had an average velocity of $2\mathrm{mm}/\mathrm{s}$ and a vessel with a diameter of $10\mu \mathrm{m}$ had an average velocity of $0.08\mathrm{mm}/\mathrm{s}$. The speed profile within each vessel was parabolic. The average vessel diameter in the model was $33\mu \mathrm{m}$ with a standard deviation of $25\mu \mathrm{m}$. The blood vessels occupied 1.0 % of the sampling volume in the model and the average flow speed was $1.0\mathrm{mm}/\mathrm{s}$. A cross-sectional view of this model is shown in Fig. 2.

Fig. 2

Cross-sectional view of the simulated model with a gamma distribution of vessel diameters. The 1-mm diameter light source is shown in the top of the image. The dark ellipsoids represent cross-sections of vessels, where the elongation of the ellipsoid depends on the angle relative to the cross-sectional plane.

2.4.

Laser Speckle Contrast Calculation

In LSCI, the speckle contrast is used to estimate blood perfusion. Fredriksson and Larsson¹⁷ have previously described how the speckle contrast can be calculated from a simulated Doppler power spectrum $P(f)$. In short, the intensity correlation function is calculated as

From this relationship, the speckle contrast $K$ as a function of exposure time $T$ is given by^24,25

2.5.

Perfusion Estimates

In commercial LDF systems, the perfusion value is calculated from the first moment of the Doppler power spectrum, i.e.,

In commercial LSCI systems, the perfusion value is calculated from the contrast $K$ at one single exposure time, for example,²

3.1.

Vessel Packaging Model Validation

Comparisons between optical Doppler spectra generated from the discrete vessel validation simulations (Sec. 2.3) and those calculated from the model described in Sec. 2.2, are found in Fig. 3, where “simulated” refers to the validation simulation and “modeled xx $\mu \mathrm{m}$” refers to calculated spectra from the model in Sec. 2.2. Corresponding comparisons of contrast as a function of exposure time are also shown in the same figure. In the first example, the validation simulation model contained 200 vessels, all with a diameter of $50\mu \mathrm{m}$. The second validation simulation contained 2000 blood vessels with a gamma distributed diameter and diameter dependent flow speed, with an average diameter of $33\mu \mathrm{m}$. To demonstrate the accuracy of the proposed vessel packaging model, spectra were calculated using a homogeneous model ($0\mu \mathrm{m}$ epidermis, same tissue fraction of blood in both dermis layer) with the same optical properties, speed distribution, and volume fraction of blood as that used in the validation simulations. To further demonstrate the vessel packaging effect, spectra were calculated from the homogeneous model using both the average diameter from the validation simulations and an infinitely small vessel diameter ($0\mu \mathrm{m}$) to mimic homogeneously distributed blood. Frequencies up to 3 kHz are shown because clear differences between spectra calculated with and without inclusion of the vessel packaging effect are foremost found at low frequencies. However, all Doppler power spectra in this study contained frequencies up to 25 kHz. The frequency resolution in the spectra is 49 Hz and the time resolution in the contrast curves is 0.02 ms. The fraction of non-Doppler shifted light was 59% in the validation simulation in the first example ($50\mu \mathrm{m}$ diameter), which should be compared to 56% in the homogeneous model including the vessel packaging effect, and 23% in the model with the homogeneously distributed blood. Corresponding numbers from the simulation containing a gamma distribution of vessel diameters in the simulation were 32%, 35%, and 13%, respectively.

Fig. 3

(a and b) Optical Doppler spectra and (c and d) speckle contrast as a function of exposure time from the discrete vessel validation simulation (solid) and calculated from homogeneous model. Spectra are given when including the vessel packaging effect (dashed) and when not, i.e., modeling homogeneously distributed blood (dash-dotted), respectively. In (a) and (c), the simulated model contained 200 vessels with $50\mu \mathrm{m}$ diameter. In (b) and (d), the simulated model contained 2000 models with a gamma distribution of vessel diameters, average diameter $33\mu \mathrm{m}$.

3.2.

Effect on Perfusion Estimates

The vessel packaging effect was studied in detail for vessel diameters ranging from 5 to $100\mu \mathrm{m}$ in the standard model described in the second last paragraph of Sec. 2.2. As shown in the aforementioned examples, the most obvious change in the Doppler spectra when changing the vessel diameter is the fraction of non-Doppler shifted light, i.e., the power of the zero frequency. For the speckle contrast, the fraction of Doppler shifted light affects the contrast more pronounced for long exposure times. In the standard model, the fraction of non-Doppler shifted light increased from 28% for a vessel diameter of $5\mu \mathrm{m}$ to 52% for a vessel diameter of $40\mu \mathrm{m}$. The results for vessel diameters between 5 and $100\mu \mathrm{m}$ can be seen in Fig. 4(a).

Fig. 4

(a) Fraction of non-Doppler shifted light as a function of vessel diameter. (b) Relative perfusion decrease compared to perfusion for $5\mu \mathrm{m}$ diameter as a function of vessel diameter.

By calculating the Doppler power spectrum and the contrast from the modeled optical Doppler spectra, perfusion estimates for LDF [Eq. (6)] as well as for LSCI [Eq. (7), $n=1$ or 2, 6 ms exposure time] were also calculated. The relative decrease in the perfusion estimate for a vessel diameter of $40\mu \mathrm{m}$ compared to the perfusion estimate for a vessel diameter of $5\mu \mathrm{m}$ was 26% for LDF perfusion, whereas it was 55% and 65%, respectively, for LSCI perfusion calculated with $1/K-1$ and $1/K^2-1$, respectively, for this particular example. The effect is presented in Fig. 4(b) for vessel diameters between 5 and $100\mu \mathrm{m}$.

In order to study the vessel packaging effect in more than one specific example, 2000 random models were generated with random epidermis thickness, scattering properties, fraction of melanin, blood tissue fractions, oxygen saturation, and speed distribution, i.e., parameters $t_{\mathrm{epi}}$, ${\mu }_{\mathrm{s}}^{\prime }$, $f_{\mathrm{mel}}$, $f_{\text{blood},1}$, $f_{\text{blood},2}$, $S_{{\mathrm{O}}_2}$, and $P_v$, see Table 1. The LDF and LSCI perfusion estimates were calculated both when the models contained homogenously distributed blood in the layers, and when the blood was modeled to be located in vessels with $40\text{-}\mu \mathrm{m}$ average vessel diameter. The average perfusion decrease when the vessel packaging effect for $40\mu \mathrm{m}$ average vessel diameter was included in the model was 25%, 43%, and 52 %, respectively, for LDF perfusion and LSCI perfusion calculated with $1/K-1$ and $1/K^2-1$. There was a correlation between the magnitude of the vessel packaging effect and the blood tissue fraction, where the effect was strongest for low blood concentrations. These findings are summarized in Fig. 5.

Fig. 5

Vessel packaging effect with a $40\text{-}\mu \mathrm{m}$ average vessel diameter in 2000 random models. The data were sorted according to blood tissue fraction and divided into 20 groups. The data points represent the average perfusion change as compared to homogeneously distributed blood, with the error bars showing the standard deviation.