2.1.

Data Acquisition

A digital multichannel LDF probe system (Figure 1) was used to estimate the perfusion generated by a tissue phantom. The LDF system consisted of a 632.8 nm He–Ne laser connected to the LDF probe via one emitting fiber. Backscattered light was guided to a detector system via seven receiving fibers arranged in a row. Multimode step index fibers [numerical aperture (NA)=0.37 ] with a diameter of 230 μm (including 30 μm of cladding) were used. The distances between the center of the emitting fiber and the center of the receiving fiber (fiber separation) was 230n μm, n=1,2,3,…,7.

Figure 1

LDF system, including a tissue phantom that simulates flow, and a custom made LDF probe with different emitting–receiving fiber separations. The light intensity detected in each receiving fiber was sampled by a data acquisition system (DAQ) in terms of the time varying signal (ac) and the light intensity signal (dc).

Backscattered light from each of the seven receiving fibers was detected by two photodetectors. By adding the two photocurrents the light intensity (dc signal) of the LDF signal was detected. The time varying part (ac signal) of the LDF signal was identified by subtracting the photocurrents, amplifying and band pass filtering with a third order Butterworth filter between 20 Hz and 12.5 kHz. By using a differential detector approach, common mode noise was reduced.⁸ Due to the low signal to noise ratio (SNR) of the ac signals of fibers 6 and 7, these fibers were not used for measurements in this study.

The filtered ac and dc signals were sampled at 50 kHz with a 12-bit data acquisition card (MIO-AT-16-10E, National Instruments Corporation) using a range of ±10 V. All data acquisition software was developed using LabView 5.0 (National Instruments Corporation).

2.2.

Tissue Phantom

The tissue phantom (Figure 2) consisted of a number of parallel rotatable disks, with thicknesses of 20–22 μm. Each of these disks was separated from each other by a static layer (approximately 95 μm thick) that contained four pieces of static structure with different optical properties. These pieces were located at the same radial distance from the center of the tissue phantom, with an offset of 90°. All together, the static layers were arranged in a way that resulted in four different stacks of static structures, each with the same optical properties. This assembly was placed in a container filled with index matching fluid (Leica index matching oil). The container was covered by a plate that had four fiber-optic face plates (windows) through which the laser Doppler probes had optical access to the tissue phantoms. These windows (w ₁–w ₄) provided a mechanical barrier, but did not affect the spatial distribution of either the incoming laser beam or light escaping from the phantom tissue. All static layers belonging to the same window had the same optical properties. The rotating disks and the static layers were made of a mixture of polyvinyl alcohol and hollow polystyrene microspheres. These microspheres, with diameters of 1 μm, served as photon scatterers. In addition, various nonscattering dyes (optical absorbers) were added to the static layers. The anisotropy ( 〈cos θ〉, θ=scattering angle) of the disks and the layers was 0.8–0.83. The scattering coefficient μ_s and absorption coefficient μ_a of the tissue phantom, for a wavelength of 633 nm, are given in Table 1. The scattering coefficient and the absorption coefficient were determined by transmission measurements in a Shimadzu UV-2101PC photospectrometer. The scattering coefficient was determined by measuring the collimated transmission through different numbers of reference samples without dyes, hence with zero absorption. An exponential decrease in the collimated light transmission with the total sample thickness was found. Hence, μ_s could be determined by an exponential fit. A similar procedure, with reference samples that only contained dyes, i.e., with zero scattering, was carried out to determine μ_a. The anisotropy was determined for a wavelength of 636 nm by measuring the angular dependence of the transmission, i.e., the phase function, through thin samples with zero absorption. A sufficiently low scattering level was used to ensure single scattering. This method for measuring the scattering phase function was described earlier by Bolt et al.;⁹ All reference samples were from the same batches of raw materials (polyvinyl alcohol solution, particles and dyes) as the layers and disks of the tissue phantom. Also, the same index matching fluid as that in the tester was used. A more thorough description of the optical tissue phantom used in this study was presented by Steenbergen and de Mul.^{10 11 12}

Figure 2

Cross section of the tissue phantom, illustrating two of the four stacks of static structure separated by a number of rotatable disks. All layers are separated by index matching oil. The magnification shows the arrangement of the LDF probe, the fiberoptic face plate (window), rotatable structures and static layers.

Table 1

Measurements were made using single moving disks at 4 different depths and 11 different velocities, equally distributed in the range of 0–3.6 mm/s. The disk depth was not the same for window 4 as it was, for window 1 and window 2, where one extra static layer was present between disk 2 and disk 3. All disk depths (the distance between the phantom’s surface and the center of the rotating disk) that were used are shown in Table 2. Due to problems with air bubbles in window 3, LDF measurements were made only on window 1 (w ₁), window 2 (w ₂), and window 4 (w ₄).

Table 2

2.3.

Signal Processing

According to laser Doppler theory, measured perfusion, Perf _M, can be estimated as^{1 13}

The Doppler spectrum was calculated as a mean over 50 power spectra, where each power spectrum was estimated from 4096 samples of the corresponding ac signal. The dc values were calculated as a mean over 10 000 samples. Calibration of all dc values was carried out by normalizing each channel with its relative amplification factor derived by emitting a constant amount of light into each fiber. This procedure made it possible to compare simulated and measured dc decay as a function of the fiber separation.

Since the tissue phantom uses a rotating rigid lattice of scattering and absorbing particles to simulate flow, the number of moving scatters is constant. Therefore, as long as the maximal Doppler frequency of the backscattered light is lower than the cut off frequency of the low pass filter (12 500 Hz), perfusion will vary linearly with the disk velocity v _disk : ¹⁴

2.4.

Simulation Model

To further understand how photons migrate through a turbid medium, such as in the tissue phantom used in this study, a mathematical model is needed. Due to the need for registering the photon Doppler shifts, the complexity of the measurement setup and the tissue phantom structure, a Monte Carlo simulation approach was chosen. With this technique, it is possible to simulate the pathway of individual photons if the scattering coefficient, absorption coefficient, and phase function of the medium are known. A short review and comparison of the Monte Carlo technique with measurements and diffusion theory was presented earlier by Flock et al.;^{15 16}

For the Monte Carlo simulations, the same geometrical configuration and optical characteristics as those for the tissue phantom were modeled using a disk velocity of 1 mm/s (model 1). The probability distribution of the photon scattering angle was modeled using the Henyey–Greenstein phase function (g=0.815). ¹⁷ To compare LDF spectra, Monte Carlo simulations using one set of optical properties (window 1) and one single rotating disk (disk 2) were carried out for two of the velocities (0.7 and 2.2 mm/s) that were used in the LDF measurements. Furthermore, an extended simulation model (model 2) with the same geometric configuration for all windows and the same optical properties as the tissue phantom was designed. This model was evaluated for both single moving disks and multiple moving disks (all disks), using a disk velocity of 1 mm/s. Simulated disk depths using model 2 are shown in Table 3. Each simulation was carried out for a fixed number of detected photons (500 000), using software developed by de Mul et al.;¹⁸

Table 3

Since a ring shaped detector was used in the simulations, all photons had to be weighted differently depending on their detection radius (see the Appendix). The photon weight factor or intensity p_i is given by

Simulated light intensity, dc _S,n, was estimated by summing all the photon intensities detected by fiber n:

The simulated Doppler power spectrum was estimated by first calculating the frequency distribution H_S,n(k) of the simulated photons, detected by fiber n. A fixed frequency resolution Δf of 12 500/512≈24.4 Hz and a frequency interval of (−12 500, 12 500) Hz were used. Hence, H _sim,n(k) is given by

The power spectrum P_S,n(k) was obtained by convoluting the frequency distribution H_S,n with itself.¹⁸ The corresponding perfusion value was estimated as

2.5.

Influence of Optical Properties

The effect of differences in optical properties on LDF perfusion was evaluated by comparing the two perfusion values Perf(OP_a,d,l) and Perf(OP_b,d,l). Both these values were simulated using the same disk depth (d) and fiber separation (l), but two different sets of optical properties ( OP_a and OP_b ). The ratio, denoted ΔPerf(OP_a,OP_b,d,l), between these two perfusion values was defined as

The maximal influence of optical properties on LDF perfusion, i.e., the maximum perfusion variation, was evaluated for each fiber separation between 0.23 and 1.61 mm by calculating the ratio

2.6.

Sampling Depth

The LDF sampling depth, as a function of the optical properties and fiber separation, was defined as the disk depth at which the perfusion had decayed to e⁻¹ of the maximal perfusion. Due to a limited number of perfusion values, linear interpolation was used to estimate the sampling depths.

3.1.

dc Decay: Measured and Simulated

A good correlation between measured and simulated (model 1) dc decay was found (Figure 3). This indicates that the optical properties and the Monte Carlo model were correct. Both the measured and simulated dc values were normalized against the dc value of fiber 3 and window 2, thereby eliminating differences in amplification between the measurements and the simulations.

Figure 3

Simulated (model 1; solid lines and open symbols) and measured (dashed lines and closed symbols) dc decay as a function of the fiber separation for the optical properties of windows 1, 2, and 4 (S=simulated; M=measured).

3.2.

Doppler Power Spectra: Measured and Simulated

For the two velocities simulated, a good agreement between the frequency distribution of measured and simulated Doppler power spectra was found (Figure 4). This indicates that the Henyey–Greenstein phase function and the anisotropy (g=0.815) of the simulation models are appropriate.

Figure 4

Simulated (model 1; open symbols) and measured (closed symbols) Doppler power spectra for two different rotating disk velocities (0.7 and 2.2 mm/s), using window 1, disk 2, and fiber 1.

3.3.

Perfusion Decay: Measured and Simulated

Simulated and measured perfusion as a function of the disk depth is presented in Figure 5 for fiber 1 and fiber 4. To eliminate differences in amplification all values were normalized against the maximum perfusion detected by each fiber. The result shows that the perfusion decay is faster for fiber 1 than for fiber 4. There is also an increase in perfusion decay with an increase in μ_s (OP_w1 →OP_w2 ) or an increase in μ_a (OP_w4 →OP_w2 ). In comparing simulated perfusion decay with measured perfusion decay a small but systematic discrepancy that increases with the disk depth is seen.

Figure 5

Simulated (model 1; solid lines and open symbols) and measured (dashed lines and closed symbols) perfusion as a function of the disk depth for fiber 1 (a) and fiber 4 (b) (S=simulated;M=measured).

3.4.

Influence of Optical Properties on Simulated LDF Perfusion

Model 2 was used to evaluate the influence of optical properties on LDF perfusion. The result, presented in Figure 6, indicates that the influence of either μ_s or μ_a on LDF perfusion increased with an increase in flow depth and decreased with an increase in fiber separation. Figure 6 also shows that the influence of μ_s on LDF perfusion increased with the magnitude of μ_a [Figure 6(a) vs 6(b)] and the influence of μ_a on LDF perfusion increased with the magnitude of μ_s [Figure 6(c) vs 6(d)].

Figure 6

Influence of the optical properties on the simulated LDF perfusion (model 2) for a range of rotating disk depths. The relative change in perfusion (ΔPerf) due to variations in μ_a (a), (b) or in μ_s (c), (d) is presented for fibers 1, 4, and 7. (a) μ_a: 0.0405→0.212 mm ⁻¹, μ_s≈15 mm ⁻¹; (b) μ_a: 0.0532→0.226 mm ⁻¹, μ_s≈45 mm ⁻¹; (c) μ_s: 14.80→45.55 mm ⁻¹; μ_a≈0.05 mm ⁻¹; (d) μ_s: 14.66→44.85 mm ⁻¹, μ_a≈0.2 mm ⁻¹.

The maximum perfusion variation in the simulation data (model 2 using both single and multiple moving disks), as a function of fiber separation, is presented in Figure 7. For the homogeneously distributed flow model (all disks moving) the maximum variation in perfusion was on average 40, ranging from 22 to 50, for fiber separations between 0.23 and 1.61 mm. In general, all single moving disk models resulted in larger variations than in the model with all disks moving. There was also an increase in the maximum perfusion variation with an increase in disk depth. However, a decrease in variation with an increase in fiber separation could be noticed for disk 3 and below (>0.3 mm).

Figure 7

Simulated maximum perfusion variation, due to variations in the optical properties, for a homogeneous distributed flow model (model 2 with all disks moving) and a discrete flow model (model 2 with single disks moving).

3.5.

Sampling Depth: Simulated

The LDF sampling depth was evaluated using simulation model 2. The result, shown in Figure 8, indicates that the LDF sampling depth increases with a decrease in μ_s ( w ₂ vs w ₁ and w ₄ vs w ₃ ) or μ_a ( w ₁ vs w ₃ and w ₂ vs w ₄ ). The sampling depth also tends to increase linearly with fiber separation. According to the simulations the sampling depth for fiber 2 (0.46 mm fiber separation) is 0.21–0.39 mm.

Figure 8

Simulated (model 2) e⁻¹ LDF sampling depth for a range of fiber separations (0.23–1.61 mm) and optical properties (w _1s –w _4s ).

4.1.

Validity of the Simulations

There was good agreement between measured and simulated dc decay (Figure 3) and LDF power spectrum (Figure 4). This strengthens the validity of the optical properties and the Monte Carlo simulation model. However, perfusion as a function of the flow depth (Figure 5) showed a small systematic mismatch between measured and simulated values. The difference tended to increase almost linearly with the flow depth. The most likely explanation for this is that the disk depths of the tissue phantom presented are slightly smaller than the actual disk depths due to the matching oil between the rotating disks and the static windows. The agreement between measured and simulated data indicates that the results from the expanded Monte Carlo simulations (model 2) reflect measurements with ideal geometry.

4.2.

Influence of the Optical Properties

The influence of optical properties on LDF perfusion was studied in a more general way by evaluating ΔPerf [Eq. (7)], using perfusion data from the expanded simulations (model 2). Figures 6(a)–6(d) indicate that an increase in either the scattering or the absorption coefficient will result in a decrease of LDF perfusion. Specifically, the findings show that the influence of the change in absorption coefficient on the LDF perfusion is rather weak for the most superficial disks, but increases rapidly for larger disk depths [Figure 6(a) vs 6(b)]. According to Figures 6(c) and 6(d), the influence of change in the scattering coefficient on LDF perfusion increases almost linearly with an increase in disk depth. Figures 6(a)–6(d) also indicate that the simulated LDF perfusion readings seem to be affected slightly more by changes in the scattering coefficient than to changes in the absorption coefficient, at least for the most superficial disks. It is also clear that the influence of optical properties on LDF perfusion is not only affected by the change in scattering or absorption coefficient, but also by the magnitude of the nonchanged coefficient. For example, a change in the absorption coefficient has a larger impact on LDF perfusion when the scattering coefficient is relatively large [Figure 6(a) vs 6(b)]. This can be understood by realizing that a larger scattering level will increase the path length of the photons detected, thus making the effect of absorption and changes in absorption larger. The same is valid for a change in the scattering coefficient [Figure 6(c) vs 6(d)]. These results show that it is not only the optical properties or the actual flow depth that affects the LDF perfusion, but rather a combination of both. One way of interpreting this result is that the optical properties affect the sampling depth and thereby make the LDF readings sensitive to the actual flow depth.

By increasing the fiber separation the influence of optical properties on LDF perfusion decreased when measuring at discrete disk depths. This effect is most pronounced when a change in the scattering coefficient occurs [Figure 6(c) vs 6(d)]. However, all perfusion readings on human skin originate from photons that have been Doppler shifted at different skin depths. This means that the depth distribution of the scattering events of the detected photons, i.e., the sampling depth, will affect the perfusion measured. This distribution is not only determined by the optical properties of the sampling volume but also by the fiber separation; an increase in fiber separation will increase the sampling depth. As mentioned before, the influence of optical properties on LDF perfusion is greater for deep blood flow than for superficial. Therefore, an increase in fiber separation does not guarantee that the influence of optical properties on LDF perfusion decreases when measuring distributed blood flow (e.g., human skin). This is illustrated in Figure 7, where a both homogeneous and discrete flow distribution model is assumed. In the case in which all disks move, the maximum perfusion variation [Eq. (7)] is somewhat independent of the fiber separation, on average 40 and ranging from 22 to 50. Figure 7 also indicates that the maximum perfusion variation increases with the flow depth. Therefore, it is logical to assume that the maximum perfusion variation is even greater when studying real tissue, where the unknown heterogeneous blood perfusion distribution is assumed to increase with the skin depth.

4.3.

Sampling Depth

Assume a homogeneous blood flow distribution, no multiple Doppler shifts and that the distribution of the photons scattering events detected decreases exponentially with the depth. The definition of sampling depth used then implies that ∼63 of the perfusion signal is generated in the region above this depth. According to Figure 8 the LDF sampling depth increases with an increase in fiber separation. It is also obvious that the optical properties have a strong influence; an increase in either scattering or the absorption coefficient will result in a decrease in sampling depth. Similar trends of the absorption coefficient for fiber separations larger than 1 mm have previously been reported by Weiss et al.;⁶ However, their predictions of the sampling depth were larger than ours, probably because they used a lattice random-walk model with a point source and a point detector. The influence of probe geometry, using Monte Carlo simulations, has previously been investigated by Jakobsson et al.;⁵ They found similar trends with regard to fiber separation but a more superficial sampling depth than we found, probably because they studied the median depth of the photons pathways detected.