2.1.

Measurement Setups

The time-resolved measurements were performed with a time-correlated single photon counting setup as shown in Fig. 1(a). A supercontinuum source (SC450-6, Fianium, Southampton, United Kingdom) provided white-light pulses with a repetition rate of 50.2 MHz, which were fed into an acousto optical tunable filter to select single wavelengths. The experiments were performed at a wavelength of $\lambda =740\mathrm{nm}$ with a spectral bandwidth of 10 nm FWHM, avoiding autofluorescence of the samples present at lower wavelengths. A computer-controlled circular neutral density filter with an optical density between 0 and 4 was used to adjust the count rate of the single photon counting system, ensuring a count rate as high as possible but $<{10}^6$ counts per second. The light was focused with an achromatic lens into a glass fiber with a core diameter of 200 μm and then injected into the sample. The average power at the sample was $\sim 3\mu \text{W}$. Transmitted or reflected photons were collected with a second fiber (core diameter 400 μm) also in contact with the sample surface at an SDS $\rho$. A hybrid photomultiplier HPM-100-50 and a single photon counting PC board SPC-130 (both Becker&Hickl, Berlin, Germany) were used for detection and recording of the time-of-flight histograms.

Fig. 1

Sketch of the time-correlated single photon counting setup. The output of the supercontinuum source was connected to the acousto optical tunable filter (AOTF) that selects light with a single wavelength out of the pulsed white light and couples it into a single-mode (SM) fiber with collimators on both ends. The count rate of the experiment was controlled with the circular variable neutral density (ND) filter. The light was coupled into a multimode step index fiber with a core diameter of 200 μm and injected into the sample for transmittance (a) or reflectance (b) measurements. The detection fiber with a core diameter of 400 μm guided the collected light to the hybrid photomultiplier (PMT) that was controlled by the PC via the detector control card (DCC-100). Detector and source were connected to the single photon counting board (SPC-130), which recorded and saved the time-of-flight histograms.

For transmission measurements, the bare ends of the glass fibers were inserted in opposite holders made of black PVC with positioning holes. One of the fibers could be displaced using fine screws to adjust the SDS. For reflectance measurements, the bare fibers were inserted in metal tubes placed in a holder of teflon with positioning holes every 2 mm [Fig. 1(b)]. In both cases, the fiber ends were in contact with the sample surface. Small pieces of black sealing strip around the fiber tips were used to prevent direct illumination or surface reflections. For measurements of the instrument response function (IRF), the fiber ends were placed face to face and were only separated by a 100-μm thin layer of white teflon. The IRF had an FWHM of $\sim 260\mathrm{ps}$ at 740 nm.

Measurements of the spatially resolved steady-state reflectance were also performed at $\lambda =740\mathrm{nm}$ using the optical system as described by Foschum et al.³² In this system, narrow spectral bands of a white-light xenon source are selected by a spectrograph and focused onto the sample with a spot size of $\sim 600\mu \text{m}$. The reflectance signal is then captured by a cooled CCD camera. Additionally, the reflectance from spruce wood samples was also measured with a modified version of the setup with a spot size of $\sim 50\mu \text{m}$ and a smaller field of view.

2.2.

Samples

Measurements were performed on two different samples: soft spruce wood and bovine ligamentum nuchae. Spruce wood as a bulk material can be approximately described by a model of cylindrical fibers, representing the tracheids that are oriented in the axial growth direction of the tree or branch, combined with additional random scatterers. The applicability of this model has been verified with Monte Carlo simulations reproducing both the time-resolved and spatially resolved experimental reflectance. In the spatial domain, the simulations clearly showed reflectance patterns of an elongated shape with different orientations to the tracheids depending on the distance from the point of light incidence.¹⁸ The samples of wood were all cut from one piece of spruce with the tracheids aligned either parallel or perpendicular to the surface. In particular, three samples of $69\times 55{\mathrm{mm}}^2$ in the $x$- and $y$-directions with the wood fibers oriented in the $z$-direction as shown in Fig. 2(a) with thicknesses of 9.9, 14.8, and 19.8 mm were used. An additional sample of $146\times 55\times 4.7{\mathrm{mm}}^3$ with the tracheids running parallel to the longest side, as in Fig. 2(b), as well as a block with $92\times 68\times 55{\mathrm{mm}}^3$, which represented an optically semi-infinitely extended sample for reflectance measurements, were investigated. All wood samples were air-dry and kept at room temperature. The refractive index of spruce wood was assumed as $n=1.19$, based on the estimation for dry soft wood by Kienle et al.¹⁸

Fig. 2

Sketch of the measurement setup for transmittance measurements. The source fiber is in contact with the bottom side of the spruce wood sample. The detection fiber on the top side can be translated with respect to the incident direction with an SDS of $\rho$. The tracheids are aligned in the $z$-direction (a), or in the $x$-direction (b). Therefore, the first case is radially symmetric, whereas in (b), the SDS can be increased parallel and perpendicular to the tracheids. For reflectance measurements, both fibers are placed on the top side. In all cases, the illumination is located at $\rho =0$.

Bovine ligamentum nuchae has to support the weight of the head and neck and also allow movement in all directions. In order to be of high tensile strength but also be flexible and elastic, it is composed mainly of a system of elastin fibers and collagen fibrils.³³ Elastin is one of the most insoluble proteins³⁴ that is, among other specific properties, durable and thermally stable, making it important as a biomaterial and in tissue engineering.^35,36 The nuchal ligament of large mammals has one of the highest contents of elastin of any tissue.³⁷ The two main fibrous constituents, elastin and collagen, are linked closely together and arranged parallel to the axis of the ligament. The elastic fibers of the adult ligament are straight cylinders with a diameter between 4 and 14 μm^35,38,39 with an amorphous core.⁴⁰ They are enveloped by collagen fibrils of $\sim 70\mathrm{nm}$ as a sheath,⁴¹ which also fills the space between the elastin fibers.^33,35 The composition of the bovine nuchal ligament was determined to be $\sim 70\%$ and 15 to 20% of the dry weight for elastin and collagen, respectively,^37,42 which agrees with numbers from histological staining.³³ The in vivo water content is assumed to be 56.5% of its original weight.^37,42 The refractive index of elastin is given by Lansing et al. as 1.534, whereas for water $n=1.33$ was determined by Hale and Querry.⁴³ Based on these two main components, we estimate a refractive index of $n=1.42$ for the nuchal ligament.

The fresh samples of ligamentum nuchae were obtained from a butcher one to three days before the experiments and were kept cooled in the refrigerator in the intermediate time. The samples were carefully freed from leftover fat and connective tissue with a scalpel, making the fibrous structure clearly visible on the whole surface. The different samples of nuchal ligament were between 10 and 20 mm thick and $\sim 300\times 60{\mathrm{mm}}^2$ in the $x$- and $y$-directions with the elastin fibers running parallel to the longest side, similar to that shown in Fig. 2(b).

2.3.

Diffusion Theory

In order to retrieve the optical properties, the solutions of the anisotropic and isotropic diffusion equations in the time domain were used. The transmittance $T(x,y,t)$ and reflectance $R(x,y,t)$ for an infinite slab with a thickness $l_z$ is given by

The relation of the diffusion tensor elements to the reduced scattering coefficients ${\mu }_s^{\prime }$ for different directions $i$ was assumed to be

3.1.

Wood

As pointed out, the time-resolved transmittance at $\rho =0\mathrm{mm}$ was shown to yield the correct result for the reduced scattering coefficient in the direction of the incident beam for a wood model with cylinders even when using the isotropic diffusion equation. ³⁰ In addition, the absorption coefficient can be determined from reflectance measurements with large SDSs.¹⁸ The values obtained for these measurements with the isotropic DT are marked with an asterisk and are given in all figures for comparison. For spruce wood, these values are the reduced scattering coefficient ${\mu }_{s\parallel }^{\prime *}$ parallel to the tracheids from transmittance measurements with $\rho =0\mathrm{mm}$ (see Fig. 4) as well as ${\mu }_{s\perp }^{\prime *}$ in the perpendicular direction (see Fig. 5) and the absorption coefficient ${\mu }_a^{*}$ obtained from reflectance measurements at large SDSs [see Fig. 6(a)].

Fig. 4

Absorption and reduced scattering coefficients obtained by fitting the time-resolved transmittance from slabs of wood with fibers aligned to the $z$-direction with increasing translational distance $\rho$ between illumination and detection fiber. The value for the isotropic diffusion theory (DT) at $\rho =0\mathrm{mm}$ in (b) is used for reference as ${\mu }_{s\parallel }^{\prime *}$ in the following. As explained in the text, the reduced scattering coefficient ${\mu }_{s\perp }^{\prime *}$ with its standard deviation $\sigma$ from transmittance perpendicular to the fibers at $\rho =0\mathrm{mm}$, as well as ${\mu }_a^{*}$ from reflectance measurements for large SDS are given for comparison.

Fig. 5

Absorption and reduced scattering coefficients derived from time-resolved transmittance measurements on slabs of wood with fibers aligned along the $x$-direction for the two cases ${\rho }_{\parallel }$ and ${\rho }_{\perp }$, where the translation $\rho$ is increased parallel or perpendicular to the wood fibers, respectively. The value for the reduced scattering coefficient at $\rho =0\mathrm{mm}$ is used as the reference value ${\mu }_{s\perp }^{\prime *}$ for comparison in the other figures. For the anisotropic model, the fit parameters are the absorption coefficient and the ${\mu }_s^{\prime }$ of the named direction, while the reduced scattering coefficient for the other direction is fixed to the value ${\mu }_s^{\prime *}$ obtained with the isotropic solution for $\rho =0\mathrm{mm}$.

Fig. 6

Absorption and reduced scattering coefficients as obtained from time-resolved reflectance measurements on a block of wood with fibers aligned in the $x$-direction for increasing distances from the illumination fiber, ${\rho }_{\parallel }$ and ${\rho }_{\perp }$, in direction of and perpendicular to the wood fibers, respectively. In (a) the mean value for the results at an SDS of 10 to 20 mm is used as reference value ${\mu }_a^{*}$.

First, the transmittance measurements from slabs of wood with the tracheids aligned along the $z$-axis [see Fig. 2(a)] are presented. Figure 4 shows the results for increasing the translational distance $\rho$ between the illumination and detection fibers. The result obtained for $\rho =0\mathrm{mm}$ is ${\mu }_{s\parallel }^{\prime *}=1.18\pm 0.12{\mathrm{mm}}^{-1}$ and is used in the following for comparison. With larger distances, ${\mu }_s^{\prime }$ increases due to the increasing influence of the higher scattering ${\mu }_{s\perp }^{\prime }$. As expected from the results for Monte Carlo data, the values obtained by the isotropic DT rise faster than that for the anisotropic solution with the value for the reduced scattering coefficient perpendicular to the fibers set to ${\mu }_{s\perp }^{\prime *}$. When using the value ${\mu }_{s\parallel }^{\prime *}$ as a fixed value, the anisotropic DT allows fitting for the reduced scattering coefficient perpendicular to the tracheids, at least for larger SDSs. The trends of the curves are in agreement with the results from Kienle et al.³⁰ Also, the results for ${\mu }_a$ show agreement with ${\mu }_a^{*}$ from reflectance measurements. The constant results for the anisotropic DT and the rising values for the isotropic model are again consistent with the comparison of the results obtained with Monte Carlo simulations.³⁰

Second, we present transmittance measurements with the wood fibers aligned with the $x$-direction, as illustrated in Fig. 2(b). There are two possibilities for the translation of the detection fiber from the illumination point, parallel and perpendicular to the wood fibers, denoted as ${\rho }_{\parallel }$ and ${\rho }_{\perp }$, respectively. The results for both measurements are shown in Fig. 5.

Again, the reduced scattering coefficient in the direction of the incident light can be obtained with the isotropic DT for $\rho =0\mathrm{mm}$, which in this case is ${\mu }_{s\perp }^{\prime *}=14.5\pm 0.9{\mathrm{mm}}^{-1}$. With increasing translational distance ${\rho }_{\parallel }$ between the glass fibers, the results obtained with the isotropic DT decrease because of the increasing dependency on light propagation along the fibers, whereas the anisotropic DT yields good results with small deviations, especially for large distances. This is similar to the results shown in Figs. 16 and 18 of Ref. 30. If illumination and detection fibers are separated in the direction perpendicular to the wood fibers, the intensity drops much faster due to higher scattering, and the maximum SDS that could be measured was ${\rho }_{\perp }=6\mathrm{mm}$. The light propagation is dominated by scattering in the plane perpendicular to the cylindrical wood fibers and, therefore, in the direction of measurement. When fitting for ${\mu }_{s\perp }^{\prime }$, the isotropic DT yields the same values as the anisotropic DT, since both models reduce to the same equation.

The obtained absorption values from these transmittance measurements show good agreement, except for the case where the isotropic DT was used for measurements with translation in the direction of the wood fibers as shown in Fig. 5(a). The obtained negative values indicate the inapplicability of the isotropic model. In the study on Monte Carlo data in Ref. 30, the obtained absorption coefficients also showed similar trends with increasing distance $\rho$, but mostly overestimated the simulation values by 10 to 20%. For the measurements, the mean values for large SDSs have relative differences up to 6% from ${\mu }_a^{*}$, but at the same time, the standard deviations for different measurement positions show differences of $\sim 30$ to 45% [see error bars in Fig. 5(a)].

Altogether, the results for ${\mu }_{s\parallel }^{\prime }=1.2{\mathrm{mm}}^{-1}$ and ${\mu }_{s\perp }^{\prime }=14.5{\mathrm{mm}}^{-1}$ are in good agreement with earlier time-resolved reflectance measurements on soft wood,¹⁸ where ${\mu }_{s\parallel }^{\prime }=1.8{\mathrm{mm}}^{-1}$ and ${\mu }_{s\perp }^{\prime }=13{\mathrm{mm}}^{-1}$ were obtained for the directions parallel and perpendicular to the tracheids, respectively. They also agree with results from time-resolved transmittance measurements on spruce by Bargigia et al., who determined the optical coefficients for wavelengths from 700 to 1200 nm.²⁰

In a last step, experiments were performed on the block of spruce wood, which can be optically treated as a semi-infinite geometry. The results for the reflectance measurements with increasing SDS are shown in Fig. 6. The scattering in the direction of the wood fibers ${\mu }_{s\parallel }^{\prime }$ can be determined quite accurately with the anisotropic DT for ${\rho }_{\parallel }$ between 8 and 14 mm with a relative difference $<5\%$ compared to ${\mu }_{s\parallel }^{\prime *}$ obtained from the transmittance measurements at $\rho =0\mathrm{mm}$. For a larger SDS, the obtained results of isotropic and anisotropic DT are very similar and the reduced scattering coefficient is slightly underestimated as compared to ${\mu }_{s\parallel }^{\prime *}$. This is similar to results of the study with Monte Carlo data,³⁰ where the anisotropic DT also approaches the expected value but the isotropic DT did not reach it within the examined distances. In contrast to transmittance measurements, it was not possible to determine ${\mu }_{s\perp }^{\prime }$ from the reflectance at an SDS ${\rho }_{\parallel }$, supposedly because of the small penetration depth due to high scattering (results not shown). For measurements in the perpendicular direction, the strong decrease in intensity allowed the retrieval of the optical properties only at smaller distances. The values of ${\mu }_{s\perp }^{\prime }$ from these measurements are 10 to 15% smaller than the reference value ${\mu }_{s\perp }^{\prime *}$ from transmittance measurements. The values for the absorption coefficient shown in Fig. 6(a) have comparable mean values as obtained by the transmittance measurements with the wood fibers aligned along the $x$-direction. However, the standard deviations are significantly smaller. Previous studies comparing the isotropic diffusion equation and Monte Carlo simulations show that the absorption coefficient can be obtained for a large SDS.¹⁸ Therefore, the absorption coefficient of the wood samples is obtained as the mean value for the measurements with an SDS between 12 and 20 mm, that is, ${\mu }_a^{*}=1.2\pm 0.2\times {10}^{-2}{\mathrm{mm}}^{-1}$. This value is used as the reference value in all other figures.

The time-resolved reflectance from the block of wood was also measured with the tracheids oriented not only parallel or perpendicular to the measurement direction. For all orientations, the SDS was chosen to be as large as possible, but to still allow sufficient count rates and, therefore, peak levels at least two orders of magnitude higher than the background. The used SDS ranged from 14 mm for the parallel direction to $\rho =6\mathrm{mm}$ for all angles $>50\mathrm{deg}$.

The results are shown in Fig. 7. As can be expected, the value of ${\mu }_s^{\prime }$ obtained with the isotropic diffusion equation increases with the angle between the wood fibers and the measurement direction. When the anisotropic DT is used, the value for ${\mu }_{s\parallel }^{\prime }$ can be determined quite accurately up to an angle of 50 deg. On the other hand, the value of ${\mu }_{s\perp }^{\prime }$ can be determined quite well for angles $>60\mathrm{deg}$, even though the results slightly underestimate the value from the transmission measurements ${\mu }_{s\perp }^{\prime *}$. This is in agreement with the results for measurements at a large SDS ${\rho }_{\perp }$ shown in Fig. 6(b). The values obtained for ${\mu }_a$ are comparatively constant over all orientations and also show only small differences regardless of which theory is used. This confirms that time-resolved reflectance measurements allow the retrieval of the absorption coefficient for a large SDS unaffected by the fiber direction. However, these measurements yield mean values that are consistently smaller than those obtained from the reflectance measurements with increasing SDSs, see Fig. 6(b), but still within one standard deviation.

Fig. 7

Absorption and reduced scattering coefficients obtained from time-resolved measurements of the reflectance from a block of spruce wood with different angles for the orientation of the wood fibers with respect to the $x$-direction. SDS was adjusted from $\rho =14\mathrm{mm}$ for 0 deg to 6 mm for all angles $>50\mathrm{deg}$. To determine the reduced scattering coefficients ${\mu }_{s\perp }^{\prime }$ or ${\mu }_{s||}^{\prime }$, the anisotropic DT was computed with the fixed values ${\mu }_s^{\prime *}$ for the respective other direction as previously determined by transmittance measurements on slabs for $\rho =0\mathrm{mm}$.

3.2.

Ligamentum Nuchae

The time-resolved transmittance on the samples of ligamentum nuchae is measured as in Fig. 2(b) with the elastic fibers in the $x$-direction and the translations ${\rho }_{\parallel }$ and ${\rho }_{\perp }$, where the SDS is increased parallel or perpendicular to the elastic fibers, respectively. The derived optical parameters are averaged over all measurements on the ligament at different positions with varying thickness from 11 to 16 mm because of the variations in size between the different samples. As for wood, the obtained value for the reduced scattering coefficients for $\rho =0\mathrm{mm}$ is again taken as a reference value ${\mu }_{s\perp }^{\prime *}=2.39\pm 0.32{\mathrm{mm}}^{-1}$ in the direction perpendicular to the elastic fibers. The results for the isotropic DT for a larger SDS decrease toward the lower value of ${\mu }_{s\parallel }^{\prime }$ that is reached with the anisotropic DT for a smaller SDS when using ${\mu }_{s\perp }^{\prime *}$ as a fixed value, see Fig. 8(b). When the mean of the results for a large SDS ${\mu }_{s\parallel }^{\prime }=0.7\pm 0.2{\mathrm{mm}}^{-1}$ is set as given, the anisotropic DT allows us to obtain good results for ${\mu }_{s\perp }^{\prime }$ for measurements in both directions. The isotropic diffusion again yields the same results as the anisotropic equation for measurements with translations ${\rho }_{\perp }$, because in that case, both models have the same equation. The obtained values of the absorption coefficient are fairly uniform except for the results obtained with the isotropic diffusion equation for ${\rho }_{\parallel }$, which again yields negative values.

Fig. 8

Absorption and reduced scattering coefficients obtained by fitting the time-resolved transmittance from ligamentum nuchae with the elastin fibers aligned to the $x$-direction with increasing distance between illumination and detection fiber, for the two cases ${\rho }_{\parallel }$ and ${\rho }_{\perp }$. For the anisotropic DT, ${\mu }_a$ is fitted together with ${\mu }_s^{\prime }$ of the named direction with the reduced scattering coefficient for the other direction fixed to the value ${\mu }_s^{\prime *}$ obtained with the isotropic solution for $\rho =0\mathrm{mm}$.

Reflectance measurements on nuchal ligament (not shown) were done for slabs instead of a semi-infinite geometry as for the spruce wood due to the finite size of the ligament samples. For measurements in the direction of the fibers, the isotropic DT results for ${\mu }_s^{\prime }$ could only be obtained for measurements with a large SDS. Nevertheless, negative absorption values were obtained, indicating the inacuraccy of the isotropic model. The values for ${\mu }_{s\parallel }^{\prime }$ determined with the anisotropic DT leveled off around $0.4{\mathrm{mm}}^{-1}$ and, thus, were smaller than for the transmittance measurements. The same behavior could be observed with the wood samples where the results from transmittance measurements at larger SDSs are 15 to 20% higher than ${\mu }_{s\parallel }^{\prime *}$ (see Fig. 5), whereas the values obtained from the reflectance measured at a large SDS but for a semi-infinite gemoetry instead of a slab are slightly smaller. For increasing ${\rho }_{\perp }$, on the other hand, the results rise to the reference value ${\mu }_{s\perp }^{\prime *}$ determined from the transmittance measurements. The results for ${\mu }_a$ show large deviations for all $\mathrm{SDS}<18\mathrm{mm}$ and yield lower values than the transmittance measurements. So for both models, the obtained optical properties have large differences compared to the expected values from transmittance measurements. With the isotropic DT, it was not even possible to yield results for any but a very large SDS and then negative values were obtained for the absorption coefficient.

Overall, the results for the reduced scattering of bovine ligamentum nuchae are comparable to the values determined from measurements on bovine achilles tendon (approximately ${\mu }_{s\parallel }^{\prime }=2.3{\mathrm{mm}}^{-1}$ and ${\mu }_{s\perp }^{\prime }=0.3{\mathrm{mm}}^{-1}$), see Ref. 29. However, in this study,²⁹ the isotropic DT for a semi-infinite geometry was used, which led to deviations because of the finite size of the achilles tendon sample in addition to its composition of different tissue structures.