2.1.

Experimental Setup for the Spatially Resolved Reflectance Method

The experimental setup is presented in Fig. 1. A multimode fiber (numerical aperture, NA = 0.22) delivers light from a 7-W tungsten-halogen light source (⊗) by means of a spherical mirror (f = 15 cm, NA = 0.13) and a flat folding mirror onto the sample. The sample's surface is positioned in the image plane of the illumination optics. The magnification equals 1. Therefore, the spot size for illumination equals the fiber diameter, which is 400 μm. An oblique incidence of 30-deg is used to avoid the Fresnel reflection entering the detection beam path. The detection optics are made up of two achromatic lenses (f = 10 cm, NA = 0.08), which image the remission from an area of the sample surface with a magnification of 1 onto the detection fiber's end face. Once again, the spot size for the collection is 400 μm. The detection fiber and the corresponding optics can be moved continuously as a whole to change the source-detector separation. It is defined as the distance between the center of the illumination and detection spot. This enables radial scans of backscattered light to be obtained. An important aspect is the ability to measure the remission for negative and positive distances so that it is possible to check if the point of maximum overlap is at the distance zero.

The spectrometer used for detection has a spectral range from 200 to 1100 nm due to an integrated order filter, a wavelength resolution of 4 nm, 14 bit dynamic range, 300:1 full signal signal-to-noise ratio (SNR), and the entrance NA is matched to the detection fiber NA (HR4000 from Ocean Optics with 100-μm slit and HC1 grating). Considering also the fiber transmissions and the lamp spectral intensity, the measuring range was 400–1000 nm.

2.2.

Spatially Resolved Reflectance Measurements

The measurements on the samples are carried out using a self-written LabVIEW program controlling the shutter of the halogen lamp, the spectrometer, and the piezomotor stages. The measurement is carried out automatically, except for the alignment of detection and illumination optics, which is done manually beforehand. The program records two spectra for every source-detector separation, one with illumination on and a dark spectrum with illumination off. The integration time is autoadjusted to receive a “light-on” signal between 90 and 100% of the saturation level. The integration time for the dark spectrum is set to the same value. The dark spectrum is subtracted from the first one, and the difference is saved to a file. The detection optics are moved according to tabulated distances (–0.5 mm, –0.4, –0.3, 0.20, –0.1, –0.05, 0, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2 mm) between the illumination spot and the recording points of the remission profile. Data points from 0.5 mm distance and above are used for the determination of the optical properties. The negative distances are included because a means is required for checking if the origin of the distance axis is really at zero. To have a higher resolution for detecting a shift of the origin, we acquired the remission with a smaller step width near the origin. The autoadjustment of the integration time to the intensity of the remission is repeated after each movement. The different integration times are accounted for by dividing the signal by them. To estimate the statistical errors, the sample is moved laterally 0.1 mm after each scan and then a completely new remission profile is recorded. This is repeated five times, and then all remission profiles are averaged. The time needed for measuring the remission for one source-detector separation strongly depends on the sample absorption and scattering coefficient. It ranges from a second for high albedo, to tens of seconds for low albedo. Statistical errors are due to the detector noise, which contributes mainly for source-detector separations of >1.5 mm. The SNR varied between 50 for source-detector separations of <1 mm and 10 for the samples with the largest amount of absorber and source-detector separations of 2 mm. The halogen lamp was tested for time variations, and this effect could be excluded as a source of error. The motor for positioning of the detection optics is made for microscope applications and is far more exact than is needed for this SRR setup. Therefore, an uncertainty in source-detector separation can also be neglected. In contrast, the stray-light contribution to systematical errors must be considered because its signal overlaps with the remission signal. This effect becomes larger for larger source-detector separations. Therefore, it is necessary to protect all optical components against dust after establishing their cleanliness (not shown in Fig. 1).

2.3.

Normalization of the Spatially Resolved Measurement Data

The measured reflectance spectra have to be normalized in order to correct them for the illumination efficiency of the light source, the collection efficiency of the fibers, and the spectrometer sensitivity, as well as being able to compare the measurement to the MCS. The aim of the normalization is to infer the power and spectral distribution of the illumination immediately before the light enters the sample. This is usually done by measuring the reflectance of a reference standard. Ideally, a nonabsorbing Lambertian surface reflector is needed for this but such a material is not available.

A pressed titanium dioxide (TiO₂) powder was used as a reference standard, and during preparation, a facial tissue was placed between the die and the powder to produce a rough surface. It is therefore assumed that the white standard is a Lambertian reflector. For a detection angle of 0 deg, and angles of incidence between 0 and 40 deg, this is validated by the measurements presented in Ref. 22 (see Chapter II-e). Titanium dioxide was measured in the ISS against Spectralon. Spectralon is a trade name for sintered polytetrafluorethylen and is a common white standard with a reflectance higher than 98% in the wavelength range of 400–1000 nm.²³ The measurement showed that titanium dioxide has a reflectance of >97% in the wavelength range between 450 and 1000 nm, with a maximum of 99% at 550 nm. From 450 to 400 nm, the reflectance decreases from 97 to 75%. In this range, the correction for the decrease in reflectance of titanium dioxide improves the determined optical properties of the phantoms by the SRR method. The calibration with the titanium dioxide standard was carried out in two steps: spectral calibration and intensity calibration. The spectral calibration is frequently carried out using the spectrometer and the same fibers that are used for the measurement of the sample. The intensity calibration is only carried out if the last measurement was taken several days prior. To perform the spectral calibration, both source and detection fibers have diameters of 400 μm. The illumination and detection fibers are positioned so that the illumination and detection spots overlap [Fig. 2]. In this way, the spectral distribution of the reference is determined but some of the light is still remitted outside of the detection area and is not collected. This loss has to be quantified by the intensity calibration, which is done by comparing the total power delivered by a 550-μm-diam detection fiber with the delivered power of the 400-μm-diam fiber using a light power meter. The 550-μm-diam fiber detects all but a negligible part of the remitted light [Fig. 2]. The ratio between the signal of the 400-μm detection fiber and the 550-μm detection fiber was 0.72. Using the titanium dioxide standard, a detection fiber with a 1000-μm diameter does not significantly increase the signal anymore, whereas using the Spectralon standard it does. The ratio between the signal of the 1000-μm detection fiber and the 550-μm detection fiber was then 0.82. A larger diameter fiber would be too stiff to be practical, which is the reason why we use titanium dioxide instead of Spectralon. The wavelength dependence of the signal from the titanium dioxide standard was measured with lasers and laser diodes of the wavelengths 425, 488, 532, 543, 633, 670, 750, and 820 nm. The ratio of the signals from the different detection fibers was 0.72 ± 0.05 for all laser wavelengths, and no systematic wavelength dependency was found.

Fig. 2

(a) Equal illumination and detection area: Light remitted off the illumination spot is not detected. (b) Larger detection area than illumination area: Light remitted off the illumination spot is detected.

Advantageously, we can determine the reference spectra from overlapping illumination and detection spot. If the reference spectrum is not measured with zero source-detector separation, then the relative contribution from different wavelengths to the signal will change, which is not desired. With fiber applicators, it is only possible to measure the remission of the white standard with a nonzero source-detector separation.

The data obtained from the phantom measurements is normalized according to Eq. 1, which is similar to the one presented in Ref. 24,

2.4.

MCS

To determine the absorption and reduced scattering coefficients from the measurement data, a numerical or theoretical model is needed to model the light transport through the sample. In our case, this is the MC simulation.

The light is regarded as a stream of particles, and so interference and polarization effects are neglected, which is valid for incoherent light sources such as halogen lamps. Although it is, strictly speaking, incorrect, here we will refer to these particles as photons. The basic principles of the MC algorithm are described in Ref. 25 (see Chapter 3). The MC algorithm launches photons at a specified position and with a specified direction distribution outside the medium and sends them onto a random walk through the medium. When the photons arrive at the sample surface, some are reflected according to Fresnel's formula and some are refracted according to Snell's law. Inside the medium, the particles can be absorbed, scattered, and/or total reflected at the surface or refracted again leaving the medium. Some of the escaped photons arrive at the detector within the acceptance angle and contribute to the signal. The main output of the MCS is the ratio between photons arriving at the detector and the total number of photons launched.

Determining the light distribution inside the sample from the absorption coefficient μ_a, reduced scattering coefficient [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ , scattering anisotropy g, refractive index n, and the experimental geometry is known as a forward problem. This is in contrast to the so-called inverse problem, where the coefficients μ_a, [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ , and g are determined from the measured detector signals.

2.5.

MCS of Spatially Resolved Reflectance

The described forward MC algorithm (Section 2.4) was adapted to the SRR setup. With the SRR setup, the remission is measured for different source-detector separations. Therefore, the MCS should be able to calculate the remission for specified source-detector separations taking into account the size of illumination and detection spots, both numerical apertures of illumination and detection, the optical properties of the sample, and the sample size and thickness. When the optical properties of the sample do not change laterally and the illumination is normal to the surface, the probability of detecting a photon is dependent only on the distance between the entrance and exit point beside the dependence on the optical properties. This distance is different from the source-detector separation because the entrance and exit points can be anywhere inside the illumination and detection spots (Fig. 3). The simulation algorithm exploits this property, and therefore, oblique incidence is not supported. To gain the remission for a specific source-detector separation, the probability p for collecting a photon must be integrated over the illumination area A and the detection area A ^′ to give the probability [TeX:] $p(r_{sd}) = \int_A {dA} \int_{A^\prime } {dA^\prime} \,p(|{\bm r} - {\bm r}^\prime |)$ $p\left(r_{sd}\right)={\int }_{A}dA{\int }_{A^{\prime }}d{A}^{\prime }p\left(\right|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)$ . Here, r is a vector to a point in A, [TeX:] ${\bm r}^\prime $ ${\mathbf{r}}^{\prime }$ to a point in A ^′ and r _sd is the source-detector separation. With a weighting function w(r), this integral can be simplified to [TeX:] $p(r_{sd}) = \int_a^b {dr} \,w(r)p(r)$ $p\left(r_{sd}\right)={\int }_a^{b}drw\left(r\right)p\left(r\right)$ , where a is the smallest and b is the largest possible distance between entrance and exit points.¹¹

Fig. 3

Geometry for spatially resolved reflectance Monte Carlo. Illumination: Numerical aperture of illuminating light beam, Detection: Numerical aperture for collecting photons, n: refractive index, r: distance between illumination and detection spot.

The described MC algorithm was compared to multi layer Monte Carlo simulation (MCML),¹² which has already been accepted as the reference MCS in the tissue optics community. The data show excellent agreement between the two simulation algorithms for the perpendicular incidence case and μ_a ranging from 0.001 to 1.2 mm⁻¹ as well as [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ ranging from 0.1 to 10 mm⁻¹.

The photons are launched with a constant probability distribution for the incidence direction within the illumination aperture. The focus is on the sample surface. The proportion of those photons that are not reflected by the surface are propagated through the sample until they exit the medium again. To calculate the probability p(r) needed for the above integral, the number of detected photons leaving the sample at a certain distance r is divided by the number of incident photons. Only the photons with a direction within the numerical aperture of the detector contribute to the signal. The simulation does this numerical integration for all source-detector area separations used in the experiment.

The MCS replaces virtually the lower lens of the detection optics by a detector with the same diameter. It does not simulate the light passing through the lenses and through the detection fiber. Nevertheless, as the acceptance angle of the collection fiber is larger than the aperture of the detection optics, and as the acceptance angle of the spectrometer equals that of the fiber, this should not lead to a discrepancy between simulation and measurement if the proper detection aperture is considered.

2.6.

Using a Lookup Table to Infer Absorption and Reduced Scattering Coefficient

The forward MCS, which was adapted to SRR, was used to calculate a lookup table with remission profiles for many combinations of μ_a and [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ . A search algorithm was made to compare the table entries with the measurement data, consecutively. The absorption and reduced scattering coefficients corresponding to the best matching profile were then read from the table. In order to have a measure as to just how well a particular simulated remission profile matches with the measurement data, the program calculates the deviation η(r) between the logarithms of simulated and measured data for each source-detector separation [Eq. 2]. Then, the average is computed over all distances in a given range [Eq. 3]. By this procedure, the noisier data points from larger source-detector separations are weighted less because the signal decreases with distance. By searching the profile with the smallest [TeX:] $\bar \eta$ $\overline{\eta }$ , μ_a and [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ of the sample can be determined. For all the simulations and measurements made in the phantom study described in Section 3, [TeX:] $\bar \eta$ $\overline{\eta }$ < 0.06,

The lookup table calculated with the forward MCS, for the SRR samples had a size of 40×40 remission profiles. The number of simulated photons was 5×10⁶. The 40 [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ values were varied within the range from 0.01 to 8 mm⁻¹ and the 40 μ_a values ranged from 0.001 to 2.3 mm⁻¹. The table was calculated for a constant g of 0.8 and extended by linear interpolation along the grid lines. This results in a higher resolution for the determination of the optical parameters (Δμ_a = 0.007 mm⁻¹, Δ [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 0.02 mm⁻¹). According to the experimental setup, the illumination beam convergence angle was set to 7.12 deg and the circular spot size on the sample surface to 400 μm. The detector is specified by the acceptance angle, which was set to 90 deg to save calculation time, although this is in contrast to the experiment (4.6-deg acceptance angle). The circular detection area was set to a diameter of 400 μm, which is again in agreement with the setup. To avoid having to create a new lookup table for every sample thickness, phantoms were made as thick as 1.5 cm so that they could be considered approximately semi-infinite for the investigated parameter range. Two MCS were conducted with μ_a set to zero and [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ set to 0.06 mm⁻¹, the lowest reduced scattering coefficient of the phantom set. In the first run, the thickness and radius of the sample were set to 1.5 and 1.75 cm, the real dimensions of the phantom. In the second run, the thickness and radius were set to 10 and 5 cm. The relative deviation between both simulations was <1% for all source-detector separations between 0 and 2 mm. All other phantoms have a higher reduced scattering coefficient and/or a higher absorption coefficient; thus, the assumption of semi-infinite samples should be true for all phantoms.

2.7.

Integrating Sphere Setup

A double-beam ISS measurement with a subsequent inverse MCS (iMCS) served as an independent method for the determination of μ_a, [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ beside SRR. The essential part of the spectrometer (Lambda 900 from PerkinElmer) is the integrating sphere, as seen in Fig. 4. The sphere has two ports with sample holders along the direction of the incoming light. The sample is placed at the left port to collect the transmitted light. For the hemispherical transmission, the right port was closed with a Spectralon standard and for the diffuse transmission it was not. The sample is placed at the right port for acquisition of the hemispherical reflection of the sample. In order to avoid uncertainties in the assignment of Fresnel reflection and diffuse reflection, the reflection measurements were performed with a closed Fresnel port. The holder at the reflection port is tilted at an angle of 8 deg to the direction of the incident light. The detector is at the bottom of the sphere and shielded against direct reflexes of the walls by a baffle (not shown). The port diameters are 2 cm, and the spot size of the beam on the sample was 5 mm. The measurements were carried out in the spectral range from 400 to 1000 nm in agreement with SRR.

Fig. 4

Overview of the sample chamber of the integrating sphere spectrometer used as an independent method to evaluate SRR.

The accuracy of the spectra measured in the ISS is limited by the dark and reference spectra with an uncertainty of 0.5% (400–850 nm) and 2% (850–1000 nm), which is already the sum of both errors. The discrepancy in the accuracy is due to a detector change at 850 nm.

2.8.

Alternative Determination of μ_a and [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ Using an ISS

An iMCS is used to relate either two spectra to μ_a and [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ or three spectra to μ_a, [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ , and g.²⁶ The iMCS algorithm was developed by the Laser-und Medizin-Technologie GmbH, Berlin especially for the PerkinElmer ISS described. The algorithm incorporates information about the sphere diameter, port sizes, beam width, and convergence, and the tilted sample holder as well, and takes account of it. The iMCS assumes a perfect glossy sample and an ideal reflecting sphere. This causes differences between the measured spectra and the simulated spectra for rough samples. In our case, the samples had a smooth surface so that the theoretical reflection coefficient agreed well with the real one.

The transmission and reflection are nonlinear functions of μ_a, μ_s, and the scattering phase function, which in our case was the one from Henyey-Greenstein.²⁸ If the selected values for the optical parameters correspond to the parameters of the sample, then the functions equal the measured transmission and reflection. Finding a solution for the optical parameters, for which the simulated and measurement spectra agree, is therefore equal to solving the system of the nonlinear equations for the transmission and reflection. For this task, the iMCS applies the Newton–Raphson root-finding algorithm.²⁷ The estimated spectra are calculated by a forward MCS. The optical parameters μ_a, [TeX:] ${{\rm \mu}^\prime }_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ and the anisotropy of the phase function are varied according to the Newton–Raphson algorithm, until the computation result coincides within the error range to the measured spectra. The basis for the ISS forward MCS is the same as for the spatially resolved MC code (Section 2.5), which was compared to MCML.¹²

In our case, the sample index of refraction was set to n = 1.41 (silicone rubber) and the initial values were set to μ_a = 0.001 mm⁻¹ and [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 0.001 mm⁻¹ for every wavelength. For some samples, the influence of the start parameters on the determined optical properties were tested and there were only negligible differences found. The threshold error level for the termination of the iteration was set to 2%, which is also roughly the accuracy for the transmission and reflection. In case of the samples with 0 or 0.1% titanium dioxide, a difference was measured between the hemispherical and the diffuse transmission. Then, three spectra were given as input to the iMCS to determine μ_a, [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ , and g. As shown in Section 3.2, the average value for g was 0.8. This value was used for the other samples where hemispherical and diffuse transmission could not be distinguished. For the sake of comparison with SRR, we calculated μ^′ _s from μ_s and g. The samples were 1 mm thick and 1.5 cm in diameter, so that while using a spot size of 5 mm, we could neglect side losses out of the sample.

3.1.

Optical Properties of Silicone Rubber

Silicone rubber is an optically clear and transparent material throughout the visible wavelength range. Transmission and reflection spectra (including Fresnel and diffuse reflection) of a solid silicone cylinder, 22 mm in height and 38 mm in diameter, were recorded with an integrating sphere spectrometer as described above. An IMCS with start values of μ_a = 0.001 mm⁻¹, [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 0.001 mm⁻¹, g = 0, and a sample index of refraction n = 1.41 was used to determine the optical parameters μ_a and [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ (Fig. 5).

Fig. 5

Absorption (●) μ_a and reduced scattering coefficient (◯) [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ of the silicone base material determined from the transmission and reflection spectra.

The material has a low absorption (μ_a < 0.001 mm⁻¹) and low scattering coefficient ( [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ < 0.004 mm⁻¹) between 400 and 1000 nm, except for 900 nm, where the silicone probably has a small absorption peak. Below 500 nm, there is a broad absorption band as well as increased turbidity in this wavelength range. With a magnifying glass, it was possible to see small amounts of impurities, such as dust particles, in the sample, which could lead to the higher scattering at shorter wavelengths. However, the absorption and scattering of the silicone rubber can be neglected compared to the absorption and scattering of the color paste and titanium dioxide in the 400–1000-nm wavelength range and the concentrations used.

3.2.

Optical Properties of the Basis Phantoms without Either Color Paste or Titanium Dioxide

The color paste is a viscous, reddish looking, opaque fluid. We used the phantoms without titanium dioxide to determine the absorption and reduced scattering coefficients of the color paste dependent on its concentration (Fig. 6).

Fig. 6

Absorption μ_a and reduced scattering coefficient [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ divided by the concentration c (1% = 0.01) of the color paste containing phantoms determined by the ISS method for different absorber concentrations (relative difference: <15%).

The measurements on the thin 1-mm phantoms were carried out with the ISS (see Section 2.8). As has already been shown in Section 3.1, the absorption and scattering of the base material can be neglected in the wavelength range between 400 and 1000 nm. Therefore, the data presented here are representative for the optical properties of the pure color paste. The material shows an absorption band below 600 nm, with the maximum absorption being located at 540 nm. The extrapolated absorption coefficient for 100% color paste is 150 mm⁻¹ in the maximum. The μ_a range covered by the color paste concentrations 0.1–0.8% is 0.15–1.2 mm⁻¹. The reduced scattering coefficient is of the same order as the absorption coefficient. It is evident from this that the scattering coefficient of the color paste must be taken into account for all subsequent phantom measurements. Furthermore, [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ is considerably lower in the spectral range of high absorption and higher in the spectral range of low absorption. This effect looks like cross talk between the absorption and reduced scattering coefficient. Despite this, the anisotropy factor is approximately constant in the wavelength range of the absorption band (Fig. 7). The value of g varies between 0.73 and 0.81 with an average of 0.76. The ISS determined absorption and reduced scattering coefficient scale linearly with the concentration of the color paste.

Fig. 7

Anisotropy factor g of the color paste determined by the ISS method for different absorber concentrations.

From the samples without color paste, we only show the reduced scattering coefficient (Fig. 8) because the residual absorption coefficient was nearly constant over the wavelength range 400–1000 nm with an average of 0.01 mm⁻¹ and independent of the concentration. The reduced scattering coefficient falls monotonically with wavelength. For the wavelength 1000 nm, it is about four times lower than for the wavelength 400 nm. Titanium dioxide at a concentration of 1% leads to a reduced scattering coefficient of 0.8 mm⁻¹ at 400 nm and 0.18 mm⁻¹ at 1000 nm. The anisotropy factor varies between 0.71 at 400 nm to 0.83 at 1000 nm with an average of 0.8 (Fig. 9).

Fig. 8

Reduced scattering coefficient of titanium dioxide containing phantoms determined by the ISS method for different concentrations of titanium dioxide. The reduced scattering coefficient was divided by the concentration. (relative error <10%).

Fig. 9

Anisotropy factor of the phantoms without color paste determined by the ISS method.

3.3.

Optical Properties of the Phantoms with Color Paste and Titanium Dioxide

After describing the basic materials, the results of the SRR and ISS measurements are described in this section for the entire set of phantoms. At first, we tested for the linearity of absorption and reduced scattering coefficients on the concentration of color paste and titanium dioxide, according to

The determined versus the expected values for the ISS measurements are shown in Fig. 10. The reduced scattering coefficient is linear to the concentration as shown by the straight solid line achieved from linear regression. The slope is 1.094 ± 0.004, which means that the determined values are, on average, 10% higher than the expected values over the range 0.2 < [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ < 10 mm⁻¹. The determined absorption coefficient is also linear to the concentration of the color paste and is about 14 ± 0.5% higher than expected in the range 0.04 < μ_a <1.2 mm⁻¹, which was also achieved by linear regression. For expected absorption coefficients of <0.04 mm⁻¹, the data are too noisy and the true values cannot be resolved anymore.

Fig. 10

Determined versus expected reduced scattering and absorption coefficients from ISS measurements. The data in the wavelength range 400–1000 nm of all but the basis phantoms were combined in these diagrams. The dashed lines indicate the ideal agreement. The solid lines are linear regression results ( [TeX:] ${{\rm \mu}^\prime}_{\rm s}^{{\rm determined}}$ ${{\mu }^{\prime }}_{\mathrm{s}}^{\mathrm{determined}}$ = 1.09 [TeX:] ${{\rm \mu}^\prime}_{\rm s}^{{\rm expected}}$ ${{\mu }^{\prime }}_{\mathrm{s}}^{\mathrm{expected}}$ , [TeX:] ${{\rm \mu}^\prime}_{\rm a}^{{\rm determined}}$ ${{\mu }^{\prime }}_{\mathrm{a}}^{\mathrm{determined}}$ = 1.14 [TeX:] ${{\rm \mu}^\prime}_{\rm a}^{{\rm expected}}$ ${{\mu }^{\prime }}_{\mathrm{a}}^{\mathrm{expected}}$ ). The vertical dotted line shows the lower limit of the measuring range.

Figure 11 shows the determined versus the expected absorption values from SRR measurements. The determined absorption coefficients of >0.04 mm⁻¹ can be fitted by a straight line, which shows their linear dependence on the concentration. The values are 26 ± 1% larger than expected, which makes a difference of 12% between ISS and SRR absorption results. Absorption coefficients of <0.04 mm⁻¹ could not be resolved anymore.

Fig. 11

Determined versus expected absorption coefficients from SRR measurements. The data in the wavelength range 400–1000 nm of all but the basis phantoms were combined in this diagram. The dashed lines indicate the ideal agreement. The solid line results from a linear regression (μ_a ^determined = 1.26μ_a ^expected).

The same diagram for the reduced scattering coefficient is shown in Fig. 12. Most of the points accumulate along the ideal line, but some of the points have a large deviation from the expected values. This diagram does not resolve the dependence of the determined [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ on the expected μ_a. Therefore, these dependencies are shown in Fig. 12. The deviation is approximately constant for μ_a < 0.03 mm⁻¹. For high μ_a and [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ , the deviation from expected is larger and negative compared to low μ_a and low [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ . There is cross talk between μ_a and [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ for expected μ_a larger than 0.03 mm⁻¹. In order to have a measure for the statistical errors, the systematic deviations can be fitted by polynomials in order to remove them (Fig. 13).

Fig. 12

(a) Determined over expected reduced scattering coefficients from SRR measurements. The data in the wavelength range 400–1000 nm of all but the basis phantoms were combined in this diagram. The dashed line indicates ideal agreement. (b) The relative deviation between determined and expected [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ values versus the expected μ_a and expected [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ . The solid line results from polynomial fits.

Fig. 13

Determined over expected reduced scattering coefficients from SRR measurements after correction for systematic deviations with a polynomial fit. The data in the wavelength range 400–1000 nm of all but the basic phantoms were combined in this diagram. The dashed line indicates ideal agreement.

After the correction, there is still a considerable deviation left, but the improvement can be clearly seen. An evaluation showed that 70% of the data points have a deviation of <10% and 96% of <20%. This agrees with an average relative error of 10% of the absolute values.

3.4.

Comparison of Simulations for Perpendicular Incidence and 90-deg Detection Aperture to Simulations for 30-deg Incidence Angle and 4.6-deg Detection Aperture

In order to investigate the influence of the incidence angle and detection aperture used in the simulation, we also conducted MCS for an oblique incidence angle of 30-deg and/or a detection solid angle of 4.6-deg, which corresponds to the geometry of the setup. With oblique incidence, the cylindrical symmetry cannot be exploited anymore resulting in worse photon statistics of the simulation result. A weighting function is also not used. The number of photons in each simulation was adjusted to similar SNR, such as in the simulation for perpendicular incidence and 90-deg detection aperture (10⁸ instead of 5×10⁶ photons).

Figure 14(a) shows a comparison of the MCS for both illumination and detection configurations for [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 0.5 mm⁻¹ and μ_a = 0.01, 0.1, 0.6, or 1.2 mm⁻¹ (The error bars belong to the simulation with 30-deg incidence and 4.6-deg detection aperture). According to the Lambert cosine law, the remission for 90-deg detection aperture was multiplied by a factor 1−cos²(4.6°) to make it comparable to the remission for 4.6-deg detection aperture. The remission for 30-deg incidence and 4.6-deg detection aperture is lower for short source detector separations, which is in agreement with the expectation that the remission becomes more and more Lambertian the higher the source detector separation is. The higher the absorption coefficient is the larger the range of distances is in which this discrepancy occurs. Figure 14(b) for [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 1 mm⁻¹ shows that the discrepancies for short source detector separations become lower than the uncertainty of the simulation results when [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ is increased. For [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 10 mm⁻¹ in Fig. 14(c) the deviation for short source detector separations disappears and there is only a deviation for μ_a = 0.6 mm⁻¹ for source detector separations of >1.2 mm. This is attributed to the bad SNR for these data points because the difference between both simulation results is lower than the uncertainty from the lack of simulated photons. For the simulation with μ_a = 1.2 mm⁻¹ and distances of >1.4, no photon was collected. Therefore, the data points are not shown.

Fig. 14

Comparison of the MCS remission for both illumination and detection configurations (perpendicular incidence, 90-deg detection aperture versus 30-deg incidence /4.6-deg detection aperture) for [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ = 0.5, 2 and 10 mm⁻¹. The data for 90-deg detection aperture was multiplied by a factor 1-cos²(4.6 deg).

This comparison of MCS with perpendicular incidence and 90-deg detection aperture versus 30-deg incidence and 4.6-deg detection aperture showed that the assumption of Lambertian remission of the samples is valid for [TeX:] ${{\rm \mu}^\prime}_{\rm s}$ ${{\mu }^{\prime }}_{\mathrm{s}}$ > 1 mm⁻¹ and μ_a <1.2 mm⁻¹. This is true for the phantoms with 0.3–0.9% titanium dioxide and 0.1–0.8% color paste, which renders our normalization procedure valid for these samples. For the samples without titanium dioxide, the precondition is not fulfilled and, therefore, we could only use the integrating sphere measurements from these samples.