4.1.

Absorption

In this section, the absorption of the matrix material is discussed. It is a very important measure since it defines the lowest absorption value possible with this phantom material. The absorption spectrum measured from a pure phantom without scatterers as described in Sec. 3 is shown including standard deviations as a faint green area in Fig. 3. The epoxy resin that is presented can be used as phantom material for biological media, since its intrinsic absorption is much lower than that of most biological tissue consisting mainly of water (black-dashed line) and fat. Furthermore, the low absorption allows for good control and adjustment using absorbing agents. Additional spectra concerning the aging process are shown that are described in the next subsection. All spectra exhibit characteristic peaks ranging from 0.0002 to $0.2{\mathrm{mm}}^{-1}$ in the considered wavelength range of 450 to 1650 nm. We found 10 of the peaks to be harmonics of ${\mathrm{CH}}_3$ and OH bonds.³² For the peaks at 710, 1190, and 1430 nm, no match could be found. The relatively large errorbars for small absorption values ($650<\lambda <850\mathrm{nm}$) are due to intrinsic scattering, which could not be completely compensated for. Around $\lambda =1\mu \mathrm{m}$, both spectrometers have a poor signal-to-noise ratio and cause a residual noise spectrum. For high absorptions, only the short cylinder phantoms could be used and the related uncertainty in length is propagated to that of ${\mu }_{\mathrm{a}}$. The absorption values determined with the transmission setup for wavelengths below 650 nm are too high due to undetected back-scattered light.

Fig. 3

Absorption value ${\mu }_{\mathrm{a}}$ of the matrix material as measured by different techniques: transmission measurement for samples without scattering agents (faint green area). Combined SRR and ToR measurements of phantom “Ref” (solid gray lines), phantom “Test” during aging process in a 60°C oven (long-dashed green lines, ${\mu }_{\mathrm{a}}$ increases with longer times in the oven) and a 1-year-old phantom with similar configuration (solid blue line). For comparison, the absorption value of water (black-dashed line, measured by Pope and Fry³⁰ and Kou et al.³¹) is presented.

4.2.

Stability in Time

The aging-related alteration of optical properties of the phantoms is discussed in this section. We assume that scattering is not affected by aging as both the mineral-based scatterers and the epoxy matrix material are not expected to change their RI with age. Most pigments are lightfast, so no change of their absorption is expected. However, the yellowing of many plastics suggests an alteration of the epoxy resin absorption with time. Therefore, we cast two phantoms using identical ingredients. The concentration of WPP was chosen such that ${\mu }_{\mathrm{s}}^{\prime }(\lambda =633\mathrm{nm})=0.8{\mathrm{mm}}^{-1}$ and no absorber was added. After curing for 5 days, both phantoms (“Ref” and “Test”) were measured with the ToR setup using the reduced scattering value from the SRR experiment. Afterwards “Test” was put into an oven with a constant temperature of 60°C for 4 weeks. It was removed only for several intermediate measurements. The high temperature in the oven accelerates the aging of the phantom “Test.” The rate of this aging cannot be precisely predicted; however, assuming a 10% increase in reaction speed for every Kelvin temperature rise, the corresponding ages of “Test” at the measurement dates are 0.75, 1.75, and 3.6 years. As expected, no drift in ${\mu }_{\mathrm{s}}^{\prime }$ could be observed (not shown). The results for ${\mu }_{\mathrm{a}}$ are presented in Fig. 3, where the absorption spectra for “Ref” (solid gray lines) and “Test” (long-dashed green lines) are shown as well as the spectrum of a 1-year-old phantom (solid blue line) stored in a cabinet and taken out regularly for performing measurements.

The results clearly show a change of the absorption spectrum of the epoxy resin with age. Both “Test” with increasing time in the oven and the 1-year old phantom show a clear rise of ${\mu }_{\mathrm{a}}$ for wavelengths shorter than 600 nm. Between 600 and 700 nm, the increase of ${\mu }_{\mathrm{a}}$ is much smaller, so it can hardly be detected when an absorber is added. At longer wavelengths, no drift can be found. The spectra of “Test” and the 1-year-old phantom qualitatively differ: since multiple chemical reactions are expected to be involved in the aging process, they most likely also differ in their acceleration rate. Thus, no phantom in the oven corresponds to a real-aged phantom for all spectral changes. Please note additionally that exposure to ultraviolet radiation, i.e., sunlight, is also expected to yellow phantoms. This absorption change is probably in the same spectral range as the reported aging. Nevertheless, exposure of an optical phantom to sunlight for longer periods should be avoided.

Interestingly, Moffitt et al.¹⁸ showed in a profound examination a drop in ${\mu }_{\mathrm{a}}$ for their PU material with increasing age. Please note that in order to compensate the error due to a slightly imprecise measurement distance in the ToR setup for determination of ${\mu }_{\mathrm{a}}$, we modified the reflectance data by a factor of 0.99 to 1.01 over the whole spectrum such that the absorption spectra match for $\lambda \approx 650–850\mathrm{nm}$.

4.3.

Scattering

The scattering of most turbid media is much higher than that of the almost transparent matrix material, therefore, scatterers have to be added. The determination of the scattering coefficient of the matrix material allows to estimate its influence and shows that it can normally be neglected. Figure 4(a) gives the result of the measurements described in Sec. 2. The contribution to the scattering coefficient of the matrix material is ${\mu }_{\mathrm{s}}<0.04{\mathrm{mm}}^{-1}$ which is very low. Since that contribution is likely to result from spatial variations in the RI or crystallized areas, there is no point in comparing the result to a spherical model such as Mie theory. Please note that insufficient stirring will give rise to flowmarks and scattering will increase in this case.

Fig. 4

(a) Intrinsic scattering coefficient ${\mu }_{\mathrm{s}}$ of the matrix material measured using a short-pass and no filter. Error bars represent the standard deviation of the measurements of samples of different lengths. (b) Experimental (exp.) data of the refractive index. Crosses (solid line) and diamonds (dashed line) correspond to the epoxy resin system and ethyl cinnamate, respectively. The lower values (blue crosses, right axis) demonstrate the precision of our measurement system by comparing measured and reported RI values of water. The circles represent values published by Daimon and Masumura.³³ The curves represent the best fit of parameters of the Cauchy model function [see Eq. (8)] to the experimental data, the values are given in the text.

4.4.

Refractive Index

In Fig. 4(b), the RI of the epoxy resin is shown (black crosses) as well as the dispersion relation of ethyl cinnamate (black diamonds). The latter one is well suited as an immersion fluid for optical coupling of epoxy resin phantoms, even though it has a stronger dispersion relation. The errorbars represent the standard deviation of measurements on all three sides of the prism. The Cauchy model equation

In the literature, RI data can be found for ethyl cinnamate in the range $n(\lambda =589.6\mathrm{nm})=1.55\dots 1.561$.^14,34 The fit curve yields $n(\lambda =589.6\mathrm{nm})=1.555$. No dispersion relation of ethyl cinnamate is yet reported and no RI data can be found for the given epoxy resin.

The RI of our optical epoxy resin strongly differs from that of biological tissue. However, this is of no relevance if the actual RI is properly considered in a method’s evaluation model. Preliminary attempts showed that by adding 10% of ethanol into the epoxy, it is possible to greatly reduce the RI value, however, this also leads to changes in absorption and scattering.

4.5.

Fluorescence

The presence of intrinsic fluorescence can render the inverse derivation of optical properties nearly impossible. It can especially falsify time-domain measurements because of the influence of the fluorescence lifetime. This underlines the necessity for measurement of the phantom’s fluorescence characteristics.

Our measurements of the intrinsic fluorescence emission spectra show that weak fluorescence occurs below 500 nm. At longer wavelengths, the noise was larger than the residual fluorescence signal. This underlines the suitability of the chosen epoxy system for optical measurements, as we could detect fluorescence at much higher wavelengths for other epoxy resins. Note the difference of intrinsic and intended fluorescence. As shown in Sec. 6.2, it is possible to add rhodamine 6G to achieve a controlled fluorescence.

5.1.

Titanium Dioxide and Aluminium Oxide

The optical properties of the epoxy resin phantoms with suspended ${\mathrm{TiO}}_2$ or ${\mathrm{Al}}_2{\mathrm{O}}_3$ scatterers are presented in the left and right columns of Fig. 5, respectively. In the first row, the most important optical property for diffusive light propagation, i.e., the reduced scattering coefficient ${\mu }_{\mathrm{s}}^{\prime }$, is depicted and has been scaled by the weight concentration $c_{\mathrm{w}}$. For ${\mathrm{TiO}}_2$, measurements of the spatially resolved (circles), time-resolved (stars), and spatial frequency (squares) reflectance are used for determination of ${\mu }_{\mathrm{s}}^{\prime }$ for various phantoms with weight concentrations of the ${\mathrm{TiO}}_2$ dispersion between 0.14% and 1.1%. Each curve represents one phantom and for clarity the error bars are not shown. A widely used empirical power law was fit (black solid curve) to the experimental data yielding ${\mu }_{\mathrm{s}}^{\prime }(\lambda )=795{\mathrm{mm}}^{-1}{(\lambda /{\lambda }_0)}^{-1.21}$, where ${\lambda }_0=633\mathrm{nm}$. This fit curve is used when casting new phantoms. We estimate the uncertainty to be 10%, then all measurements that have been performed lie in this uncertainty range. This uncertainty includes both systematic errors of the used measurement setups and analyses as well as errors in the phantoms’ manufacturing process. The main error of the latter is insufficient homogenization. First, this applies for the WPP. If the pigment paste is not stirred properly, scattering particles have sunk down and the concentration of scatterers at the surface is lower than expected. Second, the homogenization of the ingredients is very important since it prevents flowmarks and local variations in the optical coefficients and third, during curing particles sink down and agglomerate at high concentrations $c_{\mathrm{V}}>1\%$. Errors due to the weighing process and mixing stock solutions are negligible. Using pharmaceutical scales, an evaluation of the error propagation reveals less than 0.2% relative error of the final concentration. In comparison, results for ${\mathrm{Al}}_2{\mathrm{O}}_3$ of measurements of the top and bottom sides of one phantom are shown using the spatially resolved (circles) and the spatial frequency (squares) reflectance. The fit to the experimental data yields ${\mu }_{\mathrm{s}}^{\prime }(\lambda )=38.5{\mathrm{mm}}^{-1}{(\lambda /{\lambda }_0)}^{-0.68}$. Here, we assume an uncertainty of 15% due to the measurement of only one phantom.

Fig. 5

Optical properties ${\mu }_{\mathrm{s}}^{\prime }$ (a,b), ${\mu }_{\mathrm{s}}$ (c,d), $g$ (e,f) and $\gamma$ (g,h) of the ${\mathrm{TiO}}_2$ dispersion (a,c,e,g) and the ${\mathrm{Al}}_2{\mathrm{O}}_3$ powder (b,d,f,h). The methods used for obtaining these results are described in the text. Note that for the goniometer measurement of $g$ and $\gamma$, the scattering agents were dispersed in ethyl cinnamate instead of epoxy resin.

By comparison of the results for both scatterers, we draw two conclusions: first and expectedly, ${\mu }_{\mathrm{s}}^{\prime }$ for ${\mathrm{TiO}}_2$ is much higher than for ${\mathrm{Al}}_2{\mathrm{O}}_3$. This is due to the enormous RI mismatch of ${\mathrm{TiO}}_2$³⁵ and epoxy resin ($n_{\mathrm{TiO}2}/n_{\text{epoxy}}\approx 1.75$) clearly exceeding that of ${\mathrm{Al}}_2{\mathrm{O}}_3$³⁸ and epoxy resin ($n_{\mathrm{Al}2\mathrm{O}3}/n_{\text{epoxy}}\approx 1.13$). Second, the scattering power, i.e., the exponent of the assumed power law, is higher for ${\mathrm{TiO}}_2$. Zonios and Dimou³⁹ correlated the scattering power with the scatterer’s average size for a fixed RI mismatch. Their empirical fit would give an average diameter of $d\approx 272\mathrm{nm}$ for ${\mathrm{TiO}}_2$ and $d\approx 530\mathrm{nm}$ for ${\mathrm{Al}}_2{\mathrm{O}}_3$. The first value agrees surprisingly well with the size distribution derived from multiple electron microscope images $d=(250\pm 30)\mathrm{nm}$, even though a part of the scattering power is caused by the very strong RI dispersion of ${\mathrm{TiO}}_2$. A representative image can be found below in Fig. 6(a). In Rayleigh scattering, the scattering power is normally assumed to be 4, which also represents its maximum without regarding a wavelength dependence of the RI. Due to the strong dispersion relation, for small ${\mathrm{TiO}}_2$ particles, it can even exceed 4.8. The mean diameter of ${\mathrm{Al}}_2{\mathrm{O}}_3$ particles $d\approx 530\mathrm{nm}$ is a surprisingly low result, since the manufacturer’s data stated a typical $d_{50}$-value of $1.5\mu \mathrm{m}$. Here, a discussion of the estimated sizes can be found below.

For OCT and most microscopy techniques, the scattering coefficient ${\mu }_{\mathrm{s}}$ is a key parameter as it usually dominates the attenuation of a beam inside tissue. Experimental results for ${\mu }_{\mathrm{s}}$ of both scattering agents can be found in the second row of Fig. 5. The mean value of multiple phantom measurements is shown and the standard deviation is given as a shaded area. Similar to our ${\mu }_{\mathrm{s}}^{\prime }$ results, the scattering coefficient of ${\mathrm{TiO}}_2$ is much higher than that of ${\mathrm{Al}}_2{\mathrm{O}}_3$ at the same weight concentration. Additionally, the wavelength dependence is similar to that of ${\mu }_{\mathrm{s}}^{\prime }$. Below 500 nm, a systematic drift of ${\mu }_{\mathrm{s}}$ of ${\mathrm{Al}}_2{\mathrm{O}}_3$ can be observed, which is most likely caused by stray light in the spectrometer. The anisotropy value $g$ is shown in the third row of Fig. 5. It can be derived by calculation of the mean cosine of the phase function which was measured by our goniometer setup (blue solid line with error bars) and also by comparing the graphs of the first and second rows in Fig. 5 using

The results from that calculation are shown in green solid lines with a shaded area representing the error propagation. There is a large difference in $g$ between the scattering agents. For ${\mathrm{TiO}}_2$, the anisotropy value is around $0.60\pm 0.05$ in the considered wavelength range, while for ${\mathrm{Al}}_2{\mathrm{O}}_3$, $g$ is nearly constant around $0.915\pm 0.015$. This difference is the basis of the very idea of using two scattering agents and will be discussed below. Note the large error bars for the goniometer measurement of the ${\mathrm{Al}}_2{\mathrm{O}}_3$ powder. The error bars correspond to the standard deviation of three independent measurements with an added systematic error estimate. As stated above, the measured phase function has to be extrapolated for the first 10 deg. In a medium with a high anisotropy value $g$, this extrapolation becomes very important which leads to the large errorbars for the ${\mathrm{Al}}_2{\mathrm{O}}_3$ powder. In the ${\mathrm{TiO}}_2$ dispersion, the forward scattering is weak, thus the extrapolation does not have a strong influence on the analysis of $g$. The Mie model (black lines) is described below.

Results for the phase function related parameter $\gamma$ are shown in the fourth row of Fig. 5. Assuming the often used Henyey–Greenstein phase equation, it is $\gamma =1+g$. With this in mind, we can see that this is only supported in the NIR, so for the presented scatterers, the Reynolds-McCormick phase function model might be suited better. As shown by Bevilacqua and Depeursinge,³⁷ $\gamma$ is in the range from 0.9 to 2.7 for Mie scatterers. With the presented scattering agents, it is possible to cover a range from 1.4 to 2.2 for wavelengths around 500 nm which almost covers the “physiological” range from 1.3 to 2.4.²³ In contrast to the analysis of $g$, the extrapolation of the phase function hardly influences the analysis of $\gamma$ because of the weak contribution of the forward-scattered light. As already stated, the scattering agents were dispersed in ethyl cinnamate instead of epoxy resin for the goniometer measurements. The influence of the small change of the matrix material’s RI is depicted by calculations using Mie theory. When we assume a Gaussian size distribution for the ${\mathrm{TiO}}_2$ particles of $d=(330\pm 100)\mathrm{nm}$, a good agreement of the calculated value of $g$ and the experimental values is achieved. As depicted in the figure, the change due to the RI difference in the matrix material is much smaller than the measurement uncertainty. The calculated anisotropy values of ${\mathrm{TiO}}_2$ in ethyl cinnamate (black solid line) differ less than 0.01 from those in epoxy resin (black dashed line).

For ${\mathrm{Al}}_2{\mathrm{O}}_3$, a Gaussian size distribution does not agree well with both $g$ and $\gamma$, while the sum of two Gaussians does. We show calculations for 35% of the particles (with respect to volume concentration) having a diameter $d=(250\pm 35)\mathrm{nm}$ and 65% with $d=(1.9\pm 0.5)\mu \mathrm{m}$. Subsequently, we took scanning electron microscope (SEM) images [see Fig. 6(b)] that qualitatively confirmed the existence of particles with sizes around 300 nm. It can be found that a slight change of the matrix material’s RI has a significant influence on $g$ and $\gamma$ below $\lambda <600\mathrm{nm}$ due to the smaller RI mismatch of ${\mathrm{Al}}_2{\mathrm{O}}_3$ and the matrix material compared to ${\mathrm{TiO}}_2$. Calculations for other particle sizes show a similar influence of the matrix material for both scattering agents.

The assumed particle size of ${\mathrm{TiO}}_2$ is larger but in the same order as the result $d=(250\pm 30)\mathrm{nm}$ from SEM images [see Fig. 6(a)]. However, the application of Mie theory neglects two facts. First, the Mie model’s assumption of spherical particles contradicts the elongate and polygonal shape revealed in SEM images. Second, ${\mathrm{TiO}}_2$ shows strong birefringence, while Mie theory requires a scalar RI. We assumed the RI to be that of the ordinary ray.³⁵ When assuming the RI of the extraordinary ray, calculations lead to smaller anisotropy values. The assumed size distribution of ${\mathrm{Al}}_2{\mathrm{O}}_3$ is not in accordance with the $d_{50}=1.5\mu \mathrm{m}$ value provided by the distributor. Again, this deviation could be caused by the inapplicability of Mie theory for the elongated ${\mathrm{Al}}_2{\mathrm{O}}_3$ particles with rough surfaces. As they are expected to be larger than the wavelength, the particles’ shapes can strongly influence their scattering behavior. Please note that the given Mie calculations are not for comparison of experiment and theory, but for estimating the influence of the matrix material’s RI.

5.2.

Adjusting the $g$- or $\gamma$-Value

When the optical properties of both scattering agents are known, it is possible to mix them in order to adjust not only ${\mu }_{\mathrm{s}}$ or ${\mu }_{\mathrm{s}}^{\prime }$, which is usually done, but also $g$ or $\gamma$. When mixing two scatterers (subscripts 1 and 2), the scattering events are weighted with ${\mu }_{\mathrm{s}}$. The higher order Legendre moments can thus be calculated as

6.1.

Absorbing Pigments

Absorption spectra of seven pigments dispersed in epoxy resin are given in Fig. 8. The scattering of the absorbing pigments is too high to be neglected in the collimated transmission but too low to observe diffuse light propagation, so we control the scattering value by addition of a scattering agent. To obtain the given absorption spectra, we studied various scattering phantoms with and without absorbing pigments using the SRR and the ToR setups and divided the obtained difference in absorption value by the weight concentration of the pigments. Three of the shown absorbing pigments are even useful for wavelengths beyond 1000 nm. We estimate the error to be $\mathrm{\Delta }{\mu }_{\mathrm{a}}/{\mu }_{\mathrm{a}}\approx 10\dots 20\%$. A statistical analysis is not possible, since we only produced one phantom per pigment.

It can be seen that all seven pigments have quite different spectral characteristics. We use the pigments for finding the dynamic range in terms of ${\mu }_{\mathrm{a}}$ of our spectroscopic setups and for mimicking optical properties of muscle tissue. Haematite (also called “bloodstone”) has a similar characteristic to deoxyhemoglobin; however, the absorption edge at $\lambda \approx 600\mathrm{nm}$ is much steeper for blood. The cobalt colors shown in Fig. 8(b) have interesting spectra in the visible range and feature high absorption values even for wavelengths above $\lambda =1000\mathrm{nm}$. As already explained, all pigments can be incorporated into other matrix materials without changing the qualitative characteristics of their spectra. By a linear combination of pigments, custom absorption spectra may be obtained akin to the absorption properties of skin, fat, or other tissue. There are three main problems associated with pigments: first, higher density particles sink in viscous fluids. With Stokes’ law, the sinking velocity of spheres can be estimated. Second, for high concentrations $c_{\mathrm{V}}>1\%$ of pigments, the particles tend to agglomerate during the curing process, even though we observed this only for the WPP. Third, since particles are fluctuations in the RI, absorbing pigments scatter light, too. This scattering can be approximated using Mie theory. We found that particles with diameters $d=3\dots 30\mu \mathrm{m}$ give a good trade-off between weak scattering (larger pigments are preferable), sinking, and the Sieve effect (smaller pigments are preferable). When mixing the particles into the phantom, it is important to reach a high level of dispersion. To prevent the agglomeration of particles during the cure process, one can increase the phantoms’ viscosity by mixing resin and hardener some hours before the addition of the pigments.

Fig. 8

Absorption coefficient of (a) Phthalo Green, Phthalo Blue Royal, Cinnabar, and Haematite shown up to 1000 nm and (b) Cobalt Blue, Cobalt Blue Turquoise, and Cobalt Violet shown up to 1650 nm divided by their weight concentration. The numbers in the legend denote the product numbers of the distributor Kremer GmbH to uniquely identify the pigments. Uncertainties are estimated to be $\mathrm{\Delta }{\mu }_{\mathrm{a}}/{\mu }_{\mathrm{a}}\approx 10\dots 20\%$. The absorption of Cobalt Blue Turquoise for $\lambda =900\dots 1200\mathrm{nm}$ is too small to be properly determined and thus is not shown.

6.2.

Fluorophores

We successfully added rhodamine 6G into scattering phantoms. It can either be dissolved in the resin directly or in ethanol which is then mixed into the resin. Using the Cary5000, we performed measurements of the emission spectrum at different excitation wavelengths. The results show a peak at an excitation wavelength of $\lambda =534\mathrm{nm}$ and emission wavelength of $\lambda =549\mathrm{nm}$. For our fluorescent phantoms, no bleaching could be detected within 3 months.