2.1.

Materials

Porcine muscle was selected as the sample. The fresh keeping cabinet provides a temperature of $-3$ to 0°C. The tissue was sealed by preservative film to maintain freshness until it reached room temperature. Before measurement, the tissue was cut into $50\times 30{\mathrm{mm}}^2$ with about a 2-mm thickness. Then the tissue was carefully attached to the prism for CRI measurement and to a slide glass for transmittance measurement. The air gaps between the interfaces should be avoided.

2.2.

Method

Figure 1 is the schematic diagram of the CRI measurement setup, which is similar to that in Ref. 19. After passing through a beam splitter $M$, a half-wave plate $H$, a polarizer $P$, and an aperture diaphragm D1, the $P$-polarized He–Ne laser (632.8 nm) propagates into the prism and irradiates on the tissue.

Fig. 1

Schematic diagram of the complex refractive index (CRI) measurement setup.

D2 is also an aperture diaphragm. $n_p$ is the real RI of the equilateral triangle prism ($n_p=1.6166$ at 632.8 nm). $\alpha$,$\beta$, and $\theta$ are the incident angle at the air–prism interface, the apex angle of the prism and the incident angle at the prism–sample interface, respectively. A detector PD1 is used to monitor the power shift of the laser. Detector PD2 is used to detect the emergent light from the prism. The prism is mounted on a rotation stage (PI, M-038), which is controlled by a Mercury C-863 servo motor controller. $\theta$, $\beta$, $\alpha$, and $n_p$ satisfy

Based on the Fresnel equation,¹³ the reflectance $R_p$ at the prism–sample interface for the $P$-polarized wave is given by

Then the reflectance at the prism–sample interface can be calculated as

A nonlinear fitting program based on the Nelder–Mead simplex method²¹ is used to simultaneously calculate the $n_r$ and $k$. For the tissue sample, the $n_r$ is the average or effective RI of the tissue components. The consistency between the measured data and fitting curve is described by $E^2$, defined as $E^2=1-{\mathrm{\Sigma }}_{i=1}^N{(R_{m,i}-R_i)}^2/{\mathrm{\Sigma }}_{i=1}^N{(R_{m,i}-\overline{R})}^2$, where $R_{m,i}$ is the $i$’th measured reflectance calculated by Eq. (5), $R_i$ is the $i$’th calculated reflectance, and $\overline{R}$ is the mean value of measured reflectance over $N$ values of the incident angle. The value of $E^2$ ranges from 0 to 1 and it is closer to 1 when a reliable fitting is obtained. The reflectance curve of porcine muscle was measured at time points of 1, 2, 3, 4, 5, 6, 7, 8, and 10 h during dehydration. Similar measurements were repeated for three times.

The schematic diagram of the transmittance measurement setup is shown in Fig. 2. A He-Ne laser is used as the incident light. PD1 is used as a monitor to detect the reflected beam from the beam splitter $M$. The transmitted light vertically passes through the slide glass and the tissue and then is collected by an integrating sphere. PD2 is used to detect the emergent light from the integrating sphere. Transmittance measurement was also performed to investigate the OC effect of natural dehydration. The transmittance of the dehydrated porcine muscle was recorded with a time interval of 1 min for a total of 7 h. Similar measurements were repeated for three times.

Fig. 2

Schematic diagram of the transmittance measurement setup.

3.1.

Results

The measured reflectance curves of natural dehydrated porcine muscle are shown in Fig. 3. The reflectance curve of the porcine muscle shifts toward the right direction with the increase in dehydration time. Based on the nonlinear fitting program mentioned above, all curves are well fitted with $E^2>0.996$ and the CRI values are determined, as demonstrated in Fig. 4. The $n_r$ increases continuously from 1.351 to 1.376 during a 10 h measurement, while $k$ does not increase continuously. During the first stage of natural dehydration (from 0 to 4 h), $k$ increases from 0.0015 to 0.0018. During the second stage (from 4 to 10 h), $k$ decreases to 0.0011. Finally, $k$ becomes smaller than the initial value.

Fig. 3

Reflectance curves of porcine muscle during dehydration. Points: measured reflectance. Line: fitted curves.

Fig. 4

Change of CRI of porcine muscle during dehydration.

From the result of the transmittance measurement in Fig. 5, we find that the transmittance increases from about 0.15 to 0.38 in 7 h with a varied growth rate. The growth rates are about 0.003, 0.007, 0.022, 0.037, 0.061, and 0.077 at each 1-h time interval.

Fig. 5

Transmittance of porcine muscle and the growth rate during dehydration.

3.2.

Discussion

The increase of $n_r$ of porcine muscle during natural dehydration is observed, as shown in Fig. 4. A similar phenomenon can be found in Refs. 22 and 23. The optical parameters of tissue are known to depend on the water content,²⁴ and both the natural dehydration and compression-induced dehydration can lead to an increase of $n_r$.²⁵ For the fresh porcine muscle, the measured $n_r$ is 1.351, which is smaller than the value of 1.367 provided in Ref. 19. This might be because the tissue fluid exuded out from the damaged tissue and increased the fraction of fluid at the prism–tissue interface. In our early study, the tissue fluid may lead to a hump in the reflection curve.²⁰ However, the hump is not observed in this study, as shown in Fig. 3. The main reason may be the difference of tissue preparation. In this study, we used fresh porcine muscle without a freezing treatment, and high pressure was avoided when we attached the tissue to the prism.

It can be seen from Fig. 5 that the growth rate of transmittance is slow during the first few hours and then becomes faster. A similar phenomenon can be found in Ref. 4. From the Bouguer–Beer–Lambert law, collimation transmittance in tissue can be calculated by $A=\exp (-{\mu }_{t}L)$, where $L$ is the thickness.²⁶ Thus, both the thickness and the extinction coefficient influence the transmittance. The tissue will shrink due to the water loss and internal structure change, and the shrinkage will increase the transmittance. After 7 h, the thickness of the muscle reduced to about 74%. Considering the change tendency of $k$ observed in this study, we can deduce that the effects of thickness and $k$ on transmittance are opposite during the first stage and then become the same in the second stage. The opposite effects slow down the growth rate of transmittance in the first stage.

In order to explain the change tendency of $k$ during dehydration, the heuristic particle-interaction model is applied. For a dense distribution of scattering particles, the reduced scattering coefficient (${\mu }_s^{\prime }$) is related to the reduced scattering cross section (${\sigma }_s^{\prime }{\mathrm{cm}}^2$) by ${\mu }_s^{\prime }=\varphi (1-\varphi ){\sigma }_s^{\prime }/V$, where $\varphi$ is the volume fraction of the scattering particles and $V$ is the volume of a single scattering particle. The parabolic relationship between ${\mu }_s^{\prime }$ and $\varphi$ is demonstrated in Fig. 6. Therefore, the volume fraction of scatters in tissue during dehydration needs to be estimated.

Fig. 6

Relationship between the reduced scattering coefficient and scattering particle volume fraction.

Five porcine muscle samples were weighed and then put into a temperature humidity chamber at 40°C for 72 h to obtain fully dehydrated samples. The weight fractions with an average of about 28% are calculated, as listed in Table 1. According to Ref. 27, the density of dry matter from bovine muscle is $\rho (g/{\mathrm{cm}}^3)=1.551-0.349\times {10}^{-3}T$, where $T$ is the absolute temperature of the sample (measured in Kelvin). We use the data at 20°C (293 K) to estimate the density of fully dehydrated porcine muscle, and the volume fraction of water is deduced as about 0.788, which is close to the value 0.756 in rat skeletal muscle published in Ref. 28. So, the change of ${\mu }_s^{\prime }$ of the tissue should obey the parabolic form to some extent during dehydration. Assuming no change in RI of the ground substance and the size of the scattering particles in muscle, the change of the anisotropy factor $g$ is negligible;²⁹ so, ${\mu }_s$ is approximately linear with ${\mu }_s^{\prime }$. According to Ref. 26, for muscle tissue, ${\mu }_s$ is much larger than ${\mu }_a$ in the visual spectral region. We may infer that $k$ should obey the same change tendency as ${\mu }_s$, which matches well with the result obtained in this study. After 10 h dehydration, the $k$ becomes smaller than the initial value, which reveals that the tissue is optically clarified. Moreover, structural modification or dissociation of collagen can also influence the OC result.³⁰ Therefore, the CRI is a more suitable parameter to evaluate the OC effect.

Table 1

Weight fraction of fully dehydrated porcine muscle.

3.3.

Error Analysis

It is impossible to obtain two pieces of muscle tissue with exactly the same parameters and dehydration speed, but for different measurements, a similar phenomenon is observed. The time of each CRI measurement is about 1 min, which is much less than the dehydration time, so the change of the tissue can be ignored during each measurement. Therefore, most experimental errors come from the errors of incident angle and the power fluctuation. Error of the incident angle at the prism–tissue interface, $\mathrm{\Delta }\theta$, can be calculated by the differential of Eq. (1), which is

The high resolution rotation stage has a minimum incremental motion of $3.5\mu \mathrm{rad}$ and a design resolution of $0.59\mu \mathrm{rad}$; so the error caused by the rotation stage is negligible. There is an error of about 0.01 deg when we adjust $\alpha$ to zero. $\mathrm{\Delta }{n}_p$ of the prism is less than 0.0002. $\mathrm{\Delta }\beta =0.0002\mathrm{rad}$. $\mathrm{\Delta }\theta$ is calculated as 0.018 deg, which corresponds to an error of about 0.001 of $n_r$. The error of $k$ is highly related to the power fluctuation. After being calibrated, the power fluctuation measured by PD2 is less than 0.5%, which corresponds to an error of 0.0001 of $k$.