1.

Introduction

The index of refraction ($n$) is an important intrinsic optical parameter of tissue that has been recently exploited for various applications. All materials have unique $n$ values, which are associated with their electrical permittivity. Medical imaging systems use $n$ as a source of optical imaging contrast to view tissue without the need of labeling or extrinsic contrast agents. Studies have shown that the distribution of $n$ in tissue, using data from medical systems such as computed tomography and optical coherence tomography (OCT), could also be used as a biomarker for medical diagnosis.^1,2 Microbiology, hematology, and infectious disease studies show that $n$ distribution can provide valuable information about cellular growth and division and bacteria identification.^3–5 Various methods exist for the measurement of $n$ of different materials including tissues.^6–8

Using OCT, a common method to estimate $n$ is based on measuring the path-length change from a sample placed over a reflector.⁹ Although this method is only appropriate for ex vivo measurements, its accuracy and simplicity make it ideal for the verification of newly developed techniques. This method exploits the fact that the optical path-length measured by OCT is proportional to the index of refraction. Thus, measuring the same distance in air and under a known thickness of tissue ($L$), the index of refraction can easily be calculated as $n=(L+L^{\prime })/L$ (Fig. 1).

Fig. 1

Index of refraction calculation based on the path-length difference observed in the OCT imaging of samples with different indices of refraction.⁹

Other methods that have been proposed depend on very complicated experimental setups and algorithms and are also very hard to apply in vivo. They either require specific hardware to precisely track the focus and calculate the path length and/or require iterative methods to calculate the index of refraction by fitting to assumed models, which are processing-intensive and time-consuming. Focus tracking method and optical path shifting methods were evaluated for simultaneously calculating refractive index and thickness, in samples with different scattering properties. These methods required specialized hardware and multiple measurements. In addition, the optical path shifting method is limited to in vitro measurements albeit at a high precision.¹⁰ Gradient refractive index profile was also retrieved by iterative fitting of the optical path calculated by the ray tracing method, with that experimentally measured using OCT in the isolated crystalline fish eye lens.¹¹ Further studies have demonstrated the calculation of the index of refraction of isolated mouse crystalline lens to determine whether index changes occur across age, due to gene mutations or with form deprivation myopia.^12,13 Angularly resolved low-coherence interferometry to measure the phase index of refraction of pure silica by taking various thickness measurements at different incidence angles was also demonstrated but was only applied to nonbiological samples.¹⁴ Recently, a new study formulated a new method, using FD-OCT measured optical path lengths (OPLs) and properly selected spectral components of FD-OCT interference spectrum, to simultaneously measure path length and index of refraction.¹⁵ However, this method has not been demonstrated in vivo and the authors expressed concerns regarding the detrimental effects of speckle and beam aberrations caused by tissue. Even more recently, a refractive index correction algorithm that can find and correct the refractive index of all layers in a multilayer tissue, simultaneously, was also proposed. This algorithm can exploit parallel processing hardware thus achieving short processing times and is not affected by speckle.¹⁶ This technique was demonstrated on phantoms and skin tissue. A new method, appropriate for in vivo and in situ measurement of $n$ in tissues, without the need for a reflector as a reference, with minimal postprocessing and hardware modifications, is proposed here. Such a technique could be developed into a useful in vivo diagnostic tool in the future.

2.

Theory and Methods

2.1.

Theory

The proposed technique is applicable to both scattering, e.g., tissue, and nonscattering, e.g., glass, samples. When a sample, of any kind, is horizontal, the light is incident on the glass perpendicularly and reaches the bottom of the sample without changing the direction. The OPL ($L_1$) from the top to the bottom surface of the sample is equal to the sample thickness ($d$) multiplied by the index of refraction ($n$). If the incidence is at an angle, the beam is refracted and, as it passes through the medium, it follows a path length ($L_2$) that is longer than that of perpendicular incidence [Fig. 2(a)].

Fig. 2

Incidence angles and path lengths of beams perpendicular and at an angle to the samples. (a) Perpendicular (i) and at an angle (ii) incidence on the sample. (b) Two samples with different incidence angles ${\theta }_{\mathrm{1,1}}$ (i) and ${\theta }_{\mathrm{1,2}}$ (ii).

Given that the path lengths and incidence angles can be measured from the OCT images, the index of refraction can be estimated, using Snell’s law’s, as

Eq. (1)

$$\begin{gathered} \begin{array}{c}\sin {\theta }_{\mathrm{1,2}}=n\sin {\theta }_{\mathrm{2,2}}\Rightarrow n=\frac{\sin {\theta }_{\mathrm{1,2}}}{\sin {\theta }_{\mathrm{2,2}}}\\ \begin{array}{r}{L}_1=nd\\ L_2=\frac{nd}{\cos {\theta }_{\mathrm{2,2}}}\end{array}\}\Rightarrow \frac{L_1}{L_2}=\cos {\theta }_{\mathrm{2,2}}\Rightarrow {\theta }_{\mathrm{2,2}}={\cos }^{-1}(\frac{L_1}{L_2})\end{array}\}\Rightarrow \\ n=\frac{\sin {\theta }_{\mathrm{1,2}}}{\sin \left[{\cos }^{-1}\right(\frac{L_1}{L_2}\left)\right]}, \end{gathered}$$

If the beams in both images are at an angle [Fig. 2(b)], the equations become:

Eq. (2)

$$\begin{gathered} \begin{array}{r}\begin{array}{l}\sin {\theta }_{\mathrm{1,1}}=n\sin {\theta }_{\mathrm{2,1}}\Rightarrow \sin {\theta }_{\mathrm{2,1}}=\frac{\sin {\theta }_{\mathrm{1,1}}}{n}\\ \sin {\theta }_{\mathrm{1,2}}=n\sin {\theta }_{\mathrm{2,2}}\Rightarrow \sin {\theta }_{\mathrm{2,2}}=\frac{\sin {\theta }_{\mathrm{1,2}}}{n}\end{array}\\ \begin{array}{r}{L}_1=\frac{nd}{\cos {\theta }_{\mathrm{2,1}}}\\ L_2=\frac{nd}{\cos {\theta }_{\mathrm{2,2}}}\end{array}\}\Rightarrow \frac{L_1}{L_2}=\frac{\cos {\theta }_{\mathrm{2,2}}}{\cos {\theta }_{\mathrm{2,1}}}=\frac{\sqrt{1-{\sin }^2{\theta }_{\mathrm{2,2}}}}{\sqrt{1-{\sin }^2{\theta }_{\mathrm{2,1}}}}\end{array}\}\Rightarrow {\left(\frac{L_1}{L_2}\right)}^2=\frac{n^2-{\sin }^2{\theta }_{\mathrm{2,1}}}{n^2-{\sin }^2{\theta }_{\mathrm{1,1}}}\Rightarrow L_1^2(n^2-{\sin }^2{\theta }_{\mathrm{1,1}})=L_2^2(n^2-{\sin }^2{\theta }_{\mathrm{2,1}})\Rightarrow \\ n=\sqrt{\frac{L_1^2\sin ({\theta }_{\mathrm{1,1}})-L_2^2\sin ({\theta }_{\mathrm{1,2}})}{L_1^2-L_2^2}}. \end{gathered}$$

The path lengths ($L_1$ and $L_2$) can be measured directly from the images, and the incidence angles (${\theta }_{\mathrm{1,1}},{\theta }_{\mathrm{1,2}}$) can also be measured directly from the angle of the top surface of the sample. This approach was verified using samples placed over a reflector so that the path-length change could be easily measured and, at the same time, use the same reflector as a reference for estimating $n$ using the technique described by Tearney et al.⁹

However, distinct, highly reflective, structures are rarely present in tissue. To apply this technique in vivo, a different approach is required to estimate the path-length change. This can be achieved by first registering and aligning the two images, with a rigid affine transformation, and subsequently using the cross-correlation of corresponding A-scans from each image to estimate the path-length change for each A-scan. Affine registration is an intensity-based linear mapping technique that secures points, straight lines, and planes. It consists of scaling, rotation, and translation, and all pixels go through the same transformation. It is, usually, used to improve geometric distortions such as differences in image’s size. The affine transformation can be described as

$$\begin{array}{c}{x}^{\prime }=t_x+a_{1}x+a_{2}y\\ y^{\prime }=t_y+b_{1}x+b_{2}y\end{array},$$

where $t_x$ and $t_y$ refer to the translation vectors; $x^{\prime }$ and $y^{\prime }$ are the original points; and $a_1$, $a_2$, $b_1$, and $b_2$ are the affine transformation parameters.¹⁷

After affine registration, the images are matched in rotation and translation. However, the second image is distorted, in depth, relative to the first because of the elongation of the path lengths due to the larger angle of incidence [Fig. 3(b)]. The cross-correlation is, therefore, expected to have maxima, first, at the 0 lag and then, again, at the lag, which depends on the magnitude of this distortion. The path-length change, $\mathrm{\Delta }L$, is estimated from the lag of this second maximum of the cross-correlation [Fig. 3(c)]. Since the path length of the second image, $L_2$, can now be calculated by the path-length change, i.e., $L_2=L_1+\mathrm{\Delta }L$, the index is, then, calculated by Eq. (2):

Eq. (3)

$$n=\sqrt{\frac{L_1^2\sin ({\theta }_{\mathrm{1,1}})-{(L_1+\mathrm{\Delta }L)}^2\sin ({\theta }_{\mathrm{1,2}})}{L_1^2-{(L_1+\mathrm{\Delta }L)}^2}}.$$

Fig. 3

Path-length change estimate from two OCT images taken at different incidence angles. The images are first (a) registered and (b) aligned. (c) The cross-correlation of corresponding A-scans from the aligned images exhibit distinct maxima.

3.

Results

Initially, the proposed methodology was tested on glass. Images of glass were acquired at different incidence angles, ranging from 5 deg to 20 deg, with eight repetitions for each combination, for a total of 48 images [Figs. 4(a) and 4(b)]. The average results are shown in Table 1. When compared to the reference value of the index of the particular glass, $n=1.509$, the error was 1.36%. The accuracy and precision of the path-length measurements were also evaluated from the glass measurements.

Fig. 4.

(a), (b) Glass. (c), (d) Cucumber. The angles ${\theta }_{\mathrm{1,1}}$ and ${\theta }_{\mathrm{1,2}}$ for the glass were 4.5 deg and 11.8 deg, respectively, whereas for the cucumber, those angles were 0.9 deg and 12.2 deg, respectively.

Table 1

Index of refraction estimates.

Imaging of thin cucumber slices followed, in an effort to examine the validity of the method in higher scattering samples [Figs. 4(c) and 4(d)]. The average results, when using the reflector below the sample to measure the path lengths,⁹ are also shown in Table 1. The estimated $n$, compared to the reference value, exhibited a mean error of $\sim 0.7\%$ (varying from 0.5% to 2%). The index of refraction was also determined using the cross-correlation method (Fig. 5). The results are also shown in Table 1. When compared to the reference value, the error was $\sim 1.1\%$ (varying from 0.1% to 3%). The precision of the measurements of angle and distance were also evaluated from the OCT images of glass. When measuring the angle, from various locations within the same image, the values did not vary by more than a standard deviation of 0.7 deg. In addition, the measured path length, within the same image, varied by $<8\mu \mathrm{m}$ standard deviation. The accuracy of the measurements was evaluated by comparing to the expected path length, i.e., $L=nd/\cos \{a\sin [\frac{\sin ({\theta }_1)}{n}]\}$, and was found not to vary by more than $9\mu \mathrm{m}$.

Fig. 5

(a), (b) Portions of the original OCT images of cucumber (Fig. 4), at different incidence angles. (c) The registration of (b) on (a). (d) The cross-correlation of the A-scans indicated by the dashed lines. (e) A zoomed region (indicated by the dashed box) of (d). The arrow points to the first maximum after the zero lag.

Finally, to validate the applicability of the proposed technique to tissue samples, freshly excised rabbit trachea was imaged. The results are shown in Fig. 6 and Table 1. The proposed methods (measuring the path-length change with or without reflector).⁹

Fig. 6

(a), (b) The original OCT images of freshly excised trachea, at different incidence angles. The labels refer to the epithelium (e), the submucosa (sm), and cartilage (c). (c) The portion of (a). (d) Portion of (b). (e) The registered version of (d). (f) The cross-correlation of the A-scans indicated by the dashed lines in (c) and (e). The arrow points to the first maximum after the zero lag (not shown here for better visualization of the peaks).

The variation in the error, reaching in some cases 3%, is explained by but also exemplifies the most serious limitation of the proposed technique, i.e., image misalignment. Misalignment by as little as $\pm 10\mu \mathrm{m}$ between the two images could lead to up to 5% error, as measured experimentally. This limitation should be taken into account and alignment should be carefully monitored especially when measuring the index of nonuniform samples. Fortunately, angle-resolved OCT systems, which are specifically designed to image the same sample area under different incident angles, can provide angular and spatial precisions limited only by the precision of the galvanometric scanners. The performance of such systems has been validated for speckle reduction and results indicate that there is no resolution degradation implying negligible spatial shifts.¹⁸

4.

Conclusions

In this paper, we propose a measurement technique that could be deployed for in vivo estimation of $n$. This technique uses two OCT images obtained at different incidence angles and estimates $n$ by measuring the path-length changes and the incidence angles. In tissues with no distinct reflectors, the path-length change can be estimated by first registering and aligning the two images, with a rigid affine transformation, and subsequently using the cross-correlation of corresponding A-scans from each image. Experimental verification produced values of $n$ with a mean error of $\sim 1\%$ compared to the expected values. The main limitation of this technique is the effect of misalignment of the images on the results, which can result in up to 5% error for $10\mu \mathrm{m}$ misalignment. However, the use of an angular-resolved system should provide the precision required for the application of the proposed technique. These initial results are promising and provide evidence that this method should be further investigated and validated on human tissues so that, in the future, it could be developed into a clinically useful diagnostic tool.