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Abstract
Refractive index is an important optical constant that characterizes the interaction between light and specimen. A difference in refractive index between specimen and immersion medium introduces the imaging aberration and leads to a problem that the direct thickness measurement of a specimen by optical microscopy is not accurate. However, this aberration correction still requires the exact information of the refractive index of specimen and immersion medium. Herein, we propose an imaging strategy to estimate the refractive index for an unknown specimen. A simplified diffraction model is generated to obtain the relationship between axial scaling factor and refractive index. Then regular fluorescence microscopy is performed to measure the actual axial scaling factors of specimens from mouse muscle and tumor xenograft. Referring to our theoretical plot of axial scaling factor versus refractive index, the refractive index of tissue specimen is determined. For example, we obtain a mean refractive index (n) value of 1.36 for normal muscle tissues, and 1.41 for tumor xenografts. We demonstrate that this diffraction model-based estimation method is an alternative to the current techniques, improving the accurate measurement for refractive index of tissue specimen. The simple instrument requirement with an easy specimen preparation for this estimation method of refractive index may increase the image quality on tissue specimens with less aberration.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
When refractive index of the specimen is different from that of the immersion medium, focal position in microscopy will change, resulting in a focal shift with the elongated or compressed axial-image. The accurate refractive index is thus greatly required for high-resolution 3-D imaging in tissues with a certain thickness. However, due to tissue heterogenicity and optical turbidity [1–4], the refractive index values in current microscopy are often lack of experimental basis.
Several strategies have been developed to address the abovementioned problem and accurately measure the refractive index of specimen. Optical coherence tomography method was used to measure the refractive index of specimen, but MEMS mirror and complex optical setup were required before measurement [5,6]. Quantitative phase imaging was used to evaluate the refractive index of specimens like droplet, gel sphere or cells, but the method was only suitable for objects with spherical shape [7,8]. An optical fiber cladded with tissue was used to measure the refractive index of tissue, but the method was not workable when the size of tissue was small or the tissue composition was not homogeneous [9,10]. A strategy based on total internal reflection was utilized for the refractive index measurement of tissue, but the method required very complicated sample preparation to guarantee the tissue specimen with quite smooth and flat surfaces [11,12].
Herein, we report a method combining the diffraction theory with the experiments on biological samples to determine the refractive index of unknown specimen. We first established a theoretical model to obtain the relationship between axial scaling factor and refractive index. Then we performed the fluorescence microscopy to obtain the values of axial scaling factors from two different types of mouse tissues (normal muscle and tumor xenograft). Referring to the theoretical plot of axial scaling factor versus refractive index, the tissue refractive index is estimated. This biological application also demonstrates the significant difference between normal muscle and tumor xenograft in their refractive indices.
2. Theory
2.1 Theory of the simplified diffraction model for refractive index
Figure 1 shows the simplified geometry of the optical system. A water immersion objective is used to match the refractive index of sample similar to aqueous solutions. Here ${n_1}$ and ${n_2}$ represent the refractive indices before and after the cover glass, respectively. If the media are matched (${n_1} = {n_2}$), the light rays follow a straight path and are focused on apparent focus position ($\textrm{L}$). If there is a mismatch – as is the case in general when using either air, water immersion or oil immersion objectives to study real samples - for instance ${n_1} > {n_2}$, the light path (broken line) is refracted at the interface and focused at actual focus position (${\textrm{L}^\mathrm{^{\prime}}}$). The ratio of apparent focus position and actual focus position $\textrm{L/}{\textrm{L}^\mathrm{^{\prime}}}$ is defined as the axial scaling factor.
Fig. 1. Calculation for the refractive index match. A simplified model of the optical system is used for theory calculation. A homogeneous distribution of light intensity is used at the spherical wavefront on the sample side.
Several theoretical strategies have been developed for calculation of the refractive index mismatch. Török et al [13–16] developed a valid strategy to address the problem of focusing through a dielectric interface for optical systems with high value of Fresnel number. The method is suitable for conventional microscopy objectives. Hell et al [17,18] used Fermat’s principle to replace the aberration function by theorical prediction of the refractive index mismatch. Gu et al [19] investigated the refractive index mismatch and considered the effect of spherical aberration via vectorial theory, which considers high-aperture and vectorial effects including Fresnel reflections at the interface. Here we took Török’s theory as reference while simplified the theory model. The detail for the simplified theory model is below:
The electric field $\textrm{E}$ and the magnetic field $\textrm{H}$ at a point $(r,\phi ,z)$ in focal region of spherical coordinates are defined [13–16] by:
Considering that the system is illuminated by a linearly-polarized wave and fulfils the sine condition, a constant factor A is defined as below:
$f$ equals to the focal length of the system, ${l_0}$ is the amplitude of electric vector before the lens and ${\lambda _0}$ is the wavelength of the incident light.The diffraction integrals ${I_n}$ are defined by:
The aberration function $\Psi $ is given as:
where the interface is located at $z ={-} d$, and ${J_n}$ is a Bessel function of the first kind of order n.The time-averaged energy density ${\omega _e}$, the time-averaged magnetic energy density ${\omega _m}$, and the time-averaged total energy density $\omega$ can be expressed as below, each is averaged over a time interval.
Further, we see that
Thus, the distribution of time-averaged magnetic energy density equals to the distribution of time-averaged electric energy density, but the distributions are rotated with respect to each other by 90° around the optical axis. We also see that $\langle \omega (r,z,\phi )\rangle$ is independent of $\phi$, in this case the loci of constant time-averaged total energy density are surfaces of revolution about the rotation axis of the system.Since $r = 0$
we see from (11-13) that along the rotation axis in image space:2.2 Specimen preparation
All animal studies were approved by the Institutional Animal Care & Use Committee of Shanghai Institute of Materia Medica, Chinese Academy of Sciences. After mice were euthanatized by overdose CO2, one piece of muscle tissue with the size of 5 × 5 × 3 mm from a C57BL/6 mouse and one piece of tumor tissue with the same size from a MM.1S myeloma subcutaneous xenograft-bearing SCID mouse were harvested, respectively. The tissue samples were immediately fixed in 4% paraformaldehyde (PFA) (P0099, Beyotime) at 4°C overnight, and washed in PBS for three times (three minutes for each wash). The tissues were dehydrated in 30% sucrose for 48 h and embedded in Tissue-Tek O.C.T. Compound (#4583, Sakura). Then, around 30 µm-thick frozen sections were cut by a freezing microtome (CM3050 S, Leica) and washed in phosphate buffer saline (PBS) for three times (three minutes for each), followed by mounting a coverslip on a drop of 0.05 mg/mL Rhodamine B (RhoB) aqueous solution (R6626, Sigma). A glass slide-tissue-cover glass “sandwich” with a fluid layer (RhoB aqueous solution) was prepared. The final specimen slide setting for microscopy is shown in Fig. 2. It is important to perform the image acquisition immediately (< 30 minutes), otherwise the RhoB fluorescent dyes will diffuse into the tissue. In this case, the boundary between the tissue and the medium will become indistinct.
Fig. 2. Schematic of the specimen slide setting for fluorescence microscopy. A tissue section in immersion medium (RhoB aqueous solution) is sandwiched between a glass slide and a cover glass. An invert water immersion objective is positioned beneath the “sandwich”. By scanning along the z axis at indicated depth locations and recording the emitted fluorescent light at each object point in x-y plane, a 3-D image is obtained.
2.3 Measurement for the axial scaling factor
We used a Zeiss LSM 900 invert fluorescence microscope with a 40× 0.9 numerical aperture (NA) water immersion objective lens, an excitation wavelength of 561 nm, and a standard factory setting for the scanning step in z-direction of 0.2 µm. The z resolution of the scanning system is 500 nm and the position repeatability of the scanning system is 200 nm. Then we performed the tissue section fluorescence imaging by moving the specimen table in three dimensions. Regions containing the tissue or RhoB aqueous solution could be determined by eye (through microscope eyepiece). From the plots of fluorescence intensities of sectioning experiments along z-direction, the sample thicknesses of tissue and fluid region were determined.
How to measure the axial scaling factor of a tissue specimen is described below: A sample slide containing a tissue section within the immersion medium of the same physical thickness is prepared. Then fluorescence microscopy is used to measure the optical thickness of tissue specimen and immersion medium. From the ratio of physical thickness to measured optical thickness, the axial scaling factor is obtained.
3. Results and discussion
3.1 Distribution of electric energy density
The electric energy density $\langle {\omega _e}(r,z,\phi )\rangle$ in the focal regions by Eq. (22) is calculated with different parameters. Figure 3(A) shows an example of the axial electric energy density responses of a peak in meridional planes in x- or y-direction with the focusing positions d of 0, 20, 40, 60 and 100 µm, respectively, by a 40× water immersion objective lens with a NA of 0.9, an illumination wavelength of 561 nm, and the light travelling from medium (n1 = 1.33) into specimen (n2 = 1.36). The reason why we have chosen n2 = 1.36 for the specimen will be discussed in the Discussion section. The response curves in Fig. 3(A) indicate that strong oscillations occur on one side of the maximum of the curves.
Fig. 3. Theoretical calculation. A. Theoretical z-responses for different actual focal position $d$ of 0, 20, 40, 60 and 100 µm in case of a specimen-to-medium interface calculated with Eq. (23) with different parameters. B. Maximal intensity as a function of apparent focus position of the theoretical z-responses for a medium–tissue interface. C. Relationship between actual focus position ${\textrm{L}^\mathrm{^{\prime}}}$ and apparent focus position $\textrm{L}$ determined by the numerical simulation with refractive index. The red line indicates the similar relationship without refractive index. The axial scaling factor is defined as the ratio of apparent focus position $\textrm{L}$ to actual focus position ${\textrm{L}^\mathrm{^{\prime}}}$. D. The theoretical plot of axial scaling factor versus refractive index of specimen in medium immersion. The dark dots indicate the values of the axial scaling factors calculated as specimen refractive index ${n_2}$ is altered.
Figure 3(B) depicts the maximal intensity of axial response as a function of the focusing position d. It indicates that the maximal intensity decreases non-linearly with increasing position d.
Figure 3(C) shows the axial position of peak density when the depth d changes. The curve trend is substantially linear, while a least square fitting indicates a slope of 0.97. The slope reciprocal is identified as axial scaling factor with the value 1.03. In this situation, axial scaling factor is more than 1, indicating that the actual object is larger than the displayed object in an axial image.
Figure 3(D) shows the fit plot of axial scaling factor versus refractive index of specimen (${n_2}$). The curve indicates that axial scaling factor has a positive proportional relationship with refractive index of specimen. This plot enables us to obtain the refractive index of specimen indirectly when the actual axial scaling factor is accurately determined.
Our theory takes Sheppard & Török’s model for Ref. [13–16]; however, we consider the symmetrical distribution of the electric field in microscopy and greatly simplify the calculation. The equations from (1) to (14) were derived from Török’s calculation, and the equations from (15) to (22) showed our modifications. Using the same parameters from previously reported models [13,17,18,19], the axial scaling factors calculated using our model are quite comparable with theirs (Table 1). Our result was also close to the paraxial geometrical optics calculation result ${n_2}/{n_1} = 1.02$.
Table 1. Summary of the axial scaling factors calculated using the previously reported model and our model.
3.2 Measurement for axial scaling factor of mouse tissues
3-D fluorescence imaging on mouse tissue sections was performed to obtain the real axial scaling factor. Normal muscle tissue and cancer tissue with different tissue compositions were chosen as the tissue specimens for imaging. The ultra-fast tissue freezing process and specimen section by cryo-microtome ensured the minimal cellular damage avoiding the mixture of cell fluids with other mediums. The muscle and cancer tissues had sufficient autofluorescence and therefore did not require an additional staining. In this case, the osmotic effect could be ignored.
As Figs. 4(A), 4(F) show, tissue autofluorescence and RhoB dye in mounting medium were utilized for tissue and medium. Next, the intensity response profiles of mouse tissues (Figs. 4(B), 4(G)) and media (Figs. 4(C), 4(H)) were plotted, respectively. For the muscle tissue specimen, the fluorescence intensity plots exhibited a steep rise at the lower-surface, a slight increase across the object, and a sharp drop at the upper-surface of the object in both normal tissue (Fig. 4(B)) and medium (Fig. 4(C)). The similar plot patterns were observed in the cancer specimen (Fig. 4(G)) and the corresponding medium (Fig. 4(H)).
Fig. 4. Characterization of 3-D fluorescence images from mouse tissue sections. A representative section of normal muscle tissue (Section: Normal tissue-1; A) and cancer tissue (Section: Cancer tissue-1; F) were respectively imaged in 3-D by section scanning. The fluorescence intensity response profiles, achieved by projecting the intensities onto the long axis in white boxed areas in Fig. 4(A) and Fig. 4(F), were plotted for the mouse tissues (B, normal muscle tissue; G, cancer tissue) and the medium only (C, normal muscle tissue-medium; H, cancer tissue-medium) with the step by 0.2 µm. The corresponding second-order derivative intensity was shown for the mouse tissues (D, normal muscle tissue; I, cancer tissue) and the medium only (E, normal muscle tissue-medium; J, cancer tissue-medium). Specimen widths for the mouse tissues (D, normal muscle tissue; I, cancer tissue) and the medium only (E, normal muscle tissue medium; J, cancer tissue medium) were determined by the second-order derivative intensity plots in the positions of inflection points where the sudden slope changes and the second-order derivative passes through 0. Scale bars: 10 µm.
Although the full width at half maximum (FWHM) from the intensity curve is usually used to determine the thickness, this method could not work here, due to the asymmetric intensity profile as two regions of sudden slope changes at the start stage and the end stage (Figs. 4(B), 4(C), 4(G), 4(H)). Because the determination of inflection points on noisy experimental curves is not numerically robust and the second-order derivative intensity method is simple and accurate, the latter criterion was introduced for this study. The positions of the inflection points (where the second-order derivative passes through 0) was used as the boundary between ascending (left) and descending (right) edge of the thickness of specimen. The component width in Fig. 4 was then determined as 27.47 µm for normal tissue (Figs. 4(B), 4(D)), 28.30 µm for normal tissue-medium (Figs. 4(C), 4(E)), 26.70 µm for cancer tissue (Figs. 4(G), 4(I)), and 28.50 µm for cancer tissue-medium (Figs. 4(H), 4(J)), respectively. Based on the width ratio of medium to tissue, the axial scaling factors were thus obtained as 1.03 for normal tissue and 1.07 for cancer tissue. Referring to the theoretical plot of axial scaling factor versus refractive index (Fig. 3(D)), the refractive indices of specimens were estimated as 1.36 for normal muscle and 1.41 for cancer tissue.
3.3 Estimation of the refractive index of mouse tissue
Another five sections of normal muscle tissues (Supplement 1, Figs. S1-S5) and another five sections of cancer tissues (Supplement 1, Figs. S6-S10) were measured by 3-D fluorescent imaging under the same experimental setting in 3.2. The values of specimen widths, the calculated axial scaling factors, the refractive indices from the above 12 tissue sections are listed in Table 2. We notice that measured specimen width of the tissue section was smaller than that of the medium, indicating their different refractive indices. One should take into consideration, that we determined the refractive indices of normal muscle tissue and cancer tissue in a dehydrated form, so the difference we detect will reflect both, differences in structure and constitution of the tissues as well as differences in the dehydration grade achievable. The refractive index of dehydrated cancerous and especially normal muscle tissue is considerably larger compared to the normal value of fresh tissue slices under physiological conditions.
Table 2. Axial scaling factor and refractive index of normal muscle tissues (n = 6) and cancer tissues (n = 6).
The mean refractive indices (n) were determined as 1.36 for six normal muscle tissue sections, and 1.41 for six cancer tissue sections. We therefore used n2 = 1.36 as the refractive index of specimen in Figs. 3(A)–3(C) to show the electrical response in diffraction model.
3.4 Comparison with current methods for refractive index
Here are different strategies to measure the refractive index of specimen. Optical coherence tomography method was once used to evaluate the refractive index of specimen, but before experiment MEMS mirror and complex instrument setup were required. Quantitative phase imaging with algorithm was another candidate to measure the refractive index of specimens, but the method was only suitable for specimen with spherical shape. An optical fiber was once used to clad with tissue to measure the refractive index of specimen, but the difficulty of the method was that it required a homogeneous tissue and the tissues had to be blended, in the former case the homogeneous tissue was not easy to be obtained and in later case the blending process would possibly destroy tissue microstructure and density. Total internal reflection was utilized for the refractive index measurement of tissue. The method also required large amounts of tissue and direct contact between the sample and the optical component, thus it was only workable for solid specimens not for fluid specimens. Although our method is only applicable for in vitro samples, it is very simple to be implemented on a standard fluorescence microscope with easy specimen preparation. The refractive index values of different specimens measured by different methods are shown in Table 3 for reference.
Table 3. Summary of different methods on refractive index measurement.
4. Conclusions
We have developed a method based on a simplified diffraction model to estimate the refractive index of tissue specimens. The mean refractive index (n) value of 1.36 for normal muscle tissue and 1.41 for tumor xenograft is obtained. Considering the tissue heterogenicity, these values are determined by an optical imaging experiment, rather than using standard values, and might increase the image quality on tissue specimens with less aberration. An additional advantage of our approach manifests in the fact that it is conveniently implemented only using a simple and inexpensive fluorescence microscope on the easily prepared specimens. It is indicated that our diffraction model-based estimation method is a promising alternative to the current techniques, improving the accurate measurement for refractive index of tissue specimen.
Funding
National Natural Science Foundation of China (82172001); Instrument Function Development of Chinese Academy of Sciences; Natural Science Foundation of Shanghai (21ZR1474800).
Acknowledgements
We thank the technical supports from the Institutional Center for Shared Technologies and Facilities of Shanghai Institute of Materia Medica, Chinese Academy of Sciences, Yao Li and Yang Yu from Integrated Laser Microscopy System at National Facility for Protein Science in Shanghai, Zhangjiang Laboratory, China.
All animal studies were approved by the Institutional Animal Care & Use Committee of Shanghai Institute of Materia Medica, Chinese Academy of Sciences.
All authors have seen and approved this manuscript for publication.
X.F. performed theory calculation. X.F. and L.T. performed imaging experiments. X.F. and T.G. performed data analysis. X.Z. and L.T. prepared imaging samples from mouse tissues. H.H., Y.Z., X.H., J.Y., S.W. and Z.M. helped imaging. X.F., T.G. and R.H. wrote the manuscript. R.H. supervised the whole experiment. All authors read and approved the final manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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