1.

INTRODUCTION

According to World Health Organization (WHO), cancer is the second leading cause of death globally.¹ About 9.6 million deaths by cancer in 2018 is estimated, and effective treatment strategies require early and accurate diagnosis of the disease. The gold standard method for cancer diagnosis is based on the microscopic investigation of a stained tissue biopsy by a clinical pathologist. The primary stain provides the necessary contrast needed for a trained pathologist to distinguish between normal and abnormal tissue morphology. However, this type of investigation is qualitative, depends on the details of tissue processing and, as a result, often leads to interobserver variability. Thus, there is a need to provide an objective basis for evaluation based on physical metrics.

Quantitative phase imaging (QPI) is a label-free microscopy technique where contrast is generated by the optical path-length difference (OPD), which is the product of the local thickness and refractive index changes of the specimen.^2–4 For a thin specimen, such as a tissue histology, the thickness can be considered spatially invariant, in which case QPI images are proportional to a mean refractive index map, i.e., a refractive index map.^5–7 Since the refractive index is proportional to the dry mass content of cells and cellular matrix, it informs on tissue density as well as cell organization within tissue.^{8, 9} Tissue refractive index-based markers have been used in the past for medical diagnosis and prognosis of several types of cancers and diseases.^10–20 In addition to the advantages of label-free imaging, the contrast mechanism in QPI provides access to additional markers of disease, which are of value to histopathology. In particular, since QPI systems employ interferometric measurements, they are sensitive to subwavelength fluctuations in OPD in both space and time. Therefore, local fluctuations in quantitative phase images inform on nanoscale morphological alterations of cell structures, due to the dry mass accumulation as well changes in extracellular matrix components.

In this work, we introduce two types of intrinsic markers of nanoscale morphological alteration in fixed tissue biopsy referred to as disorder strength and local correlation length.^{5, 6} The disorder strength which is a marker of the local spatial fluctuations of refractive index was first used as a marker for pancreatic cancer diagnosis by Subramanian et al.²¹ Their group used a spectroscopic imaging modality to measure this marker and have subsequently employed it in diagnostic studies related to prostate, colon, breast, lung, and other cancers.^21–29 Thereafter, Eldridge et al. successfully extracted the disorder strength from quantitative phase images and demonstrated the relationship of the marker to cancer cell mechanical properties.³⁰ They applied this analysis to colon, skin, and lung cancer cells to demonstrate an inverse relationship between shear stiffness and disorder strength. Building on these results, Munoz et al. used QPI to study the on-set and progression of shear stiffness changes during malignant transformation in bronchial epithelial cells.³¹ Our group also showed that the disorder strength measured by spatial light interference microscopy (SLIM), a sensitive white light QPI method, is a quantitative marker of malignancy that can be used to classify benign and malignant breast tissue microarray (TMA) cores. On the other hand, the local correlation length is also a marker of the local spatial fluctuations of refractive index and have been proposed by our group. Although, the local correlation length map was computed by calculating the two-dimensional (2-D) correlation function over regions of an image leading to long computation times, we have proposed the efficient algorithm for calculating the local correlation length map, requiring a smaller number of calculation steps. We image the quantitative phase images of 20 benign, 16 grade 1, 16 grade 2, and 14 grade 3 malignant cores of breast TMA using SLIM, which is a high-sensitivity QPI method, able to detect sub-nanometer optical pathlength fluctuations. Then we classify these grades using the nanoscale marker(s): disorder strength, local correlation length and both of them.

2.

QUANTITAIVE PHASE IMAGING (QPI)

QPI is a label-free microscopy technique where contrast is generated by the optical path-length difference, and the phase image, ϕ(x, y), measured in QPI is given by the equation:

where n(x, y, z) is the refractive index contrast between the tissue and the surrounding medium, L(x, y) is the thickness of the tissue, and λ is the illumination wavelength. Here, in this work, we note that since we use tissue slices with constant thickness of 4 μm, the thickness can be rewritten as L(x, y) ≈ L. Several methods have been proposed so far, however the figures of merit of QPI methods such as acquisition rate, temporal and spatial sensitivities and transverse resolution basically depend on the light source, interferometer and phase calculation algorithm used in the QPI method. We use spatial light interference microscopy (SLIM) which provides subwavelength sensitivity of OPD changes in both space and time.²

2.1

Spatial Light Interference Microscopy (SLIM)

A schematic of the SLIM setup is shown in Fig. 1(a). The SLIM module is attached to a commercial phase contrast microscope (PCM). The lamp filament is imaged onto the condenser annulus (Köhler illumination conditions), which is located at the front focal plane of the condenser lens. The specimen is located at the back focal plane of the condenser lens, and front focal plane of the objective. The scattered and unscattered fields are relayed by the objective and tube lenses. As a result, the expanded phase contrast image that has the intensity distribution in accordance with the phase contrast caused by the specimen is observed at the image plane. However, because the output of PCM is qualitative, the phase image, ϕ(x,y), cannot be directly retrieved from this image. The SLIM module extracts ϕ(x,y) by phase modulating the incident light with respect to the scattered light. The field at the image plane is Fourier transformed by the lens L1, such that the unscattered light can be spatially isolated from the scattered light. Since the incident light has the ring form, by displaying the corresponding ring pattern on the reflective liquid crystal phase modulator (LCPM), we ensure that the scattered light remains unaffected. Four phase shifts are applied to the unscattered light at increments of π/2 rad, as shown in Fig. 1(b). The corresponding four images captured by the charge coupled device (CCD) are obtained. Consequently, the quantitative phase image is retrieved as described in Ref.² The most important features of SLIM image is high sensitivities: according to previous work, sensitivities in OPD of 0.3 nm and 0.03 nm can be achieved in space and time, respectively.^{2, 4} Figure 1(c) shows the quantitative phase image and its expanded view of benign and malignant breast tissue samples. Since the thickness of tissue slice is constant, the quantitative information extracted from these images of refractive index changes can be used for diagnosis and prognosis of diseases.

Figure 1.

SLIM system. (a) Optical setup. (b) Phase patterns displayed on LCPM and corresponding intensity patterns captured by CCD. (c) Example of quantitative phase images of benign and malignant breast tissue cores.

3.

NANOSCALE MORPHOLOGICAL ALTERATION MARKERS

3.1

Disorder Strength

By definition, the disorder strength map, L_d(x, y), is expressed as:

Here, < ⋯ >_w denotes the average within the window of interest, Δ means the difference from its average, i.e., Δn(x, y) = n(x, y) − < n(x, y) >_w, and l_c is the spatial autocorrelation length. Figure 2(a) shows the quantitative phase image ϕ(x, y), which contains information about the spatial variation of the refractive index change of tissues as expressed by Eq. 1. The local variance and average of the phase has the form, respectively,

Figure 2.

Calculation procedure of DS map. (a) Quantitaitve phase image. (b) Phase fluctuation map. (c) Phase fluctuation map after filtering and masking to exclude undesired pixels.

and

Here, nmean is the average of the refractive index in the tissue. Thus, the local refractive index fluctuation map, which is independent of the thickness, can be computed as

Therefore, we can rewrite Eq. 2 and obtain the final form to calculate the disorder strength map from the quantitative phase image as

In our calculation, we used a window of 5 × 5 pixels (0.125 μ m/pixel). Figure 2(b) shows the map of < Δϕ(x, y)² >_w / < ϕ(x, y) > . Since our interest is the fluctuation only in the tissue region, the background pixels were excluded as shown in Fig. 2(c). Consequently, the disorder strength can be calculated using the resulting phase image by mutiplying the constant refractive index, n_mean of 1.38 and the mean spatial autocorrelation length of all tissues, l_c.

3.2

Local Correlation Length

The local correlation length depends on the morphological disorder, i.e., local refractive index fluctuations. When the refractive index is spatially disordered, the spatial autocorrelation length within the local area will shorten. In general, the spatial autocorrelation length is calculated as the width of the spatial autocorrelation function. According to the Wiener-Khinchin theorem, the 2D spatial autocorrelation function can be obtained by taking inverse 2D Fourier transform of the spatial power spectrum. In other words, two 2D Fourier transforms for each image, leading to long computation times. Thus, to avoid this problem, we propose a new procedure that performs these calculation in the frequency-domain.

First, as shown in Fig.3, we define the local correlation function as

Figure 3.

Definition of t(x,Y; x′, y′).

where ⊗_{x, y} denotes the 2D correlation operation over (x, y). Function t(x, y; x′,y′) is a local phase function centered at (x′, y′) and is expressed as

where w(x, y; x′,y′) = rect () rect () is a local window function centered at (x′, y′), of width of a. The angular brackets denote averaging within the local window.

Next, we define the local correlation length map, ρ(x′, y′), as the variance of the probability density which can be obtained by normalizing Γ(x, y;x′, y′) by ∬ Γ(x, y;x′, y′)dxdy:

Here, ρ(x′, y′) can be related to the bandwidth map of the spatial power-spectrum, τ(x′, y′), as ρ(x′, y′)τ(x′, y′) = 2π. The local bandwidth, τ(x′, y′), itself is defined as

where t(k_x,k_y;x′, y′) is the Fourier transform of t(x, y;x′, y′) along (x, y). Using the differentiation property of Fourier transforms as well as Parseval ‘ s theorem, this equation can be rewritten as

Finally, we can obtain the final result as

Using Eq. 12, the local correlation length maps can be calculated as shown in Fig.4 which were obtained from the quantitative phase image.

Figure 4.

Calculation procedure of local correlation length map. (a) Quantitaitve phase image. (b) Local correlation length map. (c) Local correlation length map after masking to exclude undesired pixels.

4.

BREAST CANCER CLASSIFICATION USING NANOSCALE MARKER

Figure 5 compares the average of Ł_d(x, y) between benign and malignant cores (grade 1, 2 and 3). To extract only tissue regions, the pixels which satisfy < ϕ(x, y) >_w < 0.075 rad. and < Δϕ(x, y)² >_w / < ϕ(x, y) > > 2.7 were excluded. The p-values which were obtained by two-sided Wilcoxon ranksum test are listed in the row named p-value (DS) in Table 1. Although the statistically significant differences are indicated between benign and malignant cores, the results show that the inter-grade cores are hardly distiguished. On the other hand, Fig. 6 compares the average divided by the standard deviation of ρ map between benign and malignant cores (grade 1, 2 and 3). We can use any feature quantity of ρ map, however the average divided by the standard deviation was adopted because it indicated the best p-value. To extract only tissue regions, the background pixels were segmented out by setting a threshold in the ρ(x, y) map. This threshold value was determined empirically, and all pixels having correlation lengths below 1.3 μm were treated as background. The p-values which were obtained by two-sided Wilcoxon ranksum test are listed in the row named p-value (LCL) in Table 1. Since the p-value between 20 benign and 46 malignant cores was 0.000876, the local correlation length correlates with cancer grades. Furthermore, the results indicate that the statistically significant differences between cores with more than 2 inter-grade differences. However, it may need to be combined with other markers for more detail separation of grades. We conclude that the local correlation map can potentially be used by clinical pathologists as a supplementary label-free disease marker for gauging the onset of malignancy especially in borderline cases. The results of classification using both disorder strength and local correlation length will be presented at the conference.

Figure 5.

Disorder strength of benign (N = 20) and grade 1 (N = 16), grade 2 (N = 16), and grade 3 (N = 14) tissues.

Figure 6.

Local spatial autocorrelation length of benign (N = 20) and grade 1 (N = 16), grade 2 (N = 16), and grade 3 (N = 14) tissues.

Table 1.

The p-values between different grades

5.

CONCLUSIONS AND FUTURE WORKS

We have introduced two types of cancer markers of nanoscale morphological alteration referred to as disorder strength and local correlation length and demonstrated classification of breast cancer tissues. Although neighboring grades were not distingished, the effectiveness of these markers to classify more than two grades including benign and malignant tissues was shown. We need to consider tissue diagnosis using plural markers and machine learning based classification to achieve distinguishing between neighboring grades. The improved results of classification accuracy by using both disorder strength and local correlation length will be presented at the conference.