Phasor Representation Approach for Rapid Exploratory Analysis of Large Infrared Spectroscopic Imaging Data Sets

Abstract

Infrared (IR) spectroscopic imaging is potentially useful for digital histopathology as it provides spatially resolved molecular absorption spectra, which can subsequently yield useful information by powerful artificial intelligence methods. A typical analysis pipeline in using IR imaging data for chemical pathology often involves iterative processes of segmentation, evaluation, and analysis that necessitate rapid data exploration. Here, we present a fast, reliable, and intuitive method based on a phasor representation of spectra and discuss its unique applicability for IR imaging data. We simulate different features extant in IR spectra and discuss their influence on the phasor waveforms; similarly, we undertake IR image analysis in the transform space to understand spectral similarity and variance. We demonstrate the potential of phasor analysis for biomedical tissue imaging using a variety of samples, using fresh frozen surgical prostate resections and formalin-fixed paraffin-embedded breast cancer tissue microarray samples as model systems that span common histopathology practice. To demonstrate further generalizability of this approach, we apply the method to data from different experimental conditions─including standard (5.5 μm × 5.5 μm pixel size) and high-definition (1.1 μm × 1.1 μm pixel size) Fourier transform IR (FTIR) spectroscopic imaging using transmission and transflection modes. Quantitative segmentation results from our approach are compared to previous studies, showing good agreement and quick visualization. The presented method is rapid, easy to use, and highly capable of deciphering compositional differences, presenting a convenient tool for exploratory analysis of IR imaging data.

Figure 1.

Mathematical formalism of the phasor approach with representative simulated IR spectra. (A–C) Lorentzian spectrum; (A) spectrum, (B) $u$, and (C) trajectory of the phasor in $k$ space. (D–F) Gaussian spectrum; (D) spectrum, $\mathrm{€}u$, and (F) trajectory of the phasor in $k$ space. (G–I) Dirac delta spectrum; (G) spectrum, (H) $u$, and (I) trajectory of the phasor in $k$ space. (J–L) Dirac delta pair; (J) spectrum, (K) $u$, and (L) trajectory of the phasor in $k$ space.

Figure 2.

Key characteristics of phasor-based analyses and segmentation. (A) Average baseline-subtracted spectra of two different histologic classes in prostate tissue-epithelial cells (green) and stroma (pink) normalized to amide I (1650 cm⁻¹) and unnormalized background (black). (B) Absolute value of the inverse Fourier transform of the three spectra presented in (A). The epithelial and stromal data have been normalized to the $k=0$ value. The background has been scaled 500 times for visual appraisal. Inset B shows the phasor trajectory. (C) $\theta$ as defined in the main text. (D) Compass plots of the vectorial representation: (i) $k=0$, (ii) $k=0.0002747$, (iii) $k=0.003235$, and (iv) $k=0.01091$. All the vectors are normalized to $u$ ($k=0$).

Figure 3.

Unsupervised classification of IR hyperspectral images using the phasor representation. (A) Single band IR image (1650 cm⁻¹). (B) Phasor histogram plots with freeform selection areas ($k=0.0032$). (C) Classified image obtained from each of the clusters (color coded). (D) Average spectra of each of the clusters.

Figure 4.

Effect of baseline removal using phasor processing. (A) (i) Phasor representation of $k=0.0032$ without baseline subtraction and (ii) phasor representation of $k=0.0032$ with baseline subtraction. (B) (i) Phasor representation of $k=0.0002747$ without baseline subtraction and (ii) phasor representation of $k=0.0002747$ with baseline subtraction. (C) Classified area from freeform selection of B (ii). (D) Average spectra of the selected clusters normalized to amide I intensity.

Figure 5.

Unsupervised classification results. (A–D) (i and ii), (I) Results for fresh frozen prostate tissue. (A) Adjacent H&E-stained section, (B) amide I (1650 cm⁻¹) image, (C) phasor plot with the location of the clusters (color coded) highlighted, (D) classified image, (i) H&E image of ROI and (ii) classified image of ROI, and (I) average spectra of the different classes. (E–H) (iii and iv), (J) Results for FFPE breast cancer TMA cores. (E) Adjacent H&E-stained section, (F) amide I (1650 cm⁻¹) image, (G) phasor plot with the location of the clusters (color coded) highlighted, (H) classified image, (iii) H&E image of ROI and (iv) classified image of ROI, and (J) average spectra of the different classes.

Figure 6.

Supervised classification of IR hyperspectral images using phasor representation. (A) (i) H&E and (ii) class image of benign. (B) (i) H&E and (ii) class image of benign, (C) (i) H&E and (ii) class image of cancer, (D) (i) H&E and (ii) class image of cancer with desmoplasia. (E) ROC curves of classification (AUC = area under the curve) with the inset showing the extreme left end of ROC. (F) Confusion matrix.