![Left: receiver operating characteristic for the different classification methods, at 550-GHz applied to TS#2. The black line stands for TPR=FPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TPR = FPR$$\end{document}. Right: refractive index threshold as a function of the TPR-FPR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$TPR - FPR$$\end{document} measure for the different classifiers.](https://www.researchgate.net/publication/350192234/figure/fig3/AS:1003321540231171@1616221904327/Left-receiver-operating-characteristic-for-the-different-classification-methods-at_Q320.jpg)
![TS#2 tissue sample classification maps at 550-GHz for Λ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^0$$\end{document}, Λ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^1$$\end{document}, Λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^2$$\end{document}, Λ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda ^3$$\end{document} and their respective first two best thresholds. “Not applicable” refers to regions where the binary pathology classification and the binary terahertz classification image do not match spatially. The values listed in each box are respectively standing for the refractive index threshold, the true positive rate and the false positive rate.](https://www.researchgate.net/publication/350192234/figure/fig4/AS:1003321540243483@1616221904419/TS2-tissue-sample-classification-maps-at-550-GHz-for-L0documentclass12ptminimal_Q320.jpg)
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Terahertz refractive index‑based
morphological dilation for breast
carcinoma delineation
Quentin Cassar1, Samuel Caravera3, Gaëtan MacGrogan3, Thomas Bücher2, Philipp Hillger2,
Ullrich Pfeier2, Thomas Zimmer1, Jean‑Paul Guillet1 & Patrick Mounaix1*
This paper reports investigations led on the combination of the refractive index and morphological
dilation to enhance performances towards breast tumour margin delineation during conserving
surgeries. The refractive index map of invasive ductal and lobular carcinomas were constructed
from an inverse electromagnetic problem. Morphological dilation combined with refractive index
thresholding was conducted to classify the tissue regions as malignant or benign. A histology routine
was conducted to evaluate the performances of various dilation geometries associated with dierent
thresholds. It was found that the combination of a wide structuring element and high refractive index
was improving the correctness of tissue classication in comparison to other congurations or without
dilation. The method reports a sensitivity of around 80% and a specicity of 82% for the best case.
These results indicate that combining the fundamental optical properties of tissues denoted by their
refractive index with morphological dilation may open routes to dene supporting procedures during
breast‑conserving surgeries.
Terahertz imaging and spectroscopy have rapidly spread to dierent application areas thanks to the continued
development of ecient emitters and detectors between 0.1 and 7-THz1. e biomedical eld is one domain of
study that could benet from terahertz wave properties2,3. Radiations at terahertz frequencies have been shown
to be non-ionizing and non-hazardous for biological tissues at the power commonly employed to inspect the
super-cellular level4. Besides, terahertz radiations are notably sensitive to the presence of polar molecules such
as the most abundant component of the body: water5. Hence, dierent medical topics have been assessed with
terahertz imaging and spectroscopy to look for alternative and complementary methods to the existing ones.
ese investigations cover a broad range of possible surgical and clinical applications6–8. Among them, cancer
diagnosis remains the most widely investigated topic throughout the literature, covering blood9,10, brain11–13,
colorectal14,15, gastric16,17, liver6, lung18, oral19, skin20,21 and breast cancer22–26.
Investigations, conducted on breast cancer, mainly aim to develop supporting procedures for breast-conserv-
ative surgeries through breast tumour margin delineation. e success of breast-conserving surgeries is dictated
by the accuracy of delineating the concentric margins of excised breast volumes. Although there is no clear
description of what ideal margins are, it is recommended that no cancer cells remain adjacent to any inked edge/
surface of the specimen27. Conserving surgeries are usually followed by postoperative radiation management to
eradicate microscopic remains of disease28. Margin cleanliness is assessed via biopsy examinations during which
excised volumes are subsequently xed into formalin solution, embedded into paran, sliced in micrometric
sections and immersed into dierent alcohol and biological stain baths. Usually, hematoxylin and eosin stains
are used. e reason for that is that hematoxylin stains cell nuclei blue and eosin stains both the cytoplasm and
the extracellular matrix pink. e stain draws the global layout of a tissue structure so that a pathologist judges
the cleanliness of the margin29. Overall, two extreme cases of margin delineation can be observed: (1) positive
margins—malignant cells are located at the edge of the excised volume; (2) negative margins—an absence of
tumor cells at the edge or the distance of abnormal cells from the edge is at least more than 1-mm. Following
histopathologic inspection, up to 20% of excised breast samples are reported to exhibit positive margins30.
Reasons behind tumor edge delineation failure are oen presence of insitu carcinoma at close proximity to the
surgical margin, discontinuous tumor spread from the original surgery site, or inappropriate presurgical tumor
localization and inappropriate excision during surgery31. A positive margin inevitably leads to a second surgery
OPEN
*email:
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to favor low recurrence risk and to attain more widely clear surgical margins. In return, a second surgery con-
comitantly increases the morbidity rate.
So far, dierent research teams worldwide have reported the ability of terahertz imaging and spectroscopy to
discriminate between healthy and malignant breast tissues. ese studies were primarily conducted on formalin-
xed and paran-embedded breast tissue32,33. Such investigations opened the route for clinical studies on freshly
excised breast volumes18,22,24. e capabilities of terahertz radiation demarcation between normal and abnormal
tissue regions were originally attributed to free-water content. Indeed, free-water molecules have been proven
to present a specic permittivity step around 900-GHz5. Moreover cancer tissues are known to exhibit a greater
free-water content than normal tissues34. However, further studies have suggested that the origin of contrast
could not be solely attributed to water. at is because specic dielectric features exhibited by breast tissues, in the
low terahertz frequency band, were not observed in water dielectric prole35. Hence, it has been suggested that,
specic functional groups play a potential role22. Globally, the refractive index of breast cancer tissues has been
shown to be higher than the one observed for normal tissues over the terahertz band. On the contrary, the related
absorption coecient was reported as unsatisfactory parameter for demarcation35,36. Additionally, the contrast
level between healthy and malignant tissues depends on cancer cell density. In fact, while the resolution of any
light-based imager remains dictated by the diraction limit, two objects separated by a distance less than the
wavelength cannot be distinguished. For instance, the spatial resolution of a far-eld imaging system operating
at 1-THz will be limited to 0.3-mm. Hence, the respective response to the external terahertz radiation stimuli of
two biological entities, separated by a distance smaller than 0.3-mm, will have to be averaged. Considering the
typical diameter of the eukaryotic cell is at the order of tens of microns, it can be concluded that, such a terahertz
imager cannot manage to resolve entities at the cellular level. It has, however, been demonstrated that the use
of computational imaging system operating in a total internal reection geometry could resolve features with
a sub-wavelength lateral resolution37. While it can be expected that high densities of cancer cells will lead to a
well-dened demarcation, the dielectric response of isolated abnormal groups may be blurred by the healthy
surrounding and ultimately leading to recognition analysis failure. Although the diraction limit of resolution
may complicate recognition in areas sparsely populated by cancer cells38, it also raises delicate questions on the
exact frontier between two well localized normal and abnormal regions. Indeed, rather than depicting a sharp
contrast between areas, the obtained cliché may inevitably exhibit a smooth gradient from one to another area
which is a result of class-overlapping. at is particularly limiting when it comes to providing a pixel-by-pixel
diagnosis based on the information collected.
e present work proposes a new approach for the clinical classication of breast tissue pixels that overcomes
the limitations aforementioned. e method is based on the extraction of the terahertz refractive index map of
freshly excised samples followed by morphological dilation. A high value of the refractive index has been reported
as a reliable measure of the presence of cancer within a tissue22,24. Morphological dilation is a part of set-theory39
and is commonly employed to images having characteristics of ambiguity and vagueness40. It consists of expand-
ing a given shape contained in the input image. In biology, morphological processing was notably employed for
counting blood cells during blood smear test41, to isolate female gametocyte42 or for skin cancer segmentation43.
Operating dilation from regions exhibiting a higher refractive index should allow bypassing class-overlapping
limitations. Such a process is referred to as terahertz refractive index-based morphological dilation and operates
as follows: (1) the refractive index map of a freshly excised breast tissue is extracted through a specic objective
function minimization; (2) a refractive index threshold is dened such that pixels exhibiting a refractive index
higher than the threshold are classied as malignant while others are classied as benign; (3) morphological
dilation is used to spread the malignant zones to the neighborhood.
To conduct these investigations, dierent freshly excised breast tissues have been scanned in reection geom-
etry by means of a terahertz spectrometer. e refractive index maps have been extracted. Dierent refractive
index thresholds and dilation shapes have been tested. e related pixel classications have been compared to
those provided by a pathologist. Finally, the sensitivity and specicity of each combination of threshold—dila-
tion shape have been derived.
e paper is organized as follows: “Experimental framework” describes the experimental framework to
acquire raw terahertz images of the freshly excised breast tissues. “Refractive index map” describes the math-
ematical background to extract the refractive index map. “Morphological dilation” denes the morphological
dilation and the respective dilation shapes employed in the study. “Image registration” describes the registration
of obtained images with respect to the pathological cliché. “Diagnosis compliance” details the evaluation of com-
pliance between the classications provided respectively by the pathologist and the reported strategy. “Results”
presents the results for dierent samples. Finally, “Conclusions” presents the conclusions.
Experimental framework
e experimental protocol was assessed and approved by the ethics committee of the Bergonié Institute. Human
tissue analysis have been conducted in view of the fundamental ethical principles as stipulated in the Helsinki
declaration and its later revisions. Written informed consent from each patient undergoing breast surgery was
collected, stipulating their agreement regarding the use of their tissues for research purposes.
Breast tissue samples. Following surgery, breast excisions were cut into slices of a few millimeters and
kept into physiological serum before measurement to ensure the moisture content and delay the necrosis. A
maximum of one hour elapsed between the end of surgery and the terahertz acquisition starting time. Once
measurement was complete, excised tissue samples were placed in formalin-buered solution. is process ena-
bled the further histology routine to compare the diagnoses provided by the reported method and the patholo-
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gist. Biological samples analyzed using the method about to be reported were obtained from three dierent
patients. One sample was excised from each of these patients.
Measurement setup. Time-domain terahertz pulsed images were acquired with a TPS3000 spectrometer
(TeraView Ltd, Cambridge, UK) operating in reection geometry. In such systems, terahertz pulses are gener-
ated from the activation of a GaAs photoswitch. A photoswitch consists of a discontinuous metallic antenna
patterned onto a photoconductive layer. Ultra-fast near-infrared pulses with an energy greater than the semi-
conductor band gap are focused onto the gap between the two electrodes forming the photoswitch. e incident
pump laser thus propagates within the photoconductive layer and generates electron–hole pairs due to absorp-
tion. ose photocarriers are then accelerated within the electric eld of the biased antenna. e acceleration
of these charges produces a transient current that drives the metallic antenna and is eventually emitted as a
broadband terahertz pulse. e bandwidth directly depends on the lifetime of the carriers before recombination.
e carrier lifetime in the GaAs crystal is in the subpicosecond scale, hence enabling pulses with a bandwidth
ranging from 200-GHz to 2-THz.
e schematic of the experimental set-up is given in Fig.1. e route of the terahertz pulses is governed by
two planar mirrors and a knife-edge right-angle prism mirror (KERAPM). e terahertz pulses are focused on
the tissue sample supported by a 2-mm thick non-birefractive C-cut sapphire substrate (see Supplementary
Information, Supplementary Fig.1) via a polytetrauoroethylene (PTFE) lens. e maximum incident angle of
the terahertz pulses is
10◦
. Both the reections at the air-sapphire and sapphire-tissue interfaces are then focused
onto a photoconductive antenna detector. e detector is sourced from the same ultra-fast near-infrared pulses
used for terahertz wave generation with a beam splitter. e pulses are, however, delayed in time with a mechani-
cal delay line. e periodic variation of the delay line length allows a time gated detection of terahertz pulses
reected by the object. In order to reduce the natural absorption of terahertz pulses by water vapor molecules,
the terahertz route is conned within nitrogen chamber.
Refractive index map
To extract the refractive index from a raw frequency image, a reference electric eld has to be recorded. e refer-
ence electric eld
Er(ω)
refers to the electric eld generated by the acquisition system. e reference measurement
records the electric eld of the reection from a metal plate that is located where the sapphire substrate sample
holder is aimed to be positioned for tissue imaging. From the reference electric eld
Er(ω)
, the experimental
transfer function
Ts(ω)
, which is a measure of the disturbance experienced by the incident eld as a result of the
interaction with the sample, can be calculated:
with
Es(ω)
the sample frequency-dependent electric eld. e shape of transfer function
Ts(ω)
is a function of
the refractive index
n(ω)
and the extinction coecient
κ(ω)
of the sample under inspection.
Es(ω)
depends on
the Fresnel’s coecients in transmission
T(ω)
and in reection
R(ω)
, and on propagation coecients
P(ω,d)
:
with d being the thickness of the sapphire substrate. e Fresnel’s coecients
T(ω)
and
R(ω)
, as well as propaga-
tion terms
P(ω,d)
relate to the refractive index
n(ω)
and the extinction coecient
κ(ω)
through:
(1)
T
s(ω) =
Es
(ω)
E
r
(ω) ,
(2)
Es
(ω) ∝Tair−sapphire (ω) ×Rsapphire−tissue (ω) ×Tsapphire−air (ω) ×P
2
sapphire
(ω,d)
,
(3a)
T
a−b(ω) =2
ˆ
na
ˆna+ˆnb
,
Figure1. Schematic of the acquisition system. Drawn on SolidWorks 2020 SP3, www. solid works. com.
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where a and b are the indices of the respective medium,
ˆn
is the complex refractive index defined as
ˆn=n(ω) −jκ(ω)
and c is the light velocity in vacuum. Although the extinction coecient
κ(ω)
is involved
in the calculation of the transfer function
Ts(ω)
, no signicant dierences have been reported in the literature
between normal and abnormal tissue extinction22,35. Hence, solely the refractive index is further considered as
a possible intrinsic parameter for demarcation.
Map extraction. e extraction of the complex refractive index
ˆn(ω)
at each pixel from the experimental
transfer function
Ts(ω)
can be performed by solving an inverse electromagnetic problem. Inverse electromag-
netic problems usually minimize a specic convex objective function. is function denotes the discrepancies
between the experimental waveform
Es(ω)
and the waveforms
Ec
x(ω)
successively computed from a set of candi-
date parameters, where the x-index refers to the
xth
-candidate tested. e candidate waveforms
Ec
x(ω)
are com-
puted as stipulated in44. e corresponding transfer functions
Tc
x(ω)
are calculated in the same way as described
by (1). e measures of discrepancies
δMx(ω)
between the experimental transfer function
Ts(ω)
and the com-
puted transfer functions
Tc
x(ω)
are dened as:
e natural logarithmic ratio is favored here instead of standard dierence as it is more penalizing. Finally, the
objective function
χ(ω)
to be minimized is dened as:
e minimization of the transfer function is subject to the following set of candidate parameters:
It was stated before that the sample is maintained by the sapphire substrate. Instead of extracting the properties
of the sapphire substrate for each pixel, the properties were extracted upstream, in absence of a sample, and fol-
lowing the same minimization process. e sapphire properties are provided in Supplementary Information, see
Supplementary Fig.2. Finally, applying the above process to each electric eld stored in each pixel of the sample
image allows to construct the refractive index map.
Once the refractive index map is obtained, it is converted to a binary map that shows areas that are consid-
ered malignant or benign. To do so, a threshold among the refractive index vector has to be set. Depending on
the dened value for the threshold, one may progressively increase or decrease the extent of areas classied as
malignant, since pixels with a refractive index higher than the threshold are classied as cancerous. A schematic
of the process is given in Fig.2.
Operating frequency. Although the refractive index is oen referred to as optical constant, its prole var-
ies as a function of the frequency. Previous studies have reported the terahertz frequency dependent refractive
index values of abnormal and normal breast tissues36. Overall, the global dierence between these values was
shown to be the highest between 300- and 700-GHz, roughly. Hence, rather than investigating the entire band,
the classication was operated at 550-GHz, as a good trade-o between signal-to-noise ratio (SNR) and higher
frequency spatial resolution45. However, naively classifying pixels via the refractive index exhibited at 550-GHz
(3b)
R
a−b(ω) =
ˆn
a
−ˆn
b
ˆna+ˆnb
,
(3c)
Pa(ω,d)=e
−j
ωd
cˆna
,
(4)
δ
Mx(ω) =ln
|Ts
(ω)
|
|
T
c
x
(ω)|
.
(5)
χ(ω) =δM(ω) ×δM(ω).
(6)
min
n
(ω)
,
κ(ω)χ (ω), subject to
n∈[1.5;3],with �n=1.10
−2
κ∈[
0
;
1
]
,with
�κ =
1.10
−3
Figure2. resholding principle applied to the refractive index map. (a) Schematic refractive index map; (b)
binary refractive index map with a threshold set at 2.4; (c) binary refractive index map with a threshold set at
2.1; (d) binary refractive index map with a threshold set at 1.8.
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may hardly be relevant. In particular, the refractive index extracted at the edges of malignant regions with low
density may present values close to the ones of healthy tissues. erefore, morphological dilation is introduced
to overcome this limitation.
Morphological dilation
Prior to dilation, the refractive index map is converted to a binary image as it was described in the previous
section. e dilation can therefore be referred to as binary dilation. e dilation consists of a shi-invariant
addition, denoted “
⊕
”, within the meaning of Minkowski46. Mathematically, let’s dene P as an ensemble that
contains the pixels (x,y) of the tissue imaged. e binary dilation
∂�(P)
of P by a shape
∈Z2
- also referred
to as a structuring element, is given by:
where
∈
produces the translation from P to
∂�(P)
. Supposing the matrix P and the structuring element
as represented in Fig.3, the matrix
∂�(P)
is obtained by superimposing the center of
aligned with each pixel
in P that has a value of 1.
In the present work, three dierent structuring elements have been considered to dilate the binary refractive
index map. ey are referred to as
1
,
2
and
3
classiers. eir spatial properties are exposed in Fig.4. ese
specic geometries allow the classiers to act in the close vicinity of a starting pixel and with the same impact
in all directions.
erefore, depending on the classier considered, a pixel may be attributed to the malignant group if at least
one of the component
of the structuring element
n
—where
n∈N∗
—reports a pixel whose refractive index
is higher than the dened refractive index threshold. Alternatively, the structuring elements can be seen as the
area of inuence of a cancerous pixel. Consequently pixels with a refractive index lower than the threshold but
situated in such an area of inuence, are turned into malignant pixels. It is however important to note that the
process is constrained to a unique dilation and therefore, newly classied malignant pixels cannot, in turn,
exercise a zone of inuence.
In order to carry out the dilation and the registration steps that follow, it is essential to preserve the mor-
phology of the imaged sample. To do this, the dilation procedure must be carried out with respect to the initial
contour of the sample generated from a standard contouring algorithm, thus preventing the appearance of
cancerous pixels outside the original surface of the sample. A schematic of the dilation process operated on a
binary refractive index map is given in Fig.5.
Image registration
e classication images provided by the reported method and the ones given by the pathologist do not share
the same coordinate system. Image registration is the process of migrating dierent images into one common
coordinate system47. erefore, image registration is necessary to enable the comparison between the data sets.
(7)
∂
�(P)=P⊕�=
x+,y+|∈�
,
Figure3. Morphological dilation operated with a cross structuring element
on the matrix P.
Figure4. Geometry of the three dierent classiers
1
,
2
,
3
.
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Eectively, the spatial resolution of the optical microscope used to acquire pathology clichés is far greater than
the one of the employed terahertz imager. Additionally, the orientation of the tissue sample in the terahertz
image and in the clinical image are expected to be dierent, as they are not acquired with the same angle. A
simple pixel-by-pixel comparison is therefore not possible as it stands. Prior to comparison, images have to be
resized and reorientated. e registration process is feature-based and solely involves image contours to avoid
unintentional human bias. e dierent steps that are followed to register the images with respect to each other
are hereaer described.
Contouring. Contour lines, also called isolines, can be calculated by interpolating the value of the scalar eld
found at each pointel of each pixel. An innite number of isolines can however be delineated. e choice of the
contour to dene the spatial extent of the sample in the image remains therefore subjective. For each sample, the
isoline that suited the visualized tissue area best was determined by carefully comparing the terahertz image and
the dierent contour levels.
Resizing. As the resolution of the images is dierent, it is necessary to resize the histology pictures. To do
so, a bicubic interpolation is operated onto pathology images. Contrary to the previous interpolation, where it
is based on the four nearest pixels, bicubic interpolation takes into account a neighborhood of sixteen pixels.
erefore, bicubic interpolation provides a smoother histology slide than simple bilinear interpolation.
Reorientation. First, the contour of the terahertz image is manually and progressively twisted to bring it
closer to the twist angle of the pathology contour. Once the orientations approximately match, the pathology
contour is iteratively rotated to establish the correlation between the two contour matrices at each step. Basi-
cally, it consists in determining the Pearson’s correlation coecients48. e rotation angle providing the highest
positive correlation is selected and the terahertz image is correspondingly rotated. e ow chart of these three
pre-treatments, namely contouring, resizing and reorientation for image registration is provided in Fig.6.
Image discrepancies issues. Although one can resize and reorientate the two images with respect to each
other, the pathology cliché and the terahertz image may not perfectly depict the same information. First, while
terahertz imaging is performed directly on freshly excised tissues, the pathology diagnosis is established aer
the histology routine. Moreover, to obtain the pathology image, the excised tissue is rst xed in neutral buered
formalin, then dehydrated in subsequent alcohol baths with increasing concentrations, then cleared in a solvent
before being inltrated and nally embedded in paran wax. At this stage, the processed tissue is encased in a
paran block that can be sliced in sections of a few microns thickness to be deposited on glass slides. ese tis-
sue sections are deparanized, rehydrated and subsequently stained with hematoxylin and eosin dyes. Finally,
they are dehydrated in alcohol and cleared in a solvent before being mounted with a coverslip. e embed-
ding, the sectioning and the desiccation alter the global structure of the tissues. ese alterations are collectively
referred as artefacts49. Artefacts include loss of tissue area and details, folds and wrinkles or cracks and holes.
ese alterations may result in misinterpretation as they are modifying the morphological structure of tissues.
Alternatively, these artefacts may drastically limit the evaluation of the terahertz classication compliance (see
Supplementary Information Supplementary Fig.3, for an example based on one of the tissue reported by the pre-
sent work). However, histological slides remain the only available reference picture that allows one to examine
the performances of classier under-test. Overall, there are two ways to deal with such issues: (1) correcting the
histology slides at risk of adding articial information; (2) comparing directly the terahertz image with the raw
pathology image at risk of underestimating the eciency of the method. e rst way would require to morph
the pathology image to correspond to the terahertz picture. Some procedures to do so were reported in the
literature50. However, these methods are cumbersome and the evaluation of the histological cliché reconstruc-
tion is oen complicated since no perfect reference pathology image exists. As terahertz imaging remains a new
Figure5. Schematic of a morphological dilation applied to a binary refractive index map over a tissue sample.
(a) Binary refractive index map with a threshold set at 1.8; (b) morphological dilation applied to the binary
refractive index map with an arbitrary classier
n
.
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technology for breast carcinoma delineation, the second approach was favored—at risk of underestimating the
eciency of the classiers.
Diagnosis compliance
Following the histology routine, pathology images are colored in dierent shades of blue and pink. e patholo-
gist draws the contour of malignant areas based on his/her expertise. From the interpretation of the pathologist,
the images were binarized and each pixel was classied either as benign or as malignant51.
Once both diagnosis images exhibit binary information, have the same size and orientation, the compliance
between them can be evaluated. In case of discrepancies, the pathologist classication prevails over terahertz
delineation. e present section describes how the ability of classiers was evaluated with respect to the patholo-
gist one.
Performance of the classication test. As each diagnosis presents a binary information, four dierent
cases can be distinguished:
• True negative: both methods classify a pixel as benign;
• True positive: both methods classify a pixel as malignant;
• False positive: the terahertz method stands for a malignant pixel while histology stipulates a benign pixel;
• False negative: the terahertz method stands for a benign pixel while histology stipulates a malignant pixel.
Hence, for each refractive index threshold associated with a specic classier, one can ll the corresponding
confusion matrices that highlight the classication procedure performances. In such error matrices, the rows
represent the instances in the terahertz class, here the predicted class, while columns represent the actual diag-
nosis provided by histology examination52.
From these matrices, the eectiveness of the classication method is assessed by creating the receiver operatic
characteristic (ROC) curve53 for each classier. e ROC curve represents the ability of the classier to provide
the correct diagnosis as the refractive index threshold varies. e ROC curve is obtained by plotting the true posi-
tive rate (TPR) as a function of the false positive rate (FPR). e TPR is dened as the number of true positives
divided by all pixels classied by the pathologist as positives: true positives and false negatives. e FPR is dened
as the number of false negatives divided by all pixels classied by the pathologist as negatives: false positives and
true negatives. It can also be thought as a plot of the sensitivity—that is equivalent to the TPR dened in Eq. (8),
against the probability of false-alarm—that can be calculated as (1—specicity) and dened in Eq. (9)54. ese
measures of performances are favored as they are not sensitive to changes in data distributions, compared to
accuracy and to error rate. Hence, both metrics can be used with imbalanced data55.
(8)
True Positive Rate
=Sensitivity =
True Positives
True Positives +False Negatives ,
Figure6. Flow chart of the registration procedure for predicted diagnosis evaluation.
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To complement these measures, the area under each ROC curve (AUC) is calculated as it relies on the perfor-
mance of score classiers for all possible classication thresholds56.
Finally, the best discrimination thresholds are selected as the ones that provide the highest sensitivity while
preserving the healthy tissue area from false diagnosis, i.e. specicity. It is noted that the aforementioned clas-
sication procedure is studied for the specic case of breast conserving surgery. Hence, it is essential to preserve
the healthy area while removing the malignant zones. Ultimately, the best classiers are selected as the ones that
provide the highest measure of
TPR −FPR
, since higher values of this function indicates more accurate results.
Results
In this section, the classiers are employed to evaluate their eectiveness on three freshly excised breast tissues.
Two of these samples were diagnosed as invasive ductal carcinoma (IDC) and one was identied as an invasive
lobular carcinoma (ILC). ese samples are referred to as test sample TS#1, TS#2 and TS#3.
TS#1. TS#1 is an invasive ductal carcinoma. e pathology image with some enlightened pathology areas,
the pathology mask, the raw terahertz image obtained at 550-GHz and the correlated refractive index map are
presented in Fig.7.
It can be observed that the raw terahertz image as well as the refractive index map exhibits specic features
that correspond to the pathology image. Regions depicted in Fig.7a,b. correspond to brous tissues that are
included in an adipose matrix. Such regions are therefore expected to globally give rise to a lower refractive index
than the one classied as malignant as depicted in Fig.7c,d. Although such a refractive index seems overall lower
than the refractive index of the tumour, it remains relatively close to it. erefore, classifying only on the basis of
the refractive index would certainly prove to be inecient. e sensitivity and the specicity of each structuring
element classier for varying refractive index threshold were calculated for TS#1. e corresponding ROC curves
and
TPR −FPR
functions are given in Fig.8.
Each
n
-dependent ROC curve is located to the le of the
TPR =FPR
line in Fig.8, proving that the fraction
of true positives is greater than the proportion of false positives. It is clear that the use of the refractive index
alone as a classier (
0
) is shown to be less ecient than associating the refractive index with a classier. Such
a statement is not surprising as the classication does not consider the neighborhood. While on ROC graph-
ics, depicted in Fig.8, it does not seem that obvious which classier among
1
,
2
and
3
performs well, the
TPR −FPR
visualization indicates that the structuring element
3
in association with a high refractive index
threshold by about 2.6 is the most ecient rule of classication. e association provides a classication with a
sensitivity by around 80% and a specicity of 82%. What is more, the wider the structuring element, the higher
the refractive index has to be set for good performances. Eectively, starting with a high refractive index makes it
possible to identify, in a rst instance, tissue areas densely populated with cancer cells, while a broad structuring
element makes it possible to eciently spread the identication over a wide zone.
e corresponding AUC for each ROC curve, the
TPR −FPR
value, the sensitivity and the specicity for the
rst two best refractive index thresholds are given in Table1 (see Supplementary Information, Supplementary
Table1. for the complete list of performances). While
1
and
2
are less ecient than
3
for both sensitivity and
specicity, the
0
classier provides a slightly greater sensitivity for a threshold of 2.1, by about 83%. However,
the gain of 4% in sensitivity with respect to
3
costs concomitantly 20% in method specicity. Reasonably, this
gain is not worth it, considering such a drastic decrease in classication specicity. Alternatively, if one wants
(9)
False Positive Rate
=1−Specificity =
False Positives
False Positives
+
True Negatives .
Figure7. Sample TS#1. (i) Pathology image and correlated view of the respective zones (a,b,c,d); (ii) pathology
mask; (iii) raw terahertz image at 550-GHz; (iv) refractive index map at 550-GHz.
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to increase the sensitivity while maintaining specicity at a reasonable level, second best thresholds may oer a
promising substitute. On using the second best threshold provided by
3
of 2.5, an increase of 7% in sensitivity
conjointly leads to a decrease by about 12% in specicity. By doing so, one reaches a sensitivity of 86%.
e superimposition of the classication images from the reported method and the clinical one, correspond-
ing to the performances listed in Table1 are given in Fig.9.
TS#2. TS#2 sample is an invasive ductal carcinoma from which a 67 years old woman was suering. e
initial tumor site was found to be roughly 100 mm
2
. On Fig.10, the pathology image with some enlightened
pathology areas, the pathology mask, the terahertz image at 550-GHz, and the refractive index map are shown.
e pathology image as well as the pathology mask exhibit the presence of a hole, where no tissue is found. e
lack of tissue in the middle of the section is not natural and enlightens the issues, that have been previously
reported towards pathology images. Hence, this specic region is not considered for performance evaluation.
e ROC curves as well as the
TPR −FPR
function for dierent classiers with various thresholds are given
in Fig.11. Similarly to the foregoing, all ROC curves are located to the le of the
TPR =FPR
line, hence proving
that the fraction of true positives remains greater than that of false positives.
e most eective classiers towards conserving classication are
2
and
3
, both for a threshold set at 2.1.
While the combination of such a threshold with
2
provides a sensitivity of 67% and a specicity of 70%, the
same threshold operating with
3
gives rise to a sensitivity by about 78% and a specicity of 57%. Hence, tuning
the structuring element geometry would oer an interesting trade-o between specicity and sensitivity. e
respective performances of each classier applied to TS#2 are listed in Table2 (see Supplementary Information,
Supplementary Table1 for the complete list of performances).
e classication maps involving each classier and their respective best performing thresholds are exposed in
Fig.12. ese images show the improvement in classication with the use of morphological dilatation. Moreover,
they highlight the diculties of good prediction at the outer margins. Low performance at the outer margins
may come from the non-conformity of the information in these areas between the terahertz image and the
histology picture. e most convincing hypothesis for this non-conformity is the tissue deformation imposed
by the histological routine.
Figure8. Le: receiver operating characteristic for the dierent classication methods, at 550-GHz applied to
TS#1. e black line stands for
TPR =FPR
. Right: refractive index threshold as a function of the
TPR −FPR
measure for the dierent classiers.
Table 1. Statistical measure of the performance of the classiers and AUC. e sensitivity and specicity
obtained for the best performing classier-refractive index threshold association is given in bold.
Classier
0
1
2
3
AUC 0.7804 0.8149 0.8285 0.8360
RI-threshold 2.1 2.2 2.4 2.3 2.5 2.4 2.6 2.5
TPR–FPR 0.4540 0.4181 0.5227 0.4829 0.5759 0.5093 0.6068 0.5433
Sensitivity% 83 62 72 84 76 86 79 86
Specicity% 62 79 81 65 81 65 82 69
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TS#3. In contrast to the two previous samples, the TS3 sample was taken from an 83 years old patient with an
invasive lobular carcinoma. On Fig.13, the pathology image with some enlightened pathology areas, the pathol-
ogy mask, the terahertz image at 550-GHz, and the refractive index map are shown.
e ROC curves plotted in Fig.14 indicate a lower eciency towards classication than the eciencies for
TS#1 and TS#2. e cause may be found in the distribution of cancer cells within the malignant zone, in com-
parison to previously tested samples. While for other cases the malignant zone was densely populated, cancer
cells are found in small quantity and in an inhomogeneous manner over TS#3. Additionally, the histology routine
may have altered the tissue morphology as stated in “Image discrepancies issues”.
e AUC values and the performances for each classier are given in Table3 (see Supplementary Information,
Supplementary Table1. for the complete list of performances). Despite the lower eciency the most accurate
classifying strategy remains
3
when associated with a refractive index threshold of 2.3. e
TPR −FPR
measure
Figure9. TS#1 tissue sample classication maps at 550-GHz for
0
,
1
,
2
,
3
and their respective rst two
best thresholds. “Not applicable” refers to regions where the binary pathology classication and the binary
terahertz classication image do not match spatially. e values listed in each box are respectively standing for
the refractive index threshold, the true positive rate and the false positive rate.
Figure10. Sample TS#2. (i) Pathology image and correlated view of the respective zones (a,b,c,d); (ii)
pathology mask; (iii) raw terahertz image at 550-GHz; (iv) refractive index map at 550-GHz.
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is by around 0.28 with a sensitivity of 53% and a specicity of 76%. As already indicated these performances are
below the ones reached for other study cases.
2
classier oers a greater specicity of 85% but simultaneously
concedes 11% upon sensitivity, thus falling below the critical threshold of half the number of malignant pixels
correctly classied. e weak density of cancer cells within a lobular carcinoma slice may lead one to opt for
0
classier operating in association with a low refractive index threshold to maximize the sensitivity, despite
a concomitant loss in specicity.
e corresponding classication images for TS#3 for each classier and the correlated best refractive index
threshold are demonstrated in Fig.15. e global classication clearly suers from the spatial discrepancies
between the fresh state tissue and the histological state. Even though such dierences are expected to be the main
roots behind classication accuracy weakness, the histological type of TS#3 may also trigger diculties. It can
be assumed that the classication strategy may provide better performances when applied on ductal carcinoma
cases than on lobular ones. However, it is noted that the number of samples investigated does not allow to assert
such a hypothesis.
Conclusions
In this paper, a new approach to support breast carcinoma margin delineation during surgeries with terahertz
radiations was proposed. e method relies on the acquisition of the excised samples by means of a terahertz
time-domain imager followed by a segmentation based on the extracted refractive index map at 550-GHz and
its morphological dilation. Morphological dilation was introduced to overcome the weakness of the refractive
index alone as a classier in tissue regions sparsely populated with cancer cells. Dilation was used to construct a
zone of inuence of pixels. Hence, tissue areas close to regions identied as malignant were succesfully classied
as cancerous despite a refractive index suggesting benign zones.
e performances of the classications were assessed for three dierent samples. Overall, the association
of a high refractive index threshold with a wide dilation has shown to be the most appropriate combination to
maintain both method sensitivity and specicity at decent levels for invasive ductal carcinoma. e best per-
formances of the methods have been reported to stand by about 80% in sensitivity and 82% in specicity. On
the contrary, the same methodology applied onto an invasive lobular carcinoma showed lower performances.
Various hypothesis were drawn to determine the roots for classication failure. While lobular carcinoma are
Figure11. Le: receiver operating characteristic for the dierent classication methods, at 550-GHz applied
to TS#2. e black line stands for
TPR =FPR
. Right: refractive index threshold as a function of the
TPR −FPR
measure for the dierent classiers.
Table 2. Statistical measure of the performance of the classiers and AUC. e sensitivity and specicity
obtained for the best performing classier-refractive index threshold association is given in bold.
Classier
0
1
2
3
AUC 0.6976 0.7307 0.7264 0.7127
RI-threshold 1.9 1.8 2.1 2.0 2.1 2.2 2.1 2.2
TPR–FPR 0.3013 0.3006 0.3395 0.3307 0.3697 0.3113 0.3480 0.3295
Sensitivity% 62 76 54 69 67 51 78 62
Specicity% 68 54 80 64 70 80 57 71
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Figure12. TS#2 tissue sample classication maps at 550-GHz for
0
,
1
,
2
,
3
and their respective rst
two best thresholds. “Not applicable” refers to regions where the binary pathology classication and the binary
terahertz classication image do not match spatially. e values listed in each box are respectively standing for
the refractive index threshold, the true positive rate and the false positive rate.
Figure13. Sample TS#3. (i) Pathology image and correlated view of the respective zones (a,b,c,d); (ii)
pathology mask; (iii) raw terahertz image at 550-GHz; (iv) refractive index map at 550-GHz.
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globally less populated by cancer cells than the ductal histology type, pathology image alterations may also have
contribute by rendering the diagnosis evaluation tedious.
e recognition performances of malignant areas could be improved. Indeed, the terahertz classication has
localized false negatives surrounded by true positives. erefore, implementing an additional and simple pro-
cessing that classies as malignant, benign-predicted pixels that are encircled by cancerous ones would enhance
the classication accuracy.
Although investigations on higher rank classier, i.e. for
n
with
n>3
, have not been conducted, a more
ecient structuring element could be found. Nevertheless, a high rank for a structuring element is accompanied
by an equally high refractive index threshold. us, a reasonable assumption would be that the refractive index
suitable for the use of these higher-ranked classiers lies beyond the optical properties of biological tissues.
is preliminary investigation towards terahertz refractive index-based morphological dilation may open the
routes to rene strategies to improve the accuracy with which breast tumour margins are delineated. However,
the eld still stands in its early stages and suers challenges due to pathology reference image alterations that
complicate classication correctness assessment. Additionally, performance comparison with other classication
algorithms are yet to be investigated and will be needed to pursue with the proposed methodology. Finally, and
in authors’s opinion, the applicability of terahertz waves for breast carcinoma margin demarcation still requires
further studies to evaluate its feasibility in the clinical environment.
Methods
Numerical procedures were conducted with in-house soware, written with the MatLab development framework.
e soware follows mathematical procedures described in this paper and our preceding works.
Figure14. Le: receiver operating characteristic for the dierent classication methods, at 550-GHz applied
to TS#3. e black line stands for
TPR =FPR
. Right: refractive index threshold as a function of the
TPR −FPR
measure for the dierent classiers.
Table 3. Statistical measure of the performance of the classiers and AUC. e sensitivity and specicity
obtained for the best performing classier-refractive index threshold association is given in bold.
Classier
0
1
2
3
AUC 0.6478 0.6631 0.6695 0.6693
RIreshold 2.0 1.9 2.2 2.1 2.3 2.2 2.3 2.4
TPR–FPR 0.2215 0.2080 0.2484 0.2148 0.2690 0.2626 0.2845 0.2234
Sensitivity% 55 72 49 65 42 64 53 36
Specicity% 68 49 76 56 84 62 76 86
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Received: 18 December 2020; Accepted: 5 March 2021
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Acknowledgements
e authors acknowledge the German Research Foundation for funding this work as a part of the Priority Pro-
gram ESSENCE (SPP 1857). e authors gratefully acknowledge partial nancial support for this work from the
French Nouvelle-Aquitaine Region. e authors would like to acknowledge Jean-Baptiste Perraud and Dominika
Warmowska for fruitful discussions.
Author contributions
Q.C. conceived and conducted the research; S.C. prepared the freshly excised breast samples; G.MG. analyzed
the pathology images, provided the histological classication and assured the clinical background of the study;
Q.C. wrote the manuscript with input from G.MG., T.B., P.H., U.P., T.Z., J-P.G. and P.M.; P.M. supervised the
study. All authors reviewed the manuscript.
Competing interests
e authors declare no competing interests.
Additional information
Supplementary Information e online version contains supplementary material available at https:// doi. org/
10. 1038/ s41598- 021- 85853-8.
Correspondence and requests for materials should be addressed to P.M.
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