MSc Physics and Astronomy Biophysics and Biophotonics
Master Thesis
A method to find the absolute
optic axis orientation of in vivo
fibrous tissue with endoscopic,
polarization-sensitive optical
coherence tomography
by
Laura van Oosterom
10375619 March 2021 60 EC January 2020 - March 2021 Examiner/Supervisor Prof. dr. J.F. de Boer Examiner Prof. dr. A.G.J.M. van
Leeuwen
Daily supervisor MSc. M. Vaselli
Abstract
Currently endoscopic OCT is the only imaging modality that can retrieve high resolution cross section images of the airways in vivo. In addition, polariza-tion sensitive OCT (PS-OCT) can be used to retrieve the birefringent properties of the tissue in the images. Unfortunately, a PS-OCT setup itself also contains birefringent properties. From the detected signal it is not possible to determine what part of the change in polarization state was induced by the setup and what part by light-tissue interaction. This ultimately means that, without any a pri-ori knowledge, it is only possible to retrieve the relative optic axis pri-orientations in PS-OCT images. This thesis presents a method that finds the absolute optic axis orientation of birefringent tissue with an endoscopic PS-OCT setup. The method uses the absolute optic axis of the catheter sheath to calculate the optic axis orientation offset in the images. An algorithm is created that consequently corrects for the offset in the entire image. Finally, the algorithm was applied on in vivo measurements of an ILD and an asthma patient. The results show several clinical benefits, i.e. the data-processing method simplifies recognizing the same location in the lungs of a patient between independent measurements and it simplifies identifying tissue types. Finding the same location in the lungs in between measurements and identifying the tissue in the images consequently improves pre- and post-treatment comparisons of lung morphology.
Contents
1 Introduction 3
2 Theory 6
2.1 Intensity based OCT . . . 6
2.1.1 Michelson interferometer . . . 7
2.1.2 Low coherence interferometry . . . 8
2.1.3 TD-OCT . . . 9
2.1.4 FD-OCT . . . 9
2.1.5 OCT as a clinical imaging modality . . . 11
2.2 PS-OCT . . . 12
2.2.1 Polarization properties . . . 12
2.2.2 Jones formalism . . . 13
2.2.3 Stokes-Mueller formalism . . . 14
2.2.4 Differential Mueller formalism . . . 17
2.2.5 Retrieving the relative optic axis orientation . . . 18
2.3 Second harmonic generation microscopy . . . 20
3 Methodology 21 3.1 Introduction . . . 21
3.2 Setups . . . 22
3.2.1 Endoscopic PS-OCT . . . 22
3.2.2 SHG microscopy . . . 25
3.3 Chicken breast sample . . . 26
3.4 Patients . . . 26
3.5 Data processing . . . 27
3.5.1 Jones matrix modelling . . . 27
3.5.2 E-field averaging . . . 28
3.5.3 Signal roll-off correction . . . 28
3.5.4 Segmentation . . . 29
3.5.5 Chromatic dispersion compensation . . . 30
3.5.6 Polarization mode dispersion compensation . . . 30
3.5.7 Rotating mirror correction . . . 30
3.5.8 Differential Mueller matrix calculation . . . 30
3.5.10 From a rectangular to a circular image . . . 32
4 Results 34 4.1 SHG microscopy . . . 34
4.2 Endoscopic PS-OCT on sample . . . 35
4.3 Endoscopic PS-OCT in vivo . . . 39
4.3.1 ILD patient . . . 39
4.3.2 Asthma patient . . . 40
5 Discussion 44
Chapter 1
Introduction
Currently the golden standard to diagnose most lung diseases is a biopsy [1]. A biopsy is an invasive and local procedure that involves the surgical extraction of a small tissue sample. The extracted sample is examined under a microscope and this provides high resolution images of the tissue in the sub-micrometer range. Due to the invasive and local nature of a biopsy the use of optical coherence tomography (OCT) in lung disease research has attracted interest in the past decade. OCT is a widely used medical imaging technique; it reconstructs a depth profile based on the intensity of light that is back scattered by different tissue layers. OCT has a slightly lower resolution (in the micrometer range) than microscopy. It uses near-infrared light and thus has a shallow penetration depth in tissue up to several millimeters. Therefore, OCT needs to be used in an endoscopic fashion to reach the airway walls, this makes the technique minimally invasive. However, for an OCT measurement no tissue needs to be extracted and right now it is the only method that can provide high resolution cross section images of in vivo airways [2]. In addition, by pulling the endoscope out of the lungs during a measurement consecutive cross sections can be scanned, creating a three-dimensional image. This results in OCT being both less invasive and less sensitive to sampling errors than a biopsy.
Additional contrast can be created in OCT images by exploiting other prop-erties of light in the signal beyond intensity, e.g. polarization. The OCT modal-ity that exploits the polarization properties of light is called polarization sensi-tive OCT (PS-OCT). Birefringent tissue, i.e. fibrous tissue, changes the polar-ization state of light when it propagates through it. Examples of birefringent tissue are muscle and scar tissue. By exploiting the polarization properties of the OCT signal the change in polarization state due to light-tissue interaction can be measured. From the measured change in polarization state the orienta-tion of the fibers can be derived, this orientaorienta-tion is also referred to as the optic axis. These orientations can for example be color coded to create additional depth-resolved contrast in OCT images. However, unfortunately a PS-OCT setup itself also contains birefringent properties. The birefringent properties of the setup can be different every measurement due to e.g. induced stress on the
endoscope optical fiber [3]. From the detected signal it is not possible to de-termine what part of the change in polarization state was induced by the setup and what part by the tissue. This ultimately means that, without any a priori knowledge, it is only possible to find a change in birefringent properties between tissue layers in an image [4]. This is why most PS-OCT research groups show the relative optic axis orientation of tissue in their images [5][6][7]. However, finding the absolute instead of the relative optic axis orientation is clinically beneficial. Mostly because it can assist in identifying tissue types in images. For example, every lung contains airway smooth muscle (ASM) in the superfi-cial tissue layers of the airways and the fibers of this muscle type are known to be oriented circumferentially around the airways [8]. By finding the absolute optic axis orientation in images an ASM tissue layer can be identified based on its fiber orientation and this way it can be distinguished from other muscle types and scar tissue. In addition, images of the absolute orientation of fibers can sim-plify recognizing the exact same location in a patients lungs when comparing independent measurements. This is because, next to structures, the location in the airways can now also be recognized by depth-resolved absolute birefrin-gent properties. Finding the same location in the lungs consequently simplifies comparing pre- and post- treatment morphology of birefringent tissue.
Considering the additional clinical information the absolute optic axis pro-vides over the relative optic axis; this thesis elaborates on a method that finds the absolute orientation of the optic axis of tissue with an endoscopic PS-OCT setup. In 2018 Villiger et al. [9] showed in vivo absolute optic axis orienta-tion images of human coronary arteries. They used the catheter sheath that protects the endoscope as an absolute optic axis reference point in their data-processing steps. In this thesis a similar method was successfully implemented for an endoscopic PS-OCT setup designed for pulmonary measurements. In this setup the catheter sheath is located in the imaging area of the PS-OCT setup and contains birefringent properties. These birefringent properties were thus far unknown. To find the optic axis orientation of the catheter sheath a homogeneously birefringent sample was analyzed, i.e. chicken breast. The optic axis orientation of the fibers in the chicken breast sample were first found using second harmonic generation (SHG) microscopy. Then the sample was measured with the endoscopic PS-OCT setup. Because the absolute optic axis of the sam-ple was known, the absolute optic axis of the catheter sheath could be derived from the relative optic axis PS-OCT images. A post-processing algorithm was created that changes the measured optic axis orientation of the catheter sheath to its absolute optic axis value. This offset correction is applied to all pixels to find the absolute optic axis within the entire image.
Finally, the created algorithm is applied on in vivo measurements of an in-terstitial lung disease (ILD) patient and an asthma patient. ILD is a disease that can have many different causes, but always results in scarring and/or in-flammation of the superficial tissue surrounding the airways [10]. Asthma is a respiratory disease that is characterized by reversible airflow obstruction be-cause of airway inflammation and contraction of the small airways. Morphology of an asthmatic lung shows an excess amount of ASM in the superficial layers of
the airways [11]. Currently there is no cure for asthma, but patients with severe symptoms that are poorly controlled with medication are sometimes selected for a bronchial thermoplasty (BT) treatment [12]. BT is a relatively new treatment based on the release of thermal energy onto airway walls. The treatment has proven to reduce the amount of ASM and to relieve severe asthmatic symptoms for most treated patients [13]. However, the exact biological response of the human body that induces the reduction of ASM and the relieve of symptoms is yet to be determined [14]. Both ILD and asthma patients contain birefrin-gent tissue in the superficial tissue layers of the airways, therefore endoscopic PS-OCT is an appropriate tool for examination of both diseases. Finding the absolute optic axis orientation instead of the relative optic axis orientation of the birefringent tissue provides an innovative in vivo view of the morphology of the respiratory diseases illustrated in a full airway cross section. In addi-tion, finding the absolute optic axis orientation in asthma patients can assist in quantifying and qualifying the effects of BT treatments.
The next chapter of this thesis contains the theoretical background of OCT and PS-OCT. In chapter 3 the full methodology of this research is explained. Then chapter 4 contains the found results. The results are discussed in chapter 5, followed by a final conclusion in chapter 6.
Chapter 2
Theory
The first section of this chapter explains the technique behind intensity based OCT, and the development that it has gone through. Followed by a section that describes how birefringent tissue changes the polarization state of light. This section also explains how this change in polarization can be mathematically expressed with the Jones, Stokes and Mueller formalism. Finally the third section briefly introduces second harmonic generation microscopy, because this technique is used to evaluate the orientation of the fibers in the chicken sample.
2.1
Intensity based OCT
The technique of OCT is similar to that of ultrasound; a wave is sent onto tis-sue and the reflection of the wave is detected. An important difference is that OCT uses electromagnetic waves, where ultrasound uses sound waves. This is an important difference because in tissue light travels five orders of magnitude faster than sound. The reflection of sound waves is detected based on time; the measured time between emission and detection of the waves provides in-formation on where in the tissue the signal was reflected. For light waves this method is not possible, because the speed of light is too high to precisely distin-guish signals coming from different tissue layers. A solution to this problem was found in using low coherence interferometry applied in a setup that is based on a Michelson interferometer. The next two sections will elaborate on the Michel-son interferometer and the application of low coherence interferometry in such a setup. The two sections that then follow describe how the use of low coherence interferometry in a Michelson interferometer based setup results in the current two forms of OCT, i.e. time-domain OCT (TD-OCT) and Fourier-domain OCT (FD-OCT).
2.1.1
Michelson interferometer
The Michelson interferometer is a well-known interferometer employed in many scientific experiments. Probably the most famous example thereof is the Michelson-Morley experiment. The experiment was executed in 1887 by Albert Michelson, the inventor of the Michelson interferometer, and Edward Morley. The aim of the experiment was to detect the ether; this was the medium that most physicist at that time believed light waves propagated through. Instead they disproved the existence of the ether [15]. An unexpected outcome that later lead to the special theory of relativity and a revolution in physics in the early twentieth century.
Until this day the Michelson interferometer is used for scientific research. Figure 2.1 shows a schematic image of a Michelson interferometer. A monochro-matic light source shines a light beam onto a beam splitter, which splits the beam into two. The two individual beams now both travel their own indepen-dent optical path indicated in the image as Lr and Ls. Lr is the optical path
between the beam splitter and a reference mirror and Ls is the path between
the beam splitter and the sample mirror. Both the reference mirror and the sample mirror (partially) reflect the beam. This way the two beams travel back towards the beam splitter where they are recombined and will interfere with each other. The recombined beam is then directed onto a photodetector where its intensity is measured.
Figure 2.1: A schematic representation of a Michelson interferometer [16].
The use of a monochromatic light source (Figure 2.2) in a Michelson inter-ferometer will produce a repetitive sinusoidal signal as a function of optical path length difference ∆L = Lr− Ls. The translation of the sinusoidal function into
the optical path length difference is confined by phase wrapping and can only be expressed between −π and π.
Figure 2.2: Monochromatic light [16].
The intensity of the recombined beam as it is detected is expressed as: Idet = (ER∗ + E
∗
S)(ER+ ES) = |ER|2+ |ES|2+ 2Re(ER∗ES) (2.1)
Here it is only the interference term 2Re(ER∗ES) that contains information
about the the path length difference ∆L. As ∆L is the parameter of interest, the first two terms in equation 2.1 are filtered out with balanced detection. The intensity measured at the detector depends on the intensity of the light source as a function of wavelength S(k), the optical path length difference ∆L and on the reflectivity of both the sample and the reference mirror (Rs and Rr).
Finally, 2Re(ER∗ES) for a Michelson interferometer can be written as:
I(k, ∆L) = S(k)2pRrRscos(2k∆L) (2.2)
2.1.2
Low coherence interferometry
The information in the signal of a Michelson interferometer is phase wrapped, meaning that it can only be expressed as a value between −π and π. With OCT you want to be able to obtain depth-resolved scattering information up to millimeters into tissue depth. To enable this a signal which contains infor-mation of the absolute difference of traveled path is needed instead of a phase wrapped signal. A solution to this problem is found in using a polychromatic light source, also known as a broadband light source, instead of a monochro-matic light source. Figure 2.3 shows how a polychromonochro-matic light source consists of several monochromatic light beams with independent wavenumbers, added up to a localized interference pattern. If the two optical beams both travel a different distance they will only interfere if the path difference is within the coherence length of the light beam. For a Gaussian broadband light source the coherence length is expressed as [16]:
δz = 2 ln(2) nπ
λ2 0
∆λ (2.3)
Where δz is the coherence length that directly determines the axial resolu-tion, n is the refractive index of the medium, ∆λ is the bandwidth of the light source at full width at half maximum and λ0 is the center wavelength of the
Figure 2.3: Polychromatic light i.e. many monochromatic light beams with a variation of wavenumbers adding up to a localized interference pattern [16].
When using a broadband light source not one but many monochromatic interference patterns are measured, therefore equation 2.2 should be slightly adjusted to [16]: I(∆L) = S02 p RrRse−∆L 2∆k2 cos(2k0∆L) (2.4) Here S0=R ∞
0 S(k)dk is the wavenumber dependent integration of the source
power, k0is the center wavenumber of the light source and ∆k is the wavenumber
bandwidth of the Gaussian light source.
2.1.3
TD-OCT
The first implementation of OCT was time-domain OCT (TD-OCT), the imag-ing technique is directly based on a Michelson interferometer with a broadband light source. The mirror in the sample arm is now replaced with a sample, that in medical imaging will most likely be tissue. Gradually changing the position of the reference mirror causes the path length difference ∆L to vary as a function of time. If the position of the reference mirror is known as a function of time, the time-intensity signal can be translated into a depth-intensity signal. This creates a one-dimensional intensity depth profile, in OCT this profile is called an A-line. A row of consecutive A-lines form a two-dimensional B-scan and a row of consecutive B-scans form a three-dimensional image, called a C-scan. Because TD-OCT requires the mechanical movement of the reference mirror this imaging method is relatively slow. A faster way of OCT imaging is with Fourier-domain OCT (FD-OCT), which will be explained in the next subsection.
2.1.4
FD-OCT
In FD-OCT the position of the reference mirror is not mechanically adjusted, instead spectrally resolved detection is used to create a depth intensity image. This can be implemented in two forms: using a spectrometer instead of a pho-todetector (spectral-domain OCT, SD-OCT) or by using a swept source laser instead of a regular broadband laser (swept-source OCT, SS-OCT).
Figure 2.4a) shows a schematic image of an SD-OCT setup. The signal is first directed onto a grating, this is a dispersive element that will guide light in different directions as a function of wavelength. Subsequently the light is sent onto a charge-coupled device (CCD) detector where the intensity detection is done separately for multiple small wavelength bands. A schematic representa-tion of SS-OCT is illustrated in figure 2.4b). SS-OCT uses a swept-source laser, this is a narrow band light source that is rapidly tuned over a large bandwidth. Both SD- and SS-OCT have in common that the signal is an interferogram in which depth information is encoded in the form of wavenumbers. The interfer-ogram can be translated into a depth intensity profile with the aid of a Fourier transform (FT):
F (z) = Z ∞
−∞
Iint(k)e−ikzdk (2.5)
where Iint(k) is the interferogram as a function of wavenumber k and F (z) is
the intensity depth profile in the z direction.
Figure 2.4: a) Is a schematic illustration of an SD-OCT system, with a diffraction gratting component directing the light onto a CCD detector. b) Is a schematic illustration of an SS-OCT system with a swept-source laser and a photo detector [17].
Figure 2.5 illustrates an example of an SD-OCT intensity wavelength mea-surement being translated into an intensity depth profile, with the use of the Fourier transform in equation 2.5.
Figure 2.5: An example of the interferogram of an SD-OCT A-line measurement (left) being translated into a intensity depth profile (right), with the use a the Fourier transform (FT) [16].
Both forms of FD-OCT are significantly faster and have a higher signal to noise ratio (SNR) than TD-OCT. They are faster because A-lines are not depending on the mechanical movement of the reference mirror anymore. For SD-OCT an A-line can be measured in a single shot. This is not true for SS-OCT, but the swept source laser tunes over the broadband very rapidly, much faster than the mechanical movement of the reference mirror in TD-OCT. Over the years the imaging speed has gone from 2 A-lines/s with the first TD-OCT setups, to a few million A-lines/s with the newest SS-OCT systems [18]. The SNR improvement does not originate from an enhanced signal but from a noise reduction. OCT works optimally in the shot noise region and where for TD-OCT all photonic shot noise is collected in one detection, the shot noise for FD-OCT is spread over multiple detectors or multiple detecting moments in time. This results in a sensitivity increase of 3 orders of magnitude compared to TD-OCT. The huge improvement in measuring speed and sensitivity makes FD-OCT to be the wider used OCT technique [19].
2.1.5
OCT as a clinical imaging modality
OCT makes use of near-infrared light, this is a form of non-ionizing radiation. Imaging modalities like R¨ontgen photography and CT imaging do use ionizing radiation and are therefore more harmful for human tissue than OCT. However, near infrared light that propagates through tissue experiences high attenuation and this results in an OCT tissue penetration depth in the millimeter range. Figure 2.6 shows that ultrasound, CT and MRI have a deeper penetration depth than OCT, making these modalities more favorable for deeper located tissue. However, the figure also shows that OCT has a higher resolution than these modalities. Finally, the figure shows that confocal microscopy has an even higher resolution than OCT, but a lower penetration depth. This ultimately makes OCT the optimal imaging modality for high resolution 3D imaging of superficial tissue layers. Superficial tissue layers often contain important clinical information. A general example thereof is that the vast majority of cancers arise in epithelial surfaces and can be detected with high resolution imaging [20]. Specific examples of clinical information in superficial tissue layers in the airways are the presence of scar tissue in the lungs of an ILD patient and an excess amount of airway smooth muscle (ASM) in the lungs of an asthma patient.
Figure 2.6: Relative penetration depth and resolution of the different clinical imaging modal-ities: confocal microscopy, OCT, ultrasound, CT and MRI.
2.2
PS-OCT
The previous section explained that the exploitation of intensity in an OCT signal results in intensity images. However, intensity is not the only light prop-erty in the signal that contains depth-resolved clinical information of the tissue. The change of polarization state of the light induced by light-tissue interaction can provide extra contrast in OCT images. This section will explain a method to exploit the polarization properties of light in an OCT signal. It will start with the theoretical description of birefringent properties of tissue. Followed by the mathematical explanation of how the optic axis orientation can be extracted from the signal with the aid of the Jones, Stokes-Mueller and differential Mueller formalism. The final part of this section is dedicated to both the theoretical and the mathematical explanation of how, without any a priori knowledge, only the relative optic axis orientation can be found in a PS-OCT image.
2.2.1
Polarization properties
PS-OCT exploits the polarization property of light. The polarization of light refers to the direction of the electric field and is always perpendicular to the propagation direction of the beam. If light travels in the z direction, the po-larization state can be in all directions in the xy-plane. Traveling through birefringent material, different polarization orientations will experience a differ-ent refractive index. Examples of birefringdiffer-ent biological tissue are muscle and scar tissue, containing a fibrous structure. Figure 2.7 shows how beams with a horizontal and vertical polarization orientation travel a different optical path through fibrous material and that this results in a phase retardation between the two polarization states. The only situation in which polarized light is not
affected by birefringent material is when the propagation direction of the beam is in the same direction as the orientation of the fibers, this orientation is called the optic axis. In this situation the polarization of the light is perpendicular to the optic axis and because the polarization is not affected it experiences an ordinary refractive index no. Light that has its polarization in a parallel
di-rection to the optic axis of the fibers experiences an extraordinary refractive index indicated with ne. When the propagation direction of the light beam is
orthogonal in relation to the optic axis of the tissue any incident polarization direction experiences a refractive index that can be expressed as a superposition of no and ne. The mathematical relationship between an incident polarization
orientation and the resulting phase retardation ζ can be expressed as follows [21]:
ζ = 2π · ∆n · x λ0
(2.6) Where ∆n= (ne−no), x the distance traveled through the birefringent tissue
in meters and λo is the central wavelength of the light source, also in meters.
The goal of PS-OCT is to measure the phase retardation and the orientation of the optic axis of birefringent tissue. Finding the optic axis orientation of birefringent tissue and distinguishing birefringent tissue from non-birefringent tissue, ultimately provides additional clinical depth-resolved contrast.
Figure 2.7: An illustration of how a horizontal and vertical polarization orientation travel a different optical path through fibrous material, result-ing in a phase retardation [16].
2.2.2
Jones formalism
Fully polarized light can be mathematically expressed with Jones calculus. The analytical representation of the detected signal of polarized light in an OCT measurement can be written as a complex 2-element vector:
Edet(z) =
Eh(z)
Ev(z)
(2.7) Here Eh(z) and Ev(z) are the horizontal and vertical electric field
polarization state of the light beam induced by a setup component or the tissue can be mathematically described by a complex 2x2 Jones matrix. This Jones matrix can be found by filling in the component or tissue specific parameters in the general Jones matrix:
Jmatrix= e
−iζ 2
cos2θ + eiζsin2θ (1 − eiζ)e−iφcos θ sin θ
(1 − eiζ)eiφcos θ sin θ sin2θ + eiζcos2θ
(2.8) Here ζ is the phase retardation, θ the angle between the x-axis and the optic axis and φ is the circularity. The two eigenvalues of this matrix are: eiζ2 and
e−iζ2 . This means that the phase retardation ζ, can be calculated by the phase
angle between the two eigenvalues. And finally, the optic axis (OA) is defined as [22]: OA = cos θ eiφsin θ (2.9) In a PS-OCT setup, light travels through several components that inten-tionally or uninteninten-tionally change the polarization of the light beam. The total change of polarization during a measurement can be expressed as a multipli-cation of Jones matrices. For example, the optical path of the light in an endoscopic PS-OCT measurement can be split up into three parts. First, the path from the light source to the surface of the first birefringent layer in the imaging area, for endoscopic PS-OCT this is the inner surface of the catheter sheath. The signal is measured in reflection mode, thus the second part is the round-trip path of the light through the catheter sheath and the tissue and fi-nally the third path is from the catheter sheath inner surface to the detector. The induced change of polarization of the light within these three paths can be expressed by their independent Jones matrices: Jin, Jround and Jout. This
would result in a final Jones representation of the detected PS-OCT signal Edet
as:
Edet= JoutJroundJinEsource (2.10)
Where Esource is the electromagnetic field emitted by the light source.
2.2.3
Stokes-Mueller formalism
Complex Jones vectors and matrices can be translated into real value Stokes vectors and Mueller matrices. Villiger et al. [24] pointed out that the Stokes and Mueller notation allows for easier spatial averaging, because the vectors and matrices contain real instead of complex quantities. A Stokes vector consist of the 4 components: I, Q, U and V. Here I is the intensity of the light and Q, U and V describe the three possible polarization states of the light as illustrated in figure 2.8.
Figure 2.8: The three polarization states Q, U and V [23].
A Jones vector can be translated into stokes vectors as follows:
S = |Eh|2+ |Ev|2 |Eh|2− |Ev|2
2|Eh||Ev| cos ∆φ
2|Eh||Ev| sin ∆φ
(2.11)
In PS-OCT only fully polarized light will be detected, therefore I2 can be
expressed as:
I2= Q2+ U2+ V2 (2.12)
This way the stokes vector in equation 2.11 can be rewritten to:
S(z) = Q/I U/I V /I = q u v (2.13)
Another advantage of the Stokes formalism over the Jones formalism is that equation 2.13 is a 3-element vector and can be projected on a 3D sphere with radius 1; known as the Poincar´e sphere. This helps in the visualization of the optic axis orientation and the way varying polarization states are affected differently when traveling through birefringent tissue. For example, the Poincar´e sphere will be used in the next section to visualize the fact that only the relative optic axis can be found with a PS-OCT system. Figure 2.9 provides a general illustration of the Poincar´e sphere.
Figure 2.9: An illustration of the Poincar´e sphere.
Similar to how the Jones matrix describes the change in polarization of the Jones vector, the Mueller matrix describes the change in polarization of the Stokes vector. A Mueller matrix is a 4x4 real matrix and can be derived from a Jones matrix as follows:
M = A(J ⊗ J∗)A−1 (2.14)
Here ⊗ is the Kronecker tensor product and A is the 4x4 matrix of:
A = 1 0 0 1 1 0 0 −1 0 1 1 0 0 i −i 0 (2.15)
The next three equations represent the retardation Mueller matrices for pure birefringent material. Equations 2.16 and 2.17 represent the linear birefringence as a rotation around the Q- and U-axis of the Poincar´e sphere, respectively, and equation 2.18 represent the circular birefringence as a rotation around the V-axis: MrQ= 1 0 0 0 0 1 0 0 0 0 cos η sin η 0 0 − sin η cos η (2.16) MrU = 1 0 0 0 0 cos ν 0 sin ν 0 0 1 0 0 − sin ν 0 cos ν (2.17)
MrV = 1 0 0 0 0 cos µ sin µ 0 0 − sin µ cosµ 0 0 0 0 1 (2.18)
Next to the birefringent properties given in equations 2.16-2.18 a Mueller matrix normally also contains depolarization and diattenuation information. The final Mueller matrix can be rewritten as a multiplication of the birefrin-gent, depolarization and diattenuation matrices. However, these matrices do not commute and therefore the order of the matrices is ambiguous. To avoid this ambiguity a different method can be used: the differential Mueller formalism [25], which will be explained next.
2.2.4
Differential Mueller formalism
The stokes vectors vary along the direction of propagation z of the light beam. The correlation between the stokes vectors at depth z and ∆z is as follows [25]:
S(z, ∆z) = Mz,∆zS(z) (2.19)
here Mz,∆z is the Mueller matrix of a thin slice of the medium located at
depth z and of increment thickness ∆z. Subsequently S(z) is subtracted from both sides of equation 2.19, resulting in:
S(z, ∆z) − S(z) = (Mz,∆z− I)S(z) (2.20)
Where I is the identity matrix. Now both sides of equation 2.20 are divided by ∆z and the limit of ∆z → 0 is introduced leaving:
dS z = mS (2.21) with m being: m = lim dz→0 M(z + dz) − I dz (2.22)
From matrix m the birefringent properties of the tissue can be extracted. This means that, with equation 2.22, the depth-resolved birefringent properties of the tissue can be calculated as long as the Mueller matrices are known at different depths z. In the approximation of dz going to 0 the order of the bire-fringent, depolarization and diattenuation matrices within the multiplication is no longer important. In addition, with dz going to 0 only the first terms of the cosines and sines need to be taken to calculate m for MrQ, MrU and MrV, this
results in: mQ= 0 0 0 0 0 0 0 0 0 0 0 η 0 0 −η 0 , mU= 0 0 0 0 0 0 0 ν 0 0 0 0 0 −ν 0 0 , mV = 0 0 0 0 0 0 µ 0 0 −µ 0 0 0 0 0 0 (2.23)
The final Mueller matrix will include the absorption α and diattenuation parameters , τ and ω: m = α τ ω α µ ν τ −µ α η ω −ν −η α , (2.24)
The parameters , τ and ω, represent the diattenuation around the Q, U and V axis respectively. This diattenuation occurs when light attenuation is a function of polarization state in material or tissue. The parameters can be represented in a diattenuation vector D:
D = τ ω (2.25)
From which the magnitude and orientation of the diattenuation can be ex-tracted.
Similarly the parameters η, µ and ν represent the phase retardation param-eters for the three polarization states Q, U and V respectively. This can be represented in a birefringence vector γ [25]:
γ = η ν µ (2.26)
Of which the direction is the optic axis orientation and its length is equal to the phase retardation:
ζ = ||γ|| (2.27)
2.2.5
Retrieving the relative optic axis orientation
The γ vector in equation 2.26 provides the optic axis orientation in PS-OCT images. A PS-OCT system measures in reflection mode, this means that the light travels through the catheter sheath and the tissue twice. The round-trip nature of this optical path cancels the circular component V of the birefringence and diattenuation. This is because the round trip Jones matrix (Jround) of
birefringent material can be calculated from its one-way Jones matrix (Jsingle)
as follows: Jout= JTsingleJsingle [26]. Because Jround is transpose symmetric, φ
in equation 2.8 is zero and therefore Jround is a linear retarder. In other words
only the linear polarization components η and ν of the tissue can be measured. Visualizing this on the Poincar´e sphere: this means that the measured optic axis of the tissue should always lie on the QU-plane. However, birefringent properties of the setup in the path Jout do create a circular polarization component V in
the signal. The birefringent properties of the setup can be different in every measurement due to e.g. stress-induced birefringence on the fibers. Therefore,
there is no calibration method for endoscopic PS-OCT to distinguish a change in polarization state induced by the tissue and that induced by the system.
Equation 2.10 provides a Jones matrix multiplication representation of a PS-OCT measurement. This is a good place to start the explanation of finding the relative optic axis in a mathematical manner. In equation 2.10 the optical path of the light in an OCT measurement is split up into three parts. Jinrepresents
the Jones matrix for the path from the light source to the inner surface of the catheter sheath, Jround represents the round-trip of the light through the
catheter sheath and the tissue and finally Jout represents the path from the
catheter sheath surface to the detector. This way the signal from the inner reflection surface of the catheter sheath can be represented like this:
Edet = eiΦJoutJinEsource (2.28)
Where Esource is the electric field emitted by the light source and Φ is a
common phase. To include the round trip of the tissue reflected at some depth z equation 2.28 can be slightly adjusted to:
E0det= eiΦ
0
JoutJround(z)JinEsource (2.29)
Another way to write equation 2.29 is:
E0det = ei(Φ0−Φ)JoutJroundJ−1outEdet (2.30)
By assuming that Jout is non-diattenuating, it can be treated as a unitary
matrix with unit determinant after separating out a common attenuation factor [4]. Jroundcan be decomposed into a diagonal matrix Jc, surrounded by unitary
matrices Ja. JoutJroundJ−1out can now be reformed to JT = JoutJaJcJ−1a J −1 out.
Since unitary matrices with unit determinant form the special unitary group SU(2), the product JoutJa must also be a unitary matrix with unit determinant
by closure. This also means that Jc, Jround and JT are related by unitary
transforms and only differ by their respective coordinate systems. Therefore, the amount of phase retardation and diattenuation induced by JT, Jround and
Jc is the same.
Because Jc, Jround and JT differ only by a rotation of their respective
co-ordinate systems, the plane of all possible optic axes for JT can be rotated off
the QU-plane to an arbitrary plane passing through the origin. The change in the polarization state induced by the setup can be decomposed into two parts: a tilting of the plane about an arbitrary axis in the QU-plane and a rotation within the QU-plane. To find the absolute optic axis of tissue the amount of rotation and the tilting needs to be retrieved, so that the measured optic axis can be correctly rotated back onto the QU-plane.
The correction of the tilting of the plane is done by rotating it back onto the QU plane, however this rotation can be done in two directions. The sign of the orientation angle cannot be explicitly determined. This results in an π-ambiguity in the sign of the orientation angle. However, the absolute value of change from one location to the next can be determined [4]. The rotation
within the QU-plane creates an offset. Because of fast lateral scanning, it is reasonable to assume that in a PS-OCT system the birefringence of the sampling fibers does not change within a B-scan measurement [26]. This means that a depth-resolved relative distribution of the orientation of the birefringence can be retrieved within an A-line and B-scan. This way the relative optic axis orientations of the tissue can be found within an image, but not the absolute orientations.
2.3
Second harmonic generation microscopy
In this thesis second harmonic generation (SHG) microscopy is used to find the absolute optic axis orientation of the chicken sample and therefore the technique is introduced here. SHG is a non-linear, second order optical process induced by birefringent material or tissue. Two emitted photons interact with birefringent tissue in such a way that the two photons are converted into one photon with double the energy and thus half the wavelength. Figure 2.10 shows how this process is carried out via an intermediate virtual state. The SHG signal takes place at the focal point of a microscope objective in a three-dimensional optical interaction. This allows for in depth imaging of birefringent samples with a sub-micrometer resolution.
Figure 2.10: A representation of the optical process of SHG microscopy [27].
The non-linear optics theory provides a more mathematical explanation of SHG: it states that an incident electric field Eω with a frequency ω induces a
second-order polarization P2ω,i at 2ω in the i’th direction given by [28]:
P2ω,i= 0[χ(1)E + χ(2)E2+ χ(3)E3+ ...] (2.31)
where 0 is the vacuum permittivity and χ(n) is the nth-order non-linear
susceptibility tensor. The second term of Equation 2.31 describes SHG. The term χ(2) depends on the polarization of the excitation source and therefore the SHG emission is sensitive to polarization.
No excitation of the molecules in birefringent tissue is needed and therefore the molecules will not suffer from the effects of phototoxicity and photobleach-ing. This way the sample can first be measured with SHG and then with an OCT-setup without changing the biological construction of the sample by either imaging modalities.
Chapter 3
Methodology
3.1
Introduction
The previous subsection explained that it is impossible to find the absolute op-tic axis orientation of birefringent tissue with a PS-OCT setup. This is because the birefringent properties of the setup introduce a measurement specific off-set and a π-ambiguity in the signal. However, by finding a reference point in PS-OCT images of which the absolute optic axis orientation is known, the mea-surement specific offset can be found and corrected for. Therefore, the first step of this research is to find (or create) a feature with birefringent properties in the PS-OCT images of which the absolute optic axis orientation is known. This feature is found in the plastic catheter sheath protecting the endoscope. The catheter sheath contains birefringent properties, that were thus far unknown, and is located in the imaging area of the PS-OCT setup. To find the optic axis orientation of the catheter sheath a homogeneously birefringent sample is analyzed, i.e. chicken breast.
First the absolute optic axis orientation of the chicken breast sample is found with SHG microscopy. Then 6 different endoscopic PS-OCT measurements are done on the same sample. For the first endoscopic PS-OCT measurement the chicken fibers are oriented in the same direction as the light beam scanning. To create different angle-relations between the chicken fiber orientation and the light beam scanning direction: the sample is horizontally rotated in 30o incre-ments to a 150o total span. The measurements are done by manually pulling back the endoscope along the sample while maintaining the angle-relation be-tween the chicken fibers and the scanning direction. The endoscopic PS-OCT measurements are processed into images that show the relative optic axis ori-entation relation between the catheter sheath and the chicken sample. Because the absolute orientation of the chicken sample in these images is known, the absolute optic axis orientation of the catheter sheath can be determined. The absolute optic axis orientation of the catheter sheath can then function as a calibrated reference point for future endoscopic PS-OCT measurements.
In this research an algorithm is created that changes the measured optic axis value in the catheter sheath to its absolute optic axis value. The difference between the measured and absolute optic axis value of the catheter sheath is the offset created by the measurement specific birefringent properties of the setup. Because the scanning of a B-scan happens very rapidly the setup induced offset can be assumed constant within the image [26]. Therefore, the same correction as for the catheter sheath can be applied to the entire image. This creates an image that shows the absolute optic axis orientation of tissue. Finally, this absolute optic axis algorithm is applied on endoscopic PS-OCT in vivo measurements of an ILD and asthma patient.
The next sections in this chapter provide all information on the methodology of this research i.e. the used setups, samples, in vivo patient measurements, data-processing steps and the created absolute optic axis algorithm.
3.2
Setups
3.2.1
Endoscopic PS-OCT
Figure 3.1: The schematic image of the endoscopic PS-OCT setup [29].
The endoscopic PS-OCT setup used in this research is developed and built by Feroldi et al. [29], figure 3.1 shows the schematic image of the setup. It contains a swept source laser (Axsun Technologies Inc. 1310nm 50kHz) with a central wavelength of 1310nm and a frequency of 50kHz. The light source sends light to a 99/1 coupler (C), where 1 % of the light is sent to a fiber Bragg grating (FBG, O/E land Inc.). The FBG provides an A-line trigger at 1266nm to the data acquisition card (Alazar Technologies Inc., ATS9350). The other 99% of the light is sent to a 10/90% coupler where 10% of the light is sent
through a transmission-type reference arm. The other 90% goes to a passive polarization delay unit (PDU). In the PDU the light is split by a polarization beam splitter (PBS) into horizontal and vertical polarization states. The two orthogonal polarization states both travel a different path length through air before they are recombined at another PBS. This results in an imaging range of 2.5 mm in tissue, because the PDU multiplexes a 5 mm imaging range over the two polarization states. From the PDU the light is directed to a circulator from where it is sent through the optical fiber of the endoscope.
Figure 3.2 shows the schematic image of the catheter. The catheter has a total length of 120 cm and an outer diameter of 1.35 mm. The light travels through the catheter through an optical fiber. The optical fiber is angle-cleaved at 8oand glued to a 0.5 mm diameter custom GRIN lens (GRINTECH GmbH,
Jena, Germany) with both facets polished at 8o and a working distance of 1.5
mm. The light will travel through the GRIN lens from where it is directed onto a 300 µm wide reflective microprism (Edmund Optics, Barrington NJ, USA), functioning as a mirror. The mirror is positioned at a 48o angle and rotates over a full 360o circle. Therefore, the light is sent onto the tissue in the lateral direction in a light-house fashion. This ultimately results in circumferential images of the tissue surrounding the catheter. In the far end of the tip of the catheter the rotation motor is located to which the mirror is glued on. This is an alternating current-driven motor (AC motor) that is self-assembled at the mechanical workshop of the VU University by Dirck van Iperen. The motor consist of a permanent magnet of which the central axle is held into place by two conically shaped rubies and is free to rotate. The two rubies are connected by a housing, the housing has grooves in its surface in which the wires are placed to form the coils. Two copper wires are wrapped around the coils and enable the rotation. The AC currents are provided by a custom-made driver built by the electronic workshop of the VU university. The driver is connected to the motor by the two copper wires, illustrated in figure 3.2 as dotted lines. The figure shows that, to connect the motor to the driver, the wires are directed through the imaging area of the catheter. This results in a shadowed area in the OCT images, there where the wires are located. The two currents provided by the wires are two sinusoidal voltage signals with a frequency corresponding to the desired rotation speed of 960 A-lines per rotation and have a π/2 phase shift between them. To protect both the catheter and the patient the catheter is encased in a catheter sheath. The material of the sheath is polyether block amide (PEBAX 7033 SA 01 MED, Arkema, Colombes Cedex, France), also referred to as pebax. Pebax has a medical grade title, meaning the material does not react to other materials or living tissue. The catheter, including its sheath, is reusable. After using the catheter on a patient it is cleaned with a mixture of water and soap, wiped with an ethanol-wet gauze, and finally placed in a metal wire basket in which it is disinfected by a standard washing device for bronchoscopes (WD 440, Wassenburg Medical B.V., Dodewaard, The Netherlands).
Figure 3.2: The schematic representation of the catheter tip [29].
From the mirror in the endoscope the light is directed onto the tissue where it will be (partially) reflected. It will travel back through the endoscope to the circulator from where it is directed to the polarization diversity detection module (PDDM). In the PDDM the light from the sample and reference arm is split into the two orthogonal polarization states, then the two polarization states from the two different arms are combined to interfere. Lastly, this light is send onto the detectors. The axial resolution of the PS-OCT system is 12 µm in tissue (by assuming a refractive index of 1.4 in the tissue) and the maximum lateral resolution is 13 µm.
In the setup the light travels through SMF28, this is a single mode fiber (SMF). An SMF is an optical fiber that only allows light to propagate through it in one transverse electromagnetic mode. This condition is wavelength depen-dent, i.e. it is valid over a specific wavelength range. However, this range is often sufficiently wide enough for the fibers to be used as a waveguide for light emitted by a broadband light source. An SMF consists of three layers; the core, the cladding and a coating. Light travels through the core of the fiber and ex-periences total internal reflection at the core-cladding boundary. The cladding has a slightly lower refractive index than the core, this is needed to induce the total internal reflection. The coating functions as a protection layer.
3.2.2
SHG microscopy
Figure 3.3: The schematic image of the SHG setup [30]
Figure 3.3 shows the schematic representation of the, in this research used, SHG setup. The SHG setup contains a femtosecond laser light source (Amplitude Laser, Mikan) and a motorized stage (Prior). The laser emits short light pulses (<250 fs) of 1030 nm with a frequency of 54MHz. The pulses first pass a pulse picker (AA Opto Electronic) to create single pulses. These pulses are then reflected by galvo mirrors (BBE1 EO3) to enable transversely scanning. Successively, the beam is broadened by two lenses and then focused on the sample using a 40x/1.3 oil immersion objective (Nikon 5 Fluor). This results in an average power of 5-10 mW being projected on the sample. The light is reflected by the sample generating a signal containing autofluorescence, SHG and THG photons. These three different types of photons are separated from the incoming light by a dichroic mirror (Thorlabs, FF872 Di01). Then the autofluorescence signal is split of by a long pass dichroic mirror (Thorlabs, LP 580) and detected by a photomultiplier tube (PMT, Hamamatsu, H10721-210). The remaining SHG and THG signal are split by a different long pass dichroic mirror (LP 442). The two different signals then travel through a filter of 525nm (BrightLine, FF01-525/45) and 355nm (BrightLine, 355/40) for the SHG and THG signal respectively. The software settings are then adjusted to only show the image based on the SHG signal, showing fibrous tissue.
3.3
Chicken breast sample
The sample used for the catheter sheath calibration experiments is chicken breast, bought in a local supermarket. Chicken breast consists almost solely of muscle, which is birefringent tissue. Experience has shown that chicken breast muscles show relative homogeneous fiber orientations. To prepare the sample, a thin slice of chicken is cut (approximately 2 mm thick). The orientation of the muscle fibers in the chicken breast is estimated by eye when cutting the sample and then measured by SHG microscopy. The sample is then placed on a substrate and on top of a rotating plate on which it will be measured with the endoscopic PS-OCT setup. Figure 3.4 shows a picture of the sample prepared to be measured by the endoscopic PS-OCT setup. During a measurement the sample was always kept hydrated with distilled water. The water was placed on the sample by which it was absorbed. This was done to prevent the sample to denature and (partially) lose its birefringent properties.
Figure 3.4: Picture of the chicken sample on top of the substrate and the rotation plate.
3.4
Patients
The created absolute optic axis algorithm was validated on in vivo measure-ments. A total of three measurements from two different patients are used: two measurements from an ILD patient and one measurement from an asthma patient. The measurements from the ILD patient were taken one minute apart and are both from the same airway in the right lower lobe in the patients lung. The asthma patient measurement was also from the right lower lobe of the lung. The measurements were all done in the Amsterdam UMC (location AMC) hospital. The procedures were executed following protocol. The patient was anesthetized and a bronchoscope was inserted through the mouth of the patient into the lungs. First a biopsy was taken in the locations of interest. Then the
endoscope of the PS-OCT setup was inserted into the lungs through the biopsy channel of the bronchoscope. The pull-back of the endoscope during scanning was executed manually by a physician at a speed of approximately 1 mm/s.
3.5
Data processing
This section explains the data-processing steps of the endoscopic PS-OCT mea-surements. The first subsection explains how the Jones matrices are obtained per pixel. The subsections 3.5.2, 3.5.3 and 3.5.5 to 3.5.7 explain processing steps that are needed to correct for corruptions in the signal. For some of the corrections a good segmentation of the catheter sheath and the wires is needed, subsection 3.5.4 will explain the segmentation methods that are used in this research. Then subsection 3.5.8 explains how the relative optic axis orientation is retrieved from the corrected Jones matrices with an approximation of the differential Mueller method. Followed by section 3.5.9 that explains the data-processing method that retrieves the absolute optic axis orientation in images. And lastly subsection 3.5.10 describes how the measurements are processed into circular images. All data-processing steps are executed with Matlab R2017b.
3.5.1
Jones matrix modelling
Because the signal is multiplexed in depth and the horizontally and vertically polarized light is split before going to the detectors, a total of four images are obtained per cross section. An example of those four images of a lung measurement is illustrated in figure 3.5.
Figure 3.5: An example of the four images detected for one B-scan in a lung measurement.
To separate the in-depth multiplexed images H1 and H2, and V1 and V2 the autocorrelation g(z) is calculated for each A-line:
g(z) = Z ∞
−∞
E(z)E(z + δz)dz (3.1)
According to the Wiener-Khinchin theorem this is equal to [31]:
g(z) = F [|E(k)|2] (3.2)
where E(k) is the E-field signal in k-space. The A-line autocorrelations will show two peaks one where δz is zero and one where the second depth-delayed signal overlaps with the first. The difference between the peaks is used to calculate the number of pixels in between the two images. Once the total E-field signals per pixel are known the electric E-fields can be sorted into a complex 2x2 Jones measurement matrix as follows:
EJ ones,det= EH1(k) EH2(k) EV 1(k) EV 2(k) (3.3)
3.5.2
E-field averaging
The scattering of light in tissue happens at many different lateral and axial locations. Therefore, the scattered light has a random phase and amplitude, which is the origin of speckle noise. Speckle noise results in bright and dark spots in the image due to a constructive and destructive interference pattern. For intensity images speckle noise can be eliminated by taking the mean of pixel clusters throughout the image. This is because the intensity properties are extracted from the signal in a linear mathematical way. Jones calculus however is a nonlinear method of extracting information from the signal. To average the E-fields when using Jones calculus common-phase corrections are applied.
For common-phase corrections; the phase angle between the two orthogonal polarization E-fields is calculated for every pixel. These angles are averaged over the pixels inside a chosen kernel, considering a weighting factor based on the amplitudes of the pixel signals. This enables to find the noise-free phase angle that is applied to all the electric field components inside the chosen kernel. Lastly, spatial complex averaging is applied to every electric field component inside this kernel.
3.5.3
Signal roll-off correction
A signal roll-off correction is needed because the light loses intensity with in-creasing path length difference between the sample and the reference arm. In a swept source PS-OCT system this is due to a finite coherence length because of the finite bandwidth of the swept laser line. A calibration measurement is done with a mirror in the sample arm. The mirror is scanned at different depth positions in relation to the reference arm. From this calibration measurement a correction parameter for the images as a function of depth is found that will be applied throughout all C-scans.
3.5.4
Segmentation
The catheter sheath is present in every image and has two reflection surfaces, i.e. the inner and the outer surface of the sheath. A good segmentation of the inner and outer surface of the catheter sheath is needed for some of the correction discussed in the next sections of this chapter and for the absolute optic axis orientation algorithm. Moreover, since there is no signal there where the wires are located it is important that the wires are segmented correctly, this way the pixels located in that area can be excluded from data-processing procedures.
To find the right segmentation for the inner and outer surface of the catheter sheath the Dijkstra algorithm is used [32]. The algorithm finds the shortest path linking nodes in a B-scan by searching the minimum value per node among its eight neighboring nodes. The segmentation was determined using intensity images. To optimize the segmentation by the Dijkstra algorithm the 4 pixels below the found inner and outer catheter surface pixel are selected. These 5 pixels are analyzed and the pixels with the highest intensity value is then selected as the inner or outer catheter sheath surface pixel.
To find the location of the wires a different method is used. First the inner surface of the sheath is found. Then the area between pixel one (from the top of the image) to the pixel of the inner catheter surface is selected. In this area the A-line with the highest cumulative intensity value is selected. This A-line is expected to be the center of the wire area. Then the A-lines 100 pixels left and right from the center A-line of the wire area are segmented to be the wire area edges.
Figure 3.6a) shows an intensity image of an endoscopic PS-OCT measure-ment of a chicken sample, figure 3.6b) shows the same image but now the yellow, blue and red lines respectively indicate the segmentation of the inner reflection surface of the catheter sheath, the outer reflection surface of the sheath and the boundary locations of the wire region.
Figure 3.6: a) shows the intensity image of a B-scan. b) shows the same intensity image, but now the segmentation of the wires and the inner and outer surface of the catheter sheath is illustrated with red, yellow and blue lines, respectively.
3.5.5
Chromatic dispersion compensation
The SMF in the setup induce wavenumber dependent dispersion, also known as chromatic dispersion. This leads to differences in pulse size between the light coming from the sample arm and the reference arm and therefore it is important to correct for this. To compensate for this difference the required correction phase function was determined by analyzing the signal reflected at the surface of the catheter sheath beforehand and is then applied during data-processing.
3.5.6
Polarization mode dispersion compensation
Besides chromatic dispersion, also polarization mode dispersion (PMD) is in-duced in SMF. Imperfections in the core symmetry of the fiber can change the speed of light as a function of polarization mode. In this research a PMD com-pensation algorithm is used carried out in the Jones formalism. First three B-scans are selected that show a relatively large amount of birefringence and in these B-scans three regions of 100 consecutive A-lines are investigated on quality.
The 9 selected regions are the pre-processed spectra of calibration regions. The data is first binned into 15 overlapping gausian-shaped bins in the wave-number domain. The bins containing the data per A-line are then Fourier transformed into the depth domain. The E-fields of the 15 bins of the inner catheter sheath were interpolated over the whole wavenumber-spectrum of the light source. The entire image is then multiplied with the inverse of the obtained E-field at the catheter sheath surface. Then an algorithm based on a creation of Braaf et al [16] is applied to find the correction matrices for the PMD com-pensation of the path length from the light source to the catheter sheath and the path length from the catheter sheath to the detector.
3.5.7
Rotating mirror correction
The rotation of the mirror in the endoscope constantly changes the angle of incidence of the light beam on the tissue. This creates a constant shift in the polarization properties of the signal throughout one B-scan. This can be simply corrected for by multiplying the signal with the following rotation Jones matrix:
Jmirrors=
cos θ − sin θ sin θ cos θ
(3.4) Where θ is the angular position of the mirror within its 360orotation.
3.5.8
Differential Mueller matrix calculation
After the corrected Jones matrices are obtained, the depth-resolved phase retar-dation is extracted from these matrices with the differential Mueller formalism. First the Jones matrices are converted into Mueller matrices with equation 2.14. The matrices are averaged over 5 pixels in the axial and 4 pixels in
the lateral direction. To calculate the differential Mueller matrix, the Mueller matrices are compared to Mueller matrices at a different location in depth. The difference in depth dz between two Mueller matrices is always chosen to be small (≈ 55.8 µm). Because of this small value for dz, the tissue in this region can be expected to be homogeneous in the z direction. Therefore, the differential Mueller matrix can be calculated using the following approximation of the differential Mueller calculation in equation 2.22:
m ≈ M
4
pdet(M)− I (3.5)
and from the matrix m the γ-vector is derived as follows:
γ = 1 2 m34− m43 m24− m42 m23− m32 = η ν µ (3.6)
Where the first (i) and second (j) indices of mij stand for the row and
column numbers of the matrix respectively. The direction of the γ-vector is the relative optic axis orientation, and the length of the vector is the phase retardation. The γ-vectors are averaged over 4 pixels in the axial and 5 pixels in the lateral direction. Pixels that obtain a negative Mueller matrix eigenvalue are not evaluated because these are not real physical matrices.
3.5.9
Retrieving the absolute optic axis orientation
The found γ-vectors in equation 3.6 provide the relative optic axis orientation information of the tissue in the images. Because of the round trip nature of the light through the sample and the catheter sheath it is known that the optic axes of the tissue in the images should lie on the QU-plane in the Poincar´e sphere. This means that the µ-component in the γ-vectors of equation 3.6 should be zero. However, birefringent properties of the setup create a tilting of the optic axis off the QU-plane. This results in a µ-component in the γ-vectors. Most of this tilting will be corrected for by the PMD compensation data-processing steps described above. If there is any remaining value for µ in the γ-vectors after the PMD compensation, the values are simply put to 0. This changes the γ-vectors from a 3-element vector into a 2-element vector.
Next to the tilting of the optic axis orientation off the QU-plane, the bire-fringent properties of the setup also create an offset of the optic axis orientation within the QU-plane. In this thesis this offset is found by using the absolute optic axis orientation of the catheter sheath as a reference point. The catheter sheath only has two reflecting surfaces from where a signal is retained: the in-ner and the outer surface of the sheath. These two signals are used to calculate the phase retardation within the catheter sheath with the differential Mueller approximation in equation 3.5. This means that for the catheter sheath cal-culations, dz in equation 3.5 is the region between the two surfaces. This is a larger value than 55.8 µm. However, the approximation in equation 3.5 is
still valid, because the birefringent properties of catheter sheath are expected to be fully homogeneous in the z direction. The segmentation of the inner and outer surface of the catheter sheath results in the selection of one pixel for both surfaces per A-line. To limit the segmentation sampling error the Jones matrices of the segmented pixels are averaged with the Jones matrices of its adjacent pixels above and below. The average Jones matrices of the inner and outer catheter sheath surface are then used to calculate the differential Mueller matrix in between these two points, i.e. the region inside the catheter sheath.
From the found differential Mueller matrices the birefringent properties of the catheter sheath can be determined per A-line. Within every B-scan, the optic axis orientations of all A-lines are analyzed and the most present orienta-tion is selected to represent the measured optic axis orientaorienta-tion of the catheter sheath. The difference in orientation between the measured optic axis and the absolute optic axis of the catheter sheath can be expressed as a rotation by angle φ within the QU-plane of the Poincar´e sphere. To rotate this measured optic axis orientations to its absolute optic axis orientations, the measured 2-element γ-vector is multiplied by a 2x2 rotation matrix M. The matrix rotates the vector over the angle φ in the QU-plane. This rotation is applied to all the measured 2-element γ-vectors to obtain the absolute optic axis orientations in the entire image. Equation 3.7 provides the mathematical expression of the rotating procedure, here γrel represents the measured and corrected relative
birefringence vectors and γabs represents the absolute birefringence vectors.
γabs= M γrel= cos φ − sin φ sin φ cos φ η ν rel (3.7)
3.5.10
From a rectangular to a circular image
For every B-scan a total of 960 consecutive A-lines are measured. These con-secutive A-lines can be illustrated next to each other in a linear way as is done in the intensity image in figure 3.6. However, the A-lines are measured in a radial orientation in relation to each other, not linear. In this thesis a Matlab algorithm is used that illustrates the A-lines in a circular way. An example of a rotated intensity image of a in vivo airway is illustrated in figure 3.7. The two dark circles in the middle of the image are the reflecting surfaces of the catheter sheath: the inner circle is the inner reflecting surface and the outer circle is the outer reflecting surface. The white region in the middle upper part of the image is the wire area, no signal is obtained in this region. The grey colors in the rest of the circular image show the intensity signal coming from airway tissue. The result section of this thesis shows solely circular images, but of the optic axis orientation images instead of intensity. The relative and absolute optic axis orientations in the optic axis images are indicated by color. The different colors correspond to values between 0 and 180o for the optic axis orientation. The 180o range is sufficient because the optic axis has no pointing direction.
Chapter 4
Results
This chapter is subdivided into three sections. The first section presents the results of the SHG measurement on the chicken sample and shows the absolute optic axis orientation of the chicken muscle fibers. The section that follows shows the PS-OCT images of the same chicken sample. By analyzing these images the absolute optic axis orientation of the catheter sheath is determined. The absolute optic axis orientation of the catheter sheath is used in the third section of this chapter to find the absolute optic axis orientation of birefringent tissue in in vivo measurements. Results of the relative and absolute optic axis images of measurements on an ILD and asthma patient are presented. A comparison between the relative and absolute optic axis images is made and the absolute optic axis images are further analyzed.
4.1
SHG microscopy
To obtain the absolute optic axis of the chicken muscle fibers, the sample was measured with an SHG microscope. The sample was placed under the micro-scope in the orientation as illustrated in the picture in figure 3.4. Figure 4.1a) shows an SHG image of the sample with a field of view of 250 µm2. The image shows relatively homogeneous horizontally oriented muscle fibers. From this image the orientation of the fibers is determined to be horizontal. The abso-lute optic axis orientation of the sample is illustrated with brown lines in the schematic representation of the chicken sample in figure 4.1b).
Figure 4.1: a) is an SHG image of the chicken sample with a field of view of 250 µm2 and b) is a schematic representation of the chicken sample in which the brown lines indicate the absolute optic axis orientation of the muscle fibers in the sample.
4.2
Endoscopic PS-OCT on sample
On page 37 and 38 figure 4.2 shows the results of the PS-OCT measurements on the chicken sample. On the left side of the image, figure 4.2a) to f), the figure shows 6 PS-OCT cross section images of different PS-OCT measurements. The images are placed next to the schematic representation of the orientation rela-tions between the endoscope and the absolute optic axis of the chicken sample: figure 4.2g) to l). Bare in mind that the scanning of the endoscope is in the lateral direction, e.g. for measurement one (figure 4.2a) and g)) the scanning of the light is along the orientation of the absolute optic axis of the chicken fibers. The PS-OCT images show the relative optic axis orientations, but the abso-lute optic axis data-processing algorithm was already applied to these images. This was done so that the catheter sheath could be assigned the same color in all images. The color assigned to the catheter sheath corresponds to the color of 90o in the scale bar. A filter is placed over the images that shadows the
non-birefringent part of the image. The filter is based on the uniformity of the birefringent signal. This specifically means that a total of at least 50 adjacent pixels needed to have a γ-vector size in the QU-plane above 0.5 to be included into the bright part of the image.
Figure 4.2a) shows that the orientation of the catheter sheath is the same color as that of the chicken sample. This implies that the optic axis of the catheter sheath has a circumferential orientation. In the scale bar of the figures 4.2a) to f) a little red square indicates the expected color of the chicken optic axis orientation assuming that the optic axis orientation of the catheter sheath is circumferential. For most of these images the expected color indicated in the scale bar match with the color of the tissue. In this thesis the results
shown in the images 4.2a) to f) are interpreted as a validation that the optic axis orientation of the catheter sheath is circumferential. However, two of the images show unexpected features: figure 4.2b) shows relatively low signal and figure 4.2c) shows two different colors instead of one. The low signal in figure 4.2b) might be explained by the fact that the measurement was done manually and the pull back might have been unstable. An explanation for the two colors in figure 4.2c) may be found by looking at the SHG image in figure 4.1a). It shows that the muscle fibers of the chicken are not perfectly homogeneously orientated. It might be that, during measurement 4.2i), a region of the chicken sample was scanned that has a less homogeneous optic axis orientation than the regions scanned for the other 5 measurements.
The results of all measurements consisted of multiple images acquired dur-ing a pull-back. The scans were all processed into videos from which the cross section with the most signal was chosen. Because the sample is fairly homo-geneous, both the colors and the amount of signal barely vary throughout the scans. It is therefore that only one image per measurement is shown, because more images would not provide any additional information.
Figure 4.2: a) to f) show a B-scan of an endoscopic PS-OCT measurement of the chicken breast sample. g) to l) show a schematic representation of the angle relation between the endoscope and the chicken fibers (the brown lines in the sample) of the optic axis image illustrated left from it.
