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Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-3268-6ISSN 1798-5668

Dissertations in Forestry and Natural Sciences

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JENNI TICK

IMAGE RECONSTRUCTION AND MODELLING OF UNCERTAINTIES IN PHOTOACOUSTIC TOMOGRAPHY

PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND

Photoacoustic tomography is a biomedical imaging modality, where an image of an

initial pressure distribution is formed as an inverse problem based on ultrasonic boundary

measurements. This thesis introduces new computational methods to solve the inverse

problem using Bayesian formalism. The new methods provide quantitative information of

the estimated initial pressure, its uncertainties, and improve the accuracy of the estimates by modelling uncertainties in the speed of sound.

JENNI TICK

PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLANDDISSERTATIONS IN FORESTRY AND NATURAL SCIENCES

N:o 364

Jenni Tick

IMAGE RECONSTRUCTION ANDMODELLING OF UNCERTAINTIES INPHOTOACOUSTIC TOMOGRAPHY

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for publicexamination in the Auditorium SN200 in Snellmania Building at the University ofEastern Finland, Kuopio, on December 20th, 2019, at 12 o’clock.

University of Eastern FinlandDepartment of Applied Physics

Kuopio 2019

Grano OyJyväskylä, 2019

Editors: Pertti Pasanen, Jukka Tuomela,Matti Tedre, and Raine Kortet

Distribution:University of Eastern Finland Library / Sales of publications

julkaisumyynti@uef.fi

http://www.uef.fi/kirjasto

ISBN: 978-952-61-3268-6 (print)ISSNL: 1798-5668ISSN: 1798-5668

ISBN: 978-952-61-3269-3 (pdf)ISSNL: 1798-5668ISSN: 1798-5676

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Author’s address: University of Eastern FinlandDepartment of Applied PhysicsP.O. Box 162770211 Kuopio, FINLANDemail: jenni.tick@uef.fi

Supervisors: Associate Professor Tanja TarvainenUniversity of Eastern FinlandDepartment of Applied PhysicsP.O. Box 162770211 Kuopio, FINLANDemail: tanja.tarvainen@uef.fi

Docent Aki PulkkinenUniversity of Eastern FinlandDepartment of Applied PhysicsP.O. Box 162770211 Kuopio, FINLANDemail: aki.pulkkinen@uef.fi

Professor Jari KaipioUniversity of AucklandDepartment of MathematicsPrivate Bag 92019Auckland 1142, NEW ZEALANDemail: jari@math.auckland.ac.nz

Reviewers: Associate Professor Georg StadlerNew York UniversityCourant Institute of Mathematical Sciences251 Mercer StreetNew York CityNY 10012, USAemail: stadler@cims.nyu.edu

Professor Roger ZempUniversity of AlbertaDepartment of Electrical and Computer Engineering9107 – 116 StreetEdmontonAB T6G 2V4, CANADAemail: rzemp@ualberta.ca

Opponent: Research Professor Johanna TamminenFinnish Meteorological InstituteSpace and Earth Observation CentreP.O. Box 50300101 Helsinki, FINLANDemail: johanna.tamminen@fmi.fi

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Jenni TickImage reconstruction and modelling of uncertainties in photoacoustic tomographyKuopio: University of Eastern Finland, 2019Publications of the University of Eastern FinlandDissertations in Forestry and Natural Sciences 2019; 364

ABSTRACT

Photoacoustic tomography (PAT) is an imaging technique with the advantages ofhigh spatial resolution, strong optical contrast, non-invasiveness, and being basedon nonionizing radiation. This hybrid imaging modality has many potential biomed-ical applications such as imaging tissue vasculature and small animal imaging.

In PAT, a short pulse of light is used to illuminate the target of interest. Aslight propagates in the target, it is absorbed by light absorbing molecules resultingin a localised increase in pressure that is called initial pressure distribution. Thispressure propagates through the target as an acoustic wave, and can be measured onthe boundary of the target using ultrasound sensors. In the inverse problem of PAT,the initial pressure distribution is estimated from these ultrasound measurements.

Estimation of the initial pressure is an ill-posed inverse problem. Due to the ill-posedness, even small errors or uncertainties in the measurements or modelling canresult in significant errors in the solution of the inverse problem. Although manysolution methods to the inverse problem of PAT exist, reconstructed photoacousticimages do not necessarily provide quantitative information of the parameters ofinterest or their uncertainty. In addition, taking into account measurement andmodelling errors can be challenging.

In this thesis, a Bayesian approach to the inverse problem of PAT is developed.With this approach, quantitative information of the parameters and their uncertaintycan be achieved. In the Bayesian approach to PAT, the solution of the inverse prob-lem is a conditional probability distribution of the initial pressure that combinesinformation obtained through the measurements, model and prior knowledge. Fur-thermore, in the Bayesian approach, it is possible to incorporate statistical informa-tion about uncertainties and inaccuracies of the models in the solution of the inverseproblem. In this thesis, the Bayesian approximation error approach is utilised tomodel the uncertainties in the speed of sound.

All methods developed in this thesis were evaluated with simulations, and someof the methods were also validated with experimental data. The simulations and ex-periments show that the Bayesian approach is feasible for PAT producing accurateestimates of the initial pressure together with estimates for the reliability. Further-more, the simulations show that by utilising the approximation error approach, itis possible to at least partially compensate for the modelling errors due to uncer-tainties in the speed of sound and to improve the accuracy and feasibility of thesolution.

Universal Decimal Classification: 534.14, 534.22, 535.214, 517.956.27INSPEC Thesaurus: imaging; biomedical imaging; tomography; photoacoustic effect; in-verse problems; modelling; simulation; errors; error correction; Bayes methods; parameterestimation; image reconstruction; ultrasonic velocity; uncertainty handlingYleinen suomalainen ontologia: kuvantaminen; fotoakustinen kuvantaminen; tomografia;

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inversio-ongelmat; mittaus; mittausmenetelmät; laskentamenetelmät; numeeriset menetelmät;approksimointi; mallintaminen; simulointi; virheet; bayesilainen menetelmä; estimointi; ää-nennopeus; epävarmuus

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ACKNOWLEDGEMENTS

This study was carried out in the Department of Applied Physics at the Universityof Eastern Finland during the years 2015-2019. The study was financially supportedby the Academy of Finland, the doctoral school of the University of Eastern Fin-land, Alfred Kordelin Foundation, Instrumentarium Science Foundation, FinnishFoundation for Technology Promotion, Saastamoinen Foundation, Orion ResearchFoundation, and Kuopio University Foundation.

I want to express my deepest gratitude to my supervisors Associate ProfessorTanja Tarvainen, Docent Aki Pulkkinen, and Professor Jari Kaipio. It has been aprivilege to work under your guidance. You have always found me some timewhen I longed for help, advice or support. In addition, you have taught me a lotabout scientific thinking and inspired me to research. Thanks to you, I am now amuch better scientist. Furthermore, special thanks to Jari for the opportunity to visitthe University of Auckland.

Many thanks to my co-authors Dr. Felix Lucka, Dr. Robert Ellwood, SeniorLecturer Ben Cox, Professor Simon Arridge, and Dr. Jarkko Leskinen. This thesiswould not have been possible without your contribution. Special thanks to Felix,Robert, Ben and Simon for all the assistance, knowledge, and skills that I receivedduring my visit to the University College London. Jarkko, in turn, deserves greatthanks for introducing me to the wonderful world of photoacoustic measurements.

I am very thankful to the official pre-examiners Associate Professor Georg Stadlerand Professor Roger Zemp for taking the time to review and assess my thesis. Igreatly appreciate your professional comments and suggestions to improve this the-sis.

I also want to thank the past and present members of the Inverse Problemsgroup, especially those who have shared an office with me at some point. You haveprovided a pleasant and inspiring working environment where both scientific andnon-scientific conversations have been possible. Thanks for the fun time together.

Finally, I am grateful to my parents Anne and Jorma for all the support andencouragement you have given. In addition, thank you for inspiring me to get thishighly educated. I also want to thank my sister Henna and my brother Jani for thesupport and interest towards my work. My warmest thanks go to my partner Marttifor believing in me, constant support and motivating me when needed.

Kuopio, November 28, 2019

Jenni Tick

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LIST OF PUBLICATIONS

This thesis consist of the present review of author’s work in the field of photoacous-tic tomography and following selection of the author’s publications:

I J. Tick, A. Pulkkinen, and T. Tarvainen, ”Image reconstruction with uncertaintyquantification in photoacoustic tomography”, The Journal of Acoustical Societyof America 139(4), 1951–1961 (2016).

II J. Tick, A. Pulkkinen, F. Lucka, R. Ellwood, B. T. Cox, J. P. Kaipio, S. R. Arridge,and T. Tarvainen, ”Three dimensional photoacoustic tomography in Bayesianframework”, The Journal of Acoustical Society of America 144(4), 2061–2071 (2018).

III J. Leskinen, A. Pulkkinen, J. Tick, and T. Tarvainen, ”Photoacoustic tomogra-phy setup using LED illumination”, Proc. SPIE 11077, Opto-Acoustic Methodsand Applications in Biophotonics IV, 110770Q (2019).

IV J. Tick, A. Pulkkinen, and T. Tarvainen, ”Modelling of errors due to speed ofsound variations in photoacoustic tomography using a Bayesian framework”,Biomedical Physics & Engineering Express, 6(1), 015003 (2019).

The original articles have been reproduced with permission of their copyright hold-ers. Throughout the overview, these papers will be referred to by Roman numerals.

AUTHOR’S CONTRIBUTION

The publications selected in this thesis are original research papers on photoacous-tic tomography. The publications of this thesis are the result of joint work betweenthe author of this thesis and co-authors of the articles. In all publications, the au-thor implemented the numerical computations and carried out the solution of theinverse problem. However, in the numerical solution of the wave equation, a k-waveMATLAB toolbox was used, and the adjoint operator of the PAT forward problemwas developed by co-author Felix Lucka. The author was the main writer of publi-cations I, II and IV and the co-writer of publication III. The author participated inthe measurements in publication III. The photoacoustic tomography measurementdata in publication II originated from co-author Robert Ellwood.

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TABLE OF CONTENTS

1 Introduction 1

2 Photoacoustic tomography 32.1 Measurement setup .......................................................................... 32.2 Forward model ................................................................................. 52.3 Inverse problem ................................................................................ 52.4 Applications of PAI .......................................................................... 8

3 Bayesian inversion methods in PAT 113.1 Bayesian approach to PAT ............................................................... 11

3.1.1 Prior model ........................................................................... 133.1.2 Matrix-free implementation .................................................... 14

3.2 Bayesian approximation error approach for modelling of errors........... 153.2.1 Uncertainties in the speed of sound ........................................ 15

4 Summary of the results 174.1 Evaluating the Bayesian approach to PAT with simulations ............... 17

4.1.1 2D ........................................................................................ 174.1.2 3D ........................................................................................ 23

4.2 Experimental validation of the Bayesian approach to PAT ................. 264.2.1 Fabry-Perot sensor based PAT setup ...................................... 264.2.2 LED-based PAT setup ........................................................... 28

4.3 Compensation of modelling errors ..................................................... 30

5 Discussion and conclusion 35

BIBLIOGRAPHY 37

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ABBREVIATIONS

2D Two dimensional3D Three dimensionalAFM Accurate forward modelCT Computed tomographyIFM Inaccurate forward modelIFM&EM Inaccurate forward model and modelling of errorsLED Light-emitting diodeMAP Maximum a posterioriMCMC Markov Chain Monte CarloMRI Magnetic resonance imagingPA PhotoacousticPAI Photoacoustic imagingPAM Photoacoustic microscopyPAT Photoacoustic tomographyQPAT Quantitative photoacoustic tomographyTAT Thermoacoustic tomography

NOMENCLATURE

|| · || Norm∂∂s , ∂2

∂s2 Derivatives∇2 Laplacian∝ Directly proportional toarg max Maximum point· T Transpose

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SYMBOLS

β Thermal coefficient of volume expansionc Speed of soundc Fixed speed of soundCp Isobaric specific heat capacityCV Specific heat capacity at constant volumeδ Dirac delta functione Noiseek Unit vectorEp0 Relative errorε Approximation errorρ Mass densityΓ Covariance matrixΓ Grüneisen parameterH Absorbed optical energyI Identity matrixκ Isothermal compressibilityK Forward operatorl Characteristic length scaleµa Absorption coefficientM Number of measurementsn Total errorN Number of elementsNs Number of samplesN Gaussian distributionη Meanp Acoustic pressurep0 Initial pressurep0,MAP Maximum a posteriori estimate of the initial pressurept Measured pressure signalsΦ Photon fluenceπ(·) Probability densityr Position vectorR Real numbersσ Standard deviationσ2 Variancet TimeT Temperature

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1 Introduction

In biomedical imaging, one is interested in obtaining images of the internal struc-ture of the biological tissue non-invasively. The most desirable imaging techniqueswould be those that have both high contrast and high resolution. Various imagingmodalities such as computed tomography (CT), magnetic resonance imaging (MRI)and ultrasound have been widely used for biomedical imaging. However, there isstill a need for novel imaging techniques that can provide new information aboutthe imaged target. During past few decades, several hybrid or coupled physicsmodalities have been introduced. Among these is photoacoustic imaging (PAI).

PAI is based on photoacoustic (PA) effect that refers to the generation of acous-tic waves by an optical excitation [1–7]. Although the underlying physics of PAeffect was described by Alexander Graham Bell already in 1880 [8], it took morethan a century that PAI was investigated for biomedical imaging and the first PAIimages were produced [9–17]. The major PAI modalities can be divided into PAtomography (PAT) and PA microscopy (PAM). PAT refers to techniques which re-quire reconstruction algorithms to obtain PA images whereas in PAM, a PA imageis directly obtained by mechanically scanning. Depending on whether acoustic fo-cusing or optical focusing is used in the scanning, PAM can further be divided intoacoustic-resolution and optical-resolution PAM. Generally, PAT allows deep-tissueimaging with a capability of imaging entire organs while PAM is a relatively super-ficial imaging technique that can be used to image small sections of tissue. In thisthesis, the focus is on PAT.

A PAT measurement typically starts by illuminating an imaged object with ashort laser pulse. As light propagates in the object, it is partially absorbed by eitherendogenous chromophores, such as oxygenated and deoxygenated haemoglobin,melanin, lipids, and water, or exogenous contrast agents, such as organic dyes,nanoparticles and fluorescence proteins [18–20]. The absorption generates a smalltemperature rise that results in a local pressure rise through thermoelastic expan-sion. This initial pressure propagates through the object as an acoustic wave, andit is measured by ultrasound sensors on the boundary of the object. In the inverseproblem of PAT, an image of the initial pressure distribution is reconstructed fromthese boundary measurements which thus gives information on the spatial distribu-tion of chromophores.

Due to PA effect, PAT can combine benefits of optical and acoustic techniques.That is, PAT has the rich contrast of optical methods and the high spatial resolu-tion of ultrasound techniques. One feature of PAT is the scalability of the spatialresolution and the maximum imaging depth. That is, the spatial resolution can beimproved at the expense of the imaging depth and vice versa. At imaging depths ofapproximately 3 to 7 cm, resolutions of 0.5 to 5 mm can be achieved [21]. Further-more, PAT is a non-ionizing and non-invasive imaging technique. These featuresmake PAT an attractive imaging modality with great potential for biomedical imag-ing where it has found applications for example in breast [22–25], vascular [26, 27]and small animal imaging [5].

The inverse problem of PAT refers to the problem of forming an estimate of

1

the initial pressure distribution of an imaged target based on the boundary mea-surements of acoustic pressure waves. This problem has been widely studied andseveral different reconstruction algorithms have been proposed [28, 29]. Commonlyused reconstruction methods include backprojection algorithm [30–33], eigenfunc-tion expansion method [34, 35], time reversal [36–39] and penalized least squaresapproaches [40–51]. Nevertheless, there are still challenges in solving the inverseproblem. Estimation of the initial pressure is generally a moderately ill-posed prob-lem, but it becomes more ill-posed if limited view sensor geometries (sensors donot enclose the imaged target) are used. Ill-posedness means that a problem is sen-sitive to errors and uncertainties both in measurements and in modelling. Thus,in order to produce as good estimates as possible, a measurement situation hasto be modelled with sufficient accuracy. In practice, this can be difficult becauseinherent uncertainties in parameters that are required for modelling, such as mate-rial properties, sensor properties, are always present. In addition, taking modellinguncertainties or errors into account in the solution of the inverse problem can bechallenging. Furthermore, a single estimated image is typically obtained as thesolution of the inverse problem, so assessing the reliability and uncertainty of thesolution can be difficult. To overcome these challenges, improvements in solving ofthe inverse problem are needed.

This thesis develops Bayesian estimation and uncertainty quantification methodsfor PAT. The Bayesian approach is a statistical estimation method in which mea-surements, model and prior information are used to infer information about theparameters of primary interest. The aims of this thesis are

1. Formulate a Bayesian framework and computational methods for the inverseproblem of PAT. In the Bayesian approach to PAT, one is aiming at obtainingquantitative estimates of the initial pressure distribution and its uncertainties.

2. Develop a Bayesian approximation error modelling based approach to com-pensate for errors caused by an unknown speed of sound in PAT.

3. Evaluate the feasibility of the developed methods with simulations and exper-imental data.

This thesis is organized so that Chapter 2 describes the forward and inverseproblem of PAT and reviews the research. In Chapter 3, the Bayesian frameworkfor PAT and its utilisation in modelling of errors are described. The results of thepublications are reviewed in Chapter 4. Finally, Chapter 5 gives the discussion ofthe results and final conclusions.

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Figure 2.1: Principle of PAT. a) Illumination of an imaged object with a light pulse.b) Absorption of light and a corresponding localised pressure increase. c) Measure-ment of the pressure wave with ultrasound sensors on the surface of the object.

2 Photoacoustic tomography

In this chapter, the principle of PAI and the forward and inverse problems of PATare described. Furthermore, solution methods and challenges related to the inverseproblem are reviewed. Finally, applications of PAI are introduced. Further infor-mation about history of PAT, physical principles, measurement systems, solutionmethods, and applications in biomedicine can be found in several comprehensivereviews, see e.g. [1–7, 28, 52].

2.1 MEASUREMENT SETUP

The procedure of a PAT measurement situation illustrated in Figure 2.1 is as follows.First, a short (nanosecond scale) pulse of light is used to illuminate an imaged tar-get. By tuning the wavelength of light, the penetration depth inside the target andabsorption by chromophores of interest can be altered. Typically, light in the visibleor near-infrared region is utilised [3]. In this spectral region, light has its maximumdepth of penetration in tissue due to the relatively weak absorption of chromophoresin tissue. In the majority of the PAT measurement setups, the illumination is imple-mented with solid state lasers, but also laser diodes [53–56], light-emitting diodes(LEDs) [57–60] and broadband xenon lamps [61] have been used as a light source.From the light source, light is guided to the target using for example optical lenses,mirrors, optical fibers or fiber bundles.

Light propagation in a random media such as biological tissue can be describedusing analytic theory and transport theory [62]. In the analytical theory, Maxwell’s

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equations are used to describe the transportation of particles. The transport theorycan be derived using stochastic approach, which models individual particle interac-tions [63], or deterministic approach, which describes the transportation of particlesusing partial differential equations such as the radiative transport equation and itsapproximations [62, 64]. As the light pulse propagates within the imaged target,some of its energy is absorbed by chromophores. The absorbed energy at point r isgiven by

H(r) = µa(r)Φ(r), (2.1)where µa(r) is the absorption coefficient and Φ(r) is the photon fluence. The ab-sorbed energy is converted into heat and the temperature of the target increasesslightly, which temperature increase is given by

T(r) =H(r)ρCV

, (2.2)

where ρ is the mass density and CV is the specific heat capacity at constant volume.This temperature rise further causes thermoelastic expansion of the target, which inturn induces an excess pressure. If the duration of the light pulse is less than boththe thermal and stress confinement times, the local fractional volume expansion andthermal diffusion are negligible, and the pressure increase can be considered to beinstantaneous. This initial pressure can be written as

p0(r) =βT(r)

κ

=βH(r)ρCVκ

=βc2H(r)

Cp

= ΓH(r), (2.3)

where β is the thermal coefficient of volume expansion, κ is the isothermal com-pressibility, c is the speed of sound, Cp is the isobaric specific heat capacity, and Γ isthe Grüneisen parameter that indicates the efficiency of conversion of the absorbedoptical energy into pressure [2, 7]. The generated pressure relaxes and propagatesas an acoustic wave that is known as PA wave. Generally, the amplitude of the waveis relatively low but the spectral content of the wave is broad (tens of MHz).

The PA signals are acquired by ultrasound sensors at multiple locations aroundthe object. Sensors may be in contact to the surface of the imaged object or awaywhen they need to be coupled. Three most commonly used sensor geometries arespherical [23,65,66], cylindrical [67–72] and planar detection geometry [73–80] wherethe ultrasonic detection follows a (truncated) spherical surface, cylindrical surface,or plane, respectively. While spherical or cylindrical sensor geometries requires ac-cess to all sides of the object, planar sensor geometries can be utilised in a widerrange of imaged targets. However, the image quality provided by spherical andcylindrical geometries is superior to the image quality provided by a planar de-tection geometry due to their larger apertures. The sensor geometries can be im-plemented by using multiple sensors or by moving a single sensor. Multielementarrays have also been applied. Moving a single sensor around the imaged objectcan be time-consuming. Therefore, multiple sensors or arrays are preferred if a fastdata acquisition is required. Commonly used sensors can be divided into piezoelec-tric [81, 82] and optical sensors [75, 83–86].

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2.2 FORWARD MODEL

The forward problem of PAT is to solve the time-varying pressure at the sensorssurrounding the object when the acoustical properties and initial pressure distribu-tion are given. In PAT, the propagation of the PA wave can generally be modelledusing equations of linear acoustics, because PA waves are typically of small ampli-tude. For soft biological tissue, it is generally assumed that the medium is isotropicand quiescent and that the shear waves can be neglected. Further, if the mediumis assumed to be non-attenuating, the time evolution of the induced PA wave fieldscan be modeled using a wave equation

(∂2

∂t2 − c(r)2∇2)

p(r, t) = 0

p(r, t = 0) = p0(r)∂∂t p(r, t = 0) = 0

(2.4)

where p(r, t) is the acoustic pressure at point r and time t, p0(r) is the initial pressuredistribution, and c(r) is the speed of sound [2, 6].

However, in many practical applications, PA waves travel through a complexheterogeneous biological medium that cannot be considered non-attenuating. Amodel of ultrasound propagation that accounts acoustic attenuation has been usedin [39,87–91]. Furthermore, neglecting the shear waves is not valid assumption whenthe imaged target includes for example bone. Therefore, an elastic wave equationmodel has been considered [92, 93]. In this thesis, the wave equation (2.4) is used tomodel the wave propagation.

The numerical solution of the wave equation (2.4) can be obtained using forexample finite difference or finite element methods [94]. These methods are su-perb for many applications since they are simple, adaptable and easily parallelised.However, these methods can become computationally expensive when broadbandultrasound fields in the time domain are considered [95]. Alternatively, a k-spacepseudospectral method can be utilised [96]. This method enables a computation-ally efficient solving of the wave equation by combining the spectral calculation ofspatial derivatives with a temporal propagator expressed in the spatial frequencydomain or k-space. In this thesis, the k-space method implemented with the k-WaveMATLAB Toolbox [95] is used for the numerical solution of the wave equation.

2.3 INVERSE PROBLEM

In PAT, estimates of the initial pressure distribution, also called photoacoustic im-ages, are of principal interest. Estimation of the initial pressure from the measuredPA waves is the inverse problem of PAT. This problem has an unique solution al-most for all practical acoustic detection surfaces, but the solution can sometimes beunstable [29]. Thus, the problem is ill-posed. However, the inverse problem is onlymildly ill-posed in full view sensor geometries in which the imaged object is fullysurrounded by sensors on a closed surface. In this case, qualitative accurate esti-mates of the initial pressure can be achieved. Sensor geometries that do not enclosethe object are called limited view geometries. In these sensor geometries, only apart of the wavefront is recorded which can result in images that suffer from blurryedges, loss of details and reduced image quality. More specifically, sharp boundariesof the object can be reconstructed accurately if they are facing the detection surface

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Figure 2.2: Two examples of limited view imaging scenarios with sensors on anarc (left) and on a line (right). The region enclosed by the sensors (gray shadedarea), the inclusion boundaries that are reconstructed accurately (solid line), and theinclusion boundaries that are blurred (dashed line). Image is adapted from [6, 97].

but suffer from blurring if they are perpendicular to the detection surface [6, 97].This is illustrated in Figure 2.2.

The inverse problem of PAT has been widely studied and a large number ofsolution methods are available. The approaches can be classified either analytical ornumerical methods. The former aim to solve the problem analytically and the latterutilise the numerical solution of the problem.

The analytical methods include backprojection algorithm [30–33] and eigenfunc-tion expansion method [34, 35]. The backprojection approach is based on the in-verse circular/spherical Radon transform, and the initial pressure is reconstructedby summing up the backprojected measured pressure signals with appropriate timedelays. In the eigenfuction expansion method, the initial pressure is obtained asthe series solution and series coefficients are calculated from the measured pressuresignals.

The numerical methods include time reversal [36–39], penalized least squaresapproaches [40–51], and Bayesian approach [98], I and II. In the time reversal al-gorithms, an image is reconstructed by running a numerical model of the forwardproblem backwards in time. That is, the measured acoustic pressure signals arebackpropagated into the domain in the time-reversed order. The approaches basedon penalized least squares perform an image reconstruction by minimizing thesum of the least square error between the measured signal and signals predictedby the photoacoustic forward model and a regularizing penalty functional such asTikhonov [40, 42, 46] or total variation [41, 43, 45, 46, 48]. The Bayesian approach is astatistical inversion method and it is reviewed more in detail in Section 3.1.

Generally speaking, the analytical methods can provide an ease of implemen-tation and they can have a low computational cost. However, they are limited tospecific geometries such as spherical, cylindrical, and planar acoustic detection sur-faces and require that the PA signals are densely sampled on the detection surface.In contrast, the numerical methods are more versatile, albeit computationally moreintensive. In addition, the above described methods, apart from the Bayesian ap-proach, produce a single estimated image as the solution to the inverse problem.Thus, the reliability, uncertainty, and errors of the solution can be challenging toassess.

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Despite many solution methods, obtaining an accurate solution for the inverseproblem of PAT is not a straightforward task in practice. This is partly due to ill-posed nature of the problem, which is why a careful modelling of the measurementsituation is required. Next, challenges related to solving the inverse problem of PATin practice are discussed.

In practice, limited view sensor geometries are often required to be used dueto constraints associated with an experimental setup or an imaging application,although they suffer from the reduced image quality. Thus, methods for improv-ing the image quality have been developed. In the backprojection approach, theimage quality of the limited view reconstruction has been improved by adding aweighting factor from a smoothing function to backprojection signals [99]. In thepenalized least squares approaches, the artifacts resulting from the limited view sce-nario have been reduced by utilising prior information in form of a regularizationterm [100–102]. Especially, prior structural information has been shown to improvethe quality of the reconstructed images significantly [103, 104]. In the time rever-sal approaches, an enhanced version of time reversal such as iterative time reversalcan be used to improve the quality of the reconstructions in the limited view se-tups [45, 105, 106]. In the Bayesian approach, prior information can be incorporatedinto the image reconstruction procedure.

In the analytical reconstruction algorithms, it is assumed that the ultrasoundsensors are ideal point-like sensors with an infinite bandwidth. However, real ul-trasound sensors have a finite aperture and a finite bandwidth. Both these char-acteristics of the sensor can cause blurring of the reconstructed images [107, 108].However, in the numerical solution methods, the properties of sensors can be mod-elled. This has resulted in significantly reduced blurring and further the enhancedimage quality in the penalized least squares approaches [109–111].

Reconstruction algorithms rely on having accurate knowledge of the speed ofsound in the medium. This assumption is somewhat problematic in real applica-tions where the speed of sound is typically unknown. Basically, the speed of soundcould be estimated from an adjunct measurement such as ultrasound tomographymeasurements [112–115] or jointly with the initial pressure distribution [116–120].However, an implementation of adjunct imaging and non-uniqueness of the jointestimation problem cause practical challenges. Thus, a preassigned nominal valuefor the speed of sound is more commonly used, for example by setting it equal tothe speed of sound in water or to average speed of sound in tissue. Using an ap-proximate speed of sound that inaccurately addresses the true speed of sound dis-tribution can yield estimates which contain severe artefacts [38,121]. Some methodshave been proposed to mitigate these artefacts, but these approaches compensatewell mainly such acoustic heterogeneity-induced artifacts that originate from smallspeed of sound variations [122–126].

The most approaches for the inverse problem have been using the wave equation(2.4) as the forward model. In practice, this can be too simplified model for the wavepropagation in tissue, which can result in significant errors in the solution of theinverse problem due to modelling errors. Thus, models incorporating more realistictissue properties have been studied [39, 49, 88, 92, 127–129]. Although the utilisationof these models can reduce modelling errors and thus improve the accuracy of thesolution of the inverse problem, it can also make solving of the inverse problemmore challenging due to increased complexity of the model.

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2.4 APPLICATIONS OF PAI

Photoacoustic imaging is an excellent modality for imaging blood vessels due to itshigh sensitivity to haemoglobin with certain wavelenghts [26, 27]. PAI can detectboth tiny and large vessels so it can provide comprehensive imaging of the vascu-lature. In addition, a structure of vessels can be visualised using intravascular PAimaging catheters [130–133]. Since increased vascularity, high total blood concentra-tions and abnormally low haemoglobin oxygen saturation indicate the presence oftumours, PAI can potentially be used for tumour imaging [134].

PAI is well suited for molecular imaging which measures chemical and biologicalprocesses at the molecular and cellular level [18]. Molecular imaging is based onexogenous chromophores, which either change their optical properties if there arechanges in the molecular level or have specific spectral signatures and target specificmolecules.

In addition to structural and molecular imaging, PAI can also provide functionalinformation. One important functional parameter that PAI is able to measure isblood oxygen saturation [135]. Since blood oxygen saturation is determined bythe concentrations of oxyhaemoglobin and deoxyhaemoglobin, PAI is able to pro-duce an image of blood oxygen saturation. Other functional capability of PAI is themeasurement of blood flow velocity [136]. In addition, PAI can provide functionalinformation in form of temperature [137–139]. The Grüneisen coefficient changeswith temperature, thus PAI can provide images of temperature distributions in tis-sue [139].

Another major application of PAI is small animal imaging, which enables study-ing human disease processes, monitoring treatment and developing new treatments[5]. The small size of the animal means that penetration depth requirements aregenerally modest and signals can usually be recorded around the whole body ofthe animal. Thus, high-resolution images can be obtained even deeper anatomicalstructures. In small animals, for example viscera [65, 68, 140–143], heart [144, 145],skin [74, 146, 147], and brain [67, 148–150] have been studied.

Breast imaging is a promising application of PAI [22–25]. In fact, breast imagingis one of the most advanced clinical applications of PAI and clinical measurementdevices are available. PAI can provide comprehensive information of the vasculatureand tumour properties. Thus, PAI is well suited to diagnostic imaging of the breast.

Another potential application of PAI is brain imaging [151]. Imaging of brain ischallenging due to strong optical scattering and acoustic attenuation by the skull.Thus, studies have been mainly performed using small animals due to their thinskull. In these studies, both brain structure and functionality have been imaged [67,79, 149, 152]. In addition, brain tumours and disorders have been considered [153–157]. Thus far, imaging of human brains have not been performed, but a few studiespromotes its feasibility, especially if human neonates would be imaged subjects [72,158].

PAI has also been applied in dermatologic imaging [74,146,147,159,160]. Super-ficial vasculature of small animals and humans have been imaged. In addition, skincancers, burns and other skin lesions have been detected with PAI. Recently, PAI hasbeen applied in diagnosis and monitoring of psoriasis [161].

The other possible biomedical applications include gynecological [162,163], uro-logic [164, 165], endocrinological [166–168], and ocular [169] imaging. In addi-tion, PAI have been utilised in diagnosing joint diseases [170–172] and Crohn dis-ease [173].

8

In addition to imaging, PAI can be used as an assisting method. PAI can beutilised in planning and monitoring treatments and thus it could help to improvetreatments and outcomes. In addition, PAI can be an useful tool for assessing theefficacy of treatments. Examples of applications are guidance of therapy, surgeryand biopsy [134, 174–181].

9

3 Bayesian inversion methods in PAT

In this chapter, the Bayesian approach to the inverse problem of PAT is describedfollowing publication I. In addition, extending this approach to three dimensions(3D) is described following publication II. Finally, the Bayesian approximation errorapproach and its utilisation in modelling uncertainties in the speed of sound isdescribed following publication IV.

3.1 BAYESIAN APPROACH TO PAT

In the inverse problem of PAT, an unobservable parameter p0 (parameter of interest)is estimated based on the measurements pt. An observation model describes howthese depend on each other. Since the practical measured pressure signals alwayscontain noise that is commonly assumed to be additive, the discrete observationmodel of PAT corresponding to the forward model (2.4) can be written in form

pt = K(c)p0 + e, (3.1)

where pt ∈ RM is a vector composed of the acoustic pressure waves sampled at thesensors at discrete time points, p0 ∈ RN is the discrete initial pressure distribution,K(c) ∈ RM×N is the linear operator which maps the initial pressure distribution tothe measurable data by discretizing the wave equation (2.4), and e ∈ RM representsthe unknown noise. In practice, the operator K(c) can be assembled by computingthe impulse response of a discrete system approximating (2.4) for each of the ele-ments in the reconstructed area and placing the acoustic output on the columns ofK(c). In this thesis, the impulse responses were computed using the k-space timedomain method implemented in the k-Wave toolbox [95].

In the Bayesian framework [182, 183], all parameters are modelled as randomvariables, and information related to these parameters is expressed by probabilitydistributions. The solution of the inverse problem is the posterior density that is aconditional probability density for the parameters of interest in each element of thedomain. The posterior density extracts information and assesses uncertainty aboutthe variables based on information obtained through the inference of measurements,model, and prior model for unknown parameters. In the case of PAT, the posteriordensity reflects information on the unknown initial pressure distribution p0 giventhe acoustic pressure measurements pt.

According the Bayes’ theorem, the posterior density follows proportionality

π(p0|pt) ∝ π(p0)π(pt|p0). (3.2)

where π(p0) is the prior probability density and π(pt|p0) is the likelihood den-sity. The prior density models what is known of the parameters of interest beforethe measurements, while the likelihood describes the probabilities of measurementoutcomes over all choices of parameters .

Let us consider a linear observation model (3.1) and assume that p0 and e aremutually independent. Then, the likelihood becomes

π(pt|p0) = πe(pt − K(c)p0), (3.3)

11

where πe is the probability density of the noise e. The posterior density can now bewritten as

π(p0|pt) ∝ π(p0)πe(pt − K(c)p0). (3.4)

Let us further assume that the noise e and initial pressure p0 are Gaussian dis-tributed i.e. e ∼ N (ηe, Γe) and p0 ∼ N (ηp0 , Γp0) where ηe and Γe are the meanand covariance of the noise, respectively, and ηp0 and Γp0 are the mean and co-variance of the prior model, respectively. Now, in the case of a linear observationmodel and Gaussian distributed noise and prior, the posterior density is a GaussianN (ηp0|pt

, Γp0|pt) and is described by the mean and covariance

ηp0|pt= Γp0|pt

(K(c)TΓ−1

e (pt − ηe) + Γ−1p0

ηp0

)(3.5)

Γp0|pt=

(K(c)TΓ−1

e K(c) + Γ−1p0

)−1. (3.6)

It should be noted that the posterior density has this closed form solution only whenan observation model is linear and noise and prior are Gaussian distributed.

In other situations, i.e. non-linear observation model and/or non-Gaussian dis-tributions, the posterior density can in principle be explored by employing MarkovChain Monte Carlo (MCMC) technique, which provides a representative set of sam-ples from the posterior density. However, in tomography, the posterior densityis often high dimensional, and thus MCMC methods can be computationally pro-hibitively too expensive. Therefore, point estimates of the posterior density are typ-ically computed. The most commonly used point estimate is a maximum a posteriori(MAP) estimate that is defined as

p0,MAP = arg maxp0

π(p0|pt). (3.7)

In Gaussian case, the MAP estimate is same as the mean of the posterior density i.e.p0,MAP = ηp0|pt

.In addition to obtaining an estimate of the parameter of interest, the posterior

density also provides information on the uncertainty of the estimate. This informa-tion can be used to assess the accuracy of the estimate. From the posterior covarianceposterior variances σ2

p0|pt= diag{Γp0|pt

} and further standard deviations σp0|ptfor

the each unknown parameter can be extracted. In addition, the reliability of theestimates can be assessed by computing credible intervals[

ηp0|pt ,k − ασp0|pt ,k , ηp0|pt ,k + ασp0|pt ,k

], (3.8)

where ηp0|pt ,k is the value of the posterior mean ηp0|ptin the kth element, σp0|pt ,k is

the value of σp0|ptin the kth element and α = 1, 2, 3 corresponds to 68.3%, 95.5%

and 99.7% intervals, respectively. Also the posterior marginal densities can be in-formative for inspecting the uncertainty of the estimate. Since the posterior densityis a Gaussian, all marginal densities are Gaussian. The marginal density of the kthelement is defined as

p0,k|pt ∼ N (ηp0|pt ,k, Γp0|pt ,kk), (3.9)

where ηp0|pt ,k is the value of ηp0|ptin the kth element and Γp0|pt ,kk is the value of the

kth diagonal element of the matrix Γp0|pt.

12

Figure 3.1: Two sample draws of 2D spatial distributions (10 × 10 mm) from thewhite noise (first column), Ornstein-Uhlenbeck (second column) and squared ex-ponential (third column) priors with the mean ηp0 = 5 and the standard deviationσp0 = 10/6. The characteristic length scale l = 2 mm was used for the Ornstein-Uhlenbeck and squared exponential prior.

3.1.1 Prior model

A prior model describes our pre-existing knowledge of the imaged target in formof a probability density. That is, it describes what values are probable for eachparameter of interest and how these values are distributed. The prior model shouldbe chosen in such a way that the imaged target is well supported by the model.

Commonly used prior models are Gaussian priors [182]. Three examples ofGaussian priors are a white noise prior, Ornstein-Uhlenbeck prior and squared ex-ponential prior [184,185]. Sample draws of 2D spatial distributions from these priorsare shown in Figure 3.1. Gaussian priors can be described by their means ηp0 andcovariance matrices Γp0 . If the range of the unknown parameter values is roughlyknown, this information can be utilised to choose the mean ηp0 of the prior. Corre-spondingly, information related to the expected spatial distribution of the parame-ters can be utilised to choose the covariance matrix Γp0 . If the estimated parametersare assumed to be non-smooth (the parameters are independent of each other orhave no spatial correlation), the white noise prior can be an appropriate choice. Inthe white noise prior, the Γp0 is defined as

Γp0,ij = σ2p0

δij, (3.10)

where i and j are the element indices, σp0 is the standard deviation, and δij is theDirac delta function. The prior knowledge on the range of the unknown parametervalues can be utilised to choose the standard deviation σp0 . On contrary, if theparameters are assumed to be somewhat smooth, the squared exponential prior orOrnstein-Uhlenbeck prior can be an useful choice. In the squared exponential prior,

13

the Γp0 is of form

Γp0,ij = σ2p0

exp

(−‖ri − rj‖2

2l2

)(3.11)

whereas in the case of the Ornstein-Uhlenbeck prior the Γp0 is defined as

Γp0,ij = σ2p0

exp(−‖ri − rj‖

l

), (3.12)

where ri and rj are the element positions of the element indices i and j and l isthe characteristic length scale which controls the spatial range of correlation. Thecharacteristic length scale l can be chosen based on prior information about internalstructures of the imaged target and size of these structures.

In PAT, some spatial correlation in parameter values can be expected since pre-sumably properties of biological tissue types are relatively homogeneous. In addi-tion, it is possible that the target is composed of heterogeneities separated by sharperedges such as blood vessels. Thus, the squared exponential or Ornstein-Uhlenbeckprior can be regarded as more appropriate presumption for PAT than the whitenoise prior.

3.1.2 Matrix-free implementation

In a large dimensional inverse problem, the closed form matrix presentation i.e.(3.5)-(3.6) might be impractical to evaluate and store. Therefore, a matrix-free methodfor the computation of the posterior mean and covariance is needed. This can beachieved by solving linear systems corresponding to the mean and covariance us-ing iterative linear solvers, such as conjugate gradient methods, and evaluating themultiplication by the forward model K with the forward operator [96], the multipli-cation by the transpose of the forward model KT with the adjoint operator [45] andthe multiplication by the covariance of the prior Γp0 with a 3D convolution usingthe fast Fourier transform as follows. The mean of the posterior density (3.5) can becomputed by solving the system of linear equations

Cηp0|pt= d, (3.13)

where

C = Γp0 K(c)TΓ−1e K(c) + I, (3.14)

d = Γp0 K(c)TΓ−1e (pt − ηe) + ηp0 (3.15)

and I is an identity matrix. Correspondingly, the kth column of the posterior covari-ance (3.6) can be computed by solving the linear system

CΓp0|pt ,k = Γp0 ek, (3.16)

where C is as in (3.14) and ek is a unit vector with value one at the kth element andzeros elsewhere. In this thesis, the linear systems (3.13) and (3.16) are solved usinga biconjugate gradient stabilized (l) method [186] built-in MATLAB.

14

3.2 BAYESIAN APPROXIMATION ERROR APPROACH FOR MOD-ELLING OF ERRORS

A forward model that describes the dependence of unknown parameters and mea-surements is an essential part of the solution of an inverse problem. However, themodel may be inaccurate and it may contain parameters that are not well known. Inaddition, some of the unknown parameters may not be of direct interest and they arereferred to as nuisance parameters. If the model is an approximation to the accuratephysical model, this may affect unfavourably the solution of the inverse problemdue to its ill-posedness. However, a consideration of the inverse problem utilisingthe Bayesian approximation error modelling [182] allows accounting for modellingerrors.

The Bayesian approximation error modelling has been applied to handle manykinds of approximation and modelling errors in a variety of inverse problems. Ex-amples in optical and acoustic imaging modalities include quantitative PAT (QPAT)where the Bayesian approximation error approach has been utilised in the marginal-ization of scattering [187], mitigation of inaccuracies due to an acoustic solutionmethod [188] and compensation of discretisation errors [189]. In diffuse opticaltomography, a model reduction [190–193] and compensating uncertainties in mea-surement geometry, boundary shape and model parameters has been considered[194–196]. In full-waveform ultrasound tomography, errors due to domain trunca-tion and approximate boundary models were treated [197]. In PAT, the Bayesian ap-proximation error modelling has been used to model and compensate errors causedby uncertainties in the speed of sound IV (Section 3.2.1) and ultrasound sensor lo-cations [198].

3.2.1 Uncertainties in the speed of sound

In PAT, the speed of sound in the imaged object is commonly considered as a knownnuisance parameter. However, the accurate speed of sound c, that can be constantor spatially varying, is typically not available in practical applications. Therefore,some nominal value c is used in the inverse problem, which can cause modellingerrors. However, these errors can be compensated by expressing the observationmodel utilising the Bayesian approximation error modelling [182] in the form

pt = K(c)p0 + ε + e= K(c)p0 + n, (3.17)

where ε = K(c)p0 − K(c)p0 is the approximation error describing the modellingerror, or discrepancy, between the exact forward model with the accurate speed ofsound c and the approximate forward model with an inexact speed of sound c andn = ε + e is the total error.

When the total error n is approximated to be mutually independent of the un-known p0, the observation model (3.17) leads to a likelihood density

π(pt|p0) = πn(pt − K(c)p0), (3.18)

where πn is the probability density of the total error n. Further, the posterior densitycan be written as

π(p0|pt) ∝ π(p0)πn(pt − K(c)p0). (3.19)

15

Let us assume that noise e is Gaussian distributed e ∼ N (ηe, Γe), and approx-imate the modelling error ε as a Gaussian ε ∼ N (ηε, Γε). Thus, the total error nis a Gaussian distribution n ∼ N (ηn, Γn), where ηn = ηe + ηε and Γn = Γe + Γε

leading to a Gaussian approximation for the likelihood. With this likelihood, aGaussian prior, and a linear observation model, the posterior density is a GaussianN (ηp0|pt

, Γp0|pt) given by the mean and covariance

ηp0|pt= Γp0|pt

(K(c)TΓ−1

n (pt − ηn) + Γ−1p0

ηp0

)(3.20)

Γp0|pt=

(K(c)TΓ−1

n K(c) + Γ−1p0

)−1. (3.21)

The realisation of the approximation error ε is unknown since its value dependson the actual unknown p0 and inaccurately known nuisance parameter c. However,samples for the modelling error can be simulated utilising ’teaching’ distributionsof unknowns, which are probability densities including the prior information aboutthe unknowns, as follows. First, a set of samples {p(l)0 , l = 1, ..., Ns} and {c(l), l =1, ..., Ns} are drawn from the teaching distribution of the initial pressure and speedof sound, respectively. Then, samples of the approximation error are computedusing

ε(l) = K(c(l))p(l)0 − K(c)p(l)0 . (3.22)

From these samples, the mean ηε and covariance Γε of the approximation error canbe estimated as

ηε =1

Ns

Ns

∑l=1

ε(l) (3.23)

Γε =1

Ns − 1

Ns

∑l=1

(ε(l) − ηε)(ε(l) − ηε)

T. (3.24)

16

4 Summary of the results

This chapter summarizes the main results of the publications constituting this thesis.First the Bayesian approach is validated with numerical simulations in Section 4.1.These results are based on publications I and II. Secondly, results of the experimen-tal validation of the approach are presented in Section 4.2. These results are basedon publications II and III. Finally, in Section 4.3, results of numerical simulationson modelling of uncertainties in the speed of sound using the approximation errormodelling are presented. These results are based on publication IV.

4.1 EVALUATING THE BAYESIAN APPROACH TO PAT WITH SIM-ULATIONS

The feasibility of the Bayesian approach to PAT was first evaluated with numericalsimulations. In publication I, simulations were done in 2D whereas in publication IIsimulations were conducted in 3D. In the 2D simulations, the approach was inves-tigated utilising two different distributions of a Gaussian prior in various imagingsituations including limited view measurement setups. In addition, different sensorproperties such as a size and bandwidth were considered. In these simulations, theentire posterior density was determined and inspected. In addition to the poste-rior standard deviation, the reliability of the estimates was assessed by computingmarginal densities and credible intervals. In the 3D simulations, the performance ofthe approach was demonstrated in different sensor geometries, and point estimatesfor the image reconstruction and uncertainty quantification were computed. In allsimulations, the quantitative accuracy of the posterior mean estimates was evalu-ated by computing the relative errors of the estimates with respect to the true initialpressure distribution using

Ep0 = 100% · ‖p0 − p0,MAP‖‖p0‖

, (4.1)

where p0 is the simulated initial pressure distribution and p0,MAP is the estimatedvalue.

4.1.1 2D

In the 2D simulations, a square domain with a side length of 10 mm was used.The medium was assumed to be non-attenuating with a constant speed of sound1500 m/s. The simulated initial pressure distribution is shown in Figure 4.1. Sen-sors were arranged around the domain (full view), on two adjacent sides of thedomain (two side) or on one side of the domain (one side). Ideal point-like sen-sors (ideal sensors), sensors of width 0.5 mm (finite sized sensors), and sensors withlimited bandwidth (bandlimited sensors) were investigated. In the case of the ban-dlimited sensors, the frequency response of the sensors was modeled as a bandpassfilter with −6 dB bandwidth of 8 MHz and central frequency of 6 MHz. The simu-lated measurement data was generated by solving the wave equation (2.4) using the

17

Figure 4.1: The simulated (true) initial pressure distribution given in arbitraryunits. The white square and circle indicate the locations where the marginal densi-ties are plotted and the dashed line indicates the location where the credible intervalis plotted.

k-space time domain method and adding a 1% Gaussian noise to it. In the data sim-ulation, the domain was discretised into 300 × 300 square pixels (pixel side lengthof 33 µm) and 283 temporal samples (sampling frequency 20 MHz) were used.

In the inverse problem, the mean and covariance of the posterior density werecomputed using (3.5) and (3.6), respectively. In addition, credible intervals werecomputed on a diagonal cross section of the domain (a dashed line in Figure 4.1) us-ing (3.8). Furthermore, marginal densities were calculated in two locations (a squareand circle in Figure 4.1) using (3.9). In the computations, the domain was discretisedinto 120 × 120 square pixels (pixel side length of 83 µm). The Ornstein-Uhlenbeckand white noise priors were used as the prior model for the initial pressure. Forboth priors, the mean was set as the expected mean value of the initial pressureηp0 = 5 and the standard deviation was set as σp0 = 2.5, which means that 99.7%of the initial pressure values were expected to be normally distributed within therange [−2.5, 12.5]. For the Ornstein-Uhlenbeck prior, the characteristic length scalel = 1.25 mm was used. In addition, the accurate noise statistics were assumed.

The simulations show that the sensor geometry affects the initial pressure esti-mates and their uncertainties. In the full view sensor geometry, the inverse problemis only mildly ill-posed. Thus, the estimates of the initial pressure distribution arequalitatively similar with true one (first column of Figure 4.2). In addition, the es-timates are quantitatively accurate and the relative errors are smallest (Table 4.1).Furthermore, the standard deviations are small indicating small uncertainty of theestimates (first column of Figure 4.2). In limited view sensor geometries, the inverseproblem becomes more ill-posed. Thus, the estimates of the initial pressure dis-tribution suffer from artefacts and distortions (second and third column of Figure4.2). The distortion of the estimates increases at larger distances from the sensors.In addition, the quantitative accuracy of values in these distorted areas is reduced.This results into higher relative errors (Table 4.1). However, the uncertainty of theestimates also increases, especially in the distorted areas, as the number of detectionedges decreases. In all cases, the true value of the initial pressure is within the errorlimits obtained in the Bayesian approach (Figures 4.3 and 4.4). This indicates thatthe Bayesian approach can provide reliable uncertainty estimates.

The simulations also indicate that the prior has an influence on the solutionof the inverse problem (Figure 4.2). In the full view sensor geometry case, the

18

Figure 4.2: The posterior mean (top block) and standard deviation (bottom block)obtained using the Ornstein-Uhlenbeck prior (first row) and the white noise prior(second row) in the case of the ideal sensors. The columns from left to right representthe full view (first column), two side (second column) and one side (third column)sensor geometries. The red dots in the first row images indicate the locations of thesensors.

Table 4.1: The relative errors Ep0 (%) of the estimated posterior mean obtainedusing the full view, two side and one side sensor geometries in 2D. The ideal sen-sors (ideal), finite sized sensors (finite sized), and bandlimited sensors (bandlim-ited) were considered using the Ornstein-Uhlenbeck (OU) and/or white noise (WN)prior.

Ideal Finite sized BandlimitedOU WN OU OU

Full view 13 13 23 14Two side 15 16 33 19One side 34 36 56 51

19

Figure 4.3: The true initial pressure distribution (black solid line) and the posteriormean (red dashed line) with 99.7% credible intervals (purple area) on a diagonalcross-section shown in Figure 4.1 obtained using the Ornstein-Uhlenbeck prior (firstrow) and the white noise prior (second row) in the case of the ideal sensors. Thecolumns from left to right represent the full view (first column), two side (secondcolumn) and one side (third column) sensor geometries.

posterior mean estimates are almost identical and uncertainty estimates differ onlyslightly. However, the significance of the prior increases in the case of the limitedview sensor geometries. In addition, the simulations with different noise levels inpublication I indicate that proper prior information is important in the case of noisymeasurements.

In addition, the simulations show that sensor properties such as a size and band-width influence the estimates of the initial pressure and their uncertainties (Figure4.5). In the case of the finite sized sensors, the quality of the estimates is reducedand the uncertainty is increased. This can be seen very clearly in the results ob-tained using the limited view sensor geometries (second and third column of Figure4.5). The reduced quality cause significant increase in the relative errors (Table 4.1).Also, the use of the bandlimited sensors reduces the accuracy of the solution, butthe effect is not that significant as for the finite sized sensors.

20

Figure 4.4: The marginal probability densities of the posterior densities at locationsdenoted by a square (first row) and circle (second row) in Figure 4.1 in the caseof the ideal sensors. The columns presents results obtained using the full view(first column), two side (second column), and the one side (third column) sensorgeometry. Shown in the graphs are the true initial pressure (vertical black line)together with the marginal densities of the posterior density obtained using theOrnstein-Uhlenbeck prior (blue solid line) and white noise prior (red dotted line).

21

Figure 4.5: The posterior mean (top block) and standard deviation (bottom block)obtained using the finite sized sensors (first row) and bandlimited sensors (secondrow). The columns from left to right represent the full view (first column), two side(second column) and one side (third column) sensor geometries. The red dots in theimages indicate the locations of the sensors.

22

Figure 4.6: The simulated (true) initial pressure distribution given in arbitraryunits. The left image shows the contour surface that indicates the areas where theparameter has value 1 or more. The three images on the right represent maximumintensity projections along axis directions x, y, and z. The red square indicates thelocation where the marginal densities are plotted.

4.1.2 3D

In the 3D simulations, the simulation domain was a cube with a side length of 10 mmand a discretisation of 304 × 304 × 304 cubic voxels (voxel side length 33 µm). Thespeed of sound was set to 1500 m/s and attenuation was not considered. The truesimulated initial pressure distribution is illustrated in Figure 4.6. Three sensor ge-ometries of same type as in the 2D simulations but extended to 3D were considered.In the data simulation, 849 time samples (sampling frequency 60 MHz) were simu-lated using the k-space time domain method. Further, the simulated pressure signalswere corrupted by a 1% Gaussian noise.

In the inverse problem, the mean of the posterior was computed by solving thesystem of equations (3.13). In addition, the covariance of the posterior density andmarginal densities were calculated in one location (a square in Figure 4.6) using(3.16) and (3.9), respectively. For the computations, the domain was discretised into204 × 204 × 204 cubic voxels (voxel side length 49 µm). The prior model was chosento be based on the Ornstein-Uhlenbeck prior. The mean of the prior was set to thevalue of the background (ηp0 = 0), the standard deviation was set to σp0 = 2 andthe characteristic length scale was set to l = 0.49 mm. This means that 99.7% ofthe initial pressure values are expected to be in the range between −6 and 6. Inaddition, the noise statistics were assumed to be known accurately. For comparison,a time reversal solution was computed.

The simulations show that the Bayesian approach can be used to provide esti-mates of the initial pressure distribution also in 3D. In addition, the results of the3D simulations follow the trend of the 2D results. That is, the most accurate esti-mates of the initial pressure are obtained using the full view sensor geometry andthe accuracy of the estimates reduces in the limited view geometries (Figures 4.7and 4.8 and Table 4.2). In addition, the uncertainty of the estimates increases as thenumber of detection surfaces decreases (Figure 4.9). The reduction of the accuracy ismore significant in 3D. This is due to increasing ill-posedness of the inverse problemcompared to 2D case in a limited view geometry. Comparing the estimates obtainedusing the Bayesian approach and time reversal, it can be seen that the main featuresof the estimates are the same (Figures 4.7 and 4.8). However, the Bayesian approach

23

Figure 4.7: The estimate of the initial pressure distribution obtained using theBayesian approach in the full view (first row), two side (second row) and one side(third row) sensor geometries. The left image shows the contour surface that indi-cates the areas where the parameter has value 1 or more. In these images, the grayarea indicate the locations of the sensors. The three images on the right representmaximum intensity projections along axis directions x, y, and z.

24

Figure 4.8: The estimate of the initial pressure distribution obtained using the timereversal in the full view (first row), two side (second row) and one side (third row)sensor geometries. The left image shows the contour surface that indicates the areaswhere the parameter has value 1 or more. In these images, the gray area indicate thelocations of the sensors. The three images on the right represent maximum intensityprojections along axis directions x, y, and z.

25

Figure 4.9: The marginal probability densities of the posterior densities at locationdenoted by a square in Figure 4.6. Shown in the graph are the true initial pressure(vertical black line) together with the marginal densities of the posterior densityobtained using the full view (blue solid line), two side (red dotted line) and one sidesensor geometries (black dashed line).

Table 4.2: The relative errors Ep0 (%) of the estimated initial pressure distributionobtained using the full view, two side and one side sensor geometries in 3D. Theestimates were computed using the Bayesian approach and time reversal.

Bayesian Time reversalFull view 4 4Two side 25 62One side 48 80

seems to tolerate limited view artefacts better than the time reversal. The relativeerrors support these findings (Table 4.2).

4.2 EXPERIMENTAL VALIDATION OF THE BAYESIAN APPROACHTO PAT

The Bayesian approach was also evaluated with experimental data from two differ-ent measurement setups. In publication II, measurements were carried out using aFabry-Pérot sensor based photoacoustic measurement system developed in the Pho-toacoustic Imaging Group of the University College London [75,199]. In publicationIII, PAT measurements were done using an LED-based imaging setup developed atthe Department of Applied Physics in the University of Eastern Finland.

4.2.1 Fabry-Perot sensor based PAT setup

In the case of the Fabry-Pérot sensor based setup, the imaged objects were a skeletalleaf phantom and a mouse head. In the phantom measurement, an illumination ofan imaged object was done using a laser operating at the wavelength of 1064 nm,whereas for the mouse measurements two optical parametric oscillators were tuned

26

Figure 4.10: A photograph of the leaf phantom (left image), the contour surfaceof the reconstructed photoacoustic image of the leaf phantom obtained using theorthogonal sensor (middle image) and the planar sensor (right image). The rowsfrom top to bottom represent the reconstructed image obtained using the Bayesianapproach (first row) and time reversal (second row).

to the wavelength of 755 nm. PAT signals were detected with a planar or orthogonalFabry-Pérot sensor by scanning an area of approximately 10 mm × 10 mm on thesensors with a step size of 100 µm. In all measurements, the imaged targets werecoupled to the sensor using deionized water.

Before reconstructions, nuisance signal components of the measured PA signalswere removed by filtering (a bandpass filter with cutoff frequencies between 0.5 and20 MHz). This filtering was also taken into account in the forward model. The imagereconstruction was performed by solving the system of equations (3.13). The compu-tations were done in 3D, and the grid sizes were 274× 248× 242 and 304× 286× 240cubic voxels (voxel side length 50 µm) for the leaf and mouse measurements, respec-tively. In the computations, the speed of sound was assumed to be 1488 m/s. TheOrnstein-Uhlenbeck prior with the mean ηp0 = 0, standard deviation σp0 = 0.25 andcharacteristic length scale l = 0.1 mm was used as the prior model. In addition,the noise statistics at each sensor position were determined by calculating the meanand standard deviation from a time frame of the measured PA signals that is sup-posed to contain only noise. Furthermore, a time reversal solution was computedfor comparison.

The experiments with the Fabry-Pérot sensor show that reconstructions obtainedusing the Bayesian approach represents the features of the imaged target (Figures4.10 and 4.11). That is, the vein-like structure of the leaf and the vasculature of

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Figure 4.11: The photoacoustic images of the mouse head obtained using theBayesian approach (first row) and time reversal (second row). The left image showsthe contour surface of the reconstructed image. The three images on the right rep-resent maximum intensity projections along axis directions x, y, and z.

the mouse head are visible in the reconstructions. In addition, the Bayesian recon-structions are similar with the time reversal reconstructions (Figures 4.10 and 4.11).However, some differences between the reconstructions obtained with the Bayesianapproach and time reversal can be seen, especially in the case of the mouse head. Inaddition, the Bayesian approach seems to be able to detect structures deeper thanthe time reversal which is more evident in the case of the leaf phantom. Further-more, the quality of the image reconstructed from the orthogonal sensor appears tobe superior to that of the image reconstructed from the planar sensor since it givesa more complete reconstruction of the structure of the imaged target (Figure 4.10).However, the structures that are close to the sensor appear sharp with both sensors.

4.2.2 LED-based PAT setup

In the LED-based PAT measurements, an imaged object was illuminated with anLED operating at the wavelength of 617 nm. PA signals were acquired using a cir-cular piston ultrasound transducer that was rotated around the target. Differentlimited view and sparse angle measurement situations were considered. The imag-ing targets were made of plastic microcapillary tubes that were filled with an Indianink solution, and they were coupled to the sensor using degassed deionized water.

The sensors were modeled and the sensor response was included in the forwardoperator K(c). The image reconstruction was performed by solving the system ofequations (3.13). The computations were conducted in a 14.6 × 8.2 cm domain thatwas discretised into 730× 410 square pixels. The Ornstein-Uhlenbeck prior with themean ηp0 = 0.015, standard deviation σp0 = 0.005 and characteristic length scale

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Figure 4.12: A schematic picture of the measurement setup and the reconstructedphotoacoustic images from sparse angle measurements of three ink (0.1%) filledtubes when the direction of the tube pattern was in at approximately 45◦ anglerelative to the light source. From left to right and from top to bottom the separationbetween the measurement angles and the number of the measurement angles inbrakects are 1◦ (185), 2◦ (93), 5◦ (37), 10◦ (19), 20◦ (10), and 30◦ (7).

l = 0.85 mm was used as the prior model. In addition, the mean and standarddeviation of the measurement noise was calculated from a time frame preceding themeasured PA signals at each sensor position. The reconstructions were computedusing the speed of sound that corresponded the temperature of water.

The experiments with the LED-based setup show that the LED-based instrumen-tation can also be utilised in limited view and sparse angle measurement setups. Inaddition, the experiments confirm that the images reconstructed with the Bayesianapproach present the features of the imaged target (Figure 4.12). Furthermore, theresults show that the Bayesian approach enables a significant reduction of measure-ment angles without compromising the quality of the reconstructions. That is, thenumber of measurement angles can be reduced from 185 to 37 without losing thequality of the image. In fact, the tubes can be visually distinguished from the arte-facts even when only 19 measurement angles are used.

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4.3 COMPENSATION OF MODELLING ERRORS

The effect and compensation of modelling errors in the speed of sound in the so-lution of the inverse problem were studied with 2D simulations in various sensorgeometries. The modelling error was generated by modelling a heterogeneous speedof sound as constant or piece-wise constant in the inverse problem. In addition, thiserror was compensated by solving the inverse problem utilising the Bayesian ap-proximation error approach. As a reference, the solution using the accurate speedof sound distribution was computed. In each case the whole posterior density wasinspected and relative errors of the estimates were calculated. In addition, marginaldensities were computed.

In the data simulation, a square domain with a side length of 10 mm and dis-cretisation of 300 × 300 square pixels (pixel side length of 33 µm) was used. Theinitial pressure and speed of sound distribution that were employed to generatemeasurement data are shown in Figure 4.13. Sensors were arranged around the do-main (full view), on two adjacent sides of the domain (two side) or on one side ofthe domain (one side). The measurement data was simulated using the k-space timedomain method using 4001 time steps at a temporal sampling rate of 333 MHz. Themeasured signal was downsampled to 66 MHz and a 1% Gaussian noise was addedto it.

In the inverse problem, the domain was discretised into 200 × 200 square pixels(pixel side length of 50 µm). The posterior density was solved using the Ornstein-Uhlenbeck prior with the mean ηp0 = 5, standard deviation σp0 = 5 and characteris-tic length scale l = 2 mm and the correctly modelled noise statistics. If the accuratespeed of sound was used (AFM), the mean and covariance of the posterior densitywas computed using (3.5) and (3.6), respectively. In the case of the inaccurate speedof sound, the posterior density was solved using (3.20) and (3.21). If the modellingerrors were ignored (IFM) i.e. ε = 0, then ηn = ηe and Γn = Γe were used in thecomputations. In the case of error modelling (IFM&EM), an approximation of mod-elling error (mean and covariance) were computed using (3.22), (3.23) and (3.24),respectively, based on 10000 samples that were drawn from the teaching distribu-tions of the initial pressure and speed of sound. Furthermore, marginal densitieswere calculated in two locations (a square and circle in Figure 4.13) using (3.9).

The results show that the inverse problem of PAT is sensitive to errors arisingfrom uncertainties in the speed of sound (Figures 4.14 and 4.15). The sensitivityincreases as the inverse problem becomes more ill-posed i.e. sensor geometry turnsmore limited view. The inaccurate modelling of the speed of sound yields the inac-curate posterior densities. That is, the numerical values and shapes of inclusions inthe estimated posterior mean were reconstructed inadequately and the correspond-ing uncertainty estimates were not feasible meaning that the posterior uncertaintywas underestimated. In addition, the relative errors increase significantly (Table4.3). However, modelling of the errors improved the solution of the inverse prob-lem. Artefacts in the mean of the posterior density are reduced and the posteriordensity is widen such that the corresponding uncertainty estimates are meaningful.In addition, the relative errors decrease significantly. However, the accuracy of thesolution is not in the same level as in the case of the true speed of sound is used.In addition, the results of the soft tissue mimicking simulations in publication IVshow that the Bayesian approximation error modelling is able to compensate onlyfor small speed of sound variations that can be expected in practice since the speedof sound is usually relatively well known.

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Figure 4.13: The simulated (true) initial pressure distribution (left image) and thespeed of sound distribution (right image). The initial pressure distribution is givenin arbitrary units and the speed of sound distribution is given in units of m/s.The square and circle in the left image indicate the locations where the marginaldensities are plotted.

Table 4.3: The relative errors Ep0 (%) of the estimated mean of the posterior ob-tained using AFM, IFM and IFM&EM in the full view, two side and one side sensorgeometries.

AFM IFM IFM&EMFull view 4 31 12Two side 7 60 22One side 15 129 33

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Figure 4.14: The posterior mean (top block) and standard deviation (bottom block)obtained using AFM (first row), IFM (second row) and IFM&EM (third row). Thecolumns from left to right represent the full view (first column), two side (secondcolumn) and one side (third column) sensor geometries. The red dots in the firstrow images indicate the locations of the sensors.

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Figure 4.15: The marginal probability densities of the posterior densities at loca-tions denoted by a square (first row) and circle (second row) in Figure 4.13. Thecolumns presents results obtained using the full view (first column), two side (sec-ond column), and one side (third column) sensor geometry. Shown in the graphsare the true initial pressure (vertical black line) together with the marginal densitiesof the posterior density obtained using AFM (blue solid line), IFM (red dotted line),and IFM&EM ( black dashed line). Note: The true value does not lie within theprincipal support of the IFM distribution (second and third column images in thesecond row), which illustrates the infeasibility of the posterior uncertainty in thecase of modelling errors.

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5 Discussion and conclusion

In this thesis, the Bayesian approach to the inverse problem of PAT was employed.First, a solution method to the inverse problem of PAT based on the Bayesian frame-work was developed. The approach was evaluated with numerical simulations, andfurther validated using experimental data. Secondly, a method to compensate foruncertainties in the speed of sound in PAT utilising the Bayesian approximationerror approach was described and studied with numerical simulations.

One of the attractions of the Bayesian approach is the quantitative informationthat it provides. That is, the method can be used to provide a probability distribu-tion with the mean and standard deviation of the initial pressure in each elementof the domain. Thus, the uncertainty of the estimates, i.e. the reliability of the re-constructed images, can be assessed. Since the posterior density is based on themeasurements, model, and prior information, uncertainties in these will also in-fluence the uncertainty estimates. As it was shown in the thesis, uncertainties inmodelling can result in misleading uncertainty estimates. Thus, more research isrequired for interpretation of when uncertainty estimates can be regarded safe, seee.g. [200]. In the thesis, the uncertainty estimates are based on the standard devi-ations of the posterior density. In the future, an utilisation of the whole posteriorcovariance in the computation of the uncertainty estimates could be studied.

Other attractions of the Bayesian approach is an opportunity to incorporate priorinformation to estimation of the initial pressure. The relevance of proper prior in-formation increases as the inverse problem becomes more ill-posed. In the future,utilising other prior models could be studied to better incorporate, for example,anatomical information. Examples of other priors include, for example, total varia-tion, structural, anatomical and sample based prior. In fact, including total variationand structural prior information in regularization have been found to provide goodphotoacoustic images [45, 101, 104].

The results of this thesis show that by utilising the Bayesian approach, the solu-tion of the inverse problem can also be obtained when the problem is ill-posed i.e.in limited view and sparse angle measurement geometries. Thus, the approach issuitable for practical applications. In addition, the suitability of the approach wasfurther improved by compensating modelling errors using the approximation errormodelling. However, in this thesis, only modelling errors caused by the uncertaintiesand inaccuracies in the speed of sound were taken into account and uncertaintiesin other acoustical parameters were not studied. Similar procedure as describedin publication IV could be used to compensate errors caused by uncertainties inother acoustic parameters as well. Despite the promising simulation results, theexperimental verification of the approximation error approach is still required.

The Bayesian approach can be computationally expensive, since forming of largematrices or solving of a large system of equations is required. However, it couldbe possible to utilise a model reduction (for example a coarse discretisation or trun-cated expansions), for example, using the Bayesian approximation error modellingto decrease the memory requirements and speed up the computations. In addition,the utilisation of efficient optimisation algorithms could speed up solving the linear

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systems. In the approximation error modelling, the computation of the samples ofthe approximation error is time consuming, but it needs to be carried out only onceand it can be done offline (before the computation of the posterior density). On theother hand, sampling could be speeded up by utilising parallel computing. Fur-thermore, the possibility of utilising machine learning in the approximation errorapproach could be studied.

Other tomographic imaging modalities could take an advantage of the developedmethods. One of these modalities could be thermoacoustic tomography (TAT) thatis closely related to PAT [29, 34]. In TAT, an initial pressure distribution is inducedby an excitation of microwave radiation. Since the acoustic process is the same inPAT and TAT, they are nearly identical from a mathematical perspective. Thus, themethods of this thesis could be applied in TAT with relatively small modifications.Another modality that could benefit from the developed methods is QPAT, whereconcentrations of light absorbing molecules are estimated from PAT images [201].In QPAT, it is essential that PAT images are quantitative and accurate, and the meth-ods developed in this thesis can address to these requirements. It has been shownthat, an incorporation of the uncertainty information of the initial pressure into theinverse problem of QPAT improves the accuracy of the solution [188].

In conclusion, the research presented in this thesis develops computational meth-ods to the inverse problem of PAT. The developed methods not only improve thequality of the PAT images but also give the reliability of these images. In addition,the Bayesian approach enables incorporating statistical models for model errors andapproximations.

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uef.fi

PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND

Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-3268-6ISSN 1798-5668

Dissertations in Forestry and Natural Sciences

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IMAGE RECONSTRUCTION AND MODELLING OF UNCERTAINTIES IN PHOTOACOUSTIC TOMOGRAPHY

PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLAND

Photoacoustic tomography is a biomedical imaging modality, where an image of an

initial pressure distribution is formed as an inverse problem based on ultrasonic boundary

measurements. This thesis introduces new computational methods to solve the inverse

problem using Bayesian formalism. The new methods provide quantitative information of

the estimated initial pressure, its uncertainties, and improve the accuracy of the estimates by modelling uncertainties in the speed of sound.

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