Mammography Tomosynthesis Using a Coupled Source And

MAMMOGRAPHY TOMOSYNTHESIS USING A COUPLED SOURCE AND

DETECTOR IN A C-ARM CONFIGURATION

JOSEPH TOBIAS RAKOWSKI

MEDICAL COLLEGE OF OHIO

2004

ACKNOWLEDGEMENTS

I want to express my gratitude to my advisor, Michael J. Dennis, for his guidance; to Patti

McCann and Richard Lane in the Department of Anatomy for providing the specimens; to Diane Ammons for editing my dissertation; to the members of my dissertation committee; and to the Medical College for providing this opportunity. I dedicate this to my family who taught me the value of education, and to my loving and patient wife,

Linda, and my super son, Joseph Aaron.

TABLE OF CONTENTS

Acknowledgements ii

Table of Contents iii

Introduction 1

Literature 6

Materials 8

Methods 10

Results 32

Discussion 99

Summary 109

Conclusions 111

Bibliography 112

Appendix A 120

Appendix B 133

Abstract 145

iii

INTRODUCTION

Mammography is, by far, the best diagnostic tool for detecting early stage breast cancer (Baker, 1982). Tabar et al. (1987) demonstrated the importance of early detection in saving lives (Tabar, 1987). However, despite the technological and quality improvements in recent years, 10 - 30% of breast cancers are not detected, while other cancers are not detected early enough to allow effective treatment (Bassett et al., 1987;

Baines et al., 1986; Haug et al., 1987). The primary reason for missed diagnosis is that the cancer is often obscured by fibroglandular breast tissue that is radiographically dense

(Holland et al., 1982, 1983; Martin et al., 1979; Feig et al., 1977; Ma et al.,1992; Jackson et al., 1993; Bird et al., 1992; Mandelson et al., 2000; White, 2000; Boyd et al., 1998;

Rosenberg et al., 1998; van Gils et al., 1998).

A tool that could potentially prove valuable at improving the detection of early stage breast cancer, especially in radiographically dense breasts, is tomosynthesis.

Tomosynthesis is the process of reconstructing planes of interest at any level in an object from limited angle projection data in a manner similar to conventional focal plane tomography. Like conventional tomography, tomosynthesis allows the radiologist to focus on a selected image plane rather than a conventional two-dimensional (2D) static projection of all overlying tissue. Unlike conventional tomography, digital tomosynthesis makes possible two further enhancements: 1) partial removal of overlying structures that lie outside the plane of interest, and; 2) selection of any plane through the breast using a single set of projections, with only a small increase in dose over a conventional film/screen mammogram.

1 Tomosynthesis reconstruction differs from the reconstruction techniques of the widespread modality of axial computed tomography (CT) in several ways. First, the primary CT image plane is defined in the source-detector plane, while the tomosynthesis image plane is perpendicular to the source-detector plane. Second, CT uses a much greater number of projections, typically from 800 to 1500 to produce an image, and typically samples through an arc of at least 180 degrees. The tomosynthesis arc is limited to about 40 degrees, with the greatest number of projections in the literature at 64 (Sone et al., 1996), thereby producing an incomplete data set from which to reconstruct the subject. The data set is incomplete in that there are not enough data points to solve for

the attenuation coefficients of each subject voxel along the beam rays. The approximate

number of projections needed to produce a complete data set is equal to the

circumference of the scanned subject divided by the size of the smallest object one

desires to image. For example, a 10 cm diameter breast at 0.2 mm resolution would

require 1,571 projections. Third, the reconstruction method in modern commercial CT

scanners is filtered backprojection typically using the Fast Fourier Transform (FFT), an

approach that has not been applied to tomosynthesis. The methods better suited for tomosynthesis are Algebraic Reconstruction Technique (ART) and pixel shifting with iterative summation/subtraction techniques, both of which are used in this project. The

algebraic/iterative techniques produce higher quality images than the filtered

backprojection technique when working with incomplete sets of data, or when there is a

large statistical uncertainty from a relatively small number of photons contributing to the

projection images. Fourier based reconstruction methods improve contrast, i.e., soft tissue

discrimination, but generate significant reconstruction artifacts when used with a limited

2 number of projections over a limited angular range. The ART approach was used in the early scanners; However, because of its computational complexity, it was replaced by the filtered backprojection technique. Fourth, CT reconstruction creates a cross sectional map of the attenuation coefficients of the voxels in the subject tissue, which are translated proportionately to Hounsfield numbers for display purposes, scaled relative to the attenuation of water. This characteristic creates tremendous contrast relative to radiographic imaging, allowing discrimination of soft tissue. Conversely, radiography displays only the amount of radiation transmitted through the subject along each ray from the source to detector. Tomosynthesis, however, can improve contrast by focusing on individual planes in a subject and partially removing interfering out-of-plane structures, commonly referred to as blur in conventional film tomography. Finally, the quality and independence of tomosynthesis planes are compromised by blurring artifacts produced from unregistered details located outside the plane of reconstruction.

Digital tomosynthesis methods will be implemented for use with isocentric stereotactic breast biopsy units with digital imaging capabilities. The methods used in this project provide for planar reconstruction orthogonal to the source-receptor axis, as well as tilted relative to that axis. The images are collected on a Lorad stereotactic prone breast biopsy unit. The imaging system comprises an x-ray source and image receptor coupled in a c-arm configuration. The image receptor comprises a scintillation screen that is lens-coupled to a Charged-Coupled Device (CCD) array with a 56 mm x 56 mm effective field of view. The CCD array size selections are 512 x 512 or 1024 x 1024.

The general objective of this project was to quantify the imaging performance of the tomosynthesis process performed on an isocentric digital stereotactic digital breast

3 imaging system (Lorad Stereoguide with DSM digital image receptor). The image quality will be quantified in terms of the line spread function, the Modulation Transfer

Function (MTF), the appearance of the ACR digital mammography phantom, and the

Signal to Noise Ratio (SNR), Contrast to Noise Ratio (CNR) and Signal to Background

Ratio (SBR) of a low contrast detectability tool developed for this project.

Specific objectives were to:

1. Design C++ code to implement the tomosynthesis reconstruction

methodology for the C-arm configuration.

2. Design C++ code to implement the tomosynthesis plane subtraction

methodology for the c-arm configuration. The goal of plane

subtraction is to remove the blur artifacts produced by structures

positioned outside the reconstructed plane.

a. Demonstrate a linear technique.

b. Demonstrate a logarithmic technique.

c. Demonstrate an iterative multiplane technique.

d. Evaluate the effectiveness of each technique using the test

phantoms and cadaver breast specimen.

3. Design C++ code to implement the iterative reconstruction

methodology for the C-arm configuration, similar to ART.

a. Evaluate the effectiveness of the ART using the test phantoms

and cadaver breast specimen.

4. Evaluate the effect of reconstruction on system resolution through the

Line Spread Function (LSF) and MTF using an edge phantom.

4 a. Compare the results from objective 4 with performance of the

zero angle single projection image.

5. Develop a low contrast detectability phantom.

a. Evaluate the effect of iterative and non-iterative reconstruction

on Contrast, CNR, SNR, and SBR without structured noise.

b. Modify the low contrast phantom with in-plane and out-of-

plane structures designed to mimic fibroglandular breast tissue,

i. e., structured noise.

c. Evaluate the effect of iterative and non-iterative reconstruction

with structured noise subtraction on Contrast, CNR, SNR, and

SBR.

d. Compare the results from objectives 5a. and c. with the

performance of the zero angle single projection image.

6. Image a cadaver breast specimen. Perform both iterative and non-

iterative reconstruction. Compare results with a single projection

image.

7. Develop and image a step phantom to measure the effective slice

thickness of a tomosynthesis plane.

5 LITERATURE

Niklason et al. (1997), at Massachusetts General Hospital, using a full field digital

General Electric (GE) mammographic system with a stationary detector geometry, demonstrated that tomosynthesis may improve the specificity of mammography with improved lesion margin visibility and may improve early breast cancer detection, especially in women with radiographically dense breasts. Webber, et al. (2000), at Wake

Forest University in Winston-Salem, North Carolina, have developed an extension of tomosynthesis in their method of Three Dimensional Mammography Using Tuned

Aperture Computed Tomography (TACT™). Webber et al. (1997) first presented their method in an earlier publication on dento-alveolar imaging. The TACT™ method produces a three-dimensional image of a limited slab by layering the 2D data. Webber, et al. (2000) conducted a factorially balanced multivariate evaluation of the detection accuracy of TACT™ with that of conventional 2D digital spot mammograms. The

TACT™ reconstruction method yielded significantly higher detection scores for all tasks than did 2D digital spot mammograms. Instrumentarium Corporation has applied the

TACT™ method to small field mammographic spot imaging and stereotactic biopsy.

Both the GE and Instrumentarium systems use stationary detector/moving source geometry, unlike the c-arm configuration that was used in this project.

Others have investigated the viability of tomosynthesis in mammography

(Suryanarnyannan et al., 1999, 2000, 2001; Webber et al., 1999; Webber and Messura,

1999; Wu et al., 2003), dentistry (Webber, et.al., 1997, 1999; Yamamoto et al., 1998,

1999, 2000; Ruttimann et al., 1984, 1989) orthopedics (Duryea et al., 2003; Fahey et al.,

6 2003), radiation therapy simulators and isocentric fluoroscopy (Kolitsi et al., 1992, 1993,

1996; Messaris et al., 1999; Zwicker and Atari, 1997), linear tomography (Sone et al.,

1991; Takahashi et al., 1990), chest imaging (Sone et al., 1993, 1996; Godfrey et al.,

2001), angiography (Badea et al., 1998, 2001; Kruger et al., 1984; deVries et al., 1985;

Haaker et al., 1985; Maravilla et al., 1984; Woelke et al., 1982) and the inner ear

(Chakraborty et al., 1984). There is potential for widespread use of tomosynthesis in many radiographic exams with the growth in use of direct digital image receptors.

7 MATERIALS

Images were produced on a Lorad Stereoguide stereotactic breast biopsy system that uses a CsI scintillating screen lens-coupled to a CCD digital camera, with a 56 mm x

56 mm effective image area, and 512 x 512 or 1024 x 1024 pixel matrix at 16 bits per pixel. This produces pixel pitches of 0.11 mm and 0.055 mm, respectively. The Lorad system is a C-arm configuration in which the detector is always diametrically opposite the source. Source to detector distance is 880 mm. Source to isocenter distance is 663 mm. Focal spot size is 0.3 mm. Filtration is 0.03 mm molybdenum. Images were transferred to a desktop PC for tomosynthesis reconstruction. Reconstruction algorithms were written in the C++ programming language and run on an ordinary 1.6 GHz desktop

PC. Images were viewed and analyzed on a desktop PC using Scion Image, freeware published by Scion Corporation. This is the PC version of Mac-based NIH Image, a freeware program from the National Institutes of Health. Test objects were the following:

1. Nuclear Associates Digital Stereotactic Mammography Evaluation Phantom;

2. 0.38 mm lead foil to generate the edge spread function;

3. Low contrast detectability phantom, developed for this project, consisting of

water filled cylindrical cavities in 70/30 BR12 (70% glandular, 30% adipose)

breast equivalent material, 4 cm thick, 0.989 g/cm3. The cylinder diameters in

inches, according to the drill bit manufacturer, are 0.3906, 0.3480, 0.3125,

0.2720, 0.2344, 0.1960, 0.1570, 0.1160, 0.0785, and 0.0400. These diameters

in millimeters are 9.92, 8.84, 7.94, 6.91, 5.95, 4.98, 3.99, 2.95, 1.99, and 1.02.

8 Only the five smallest diameters were imaged. Type A very fine

Scotchbrite™, a ¼ inch thick plastic mesh, to simulate overlying and

underlying tissue (out-of-plane structures);

4. A preliminary phantom used in the early stages of algorithm development,

made of a lead Beekley Spot Marker and Playdough overlapping the Digital

Phantom, to demonstrate artifacts created by out-of-plane structures, and their

removal from the image plane.

5. Phantom to further examine the effectiveness of the reconstruction schemes at

removing out-of-plane structures. The phantom consists of 6.5 mm thick

interchangeable layers of wax impregnated with chalk dust to simulate

calcifications, and cotton string to simulate fibers. Clear wax can be inserted

between the structure layers for any desired thickness.

6. Rose model low contrast detectability phantom constructed for this project

consisting of cylindrical voids in plastic, aligned in a single plane,

perpendicular to the zero angle projection axis, parallel with image

reconstruction plane. The void diameters in mm are 6.35, 4.76, 3.18, and

1.59. The depths in mm are 5.0, 2.50, 1.34, and 0.77. Total depth of phantom

is 62 mm.

7. Cadaver breast specimen, approximately 4 cm thick.

8. Step phantom to examine slice thickness. Comprises staggered 2.4 mm

diameter holes in 1 mm layers of plastic.

9 METHODS

Our non-iterative image reconstruction algorithm is based on the work of Kolitsi,

et al. (1992, 1993). Their digital tomosynthesis multiple projection method was

developed for use with isocentric fluoroscopy units. It is capable of performing digital

tomosynthesis of anatomical planes of user selected orientation and distance from the

isocenter. Their method of reconstruction involves shifting the locations of pixels in each

projected image so that they are aligned with an abstract image formation plane. The

image formation plane represents a flat image receptor appropriately coupled in motion to

the x-ray source such that integration of the signal at the receptor over an acquisition arc would yield a tomographic image of a selected plane in the scanned object. The shifted

images from all projections then are summed to produce the final reconstructed image.

The image acquisition and reconstruction geometry is shown schematically in Figure 1.

Figure 1 - Geometric Configuration for the Acquisition and Reconstruction of Tomosynthesis Data.

11 The subscript “i” denotes the column number in the acquired image matrix. The subscript “r” denotes the position on the reconstructed image matrix to which the corresponding matrix column “i” is moved.

S = position of source at zero angle

S′ = position of source at angle ω

L = distance from isocenter to anatomic structure

B = source to isocenter distance

D = source to image receptor distance

IA = horizontal plane through isocenter

IiAi = acquired image of IA on image receptor

IhAh = projection of IiAi onto horizontal plane passing through the intersection of the beam’s central axis and the detector plane. This is the image formation plane for an anatomic structure positioned at isocenter. Addition of these projections over all sampling

angles will yield the reconstructed image of IA.

QP = structure lying on a horizontal plane at a distance L from isocenter

QiPi = acquired image of QP on image receptor

QhPh = projection of QiPi onto the horizontal plane passing through the intersection of the

beam’s central axis and the detector plane

QrPr = size modified, focused image of QP on its image formation plane. Addition of

these projections over all sampling angles will yield the reconstructed image of QP.

For horizontal planes lying away from isocenter, the image formation plane is a

horizontal plane suspended from point Qr. During rotation and acquisition, the distance

IiQr = Ld/b remains constant for a given L. The loci of the suspension point Qr therefore

12 lies on a circle of radius (d-b), but with the center displaced by an amount equal to IiQr from isocenter.

The reconstruction of a horizontal plane uses the following equations for column shifts in the matrix: h = id/(d cos ω - i sin ω) r = (h – (Ld sin ω)/(b cos ω - L)) (1 – L/(b cos ω))

Planes that are tilted with respect to the x axis, to form an angle α to the vertical, can be regarded as horizontal, if the coordinate system is rotated by an angle α-90°

around the y axis. In that case, the above expressions become the following:

h = id/(d sin (ω+α) + i cos (ω+α))

r = (h + (Ld sin α cos (ω+α))/(b sin (ω+α) – L sin α))*

(1 – ((L sin α)/(b sin (ω+α))).

A plane that is additionally tilted with respect to plane of rotation, i.e., the y axis,

presents a situation where each element in the plane corresponds to a different distance L

from the isocenter. The new tilt relative to the plane of rotation will be described by the

angle β with the vertical (z). Now rows must be shifted in addition to columns.

Let “j” denote the position of a given row in the acquired image matrix. Let “k”

denote the projected position of “j” on the plane through Ii and parallel to the plane of

reconstruction. Each row is slid along the x axis by an amount ∆h defined as follows

∆h = (Lkd sin ω)/(b cos (ω - Lk))

where

Lk = ((b – L)k cos β)/d

13 and

K = jd/(j cos β + d sin β).

And finally, when planes outside the isocenter are reconstructed, the tomosynthesized image must be appropriately magnified in the direction perpendicular to the plane of rotation by shifting rows by the factor

(b – L)/b.

Our method used a small modification to eliminate vertical bands, or aliasing artifacts that occurred with direct application of the Kolitsi, et al. (1992) method to our imaging system. Their original method shifts groups of projection image pixels (columns denoted by subscript “i”) to locations in the image formation plane (positions denoted by subscript “r”), which almost consistently lie between predefined pixel locations in the reconstructed image. The shifted pixels then must be distributed with weighting to the nearest two adjacent final reconstructed image pixel locations. Our modification reversed this process in that we started with the image formation plane pixel location and selected pixel values from the projection image data for weighting and summation in the target image formation plane pixel, thus assuring that there would be no gaps in the image formation plane.

We explored ways of removing unwanted structured noise from the plane of interest through linear and logarithmic subtraction. We used the Kolitsi, et al. (1993) removal technique. This involved selecting the plane that contained objects that appear as structured noise in the targeted plane of interest. This structured noise is referred to as blur in conventional film tomography. The structured noise plane is reconstructed in the

same manner as the target plane. The noise plane then is projected onto the target plane

14 for all acquisition angles, synthesized as it would appear in the target plane, and then subtracted from the target plane image, with weighting factors applied. Selection of the appropriate weighting factor is a crucial step in the subtraction process. The subtraction has been done both linearly and logarithmically.

Logarithmic subtraction eliminates negative pixel values in the subtracted image and reduces the number of gray scale levels needed to view the image, but also reduces the gray scale fidelity. Also, the pixel values resulting from a logarithmic versus linear subtraction more closely approximate the radiologic properties of the subject tissue since the logarithm of a pixel value is an approximation of the product µx, µ being the linear attenuation coefficient for the tissue traversed along distance x for a given energy. It is an approximation because the linear attenuation coefficient depends on the x-ray energy and tissue density, neither of which is discreet. The x-ray energy comprises a spectrum that is characteristic of the Molybdenum target and Molybdenum filter combination. The tissue densities vary throughout the breast. This subtraction technique is employed in digital subtraction angiography in which a pre-contrast mask is logarithmically subtracted from an image of the identical region with contrast, leaving an image of the vessel with contrast, and void of all other visible tissues.

We extended this subtraction scheme to an iterative subtraction process. A predetermined number of N evenly spaced image planes through the subject volume will be reconstructed and stored. A second set of N image planes then will be created by weighted subtraction of N-1 planes from each individual plane from the previous set, i.e., given N = 5, planes 2 through 5 will be subtracted from 1 to produce 1′, planes 1 and 3

through 5 will be subtracted from 2 to produce 2′, etc., until the full set of 1′ through 5′ is

15 produced. This will be repeated with the primed arrays to produce a third set (double primed), i.e., subtracting primed from primed, and so on until a satisfactory outcome is reached. We are investigating both linear and logarithmic subtraction in this iterative

process. Again, selection of appropriate weighting factors is crucial.

The iterative reconstruction technique is outlined in the following pages,

accompanied by algorithm flow charts in Figures 2 and 3. The computer code is in

Appendix A.

16 Figure 2: Non-iterative Algorithm Flow

Tomosynthesis Core Structured Noise Subtraction

retrieval of acquired projection selection matrix of “noisy” plane

w = 1, n projection on image formation

plane derive retrieval of projection on reconstructed plane of image matrix interest

shifting

and magnification w = 1, n

summing of planes q is the number of “noisy” planes w = 1, n selected for removal.

n is the number of projection data sets. storage of reconstructed

image matrix

subtract if mask continue if I = 1, q is ready subtracting

17 Figure 3: Iterative Algorithm Flow

Tomosynthesis Core

retrieval of acquired projection matrix

w = 1, n projection on image formation plane

shifting and magnification

summing of planes w = 1, n

storage of reconstructed image matrix

Figure 3 continuation: Iterative Algorithm Flow - Iterative Subtraction Portion

select Loop from 1 to N reconstructed target plane i

select reconstructed plane to be

subtracted repeat for designated number of iterations

project onto target plane for each projection angle

N-1 loops

sum projections and subtract from target plane with appropriate weighting

update reconstructed N planes image array

output final image array

19 Iterative Linear Subtraction Technique

The process of the linear subtraction method can be described as follows:

The initial plane construction from raw projections can be represented by

P pi = Sij ∑ n 0 j j=1

where pi is a pixel value in a reconstructed plane,

i is the plane number,

Sij is the shift operator that projects plane j to plane i,

S′ti is the shift operator that projects plane i to target plane t,

n0j is the pixel value contributed from projection j, j is the projection number,

W is the user-defined weighting factor,

I is the number of iterations.

The first iteration reconstruction can be represented by

1 N ′ ′ pi = p t − ∑Sti pi PNWI i=1,i≠t

The second iteration reconstruction can be represented by

1 N ′′ ′ ′ ′ pi = p t − ∑Sti pi PNWI i=1,i≠t and so on.

Iterative Logarithmic Subtraction Model

The process of the logarithmic subtraction method can be described as follows:

The initial plane construction from raw projections can be represented again by

20 P pi = Sij ∑ n 0 j j=1

The first iteration reconstruction can be represented by

⎡ 1 N ⎞⎤ ′ ⎛ ′ ⎟ pt = exp⎢ln(p t ) − ln⎜ ∑Sti pi ⎟⎥ ⎣⎢ ⎝ PNWI i=1,i≠t ⎠⎦⎥

The second iteration reconstruction can be represented by

⎡ 1 N ⎞⎤ ′′ ′ ⎛ ′ ′ ⎟ pt = exp⎢ln(p t ) − ln⎜ ∑Sti pi ⎟⎥ ⎣⎢ ⎝ PNWI i=1,i≠t ⎠⎦⎥ and so on.

Algebraic Reconstruction Technique

The ART uses the following method, which takes advantage of the Kolitsi pixel shifting scheme for backward and forward projection:

1. Create an image data set at all levels of interest in the subject.

2. Forward project the image data to the detector plane for each angle of acquisition.

3. Determine the error function for that angles projection data set.

4. Backproject that angle’s error function to all image pixel locations.

5. Repeat for the next angle.

6. Repeat for the next iteration.

The following defines the ART terms and their relationships.

Image pixel I(i,j) is located at (x,y,k) in level “k” of the subject.

Projection values are the measured data, after normalization and logarithmic scaling.

Projection value p(i′,j′) at angle “φ” is located at p(x′,y′,φ).

21 Note that (x′,y′) indicates a specific location which is likely not to align with the detector element centers. Estimate of a functional value at location (x′,y′) is be done by interpolation.

G is the backprojection mapping function that will associate a particular image pixel location, I(i,j,k) with a measurement plane value viewed at angle φ, pφ(x′,y′,φ).

I(i,j,k,φ) = G{pφ(x′,y′,φ),k} (interpolate in data plane)

Also, H is the forward projection mapping function that will associate a particular data plane pixel location pφ(i′,j′,φ) with an image plane location and value, I(x,y,k).

pφ(i′,j′,φ) = H{Ik(x,y,k),φ} (interpolate in the image plane)

Then, in order to iteratively reconstruct a set of image planes from a set of projection data sets:

1) Create an initial estimate for all I(i,j,k)

a) consider uniform density object, or

b) consider the zero degree projection data divided by the number

of image levels

2) Forward project to determine an estimated projection value, pφ′ (i′,j′,φ), for each

detector element location.

pφ′ (i′,j′,φ) = ∑ H{Ik(x,y,k),φ} summed over all levels, k.

3) Determine the error map, ε(i’,j’,φ) for all detector element locations,

εφ(i′,j′,φ) = pφ, measured(i′,j′,φ) – p′φ,estimated(i′,j′,φ)

4) Backproject error correction (with weighting and damping factors) into each of

the imaging planes and add to current image estimate value

22 Ik,(n+1)(i,j,k) = Ik,n(i,j,k) + (D/nk) G{εφ(x′,y′,φ),k}

th where n is the n iteration and nk is the number of image plane levels being

reconstructed, and D is a damping factor (which may vary depending on the value

of ∑ε).

Repeat for the next projection angle. Repeat for the next iteration. Evaluate ∑ε.

Done? If not, then go to 2). Note that the G and H mapping functions make use

of the Kolitsi pixel shifting scheme.

The process is described further through the algorithm flowchart in Figure 4. The

computer code is in Appendix B.

23 Figure 4: Algebraic Reconstruction Technique (ART) Algorithm Flow

Estimate initial Ik(I,j,k) at all levels k, with a uniform array or zero angle

projection array

For all levels k, call forward projection mapping function forward(), represented here by H. Sum the output of each application of H in the estimated projection data array

p′ (i′,j′,φ) = H{I (x,y,k),φ}. φ,estimated k

Determine the error map

ε (i′,j′,φ) = p (i′,j′,φ) – p′ (i′,j′,φ) φ φ, measured φ,estimated

Call back projection mapping function back(), represented here by G. Sum the output of each application of G into the image data array Ik,(n+1)(i,j,k) = Ik,n(i,j,k) + (D/nk) G{εφ(x′,y′,φ),k}, Where D is a damping factor and nk is the number of image plane levels

Proceed with next All projections completed? projection angle data

Proceed with next All iterations completed? iteration

Generate output files and stop

For modeling purposes, the imaging chain can be considered a cascading

linear system, which is most easily analyzed in the frequency domain.

Diagrammatically this is shown in Figure 5.

Figure 5 - Diagram Representation of the Source and Detector in Image Formation.

S1 S2

Source Point object being radiographed Detector

The detector output can be represented by the following (Barrett and Swindell, equation

4.51 (1981)):

D(aρ′)H(aρ′) = (C/a2)D(aρ′)F(-bρ′)G(ρ′)

Where

D = the Fourier transform of d(r″), the transfer function of the detector

H = the Fourier transform of h(r″), the input signal to the detector system

F = the Fourier transform of f(r), the mean number of photons per unit area per unit time emitted into all space (f(r) will be assumed constant in time) 25 G = the Fourier transform of g(r′), the transmittance at the object plane r = the vector in the source plane r′ = the vector in the subject plane r″ = the vector in the detector plane

ρ = the spatial frequency vector conjugate to r

ρ′ = the spatial frequency vector conjugate to r′

ρ″ = the spatial frequency vector conjugate to r″

2 C = T/(4π(S1 + S2) )

T = the time over which the exposure is made a = S1/(S1 + S2) b = S2/(S1 + S2)

S1 = source plane to object plane distance

S2 = object plane to detector plane distance

S1 and S2 must be defined for the c-arm configuration. Referring to the schematic system drawing at page 5, we have the following:

2 2 ½ S1 = [(B - L(1 - cos ω)) + (L sin ω) ]

and

S2 = (D/cos Φ) – S1

where

-1 Φ = Sin ((L sin ω)/S1)

The overall transfer function of the system is given by the coefficient of G(ρ′):

2 TFtot = Ptot(ρ′) = (C/a )D(aρ′)F(-bρ′)

The overall Point Spread Function (PSF) in the space domain is given by

2 -1 PSFtot = ptot(r′) = (C/a )ƒ {D(aρ′)F(-bρ′)},

= pfs(r′) ** pdet(r′), where pfs(r′) is the PSF due to the focal spot alone,

2 -1 2 2 pfs(r′) = (C/a )ƒ {F(-bρ′)} = (C/(a b ))ƒ(-r′/b).

The detector contribution to the PSF is

-1 2 pdet(r′) = ƒ {D(aρ′)} = (1/a )d(r′/a).

Note that d, the inverse transform of D, will probably be a comb function. The focal spot distribution, represented by function f, may be estimated. When modeling the system’s edge response, G will assume a step function offset a distance L sin ω from the central axis. Initially we will assume an infinitely thin physical edge. The stationary frame of reference is the entire C-arm assembly, so any imaged object will be changing position as ω changes with each projection.

Images were made of the following test tools, phantoms, and specimen at the indicated techniques:

1. Nuclear Associates Digital Stereotactic Mammography Evaluation Phantom,

with artifacts created by out-of-plane Beekley spot marker and Playdough.

a. Single projection at zero angle, 28 kVp, 80 mA, 208 mAs.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 80 mA, 24 mAs per

projection.

2. Low contrast detectability phantom, developed for this project, consisting of

water filled cylindrical cavities in 70/30 BR12 (70% glandular, 30% adipose)

breast equivalent material, 4 cm thick. The cylinder diameters in millimeters

27 are 9.92, 8.84, 7.94, 6.91, 5.95, 4.98, 3.99, 2.95, 1.99, and 1.02. Only the five

smallest diameters were imaged.

a. Single projection at zero angle, 28 kVp, 224 mAs, with and without

Scotchbrite™.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 80 mA, 32 mAs per

projection, with Scotchbrite™.

3. Multilayer wax/chalk/string phantom

a. Zero angle single projection, ±15°, 28 kVp, 80 mA, 208 mAs.

b. Nine projection tomosynthesis. ±15°, 28 kVp, 80 mA, 24 mAs per

projection. Reconstruction via iterative linear subtraction, iterative log

subtraction, and ART.

4. Rose model low contrast detectability phantom constructed for this project

consisting of cylindrical voids in plastic, aligned in a single plane,

perpendicular to the zero angle projection axis, parallel with image

reconstruction plane. The void diameters in mm are 6.35, 4.76, 3.18, and

1.59. The depths in mm are 5.0, 2.50, 1.34, and 0.77. Total depth of

phantom is 62 mm. Structured noise was introduced as grains of rice

embedded uniformly in 2 cm thick epoxy slabs layered on both sides of the

low contrast phantom.

a. Single projection at zero angle, 28 kVp, 328 mAs, 80 mA, 512 x 512 field.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 32 mAs per projection, 80

mA, 512 x 512 field. Reconstruction via simple tomosynthesis, iterative

linear subtraction, iterative log subtraction, and ART. 28 c. Forty one projection tomosynthesis, ±15°, 28 kVp, 8 mAs per projection,

80 mA, 512 x 512 field. Reconstruction via simple tomosynthesis,

iterative linear subtraction, iterative log subtraction, and ART.

5. 0.38 mm lead foil to generate the edge spread function.

a. Single projection at zero angle, 28 kVp, 240 mAs.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 32 mAs per projection with

edge at isocenter, and 10 mm away from isocenter.

6. Cadaver breast specimen, approximately 4.0 cm thick. Since the minimum

available mAs is 8, 2.2 cm Acrylic was added to allow the use of 8 mAs per

projection. This maintains the recommended histogram value of 4K for the

single projection (8 x 41 = 328).

a. Single projection at zero angle, 28 kVp, 328 mAs, 80 mA, 512 x 512 field.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 32 mAs per projection, 80

mA, 512 x 512 field. Reconstruction via simple tomosynthesis, iterative

linear subtraction, iterative log subtraction, and ART.

c. Forty one projection tomosynthesis, ±15°, 28 kVp, 8 mAs per projection,

80 mA, 512 x 512 field. Reconstruction via iterative linear subtraction,

iterative log subtraction, and ART.

7. Step phantom to examine slice thickness. Consists of staggered 2.4 mm

diameter holes in 1 mm layers of plastic.

Results were grouped according to test object as presented in the preceding paragraph rather than by test category.

29 The single projection mAs values were selected to produce adequate pixel values per Lorad recommendations. In the later stages of testing with the specimen and Rose model phantom, the projection mAs values were based on the single projection mAs values, adjusted so that the total glandular dose from N projections was as close as possible to 100% of the single projection dose. This Lorad unit produces an average clinical glandular dose to a 4.2 cm breast of approximately 3 mGy for the 1024 x 1024 matrix, and approximately 1.5 mGy for the 512 x 512 matrix, using the Lorad recommended 160 mAs at 28 kVp for the 1024 x 1024 field, and 80 mAs at 28 kVp for the 512 x 512 field. Lorad recommendation regarding exposure is to deliver enough mAs to center the image histogram peak at 4000.

Low contrast performance was measured as SNR, CNR and SBR. Signal-to-

Noise Ratio is defined as (N1-N0)/σ0, where N1 is the mean pixel value in the Region of

Interest (ROI) within the faintest large diameter disk in the contrast phantom, N0 is the mean pixel value in the (ROI) in a background area adjacent to the disc, and σ0 is the standard deviation of pixel values in the background ROI. Throughout these results, σ0 includes structure noise that can obscure the object, not just photon statistics. Contrast- to-Noise Ratio is defined as 2(N1-N0)/(σ0(N1+N0)). Finally, SBR is defined as (N1-

N0)/N0.

The goal of tomosynthesis is to remove interfering structures from a target plane image, and to improve the contrast of target plane structures against a background of interfering out-of-plane structures. This equates to improving existing contrast, or creating contrast where there would otherwise be no measurable contrast in a single projection image. The ability of tomosynthesis to improve contrast suggests that 30 measured contrast should be the metric of choice for evaluating performance. However, because of the post processing capabilities available in digital imaging, as well as the variables associated with tomosynthesis such as number of projections, exposure per projection, number of reconstructed planes, weighting, etc., all of which affect noise in the final image, it makes sense to include noise in the evaluation. Therefore, in the absence of Contrast-Detail and ROI analysis, the CNR wins out over SNR and SBR as the more relevant descriptor of tomosynthesis imaging performance.

Spatial Resolution was evaluated using the Line Spread Function (LSF) Full

Width at Half Maximum (FWHM) and Modulation Transfer Function (MTF). The LSF is generated as the first derivative of the edge function. The MTF is generated as the

Fourier Transform of the first spatial derivative along a row of pixels spanning the edge.

The identical row was used in each image.

31 RESULTS

The brightness and contrast of all images in the document have been manipulated in order to maximize visibility.

American College of Radiology (ACR) Phantom

The images in Figures 6 through 10 were produced with the Nuclear Associates

Digital Stereotactic Mammography Evaluation Phantom, with artifacts created by out-of- plane Beekley spot marker and Playdough. A schematic of the phantom is shown below in Figure 6. The thickness and density of the fibers and specks vary, as do the masses, which is not obvious in the diagram. The imaging techniques were the following:

a. Single projection at zero angle, 28 kVp, 80 mA, 208 mAs.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 80 mA, 24 mAs per

projection.

The ACR Phantom images are presented in Figures 6 through 10. Image content does not come across well on paper. Also note that the left and right edges are obscured by the compression paddle when it is in the beam, as shown in Figures 8 through 10 and

13 through 18. The paddle was removed for all latter imaging. Nonetheless, we can note the following about the artifact demonstration. When nine discrete single projections are used, we do not see the traditional continuous blurring associated with film tomography, but instead we see a discrete, discontinuous "blur," especially for high density objects, as demonstrated by the out-of-plane lead marker. In Figures 9 and 10, the darkening of the brightness level of the high contrast lead marker, with moderate reduction in the visibility of the low-density Playdough, demonstrates the advantage in

32 applying the plane removal process for reducing interference from high contrast objects.

The best outcome in terms of phantom visibility through the obscuring structures is shown in Figure 14, the result of Log Subtraction.

Figure 6. Schematic of ACR Digital Mammography Phantom

Smallest Specks Masses

Largest Specks

Thickest Fiber

Thinnest Fiber

33 Figure 7. Artifact Demonstration, Single Projection Without Tomosynthesis or Filtering

Figure 8. Artifact Demonstration, Reconstruction of ACR Digital Stereotactic Phantom

34 Figure 9. Artifact Demonstration, Reconstruction of ACR Phantom and Linear Subtraction of Plane Containing Overlying Structures.

Figure 10. Artifact Demonstration, Reconstruction of ACR Phantom and Logarithmic Subtraction of Plane Containing Overlying Structures.

35 Low Contrast Detectability With BR-12 Phantom

Here the Low Contrast Detectability phantom was imaged. It comprises water filled cylindrical cavities in 70/30 BR12 (70% glandular, 30% adipose) breast equivalent material, 4 cm thick. The cylinder diameters in millimeters are 9.92, 8.84, 7.94, 6.91,

5.95, 4.98, 3.99, 2.95, 1.99, and 1.02. Only the five smallest diameters were imaged.

Exposure techniques were the following:

a. Single projection at zero angle, 28 kVp, 224 mAs, with and without

Scotchbrite™.

b. Nine projection tomosynthesis, ±15°, 28 kVp, 80 mA, 32 mAs per projection,

with Scotchbrite™.

We can describe the intrinsic contrast in the phantom in terms of density and thickness of the phantom material, i.e., g/cm2. Since the density of the BR12 material is

0.989 g/cm3, and the density of water is 1.000 g/cm3, we have the following:

Cylinder Diameter (mm) % Intrinsic Contrast

1 0.025%

2 0.050%

3 0.076%

4 0.101%

5 0.139%

36 In the following Table, Percent Contrast is calculated as (100)(N1 - N0)/N0. The

CNR is calculated as 2(N1-N0)/(σ0(N1+N0)). In both cases, N1 is the mean pixel value in the ROI within the target structure in the contrast phantom, N0 is the mean pixel value in the ROI in a background area adjacent to the target, and σ0 is the standard deviation of pixel values in the background ROI. Throughout these results, σ0 includes structure noise that can obscure the object, not just photon statistics. In the following Table, we are comparing the efficacy of the reconstruction techniques at improving the CNR relative to that of a single projection. These reconstructions were non-iterative subtractions of interfering structures. The brightness and contrast of all images in this document have been manipulated in order to show details.

Tomosynthesis improved the visibility of the phantom obscured by the overlying and underlying layers of Scotchbrite™, as shown in Table I. There was an overall improvement in CNR, and an improvement in Contrast for the reconstructions in which the obscurring layers were removed.

37 Table I - 4 mm Cylinder Low Contrast Detectability

% change in Contrast % change in CNR relative Image relative to single to single projection with projection with interfering interfering structures structures single projection with no interfering structures ------Figure 16 single projection with interfering structures ------Figure 17 9 projection simple tomo with interfering structures -34.6% 127.8% Figure 18 9 projection simple tomo with high pass filter -28.8% 115.6% Figure 19 9 projection tomo with 1 plane removed via non-iterative 428.8% 166.1% subtraction Figure 20 9 projection tomo with 1 plane removed via non-iterative 334.6% 130.4% subtraction and high pass filter applied Figure 21 9 projection tomo with 2 planes removed via non- 513.5% 278.3% iterative subtraction Figure 22 9 projection tomo with 2 planes removed via non- 342.3% 207.8% iterative subtraction and high pass filter applied Figure 23

Figure 11. Low Contrast Phantom, Single Projection, 224 mAs

Figure 12. Low Contrast Phantom with Overlying and Underlying Structures, Single Projection, 224 mAs

Figure 13. Low Contrast Phantom with Overlying and Underlying Structure, 9 Projections, 32 mAs per Projection

Figure 14. Low Contrast Phantom with Overlying and Underlying Structure, 9 Projections, 32 mAs per Projection, High Pass Filter Applied

40 Figure 15. Low Contrast Phantom with Overlying and Underlying Structure, 9 Projections, 32 mAs per Projection, 1 Plane Removed

Figure 16. Low Contrast Phantom with Overlying and Underlying Structure, 9 Projections, 32 mAs per Projection, 1 Plane Removed, High Pass Filter Applied

41 Figure 17. Low Contrast Phantom with Overlying and Underlying Structure, 9 Projections, 32 mAs per Projection, 2 Planes Removed

Figure 18. Low Contrast Phantom with Overlying and Underlying Structure, 9 Projections, 32 mAs per Projection, 2 Planes Removed, High Pass Filter Applied

42 Tissue Mimicking Phantom

The following images are presented in an inverted grayscale, i.e., denser objects appear darker than their surroundings. In the clinical digital mammographic images, denser objects typically appear lighter than their surroundings, as in the preceding sections. The clinical grayscale will be used in the presentation of the later stage results to follow. The brightness and contrast of all images in this document have been manipulated in order to show details.

This phantom was designed to further examine the effectiveness of the reconstruction schemes at removing out-of-plane structures. The phantom consists of 6.5 mm thick interchangeable layers of wax impregnated with chalk dust to simulate calcifications, and cotton string to simulate fibers. Clear wax can be inserted between the structure layers for any desired thickness.

Multilayer wax/chalk/string phantom techniques were the following:

a. Zero angle single projection, ±15°, 28 kVp, 80 mA, 208 mAs.

b. Nine projection tomosynthesis. ±15°, 28 kVp, 80 mA, 24 mAs per

projection. Reconstruction via iterative linear subtraction, iterative log

subtraction, and ART.

The iterative subtraction and ART methods were applied to the tissue mimicking wax/chalk/string phantom. Sample images are shown in Figures 19 through 28.

Comparing the iterative subtraction and ART images with the zero projection image

43 Figure 19. Tissue Mimicking Phantom, Zero Projection

Figure 20. Shift and Sum Tomosynthesis, No Subtraction, No Iterations. Focus on Diagonal String

Note the disappearance of the out-of-plane string structures and the blurring of the dense object (calcium) structures. White streak is a crack in the wax, coplanar with diagonal string. 44 Figure 21. Iterative Linear Subtraction. Weight = 0.5, Iterations = 5, Planes = 20. Focus on Diagonal String

Figure 22. Iterative Linear Subtraction. Weight = 1.0, Iterations = 5, Planes = 20. Focus on Diagonal String

45 Figure 23. Iterative Logarithmic Subtraction. Weight = 0.025, Iterations = 5, Planes = 20. Focus on Diagonal String

Figure 24. Iterative Logarithmic Subtraction. Weight = 1.0, Iterations = 5, Planes = 20. Focus on Diagonal String

46 Figure 25. ART. Damping Factor = 0.2, Iterations = 5, Planes = 20. Focus on Diagonal String

Figure 26. ART. Damping Factor = 1.0, Iterations = 5, Planes = 20. Focus on Diagonal String

47 Figure 27. ART. Damping Factor = 0.2, Iterations = 5, Planes = 40. Focus on Diagonal String

Figure 28. ART. Damping factor = 1.0, Iterations = 5, Planes = 40. Focus on Diagonal String

48 at the diagonal string. The level of the string was near the phantom surface, approximately 15 mm from the chalk artifacts at the phantom’s center. Notice in Figure

19 how the chalk clearly overlaps and obscures the string. The chalk plane was most effectively removed in Figure 21 (Iterative Linear Subtraction, 0.5 weight, 5 iterations, 20 planes), Figure 25 (ART, 0.2 damping factor, 5 iterations, 20 planes), Figure 26 (ART,

1.0 damping factor, 5 iterations, 20 planes), and Figure 28 (ART, 1.0 damping factor, 5 iterations, 40 planes). These results demonstrate a heavy dependence of the Iterative

Subtraction method on weighting factors, and at first glance, suggest superiority of the

ART method over Iterative Subtraction.

All of the following images were generated with 512 x 512 matrix projections in the interest of minimizing reconstruction time. Using a 1024 x 1024 matrix will improve high contrast spatial resolution and increase the dose required to achieve equivalent SNR and CNR. The brightness and contrast of all images in this document have been manipulated in order to maximize visibility.

Low Contrast Detectability with Embedded Rice Phantom

The Rose model low contrast detectability phantom constructed for this project consisting of cylindrical voids in plastic, aligned in a single plane, perpendicular to the zero angle projection axis, parallel with image reconstruction plane. The void diameters in mm are 6.35, 4.76, 3.18, and 1.59. The depths in mm are 5.0, 2.50, 1.34, and 0.77.

Total depth of phantom is 62 mm. Structured noise was introduced as grains of rice embedded in 2 cm thick epoxy slabs layered on both sides of the low contrast phantom.

Exposure techniques were the following:

a. Single projection at zero angle, 28 kVp, 328 mAs, 80 mA, 512 x 512 field.

49 b. Nine projection tomosynthesis, ±15°, 28 kVp, 32 mAs per projection, 80 mA,

512 x 512 field. Reconstruction via simple tomosynthesis, iterative linear

subtraction, iterative log subtraction, and ART.

c. Forty one projection tomosynthesis, ±15°, 28 kVp, 8 mAs per projection, 80

mA, 512 x 512 field. Reconstruction via simple tomosynthesis, iterative

linear subtraction, iterative log subtraction, and ART.

Low contrast performance test results are in Tables III through VII and Figures 29 through 40. Performance was measured as SNR, CNR and SBR. Signal-to-Noise Ratio is defined as (N1-N0)/σ0, where N1 is the mean pixel value in the ROI within the faintest large diameter disk in the contrast phantom, N0 is the mean pixel value in the ROI in a background area adjacent to the disc, and σ0 is the standard deviation of pixel values in the background ROI. Throughout these results, σ0 includes structure noise that can obscure the object, not just photon statistics. Contrast-to-Noise Ratio is defined as 2(N1-

N0)/(σ0(N1+N0)). Finally, SBR is defined as (N1-N0)/N0.

Results are categorized according to the reconstruction method, i.e., single projection with no tomo, ART, iterative logarithmic subtraction, and iterative linear subtraction. The reconstructions were performed over combinations of different weights or damping factors, number of planes, and number of iterations for both 9 and 41 projections. The low contrast was imaged in two ways, once with overlying and underlying structures (rice in cast resin), and once with the structures replaced by clear plastic. The linear reconstruction technique test results are presented here only for the

50 Table III - Percent Change in Metrics Relative to Single Projection, Calculated with

Median Values

9 Projection Median Values 41 Projection Median Values

No Overlying Structure No Overlying Structures

Simple ART Log Lin Simple ART Log Lin

SNR -9.3% -56.9% -5.7% -47.4% SNR -4.4% -59.6% -20.2% -41.1%

CNR 8.3% -82.1% 213.3% 115.8% CNR 38.0% 457.8% 12.4% 96.8%

SBR -9.2% 251.0% -11.8% 66.6% SBR 2.0% 24.0% 0.7% 5.1%

9 Projection Median Values 41 Projection Median Values

With Overlying Structure With Overlying Structure

Simple ART Log Simple ART Log

SNR 384.8% 177.2% 398.9% SNR 156.1% 50.0% 126.4%

CNR 385.7% 7.1% 1023.8% CNR 158.1% 100.0% 2177.9%

SBR 335.1% 412.7% 120.6% SBR -0.7% 541.4% 26.8%

Table IV - 9 Projections, No Structure Mode: Single Simple ART ART ART ART ART Log Log Log Log Log Log Iterations: N/A N/A 1 10 20 20 40 1 5 10 10 10 20 Damping/Weight: N/A N/A 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Planes: 1 1 9 9 9 5 5 9 9 5 7 9 9 Projections: 1 9 9 9 9 9 9 9 9 9 9 9 9 mAs/projection: 328 32 32 32 32 32 32 32 32 32 32 32 32 Disk A1: SNR: 5.04 4.57 3.06 1.59 1.27 2.16 3.07 4.52 4.86 4.88 4.93 4.89 4.99 CNR: 0.00116 0.00129 0.00032 0.00015 0.00012 0.00011 0.00017 0.00275 0.00290 0.00477 0.00353 0.00291 0.00296 SBR: 0.0553 0.0502 0.0671 0.1250 0.1320 0.1566 0.3211 0.0484 0.0485 0.0490 0.0485 0.0490 0.0492 Disk A2: SNR: 8.15 8.08 5.25 2.68 2.27 3.34 2.92 8.36 8.82 8.56 8.44 8.79 8.48 CNR: 0.0018 0.0022 0.0005 0.0002 0.0002 0.0002 0.0001 0.0049 0.0051 0.0081 0.0059 0.0051 0.0049 SBR: 0.0886 0.0846 0.1368 0.2001 0.2238 0.2399 0.2757 0.0839 0.0802 0.0848 0.0829 0.0804 0.0843 Disk A3: SNR: 17.46 16.72 12.84 5.20 4.38 6.61 5.38 16.02 15.71 19.04 19.16 15.97 15.98 CNR: 0.0038 0.0043 0.0012 0.0004 0.0004 0.0003 0.0002 0.0090 0.0086 0.0172 0.0127 0.0088 0.0088 SBR: 0.1729 0.1605 0.2252 0.3889 0.4352 0.4517 0.4822 0.1537 0.1542 0.1599 0.1628 0.1549 0.1534 Disk A4: SNR: 32.20 35.78 38.63 11.18 9.16 12.69 10.87 34.09 33.85 35.96 36.34 27.94 33.82 CNR: 0.0065 0.0083 0.0032 0.0007 0.0006 0.0005 0.0004 0.0172 0.0168 0.0295 0.0218 0.0138 0.0167 SBR: 0.3892 0.3408 0.5103 0.8489 0.9214 0.9316 0.9972 0.3303 0.3289 0.3393 0.3408 0.3274 0.3302

Table IV - 9 Projections, No Structure Mode: Single Simple Linear Linear Linear Linear Linear Linear Iterations: N/A N/A 1 5 10 20 20 20 Damping/Weight: N/A N/A 0.5 0.5 0.5 0.5 1 0.5 Planes: 1 1 9 9 9 9 3 3 Projections: 1 9 9 9 9 9 9 9 mAs/projection: 328 32 32 32 32 32 32 32 Disk A1: SNR: 5.04 4.57 2.84 2.03 2.26 2.42 3.27 2.04 CNR: 0.00116 0.00129 0.00005 0.00513 0.00449 0.00436 0.00179 0.00229 SBR: 0.0553 0.0502 0.0014 0.1829 0.1543 0.1472 0.0858 0.1398 Disk A2: SNR: 8.15 8.08 -0.07 3.59 3.01 3.99 4.06 2.92 CNR: 0.0018 0.0022 0.0000 0.0084 0.0058 0.0068 0.0021 0.0031 SBR: 0.0886 0.0846 0.0000 0.3074 0.2024 0.2382 0.1265 0.2116 Disk A3: SNR: 17.46 16.72 -0.88 6.94 7.54 7.58 10.12 7.13 CNR: 0.0038 0.0043 0.0000 0.0143 0.0127 0.0117 0.0050 0.0070 SBR: 0.1729 0.1605 -0.0002 0.6072 0.4843 0.4476 0.2484 0.4267 Disk A4: SNR: 32.20 35.78 -1.34 13.93 12.30 14.10 23.84 19.90 CNR: 0.0065 0.0083 0.0000 0.0222 0.0165 0.0175 0.0102 0.0158 SBR: 0.3892 0.3408 -0.0016 1.2831 1.0289 0.9631 0.5446 0.9486

Table V - 41 Projections, No Structure Mode: Single Simple ART ART ART ART ART Log Log Log Log Log Log Iterations: N/A N/A 1 10 20 20 40 1 5 10 10 10 20 Damping/Weight: N/A N/A 0.2 0.2 0.2 0.2 0.2 0.02 0.02 0.02 0.02 0.02 0.02 Planes: 1 1 9 9 9 5 5 9 9 5 7 9 9 Projections: 1 41 41 41 41 41 41 41 41 41 41 41 41 mAs/projection: 328 8 8 8 8 8 8 8 8 8 8 8 8 Disk A1: SNR: 5.04 4.82 2.33 0.79 1.38 3.28 3.21 4.79 4.94 2.53 2.59 5.09 5.51 CNR: 0.00116 0.00114 0.00082 0.00031 0.00056 0.00070 0.00072 0.00770 0.00718 0.01132 0.00568 0.00730 0.00789 SBR: 0.0553 0.0763 0.1733 0.1217 0.2347 0.3554 0.4952 0.0695 0.0666 0.0523 0.0509 0.0677 0.0734 Disk A2: SNR: 8.15 7.89 3.99 0.21 2.03 4.46 4.33 7.93 7.96 5.34 6.14 8.33 8.88 CNR: 0.0018 0.0017 0.0013 0.0001 0.0007 0.0009 0.0009 0.0118 0.0107 0.0224 0.0126 0.0111 0.0118 SBR: 0.0886 0.0977 0.2767 0.0297 0.3091 0.4437 0.6016 0.0905 0.0887 0.0843 0.0861 0.0914 0.0975 Disk A3: SNR: 17.46 12.28 6.40 2.67 2.07 6.73 6.22 12.74 13.05 12.97 14.86 13.76 13.51 CNR: 0.0038 0.0024 0.0017 0.0008 0.0006 0.0011 0.0011 0.0174 0.0161 0.0504 0.0282 0.0168 0.0164 SBR: 0.1729 0.1659 0.3927 0.3570 0.2475 0.6337 0.8353 0.1520 0.1547 0.1561 0.1642 0.1548 0.1532 Disk A4: SNR: 32.20 22.62 12.05 5.87 6.67 11.47 9.64 22.94 23.73 22.14 21.60 25.82 25.45 CNR: 0.0065 0.0038 0.0026 0.0014 0.0016 0.0015 0.0013 0.0274 0.0256 0.0757 0.0357 0.0276 0.0270 SBR: 0.3892 0.3409 0.7699 0.7722 0.9540 1.1459 1.4171 0.3047 0.3075 0.2836 0.2996 0.3125 0.3103

Table V - 41 Projections, No Structure Mode: Single Simple Linear Linear Linear Linear Linear Linear Iterations: N/A N/A 1 5 10 20 20 20 Damping/Weight: N/A N/A 0.5 0.5 0.5 0.5 1 0.5 Planes: 1 1 9 9 9 9 3 3 Projections: 1 9 41 41 41 41 41 41 mAs/projection: 328 8 8 8 8 8 8 8 Disk A1: SNR: 5.04 4.82 0.60 2.12 2.31 0.52 5.42 2.21 CNR: 0.00116 0.00114 0.00001 0.00470 0.00398 0.00079 0.00248 0.00205 SBR: 0.0553 0.0763 0.0012 0.1014 0.0953 0.0240 0.1636 0.2165 Disk A2: SNR: 8.15 7.89 0.91 4.90 4.32 9.34 5.28 2.61 CNR: 0.0018 0.0017 0.0000 0.0091 0.0064 0.0125 0.0022 0.0022 SBR: 0.0886 0.0977 0.0014 0.4652 0.3874 0.2838 0.1574 0.2169 Disk A3: SNR: 17.46 12.28 0.81 3.12 2.66 4.48 6.91 9.02 CNR: 0.0038 0.0024 0.0000 0.0053 0.0035 0.0055 0.0026 0.0067 SBR: 0.1729 0.1659 0.0013 0.5498 0.4336 0.4121 0.2561 0.3962 Disk A4: SNR: 32.20 22.62 0.20 4.83 -3.68 5.77 28.57 11.55 CNR: 0.0065 0.0038 0.0000 0.0060 -0.0074 0.0054 0.0090 0.0068 SBR: 0.3892 0.3409 0.0004 1.0612 -0.5885 0.8370 0.5242 0.8311

Table VI - 9 Projections, With Structure Mode: Single Simple ART ART ART ART ART ART ART ART ART ART Iterations: N/A N/A 1 3 10 10 10 10 10 20 20 40 Damping/Weight: N/A N/A 0.2 0.2 0.2 0.1 0.2 1 0.2 0.2 0.2 0.2 Planes: 1 1 9 9 5 9 9 9 19 9 5 9 Projections: 1 9 9 9 9 9 9 9 9 9 9 9 mAs/projection: 288 32 32 32 32 32 32 32 32 32 32 32 Disk A1: SNR: 0.46 2.23 2.12 1.19 1.05 0.97 0.60 0.56 0.43 0.52 0.93 0.65 CNR: 0.0002 0.0010 0.0004 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 SBR: 0.0228 0.0992 0.0865 0.1232 0.1057 0.0853 0.0734 0.1298 0.0612 0.0822 0.1137 0.1726 Disk A2: SNR: 2.40 1.23 5.10 2.70 3.90 2.20 2.59 1.61 2.20 1.93 2.74 1.01 CNR: 0.0010 0.0005 0.0008 0.0004 0.0003 0.0003 0.0004 0.0003 0.0007 0.0003 0.0002 0.0002 SBR: 0.1186 0.0629 0.1719 0.1679 0.4078 0.1926 0.3508 0.4560 0.3198 0.3496 0.3944 0.2106 Disk A3: SNR: 2.79 3.14 6.02 5.59 4.88 4.78 3.74 2.13 3.04 3.00 3.76 2.33 CNR: 0.0011 0.0012 0.0009 0.0007 0.0003 0.0006 0.0005 0.0003 0.0008 0.0004 0.0003 0.0003 SBR: 0.1813 0.1427 0.2523 0.3279 0.4377 0.3572 0.3890 0.4847 0.3799 0.4287 0.4516 0.4383 Disk A4: SNR: 5.77 5.43 11.83 7.24 6.65 6.30 5.35 3.03 4.89 4.52 3.58 3.51 CNR: 0.0022 0.0020 0.0015 0.0008 0.0004 0.0007 0.0006 0.0004 0.0011 0.0005 0.0002 0.0004 SBR: 0.3494 0.2871 0.4337 0.5408 0.7600 0.5844 0.6651 0.7982 0.6503 0.7173 0.4272 0.7452

Table VI - 9 Projections, With Structure Mode: Single Simple Log Log Log Log Log Log Log Iterations: N/A N/A 5 10 10 10 10 20 20 Damping/Weight: N/A N/A 0.2 0.2 0.2 0.2 0.2 0.2 0.2 Planes: 1 1 9 5 7 9 19 9 5 Projections: 1 9 9 9 9 9 9 9 9 mAs/projection: 288 32 32 32 32 32 32 32 32 Disk A1: SNR: 0.46 2.23 2.14 2.45 2.29 2.29 2.36 2.33 2.36 CNR: 0.0002 0.0010 0.0011 0.0014 0.0013 0.0012 0.0012 0.0012 0.0036 SBR: 0.0228 0.0992 0.0464 0.0522 0.0489 0.0504 0.0504 0.0493 0.0484 Disk A2: SNR: 2.40 1.23 5.07 5.17 5.62 5.61 5.37 5.47 5.24 CNR: 0.0010 0.0005 0.0026 0.0029 0.0029 0.0028 0.0025 0.0028 0.0075 SBR: 0.1186 0.0629 0.0937 0.0952 0.0971 0.0996 0.1000 0.0995 0.0941 Disk A3: SNR: 2.79 3.14 11.19 11.79 10.93 11.18 11.52 10.22 7.37 CNR: 0.0011 0.0012 0.0052 0.0061 0.0053 0.0052 0.0050 0.0048 0.0099 SBR: 0.1813 0.1427 0.1668 0.1662 0.1668 0.1674 0.1672 0.1649 0.1743 Disk A4: SNR: 5.77 5.43 11.34 11.33 11.66 11.51 11.12 12.65 13.71 CNR: 0.0022 0.0020 0.0047 0.0053 0.0051 0.0048 0.0043 0.0053 0.0167 SBR: 0.3494 0.2871 0.2922 0.2903 0.2938 0.2960 0.2970 0.2984 0.2957

Table VII - 41 Projections, With Structure Mode: Single Simple ART ART ART ART ART ART ART ART ART ART ART Iterations: N/A N/A 1 3 3 10 10 10 10 10 20 20 40 Damping/Weight: N/A N/A 0.2 0.2 0.2 0.2 0.1 0.2 1 0.2 0.2 0.2 0.2 Planes: N/A 1 9 9 19 5 9 9 9 19 9 5 9 Projections: 1 41 41 41 41 41 41 41 41 41 41 41 41 mAs/projection: 328 8 8 8 8 8 8 8 8 8 8 8 8 Disk A1: SNR: 1.23 3.15 2.34 1.70 1.13 1.02 0.89 0.57 0.60 0.41 0.58 1.28 0.96 CNR: 0.0004 0.0011 0.0013 0.0012 0.0018 0.0004 0.0007 0.0004 0.0004 0.0007 0.0004 0.0006 0.0006 SBR: 0.0607 0.0603 0.2822 0.4942 0.4778 0.2722 0.2584 0.1896 0.3012 0.2376 0.2119 0.5891 0.4406 Disk A2: SNR: 1.50 1.99 2.14 1.98 1.48 1.25 0.99 0.82 0.47 0.54 0.66 1.64 0.96 CNR: 0.0005 0.0006 0.0010 0.001 0.0017 0.0004 0.0005 0.0004 0.0003 0.0006 0.0003 0.0005 0.0005 SBR: 0.0788 0.0630 0.2030 0.3545 0.3557 0.2373 0.2139 0.2116 0.1991 0.2322 0.2045 0.4685 0.3746 Disk A3: SNR: 3.21 2.98 2.68 2.11 1.59 1.65 1.31 1.15 0.80 0.64 0.93 1.79 1.14 CNR: 0.0011 0.0008 0.0010 0.0009 0.0015 0.0004 0.0006 0.0005 0.0004 0.0006 0.0004 0.0005 0.0005 SBR: 0.1393 0.1141 0.2622 0.2950 0.3179 0.2746 0.2554 0.2612 0.2789 0.2472 0.2397 0.3846 0.3633 Disk A4: SNR: 9.13 9.86 6.41 4.18 3.12 4.23 3.19 2.68 1.77 1.59 2.27 3.63 2.68 CNR: 0.0029 0.0023 0.0019 0.0014 0.0023 0.0008 0.0011 0.0009 0.0006 0.0012 0.0008 0.0007 0.0009 SBR: 0.3846 0.2401 0.5630 0.6532 0.6827 0.6704 0.5936 0.5966 0.5874 0.5793 0.5694 0.8463 0.7384

Table VII - 41 Projections, With Structure Mode: Single Simple Log Log Log Log Log Log Log Iterations: N/A N/A 5 10 10 10 10 20 20 Damping/Weight: N/A N/A 0.02 0.02 0.02 0.02 0.02 0.02 0.02 Planes: N/A 1 9 5 7 9 19 9 5 Projections: 1 41 41 41 41 41 41 41 41 mAs/projection: 328 8 8 8 8 8 8 8 8 Disk A1: SNR: 1.23 3.15 4.08 1.49 4.01 3.94 3.98 3.78 2.69 CNR: 0.0004 0.0011 0.0083 0.0091 0.0123 0.0080 0.0034 0.0076 0.0162 SBR: 0.0607 0.0603 0.0807 0.0759 0.0796 0.0776 0.0810 0.0729 0.0787 Disk A2: SNR: 1.50 1.99 3.32 2.84 3.14 3.25 3.19 3.15 2.52 CNR: 0.0005 0.0006 0.0062 0.0159 0.0088 0.0060 0.0025 0.0057 0.0140 SBR: 0.0788 0.0630 0.0642 0.0575 0.0634 0.0645 0.0655 0.0620 0.0775 Disk A3: SNR: 3.21 2.98 3.41 3.54 3.65 3.58 3.54 3.88 6.58 CNR: 0.0011 0.0008 0.0056 0.0177 0.0090 0.0058 0.0024 0.0063 0.0330 SBR: 0.1393 0.1141 0.0911 0.0833 0.0907 0.0917 0.0933 0.0929 0.1327 Disk A4: SNR: 9.13 9.86 9.71 9.95 9.84 9.81 9.58 10.69 7.32 CNR: 0.0029 0.0023 0.0136 0.0431 0.0209 0.0136 0.0056 0.0147 0.0320 SBR: 0.3846 0.2401 0.2090 0.1896 0.2014 0.2080 0.2136 0.2116 0.2709

Figure 29 - SNR Performance, 9 Projections, No Structure

4 Single Simple 3 ART SNR Log 2

0 012345 Reconstruction Method

Figure 30 - CNR Performance, 9 Projections, No Structure

0.006

0.005

0.004 Single Simple 0.003 ART CNR Log 0.002

0.001

0 012345 Reconstruction Method

60 Figure 31 - SBR Performance, 9 Projections, No Structure

0.35

0.3

0.25

Single 0.2 Simple ART SBR 0.15 Log

0.1

0.05

0 012345 Reconstruction Method

Figure 32 - SNR Performance, 41 Projections, No Structure

4 Single Simple 3 ART SNR Log 2

0 012345 Reconstruction Method

Figure 33 - CNR Performance, 41 Projections, No Structure

0.6

0.5

0.4 Single Simple 0.3 ART CNR Log 0.2

0.1

0 012345 Reconstruction Method

Figure 34 - SBR Performance, 41 Projections, No Structure

1.6

1.4

1.2

1 Single Simple 0.8 ART SBR Log 0.6

0.4

0.2

0 012345 Reconstruction Method

62 Figure 35 - SNR Performance, 9 Projections, With Structure

3.00

2.50

2.00 Single Simple 1.50 ART

SNR Log

1.00

0.50

0.00 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Reconstruction Method

Figure 36 - CNR Performance, 9 Projections, with Structure 0.004

0.0035

0.003

0.0025 Single Simple 0.002 ART CNR Log 0.0015

0.001

0.0005

0 012345 Reconstruction Method

63 Figure 37 - SBR Performance, 9 Projections, with Structure

0.2

0.18

0.16

0.14 Single 0.12 Simple 0.1 ART SBR 0.08 Log

0.06

0.04

0.02

0 012345 Reconstruction Method

Figure 38 - SNR Performance, 41 Projections, with Structure 6

4 Single Simple 3 ART SNR Log 2

0 012345 Reconstruction Method

64 Figure 39 - CNR Performance, 41 Projections, with Structure

0.018

0.016

0.014

0.012 Single 0.01 Simple

0.008 ART CNR Log 0.006

0.004

0.002

0 012345 Reconstruction Method

Figure 40 - SBR Performance, 41 Projections, with Structure

0.6

0.5

0.4 Single Simple ART 0.3 SBR Log

0.2

0.1

0 012345 Reconstruction Method

65 phantom in clear plastic because the linear technique failed to produce a discernible image of the low contrast target disc when obscured by the interfering structures.

The low contrast charts display the SNR, CNR, and SBR for each technique. Range bars are displayed for the ART, iterative logarithmic and iterative linear because the reconstructions were performed using different combinations of weights or damping factors, numbers of planes and numbers of iterations, which yielded a range of results for each technique. The central data point of each is the median value of the results for that technique. The comprehensive data is presented after the charts, followed by the low contrast phantom reconstruction images for the best of each metric, along with the single projection for comparison.

Tomosynthesis reconstruction of the clear low contrast phantom produced some improvement in CNR and SBR with degradation in SNR for both 9 projection and 41 projection acquisitions; however, the technology demonstrated its greatest advantages in the reconstruction of the obscured phantom. With the obscured phantom, all metrics were dramatically improved with each reconstruction technique, with the exception of

SBR in the 41 projection Simple Non-Iterative Reconstruction. Looking at the median values in Figures 35 through 40 for each metric, we find the following. Using nine projections, Iterative Log Subtraction produced the greatest improvement in SNR and

CNR, ART produced the greatest improvement in SBR. Using 41 projections, we find the Iterative Log Subtraction produced the greatest improvement in SNR and CNR, and the ART technique produced the greatest improvement in SBR. Note that the range bars show that 41 projection reconstructions produced higher scores than the 9 projection reconstructions for all metrics.

The first image in Figure 41, shown for comparison, is a single projection taken with the full 328 mAs with no tomosynthesis. The other three images, in Figures 42 through 44, demonstrate the best of the SNR, CNR and SBR categories, regardless of number of projections. It is worth noting that 41 projections won out over 9 projections in each case.

Figure 41 - Single Projection, 328 mAs Figure 42 - Best CNR. Iterative Log Subtraction. 41 projections, 8 mAs per projection, 20 iterations, 5 planes, weight = 0.02.

Figure 43 - Best SNR. Iterative Log subtraction. Figure 44 - Best SBR. ART Reconstruction. 41 41 projections, 8 mAs per projection, 10 projections, 8 mAs per projection, 20 iterations, iterations, 7 planes, weight = 0.02. 5 planes, damping factor = 0.2.

Spatial Resolution

The 0.38 mm lead foil edge was imaged to generate the ESF, and LSF. The LSF

was generated as the first derivative across the ESF. The MTF was determined as the

Fourier Transform of the LSF along a row of pixels spanning the edge. The identical row was used in all images. The single projection at zero angle technique was 28 kVp, 240

mAs. Nine projection tomosynthesis images were collected for ±15°, 28 kVp, 32 mAs per projection, with the edge at isocenter, and also 10 mm away from isocenter. Note in

Table VII that the FWHM performance improves with the edge closer to the image receptor, which creates a reduction in penumbra size as well as a reduction in geometric blur caused by mechanical imperfections in the C-Arm. Line Spread Function plots are in Figures 45 through 49.

Tomosynthesis degrades high contrast spatial resolution, as is evident in Table

VIII. Probably the greatest factor in MTF degradation is the slight misalignment of the

projection images in the reconstruction caused by inaccuracies in mechanical distances

and angles. Another factor may be the increased obliquity of the x-ray beam with the

image formation plane as the projection angle ω increases. Application of the simple

high pass ramp filter slightly improved spatial resolution, as expected.

All reconstruction techniques produced degradation in high contrast spatial

resolution in terms of MTF relative to the MTF of the single projection image, for MTF

values of 0.2 through 0.8. The MTF evaluations were done with the 512 x 512 matrix

exclusively, with a 0.11 mm pixel pitch. Distinct frequencies corresponding to MTF

69 values of 0.1, 0.2, 0.5 and 0.8 are given in Table VIII below. The MTF is displayed in

Figures 50 through 63 below.

Table VII – Edge Response, 1024 x 1024 Matrix

Edge Position Tomosynthesis High Pass Filter FWHM Figure

Performed? Applied? (mm)

Isocenter No No 0.169 6

Isocenter Yes No 0.273 7

Isocenter Yes Yes 0.258 8

10 mm beyond Yes No 0.199 9

isocenter, toward

image receptor

10 mm beyond Yes Yes 0.175 10

isocenter, toward

image receptor

70 Figure 45. Line Spread Function for Edge At Isocenter, No Tomosynthesis, No Filter Applied 1024 x 1024 Matrix, FWHM = 0.169 mm

4000 3500 3000 2500 2000 1500 1000 500 0 -1.5 -1 -0.5 -500 0 0.5 1 1.5 Distance (mm)

71 Figure 46. Line Spread Function for Tomosynthesis of Edge at Isocenter, No Filter Applied 1024 x 1024 Matrix, FWHM = 0.273 mm 3000

2500

2000

1500

1000

500

0 -1.5 -1 -0.5 0 0.5 1 1.5 -500 Distance (mm)

72 Figure 47 Line Spread Function for Tomosynthesis of Edge at Isocenter, High Pass Filter Applied, 1024 x 1024 Matrix FWHM = 0.258 mm 800 700 600 500 400 300 200 100 0 -1.5 -1 -0.5 -100 0 0.5 1 1.5 Distance (mm)

Figure 48 Line Spread Function for Tomosynthesis of Edge 1 cm from Isocenter toward receptor, No Filter Applied, 1024 x 1024 Matrix, FWHM = 0.199 mm

4000 3500 3000 2500 2000 1500 1000 500 0 -1.5 -1 -0.5 -500 0 0.5 1 1.5 Distance (mm)

73 Figure 49 Line Spread Function for Tomosynthesis of Edge 1 cm from Isocenter Toward Receptor, High Pass Filter Applied, 1024 x 1024 Matrix, FWHM = 0.175 mm

1200

1000

800

600

400

200

0 -1.5 -1 -0.5 0 0.5 1 1.5 -200 Distance (mm)

Table VIII - MTF Performance

Mode: Single Simple ART ART ART ART

Iterations: N/A N/A 10 10 10 10

Damping or Wt: N/A N/A 0.1 0.2 0.1 0.2

Planes: N/A 1 5 5 9 9

Projections: 1 9 9 9 9 9

0.1 MTF Freq: 4.98 3.79 4.87 4.44 4.53 6

0.2 MTF Freq: 4.46 3.31 3.91 3.24 3.38 3.39

0.5 MTF Freq: 2.4 1.82 1.4 1.71 1.53 1.66

0.8 MTF Freq: 0.88 0.64 0.62 0.6 0.46 0.53

Mode: Single Simple ART ART ART ART ART

Iterations: N/A N/A 10 20 20 20 40

Damping or Wt: N/A N/A 1 0.1 0.2 0.2 0.2

Planes: N/A 1 9 5 9 5 5

Projections: 1 9 9 9 9 9 9

0.1 MTF Freq: 4.98 3.79 5.56 5 4.99 3.72 4.94

0.2 MTF Freq: 4.46 3.31 3.32 3.94 3.86 2.99 4.21

0.5 MTF Freq: 2.4 1.82 1.69 1.5 1.36 2.02 1.61

0.8 MTF Freq: 0.88 0.64 0.72 0.58 0.32 0.86 0.6

Mode: Single Simple Log Log Log Log

Iterations: N/A N/A 10 10 20 20

Damping or Wt: N/A N/A 0.2 0.2 0.2 0.2

Planes: N/A 1 5 9 9 5

Projections: 1 9 9 9 9 9

0.1 MTF Freq: 4.98 3.79 3.61 4.6 4.76 4.67

0.2 MTF Freq: 4.46 3.31 2.66 3.44 3.76 3.5

0.5 MTF Freq: 2.4 1.82 1.6 1.73 1.49 1.8

0.8 MTF Freq: 0.88 0.64 0.63 0.63 0.44 0.68

75 Figure 50 - Edge Single Projection

1.2

0.8 F 0.6 MT

0.4

0.2

0 012345678910 Spatial frequency (cycles/mm)

Figure 51 - Edge Simple Tomo

1.2

0.8 F 0.6 MT

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 52 - Edge Log 10 Iterations, 5 Planes, Wt 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 53 - Edge Log 10 Iterations, 9 Planes, Wt 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

77 Figure 54 - Edge Log 20 Iterations, 5 Planes, Wt 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 55 - Edge Log 20 Iterations, 9 Planes, Wt 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

78 Figure 56 - ART 10 Iterations, 5 Planes, DF 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 57 - ART 10 Iterations, 9 Planes, DF 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 58 - ART 20 Iterations, 5 Planes, DF 0.2

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 59 - ART 10 Iterations, 9 Planes, DF 1.0

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 60 - ART 10 Iterations, 9 Planes, DF 0.1

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 61 - ART 20 Iterations, 5 Planes, DF 0.1

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

81 Figure 62 - ART 10 Iterations, 5 Planes, DF 0.1

1.2

0.8

0.6 MTF

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

Figure 63 - ART 20 Iterations, 9 Planes, DF 0.2

1.2

0.8 F 0.6 MT

0.4

0.2

0 012345678910 Spatial Frequency (cycles/mm)

82 Cadaver Breast Specimen

Cadaver breast specimen reconstructions are presented below in Figures 64 through 135. The specimen thickness was 40 mm. Each group was produced using 41 projections, 8 mAs per projection, over an angle of 30 degrees. The single projection, full mAs image is shown as the first image in each group for comparison. The groups differ in the number of reconstructed planes, number of iterations, and reconstruction technique of Log Iterative Subtraction or ART.

The 41 projection reconstructions are superior at separating the layers of the breast and removing structures outside the target plane, relative to the 9 projection reconstructions. The use of a higher number of projections, 41 versus 9, improves the removal of high contrast artifacts. This can be seen most clearly by looking at the shadow of the lead projection angle marker across the top of each image.

We can see the effect of variation in the number of reconstructed planes most obviously in the ART technique. The ART images reconstructed with a higher number of planes seem to have greater clarity, or sharpness, the result of a greater number of planes being iteratively removed from interfering with the target plane. The iterative log subtraction images of the cadaver breast specimen in Figures 121 through 129 and 131 through 135, exhibit greater blur than ART images, similar to what is seen in screen/film tomography.

Figure 64 - Zero Projection Figure 65 - Plane 1 of 19, ART, 3 iterations

Figure 66 - Plane 2 of 19, ART, 3 iterations Figure 67 - Plane 3 of 19, ART, 3 iterations

Figure 68 - Plane 4 of 19, ART, 3 iterations Figure 69 - Plane 5 of 19, ART, 3 iterations 84

Figure 70 - Plane 6 of 19, ART, 3 iterations Figure 71 - Plane 7 of 19, ART, 3 iterations

Figure 72 - Plane 8 of 19, ART, 3 iterations Figure 73 - Plane 9 of 19, ART, 3 iterations

Figure 74 - Plane 10 of 19 - center plane Figure 75 - Plane 11 of 19, ART, 3 iterations 85

Figure 76 - Plane 12 of 19, ART, 3 iterations Figure 77 - Plane 13 of 19, ART, 3 iterations

Figure 78 - Plane 14 of 19, ART, 3 iterations Figure 79 - Plane 15 of 19, ART, 3 iterations

Figure 80 - Plane 16 of 19, ART, 3 iterations Figure 81 - Plane 17 of 19, ART, 3 iterations 86

Figure 82 - Plane 18 of 19, ART, 3 iterations Figure 83 - Plane 19 of 19, ART, 3 iterations

Figure 84 - Zero Projection Figure 85 - Plane 1of 19, ART, 10 iterations

Figure 86 - Plane 2 of 19, ART, 10 iterations Figure 87 - Plane 3 of 19, ART, 10 iterations

Figure 88 - Plane 4 of 19, ART, 10 iterations Figure 89 - Plane 5 of 19, ART, 10 iterations 88

Figure 90 - Plane 6 of 19, ART, 10 iterations Figure 91 - Plane 7 of 19, ART, 10 iterations

Figure 92 - Plane 8 of 19, ART, 10 iterations Figure 93 - Plane 9 of 19, ART, 10 iterations

Figure 94 - Plane 10 of 19 - center plane Figure 95 - Plane 11 of 19, ART, 10 iterations 89

Figure 96 - Plane 12 of 19, ART, 10 iterations Figure 97 - Plane 13 of 19, ART, 10 iterations

Figure 98 - Plane 14 of 19, ART, 10 iterations Figure 99 - Plane 15 of 19, ART, 10 iterations

Figure 100 - Plane 16 of 19, ART, 10 iterations Figure 101 - Plane 17 of 19, ART, 10 iterations 90

Figure 102 - Plane 18 of 19, ART, 10 iterations Figure 103 - Plane 19 of 19, ART, 10 iterations

Figure 104 - Zero Projection Figure 105 - Plane 1 of 9, ART, 20 iterations

Figure 106 - Plane 2 of 9, ART, 20 iterations Figure 107 - Plane 3 of 9, ART, 20 iterations

Figure 108 - Plane 4 of 9, ART, 20 iterations Figure 109 - Plane 5 of 9 - center plane 92

Figure 110 - Plane 6 of 9, ART, 20 iterations Figure 111 - Plane 7 of 9, ART, 20 iterations

Figure 112 - Plane 8 of 9, ART, 20 iterations Figure 113 - Plane 9 of 9, ART, 20 iterations

Figure 114 - Zero Projection Figure 115 - Plane 1 of 5, ART, 20 iterations

Figure 116 - Plane 2 of 5, ART, 20 iterations Figure 117 - Plane 3 of 5 - center plane

Figure 118 - Plane 4 of 5, ART, 20 iterations Figure 119 - Plane 5 of 5, ART, 20 iterations 94

Figure 120 - Zero Projection Figure 121 - Plane 1 of 9, Iterative Log, 20 its.

Figure 122 - Plane 2 of 9, Iterative Log, 20 its. Figure 123 - Plane 3 of 9, Iterative Log, 20 its.

Figure 124 - Plane 4 of 9, Iterative Log, 20 its. Figure 125 - Plane 5 of 9, center plane 95

Figure 126 - Plane 6 of 9, Iterative Log, 20 its. Figure 127 - Plane 7 of 9, Iterative Log, 20 its.

Figure 128 - Plane 8 of 9, Iterative Log, 20 its. Figure 129 - Plane 9 of 9, Iterative Log, 20 its.

Figure 130 - Zero Projection Figure 131 - Plane 1 of 5, Iterative Log, 20 its

Figure 132 - Plane 2 of 5, Iterative Log, 20 its Figure 133 - Plane 3 of 5, center plane

Figure 134 - Plane 4 of 5, Iterative Log, 20 its Figure 135 - Plane 5 of 5, Iterative Log, 20 its

97 Slice Thickness

Figure 136 is the simple, non-iterative reconstructed image of the thickness phantom, demonstrating agreement with the predicted focused section thickness of 2.5 mm with a total tomographic angle of 30 degrees. Each disk is 1 mm thick.

Figure 136. Thickness Phantom

98 DISCUSSION

ACR Phantom

The ACR Phantom images are presented in Figures 6 through 10. Image content

does not come across well on paper. Also note that the left and right edges are obscured

by the compression paddle when it is in the beam, as shown in Figures 8 through 10 and

The best outcome in terms of phantom visibility through the obscuring structures is shown in Figure 14, the result of Log Subtraction.

Low Contrast Detectability With The BR-12 Phantom

A striking result of tomosynthesis was the improved visibility of the preliminary

BR-12 low contrast phantom obscured with overlying and underlying structures consisting of one-quarter inch thick layers of Scotchbrite™, a plastic mesh material. As seen in Table I, although the percent image contrast was greater with a single projection without tomosynthesis, simple tomosynthesis increased the CNR by 127.8%. This demonstrates the ability of tomosynthesis to reduce structured noise. The non-iterative

99 removal of overlying and underlying structures using the method of Kolitsi (1993) substantially further improved the percent image contrast 428.8% over a single ordinary projection for one structure plane removed, and 513.5% over a single ordinary projection for two structure planes removed. Likewise, the CNR also increased by 166.1% over a single ordinary projection for one structure plane removed, and 278.3% over a single ordinary projection for two structure planes removed. Application of a simple high pass filter slightly reduced the advantages of low contrast enhancement, as would be expected.

Tissue Mimicking Phantom

The iterative subtraction and ART methods were applied to the tissue mimicking wax/chalk/string phantom. Sample images are shown in Figures 19 through 28.

Comparing the iterative subtraction and ART images with the zero projection image

(Figure 19), we can see the effectiveness of these methods at removing dense overlying and underlying structures. The reconstruction concentrated on the 32 mm thick center portion that contained the chalk and string artifacts, with the target plane, or focus plane, at the diagonal string. The level of the string was near the phantom surface, approximately 15 mm from the chalk artifacts at the phantom’s center. Notice in Figure

1.0 damping factor, 5 iterations, 20 planes), and Figure 28 (ART, 1.0 damping factor, 5 iterations, 40 planes). These preliminary results demonstrate a heavy dependence of the

Iterative Subtraction method on weighting factors, and suggest superiority of the ART method over Iterative Subtraction.

100 Low Contrast Detectability with the Embedded Rice Phantom

The goal of tomosynthesis is to remove interfering structures from a target plane image, and to improve the contrast of target plane structures against a background of interfering out-of-plane structures. This equates to improving existing contrast, or creating contrast where there would otherwise be no measurable contrast in a single projection image. This can be seen in Figures 41 through 44. The ability of tomosynthesis to improve contrast suggests that measured contrast should be the metric of choice for evaluating performance. However, because of the post processing capabilities available in digital imaging, as well as the variables associated with tomosynthesis such as number of projections, exposure per projection, number of reconstructed planes, etc., all of which affect noise in the final image, it makes sense to include noise in the evaluation. Therefore, in the absence of Contrast-Detail and ROI analysis, CNR wins out over SNR and SBR as the more relevant descriptor of tomosynthesis imaging performance.

Tomosynthesis reconstruction of the clear low contrast phantom produced some improvement in CNR and SBR with degradation in SNR for both 9 projection and

41 projection acquisitions. However, the technology demonstrated its greatest advantages in the reconstruction of the obscured phantom. With the obscured phantom, all metrics were dramatically improved with each reconstruction technique, with the exception of SBR in the 41 projection Simple Non-Iterative Reconstruction. Looking at the median values in Figures 29 through 40 for each metric, we find the following. Using nine projections, Iterative Log Subtraction produced the greatest improvement in SNR and CNR, while the Simple Non-Iterative technique produced the greatest improvement

101 in SBR. Using 41 projections, we find the Iterative Log Subtraction produced the greatest improvement in SNR and CNR, and the ART technique produced the greatest improvement in SBR. Note that 41 projection reconstructions produced higher scores than the 9 projection reconstructions for every metric.

It is also important to look at the ranges in the SNR, CNR and SBR in the obscured phantom. Weighting and damping factors, number of reconstructed planes, number of iterations, number of projections and reconstruction techniques were varied widely to explore these ranges. Again, looking at the Figures 29 through 40, we see that

SNR varies widely with ART for 9 projections, and widely for both ART and Iterative

Log Subtraction at 41 projections. The CNR varies widely with Iterative Log Subtraction for both 9 and 41 projections. The CNR varies very little with ART, for both 9 and 41 projections. The SBR varies widely with ART for both 9 and 41 projections. The SBR varies very little with Iterative Log Subtraction, for both 9 and 41 projections.

High Contrast Spatial Resolution

Tomosynthesis degrades high contrast spatial resolution, as is evident in Table

VII and in Figures 45 through 49. Application of the simple high pass ramp filter slightly improved spatial resolution, as expected.

As seen in Table VIII and Figures 50 through 63, all reconstruction techniques produced degradation in high contrast spatial resolution in terms of MTF relative to the

MTF of the single projection image, for MTF values of 0.2 through 0.8. A contributor to the degradation in spatial resolution is the increased obliquity of the x-ray beam with the image formation plane is the projection angle ω increases, as can be seen in the schematic diagram in Figure 1, i.e., object shadows are elongated in the image formation plane as ω

102 increases. Probably the greatest factor in MTF degradation is the slight misalignment of

the projection images in the reconstruction caused by inaccuracies in mechanical

distances and angles. An anomaly is seen in ART reconstruction where three of the ART reconstructions produced a 0.1 MTF frequency that met or exceeded the single projection

MTF frequency. This may be a result of the crude edge enhancing properties of the ART technique, as can be seen in the following plots in Figures 137 through 141. Shown for comparison are edge profiles for the single projection edge image, the simple nine projection reconstruction of the edge, an iterative log subtraction reconstruction. The

MTF evaluations were done with the 512 x 512 matrix exclusively, with a 0.11 mm pixel pitch, in order to minimize reconstruction time.

103 Figure 138

ART Edge Profile of 0.1 MTF at 6.00

Figure 139

Iterative Log, 20 Iterations, 9 Planes 9 Projections, 0.2 weight 0.1 MTF at 4.76

Figure 140

Simple Tomo Edge Profile 0.1 MTF at 3.79

Figure 141

Single Projection Edge Profile 0.1 MTF at 4

104 Cadaver Breast Specimen

All of the cadaver breast reconstructions presented in Figures 64 through 135 were

done with 41 projections. The 41 projection reconstructions are superior at separating the

layers of the breast and removing structures outside the target plane, relative to the 9

projection reconstructions. The use of a higher number of projections, 41 versus 9, improves

the removal of high contrast artifacts. This can be seen most clearly by looking at the projections of the lead projection angle marker across the top of each image. The projection

angle marker is nearly invisible in the 41 projection reconstructions, but is noticeable in the 9

projection reconstructions. This quality also is apparent in the 41 projection reconstructions

of the breast specimen in that dense features that are clearly visible at one level are not present at another. This demonstrates the efficacy of this technology at removing the more dense features located in planes outside the target plane, that would otherwise obscure the target plane anatomy as in standard two-view projection mammography.

Upon further review of the cadaver breast specimen images, we can see the effect of

variation in the number of reconstructed planes most obviously in the ART technique. The

ART images reconstructed with a higher number of planes, as seen in Figures 85 through

103, seem to have greater clarity, or sharpness, the result of a greater number of planes being

iteratively removed from interfering with the target plane. This is best seen at the center

plane of each group, labeled accordingly. The iterative log subtraction images of the cadaver

breast specimen in Figures 121 through 129 and 131 through 135, exhibit greater blur than

ART images, similar to what is seen in screen/film tomography.

105 In tomosynthesis, the discrete blur artifacts caused by high-contrast features arise because of a limited number of projections used for reconstruction. However, the artifacts are far less visible for the 41 projection reconstructions than the 9 projection reconstructions.

This is most obvious by observing the projections of the lead projection angle marker at the top of each image, which are practically invisible in the 41 projection images.

Slice Thickness

Future Applications

Tomosynthesis reconstructions from a limited number of low dose x-ray projections are useful for detecting anatomic features in three dimensions. This technique can partially separate overlapping structures that make conventional projection mammography so difficult to perform with dense breasts. Other investigators (Wu, et al., 2003, Chen and Ning, 2002) are going a step further with cone-beam tomographic mammography, a technique that uses the same C-arm configuration. Developing tomosynthesis for a C-Arm configuration will allow a powerful, cost saving flexibility of being able to perform either tomosynthesis or cone-beam CT on a single platform. The appropriate application of each deserves further investigation. Cone-beam CT probably will not be necessary in a majority of patients, with tomosynthesis probably applied to a larger fraction of patients.

The C-arm configuration is favorable over the flat panel stationary detector. It offers an air gap between the subject and detector, which helps reduce the amount of scatter 106 reaching the detector. It also offers the option of a scatter-reduction grid, since the source to

detector alignment is constant. With a flat panel detector, because the angle between the

source and detector face changes as the source rotates, the use of a grid is not practical. Also,

with a flat panel stationary detector, the range of available projection angles is reduced. The

C-Arm also offers variable magnification, reduced detector blurring because the radiation central axis is normal to the detector, and the possibility of non-linear motion.

As for which of the reconstruction schemes works best, the answer depends on the

desired outcome. Logarithmic subtraction with a large number of projections, which in this

study was 41, is best for enhancing low contrast objects that are obscured by structured noise.

The ART approach with a large number of projections is best when image sharpness is the

desired outcome, for example when searching for calcifications. Either technique, or some

combination, may be applied in practice. A drawback to tomosynthesis, which may be

exacerbated by more than one reconstruction method, will be the large number of images that

the radiologist must read in place of the current two projections per breast.

Research Topics

Suggestions for additional work include the following:

a. Quantify, in greater detail with finer increments, the dependence on weighting

and damping factors, number of planes, number of projections, exposure per projection, breast thickness and composition, and distribution of angle increments, i.e., uniform or clustered around ω = 0.

b. Investigate the use of different x-ray spectra, or multiple energy imaging

combined with tomosynthesis. 107 c. Investigate the effectiveness of Computer Aided Detection combined with tomosynthesis.

d. Compare tomosynthesis with cone-beam CT and try to determine the appropriate

use of each. Possibly screening mammography of medium to upper medium density breasts

will be performed with tomosynthesis. The total dose should be less than the total dose for a

conventional two-view mammogram. That would leave cone-beam CT for the difficult

dense breasts that are beyond the capability of tomosynthesis to penetrate.

e. Investigate other reconstruction schemes in concert with the C-Arm

configuration, such as minimum or maximum intensity pixel projection, Filtered

backprojection, and matrix inversion.

108 SUMMARY

Tomosynthesis shows great promise as a diagnostic tool for radiographically dense breasts. The strength of this technology is its ability to focus on individual planes in the breast and to partially remove structures outside the target plane that would otherwise obscure the target structures in a conventional single projection mammogram. It will accomplish this without an increase in radiation dose relative to the conventional two views per breast mammogram.

This technology enhances the contrast of less dense features while removing the features of greater density located in planes outside the target plane, that would otherwise obscure the target plane anatomy as in standard two-view projection mammography. Forty one projection reconstructions are superior at separating the layers of the breast and removing structures outside the target plane, relative to the nine projection reconstructions.

The ART method produces greater clarity than the iterative subtraction method, especially as

the number of evenly spaced planes through the breast is increased. The iterative log

subtraction method produces a blurring effect when the 41 projections are used, an effect that

is absent from the ART results.

Tomosynthesis degrades high contrast spatial resolution as seen in Tables VII and

VIII and Figures 45 through 49 and 50 through 63. The greatest contributor to the reduction

in spatial resolution is the slight misalignment of the projection images used in the

reconstruction, caused by inaccuracies in mechanical distances and angles. The ART method

produces an edge enhancing effect that raises the 0.1 MTF frequency up to and above the 0.1

109 MTF frequency of the single projection image, while not affecting corresponding frequencies at higher MTF values.

The greatest advantage of tomosynthesis is its ability to remove structured noise and to improve CNR in the target plane. With the obscured phantom, all metrics were dramatically improved with each reconstruction technique, with the exception of SBR in the

41 projection Simple Non-Iterative Reconstruction. Looking at the median values in Figures

29 through 40 for each metric, we find the following. Using nine projections, Iterative Log

Subtraction produced the greatest improvement in SNR and CNR, while the Simple Non-

Iterative technique produced the greatest improvement in SBR. Using 41 projections, we find the Iterative Log Subtraction produced the greatest improvement in SNR and CNR, and the ART technique produced the greatest improvement in SBR. Note that 41 projection reconstructions produced higher scores than the 9 projection reconstructions for all metrics.

110 CONCLUSION

We have demonstrated that tomosynthesis can be successfully applied to breast

imaging using a C-Arm configuration. Tomosynthesis is able to focus on individual selected planes throughout the breast and partially remove structures outside the target plane that would otherwise obscure the target structures in a single projection mammogram. This is accomplished without an increase in dose above that of a two view mammogram.

Tomosynthesis also enhances the contrast of less dense features while removing denser features located in planes outside the target plane.

The ART method produces greater clarity than the iterative subtraction method, especially as the number of iterative planes through the breast is increased. The iterative log method produces a blurring effect when the 41 projections are used, an effect that is absent from the ART images.

The greatest advantage of tomosynthesis over single projection mammography is its

ability to partially remove structured noise and to improve CNR in the target plane.

111

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APPENDICES

119 Appendix A: Iterative Subtraction Code

/*Iterative Subtraction.cpp backprojection, allowing both x and y tilt of image plane. Removes overlying structures from each reconstruction plane by subtracting each plane from every other plane for a user defined number of iterations.*/ #include #include #include #include #include #include #include

/*file_list is the pointer to "inputs", and "outputs" the files containing input projection image names and output file names*/ FILE *infile,*outfile; /*array indices*/ int g,i,j,k,p,q,n,w,z,iterate,inew1,inew2,jnew1,jnew2,rnew1,rnew2; /*temp1 and temp2 are temporary variables used in the aray subtraction.*/ double temp1,temp2; /*file_name is the character array that holds the open file name*/ char file_name[20]; /*NPROJ is the number of projections*/ const short int NPROJ=9; /*NPLANES is the number of planes to be reconstructed*/ const short int NPLANES=5; /*ndet is the number of pixels per row*/ const short int ndet=512; /*Array M receives the pixel data from the image file*/ unsigned short int M[NPROJ][ndet][ndet];

120 /*IT is the number of iterations*/ int IT; /*thick is the total anatomical thickness*/ double thick; /*Array Mnew holds shifted and summed image data for the output image*/ float Mnew[NPLANES][ndet][ndet]={0}; /*Array Mnew2 holds the summed data from the remove() function*/ float Mnew2[ndet][ndet]={0}; /*Array Mnew3 holds the result of Mnew-Mnew2*/ float Mnew3[NPLANES][ndet][ndet]={0}; /*2D array input receives the raw data from the projection files*/ short int input[ndet][ndet]; /*2D array output holds the final output in unsigned short int format*/ short int output[ndet][ndet]; /*WEIGHT is the subtraction weighting factor.*/ const double WEIGHT=0.2; double WT; /*Beta is the angle of tilt with respect to the y axis. Horizontal plane is 90 degrees.*/ double beta=1.570796; /*Alpha is the angle of tilt with respect to the x axis. Horizontal plane is 90 degrees.*/ double alpha=1.570796; /*L is the distance between the target plane and isocenter*/ /*Lr is the distance between the subtraction plane and isocenter*/ /*dL is the distance between layers*/ double dL; double L,Lr,TPID,TPIDr; /*offL is the offset error, for shifting phantom position relative to isocenter*/ double offL=0;

121 /*b is the source to isocenter distance in mm*/ double b=663.0; /*d is the SID in mm*/ double d=880.0; /*det_size is the detector size in mm*/ double det_size=56.0; /*omega is the projection angle*/ /*double omega[9]={0.2618, 0.2023, 0.1428, 0.0714, 0.0, -0.0833, -0.1428, -0.2023, -0.2618};*/ /*double omega[9]={0.2618, 0.2020, 0.1406, 0.0711, 0.0, -0.0682, -0.1317, -0.1745, -0.2618};*/ double fractioni,fractionj,ireal,jreal,integer; /*char *in_file_name[NPROJ]={"175418.img","175504.img","175541.img","175744.img","175818.img", "175856.img","175934.img","180015.img","180111.img"};*/ /*6-18-03 specimen angles:*/ /*double omega[NPROJ]={0.2618, 0.2490, 0.2331, 0.2203, 0.2075, 0.1937, 0.1809, 0.1670, 0.1540, 0.1422, 0.1294, 0.1166, 0.1047, 0.0919, 0.0780, 0.0642, 0.0493, 0.0365, 0.0248, 0.0118, 0.0, -0.0134, -0.0287, -0.0421, -0.0553, -0.0687, -0.0811, -0.0964, -0.1098, -0.1243, -0.1365, -0.1520, -0.1662, -0.1807, -0.1910, -0.2073, -0.2186, -0.2321, -0.2422, -0.2577, -0.2618};*/ /*6-18-03 low contrast angles:*/ /*double omega[NPROJ]={0.2618, 0.2432, 0.2270, 0.2146, 0.2010, 0.1886, 0.1762, 0.1613, 0.1476, 0.1352, 0.1216, 0.1079, 0.0955, 0.0831, 0.0682, 0.0546, 0.0397, 0.0372, 0.0248, 0.0100, 0.0, -0.0124, -0.0247, -0.0359, -0.0461, -0.0595, -0.0719, -0.1067, -0.1191, -0.1292, -0.1461, -0.1618, -0.1753, -0.1865, -0.1989, -0.2101, -0.2236, -0.2371, -0.2438, -0.2550, -0.2618};*/ /*10-2-03 low contrast angles:*/ /*double omega[NPROJ]={0.2618, 0.1959, 0.1337, 0.0659, 0, -0.0693, -0.1386, -0.2002, -0.2618};*/ /*Oct-03 low contrast with rice angles:*/ /*double omega[NPROJ]={0.2618, 0.2430, 0.2310, 0.2139, 0.2053, 0.1916, 0.1762, 0.1643, 0.1506, 0.1369, 0.1232, 0.1112, 0.0992, 0.0856, 0.0719, 0.0582, 0.0479, 0.0359, 0.0222, 0.0120, 0.0, -0.0072, -0.0269, -0.0484, -0.0610, -0.0735, -0.0861, -0.0986, -0.1130, -0.1255, -0.1363, -0.1470, -0.1632, -0.1775, -0.1901, -0.2044, -0.2170, -0.2295, -0.2421, -0.2510, -0.2618};*/ /*11-4-03 low contrast with rice 9 projection angles:*/

122 /*double omega[NPROJ]={0.2618, 0.1920, 0.1309, 0.0625, 0, -0.0687, -0.1360, -0.2018, -0.2618};*/ /*Oct-03 low contrast with rice angles:*/ /*Thickness phantom angles.*/ double omega[NPROJ]={0.2618, 0.1902, 0.1224, 0.0509, 0, -0.0727, -0.1382, -0.2054, -0.2618}; /*Edge phantom angles.*/ /*double omega[NPROJ]={0.2618, 0.1900, 0.1293, 0.0607, 0, -0.0718, -0.1373, -0.1995, -0.2618};*/ /*6-18-03 specimen files:*/ /*char *in_file_name[NPROJ]={"221613.IMG","221703.IMG","221755.IMG","221827.IMG","221852.IMG","221919.IMG","221941.I MG","222009.IMG","222032.IMG","222055.IMG","222125.IMG","222148.IMG","222213.IMG","222235.IMG","222257.IMG", "222317.IMG","222340.IMG","222434.IMG","222458.IMG","222536.IMG","222556.IMG","222621.IMG","222659.IMG","2227 36.IMG","222759.IMG","222821.IMG","222843.IMG","222904.IMG","222925.IMG","222946.IMG","223011.IMG","223034.I MG","223056.IMG","223116.IMG","223138.IMG","223158.IMG","223219.IMG","223241.IMG","223305.IMG","223412.IMG", "223433.IMG"};*/ /*10_17_03 specimen angles.*/ /*double omega[NPROJ]={0.2618, 0.2531, 0.2387, 0.2271, 0.2155, 0.1996, 0.1895, 0.1736, 0.1620, 0.1475, 0.1345, 0.1201, 0.1056, 0.0926, 0.0796, 0.0651, 0.0521, 0.0376, 0.0246, 0.0116, 0.0, -0.0156, -0.0272, -0.0428, -0.0531, -0.0635, -0.0765, -0.0907, -0.1011, -0.1141, -0.1257, -0.1374, -0.1465, -0.1607, -0.1737, -0.1996, -0.2087, -0.2216, -0.2359, -0.2450, -0.2618};*/ /*10_17_03 specimen 9 projection angles.*/ /*double omega[NPROJ]={0.2618, 0.1943, 0.1295, 0.0620, 0, -0.0717, -0.1364, -0.2026, -0.2618};*/ /*6-18-03 low contrast files:*/ /*char *in_file_name[NPROJ]={"210300.IMG","210754.IMG","210823.IMG","210853.IMG","210925.IMG","210949.IMG","211015.I MG","211043.IMG","211104.IMG","211129.IMG","211154.IMG","211227.IMG","211252.IMG","211320.IMG","211402.IMG", "211431.IMG","211459.IMG","211650.IMG","211721.IMG","211749.IMG","211822.IMG","211845.IMG","212044.IMG","2121 36.IMG","212216.IMG","212248.IMG","212320.IMG","212504.IMG","212536.IMG","212612.IMG","212704.IMG","212736.I MG","212808.IMG","212840.IMG","212912.IMG","212942.IMG","213011.IMG","213043.IMG","213120.IMG","213214.IMG", "213233.IMG"};*/ /*10-2-03 low contrast files:*/

123 /*char*in_file_name[NPROJ]={"171341.IMG","171623.IMG","171716.IMG","171802.IMG","171830.IMG","171909.IM G","171944.IMG","172016.IMG","172046.IMG"};*/ /*Oct-03 low contrast with rice files:*/ /*char*in_file_name[NPROJ]={"183413.IMG","183528.IMG","183616.IMG","183718.IMG","183858.IMG","183939.IM G","184024.IMG","184207.IMG","184242.IMG","184322.IMG","184357.IMG","184438.IMG","184512.IMG","184549.IMG","1 84626.IMG","184658.IMG","184729.IMG","185121.IMG","185237.IMG","185310.IMG","185356.IMG","185814.IMG","19000 0.IMG","190123.IMG","190153.IMG","190220.IMG","190300.IMG","190333.IMG","190415.IMG","190511.IMG","190539.IM G","190608.IMG","190639.IMG","190707.IMG","190732.IMG","190757.IMG","190822.IMG","190844.IMG","190907.IMG","1 90941.IMG","191011.IMG"};*/ /*11-4-03 low contrast with rice 9 projection files:*/ /*char*in_file_name[NPROJ]={"124759.IMG","124839.IMG","124914.IMG","124943.IMG","125007.IMG","125040.IM G","125106.IMG","125133.IMG","125204.IMG"};*/ /*10_17_03 specimen files.*/ /*char*in_file_name[NPROJ]={"171953.IMG","172130.IMG","172159.IMG","172234.IMG","172256.IMG","172319.IM G","172345.IMG","172410.IMG","172430.IMG","172450.IMG","172512.IMG","172533.IMG","172556.IMG","172617.IMG","1 72658.IMG","172721.IMG","172742.IMG","172804.IMG","173411.IMG","173441.IMG","173524.IMG","173544.IMG","17361 0.IMG","173701.IMG","173805.IMG","173850.IMG","173916.IMG","173939.IMG","173959.IMG","174020.IMG","174040.IM G","174101.IMG","174122.IMG","174144.IMG","174207.IMG","174250.IMG","174312.IMG","174332.IMG","174355.IMG","1 74420.IMG","174452.IMG"};*/ /*10_17_03 specimen 9 projection files.*/ /*char *in_file_name[NPROJ]={"174728.IMG","174807.IMG","174832.IMG","174856.IMG","174918.IMG", "174941.IMG","175004.IMG","175025.IMG","175046.IMG"};*/ /*Thickness phantom files.*/ char*in_file_name[NPROJ]={"173358.IMG","173443.IMG","173530.IMG","173608.IMG","173639.IMG","173721.IMG ","173753.IMG","173822.IMG","173847.IMG"}; /*Edge phantom files.*/ /*char*in_file_name[NPROJ]={"132935.IMG","133155.IMG","133401.IMG","133431.IMG","133456.IMG","133525.IM G","133552.IMG","133620.IMG","133643.IMG"};*/ char *out_file_name[NPLANES]={"Thick_Log20_1of5","Thick_Log20_2of5","Thick_Log20_3of5", "Thick_Log20_4of5","Thick_Log20_5of5"};

124 void shift(); void remove(); main() { n=0; cout<<"Please specify the total phantom thickness:"; cin>>thick; cout<<"Please specify the number of iterations:"; cin>>IT; WT=NPROJ*NPLANES*IT*WEIGHT; if(NPLANES==1) dL=0; else dL=thick/(NPLANES-1);

/*Read in projection data*/ for(w=0;w

125 fread(&input[i][0],2,ndet,infile); fclose(infile); for (i=0;i

/*Create initial set of planes*/ for(n=0;n

L=(NPLANES-1)*dL/2-(z*dL)+offL; for(n=0;n

126 {if(n!=z) {Lr=(NPLANES-1)*dL/2-(n*dL)+offL; cout<<"Calling remove()."; cout<

cout<<"Performing subtraction."; cout<

}

127 /*Update final image array*/ cout<<"Updating final image array."; cout<

/*Able to open output file? If not, print error message*/ if((outfile=fopen(out_file_name[g],"wb"))==NULL) {cout<

fclose(outfile); } return 0; }

/*Start of the pixel shifting function for target plane*/ void shift() { double JR,IR,H,H1,K,LK,DELTAH,REALI,REALJ; double SA=sin(alpha); double SB=sin(beta); double CB=cos(beta); double CO=cos(omega[p]);

128 double SO=sin(omega[p]); double SAO=sin(alpha+omega[p]); double CAO=cos(alpha+omega[p]); double B=L*d*SA*CAO/(b*SAO-L*SA); double A=(1-L*SA/b*SAO);

/*Loop for all rows*/ for(i=0;i=0 && IR<=(ndet-1) && IR>=0) { fractionj=modf(JR,&integer); jnew1=(int)integer; jnew2=(int)integer+1; fractioni=modf(IR,&integer); inew1=(int)integer; inew2=(int)integer+1;

129 Mnew[n][i][j]= (M[p][inew1][jnew1]*(1-fractioni)*(1-fractionj)+ M[p][inew1][jnew2]*(1-fractioni)*(fractionj)+ M[p][inew2][jnew1]*(fractioni)*(1-fractionj)+ M[p][inew2][jnew2]*(fractioni)*(fractionj))+Mnew[n][i][j]; }}}}

/*Start of the structure removal function. The matrix Mnew2 is now projected back onto the plane of image formation for each one of the viewing angles omega[n]. These projections are used to tomosynthesize an image in the plane of interest.*/

void remove() { int q; double JR,IR,H,H1,K,Kr,LK,LKr,DELTAH,DELTAHr,REALI,REALJ; double SA=sin(alpha); double CB=cos(beta); double SB=sin(beta);

/*Loop for all projections.*/ for(q=0;q

130 for(i=0;i

K=REALI*d/(REALI*CB+d*SB); LK=(b-L)*K*CB/d; DELTAH=LK*d*SO/(b*CO-LK); ireal=K*(b-L)/b; IR=(ireal+det_size/2)*ndet/det_size;

/*Loop for all columns.*/ for(j=0;j

H1=REALJ*d/(d*SAO+REALJ*CAO); H=H1+DELTAH; jreal=(H+B)*A; JR=(jreal+det_size/2)*ndet/det_size;

if(JR<=(ndet-1) && JR>=0 && IR<=(ndet-1) && IR>=0) {fractionj=modf(JR,&integer); jnew1=(int)integer; jnew2=(int)integer+1; fractioni=modf(IR,&integer); inew1=(int)integer;

131 inew2=(int)integer+1;

Mnew2[i][j]=Mnew2[i][j]+Mnew[n][inew1][jnew1]*(1-fractioni)* (1-fractionj)+Mnew[n][inew1][jnew2]*(1-fractioni)* (fractionj)+Mnew[n][inew2][jnew1]*(fractioni)*(1-fractionj) +Mnew[n][inew2][jnew2]*(fractioni)*(fractionj);

}}}}}

132 Appendix B: Algebraic Reconstruction Technique Code

/*ART Recon.cpp Algebraic Reconstruction, allowing both x and y tilt of image plane.*/ #include #include #include #include #include #include #include

/*file_list is the pointer to "inputs", the file containing input projection image names*/ FILE *infile,*outfile; /*array indices*/ int g,i,j,k,n,w,u,inew1,inew2,jnew1,jnew2,rnew1,rnew2; double ireal,fraction1,integer,jreal,fractioni,fractionj; /*ndet is the number of pixels per row and column*/ const int ndet=512; /*NPROJ is the number of projections*/ const int NPROJ=9; /*nk is the number of layers to be reconstructed*/ const int nk=5; /*index is the raw data projection number for array Ik initialization*/ const int index=4; /*2D array input receives the raw data from the projection files*/ unsigned short int input[ndet][ndet]; /*3D Array M receives the pixel data from the input array*/ double M[NPROJ][ndet][ndet];

133 /*3D array Ik holds holds the output from function back()*/ double Ik[nk][ndet][ndet]={0}; /*2D array "output" holds the final output in unsigned short int format*/ unsigned short int output[ndet][ndet]; /*Array err holds the error maps for each projection angle*/ double err[NPROJ][ndet][ndet]={0}; /*Array p holds the summation of all forward projections of image planes to the projection data plane*/ double p[ndet][ndet]={0}; /*T is the breast thickness*/ double T; /*dL is the distance between layers*/ double dL; /*L is the distance of a layer from isocenter*/ double L; /*offL is the L offset error, for phantoms not perfectly at isocenter*/ double offL=0; /*N is the number of iterations*/ int N; /*D is a damping factor in the backprojection error correction. 2.0 is poor, with excessive vertical stripes. 0.01 too small, with very limited range of densities.*/ const double D=0.2; /*Beta is the angle of tilt with respect to the y axis. Horizontal plane is 90 degrees.*/ const double beta=1.570796; /*Alpha is the angle of tilt with respect to the x axis. Horizontal plane is 90 degrees.*/ const double alpha=1.570796; /*b is the source to isocenter distance in mm*/

134 const double b=663.0; /*d is the SID in mm*/ const double d=880.0; /*det_size is the detector size in mm*/ const double det_size=56.0; /*omega is the projection angle*/ /*double omega[NPROJ]={0.2618, 0.2023, 0.1428, 0.0714, 0.0, -0.0833, -0.1428, -0.2023, -0.2618};*/ /*double omega[NPROJ]={0.2618, 0.2020, 0.1406, 0.0711, 0.0, -0.0682, -0.1317, -0.1745, -0.2618};*/ /*Thickness phantom angles.*/ double omega[NPROJ]={0.2618, 0.1902, 0.1224, 0.0509, 0, -0.0727, -0.1382, -0.2054, -0.2618}; /*6-18-03 specimen angles:*/ /*double omega[NPROJ]={0.2618, 0.2490, 0.2331, 0.2203, 0.2075, 0.1937, 0.1809, 0.1670, 0.1540, 0.1422, 0.1294, 0.1166, 0.1047, 0.0919, 0.0780, 0.0642, 0.0493, 0.0365, 0.0248, 0.0118, 0.0, -0.0134, -0.0287, -0.0421, -0.0553, -0.0687, -0.0811, -0.0964, -0.1098, -0.1243, -0.1365, -0.1520, -0.1662, -0.1807, -0.1910, -0.2073, -0.2186, -0.2321, -0.2422, -0.2577, -0.2618};*/ /*6-18-03 low contrast angles:*/ /*double omega[NPROJ]={0.2618, 0.2432, 0.2270, 0.2146, 0.2010, 0.1886, 0.1762, 0.1613, 0.1476, 0.1352, 0.1216, 0.1079, 0.0955, 0.0831, 0.0682, 0.0546, 0.0397, 0.0372, 0.0248, 0.0100, 0.0, -0.0124, -0.0247, -0.0359, -0.0461, -0.0595, -0.0719, -0.1067, -0.1191, -0.1292, -0.1461, -0.1618, -0.1753, -0.1865, -0.1989, -0.2101, -0.2236, -0.2371, -0.2438, -0.2550, -0.2618};*/ /*Oct-03 low contrast with rice angles:*/ /*double omega[NPROJ]={0.2618, 0.2430, 0.2310, 0.2139, 0.2053, 0.1916, 0.1762, 0.1643, 0.1506, 0.1369, 0.1232, 0.1112, 0.0992, 0.0856, 0.0719, 0.0582, 0.0479, 0.0359, 0.0222, 0.0120, 0.0, -0.0072, -0.0269, -0.0484, -0.0610, -0.0735, -0.0861, -0.0986, -0.1130, -0.1255, -0.1363, -0.1470, -0.1632, -0.1775, -0.1901, -0.2044, -0.2170, -0.2295, -0.2421, -0.2510, -0.2618};*/ /*11-4-03 low contrast with rice 9 projection angles:*/ /*double omega[NPROJ]={0.2618, 0.1920, 0.1309, 0.0625, 0, -0.0687, -0.1360, -0.2018, -0.2618};*/ /*10_17_03 specimen angles.*/ /*double omega[NPROJ]={0.2618, 0.2531, 0.2387, 0.2271, 0.2155, 0.1996, 0.1895, 0.1736, 0.1620, 0.1475, 0.1345, 0.1201, 0.1056, 0.0926, 0.0796, 0.0651, 0.0521, 0.0376, 0.0246, 0.0116,

135 0.0, -0.0156, -0.0272, -0.0428, -0.0531, -0.0635, -0.0765, -0.0907, -0.1011, -0.1141, -0.1257, -0.1374, -0.1465, -0.1607, -0.1737, -0.1996, -0.2087, -0.2216, -0.2359, -0.2450, -0.2618};*/ /*10_17_03 specimen 9 projection angles.*/ /*double omega[NPROJ]={0.2618, 0.1943, 0.1295, 0.0620, 0, -0.0717, -0.1364, -0.2026, -0.2618};*/ /*Edge phantom angles.*/ /*double omega[NPROJ]={0.2618, 0.1900, 0.1293, 0.0607, 0, -0.0718, -0.1373, -0.1995, -0.2618};*/ /*Low contrast 9 projections.*/ /*double omega[NPROJ]={0.2618, 0.1959, 0.1337, 0.0659, 0, -0.0693, -0.1386, -0.2002, -0.2618};*/ /*double fractioni,fractionj,ireal,jreal,integer;*/ /*char *in_file_name[NPROJ]={"175418.IMG","175504.IMG","175541.IMG","175744.IMG","175818.IMG", "175856.IMG","175934.IMG","180015.IMG","180111.IMG"};*/ /*Thickness phantom files.*/ char *in_file_name[NPROJ]={"173358.IMG","173443.IMG","173530.IMG","173608.IMG","173639.IMG","173721.IMG","173753.I MG","173822.IMG","173847.IMG"}; /*6-18-03 specimen files:*/ /*char*in_file_name[NPROJ]={"221613.IMG","221703.IMG","221755.IMG","221827.IMG","221852.IMG","221919.IM G","221941.IMG","222009.IMG","222032.IMG","222055.IMG","222125.IMG","222148.IMG","222213.IMG","222235.IMG","2 22257.IMG","222317.IMG","222340.IMG","222434.IMG","222458.IMG","222536.IMG","222556.IMG","222621.IMG","22265 9.IMG","222736.IMG","222759.IMG","222821.IMG","222843.IMG","222904.IMG","222925.IMG","222946.IMG","223011.IM G","223034.IMG","223056.IMG","223116.IMG","223138.IMG","223158.IMG","223219.IMG","223241.IMG","223305.IMG","2 23412.IMG","223433.IMG"};*/ /*6-18-03 low contrast files:*/ /*char*in_file_name[NPROJ]={"210300.IMG","210754.IMG","210823.IMG","210853.IMG","210925.IMG","210949.IM G","211015.IMG","211043.IMG","211104.IMG","211129.IMG","211154.IMG","211227.IMG","211252.IMG","211320.IMG","2 11402.IMG","211431.IMG","211459.IMG","211650.IMG","211721.IMG","211749.IMG","211822.IMG","211845.IMG","21204 4.IMG","212136.IMG","212216.IMG","212248.IMG","212320.IMG","212504.IMG","212536.IMG","212612.IMG","212704.IM G","212736.IMG","212808.IMG","212840.IMG","212912.IMG","212942.IMG","213011.IMG","213043.IMG","213120.IMG","2 13214.IMG","213233.IMG"};*/ /*Oct-03 low contrast with rice files:*/

136 /*char*in_file_name[NPROJ]={"183413.IMG","183528.IMG","183616.IMG","183718.IMG","183858.IMG","183939.IM G","184024.IMG","184207.IMG","184242.IMG","184322.IMG","184357.IMG","184438.IMG","184512.IMG","184549.IMG","1 84626.IMG","184658.IMG","184729.IMG","185121.IMG","185237.IMG","185310.IMG","185356.IMG","185814.IMG","19000 0.IMG","190123.IMG","190153.IMG","190220.IMG","190300.IMG","190333.IMG","190415.IMG","190511.IMG","190539.IM G","190608.IMG","190639.IMG","190707.IMG","190732.IMG","190757.IMG","190822.IMG","190844.IMG","190907.IMG","1 90941.IMG","191011.IMG"};*/ /*10-2-03 low contrast files:*/ /*char*in_file_name[NPROJ]={"171341.IMG","171623.IMG","171716.IMG","171802.IMG","171830.IMG","171909.IM G","171944.IMG","172016.IMG","172046.IMG"};*/ /*11-4-03 low contrast with rice 9 projection files:*/ /*char*in_file_name[NPROJ]={"124759.IMG","124839.IMG","124914.IMG","124943.IMG","125007.IMG","125040.IM G","125106.IMG","125133.IMG","125204.IMG"};*/ /*Edge phantom files.*/ /*char*in_file_name[NPROJ]={"132935.IMG","133155.IMG","133401.IMG","133431.IMG","133456.IMG","133525.IM G","133552.IMG","133620.IMG","133643.IMG"};*/ /*10_17_03 specimen files.*/ /*char*in_file_name[NPROJ]={"171953.IMG","172130.IMG","172159.IMG","172234.IMG","172256.IMG","172319.IM G","172345.IMG","172410.IMG","172430.IMG","172450.IMG","172512.IMG","172533.IMG","172556.IMG","172617.IMG","1 72658.IMG","172721.IMG","172742.IMG","172804.IMG","173411.IMG","173441.IMG","173524.IMG","173544.IMG","17361 0.IMG","173701.IMG","173805.IMG","173850.IMG","173916.IMG","173939.IMG","173959.IMG","174020.IMG","174040.IM G","174101.IMG","174122.IMG","174144.IMG","174207.IMG","174250.IMG","174312.IMG","174332.IMG","174355.IMG","1 74420.IMG","174452.IMG"};*/ /*10_17_03 specimen 9 projection files.*/ /*char *in_file_name[NPROJ]={"174728.IMG","174807.IMG","174832.IMG","174856.IMG","174918.IMG", "174941.IMG","175004.IMG","175025.IMG","175046.IMG"};*/

char*out_file_name[20]={"Spec3_ART20_1of5","Spec3_ART20_2of5","Spec3_ART20_3of5","Spec3_ART20_4of5","S pec3_ART20_5of5"};

void back();

137 void forward(); main()

{ cout<<"Specify the breast thickness in mm:"; cin>>T; cout<<"Specify the number of iterations:"; cin>>N; if(nk==1) dL=0; else dL=T/(nk-1);

/*Read in projection data*/ for(w=0;w

138 for (i=0;i

/*Initialize Ik cout<<"Initializing Ik."; cout<

/*Iterations loop*/ cout<<"Entering iterations loop."; cout<

139 /*Layers loop*/ cout<<"Entering layers loop 1."; cout<

/*Layers loop again*/ cout<<"Entering layers loop 2."; cout<

/*Update final image array*/ for(g=0;g

/*Able to open output file? If not, print error message*/ if((outfile=fopen(out_file_name[g],"wb"))==NULL)

140 {cout<<"Output file open error."; cout<

} fclose(outfile);

return 0; }

/*Back projection function*/ void back() { double JR,IR,H,H1,K,LK,DELTAH,REALI,REALJ; double SA=sin(alpha); double SB=sin(beta); double CB=cos(beta); double CO=cos(omega[w]); double SO=sin(omega[w]); double SAO=sin(alpha+omega[w]); double CAO=cos(alpha+omega[w]); double B=L*d*SA*CAO/(b*SAO-L*SA); double A=(1-L*SA/b*SAO);

/*Loop for all rows*/ for(i=0;i=0) { fractioni=modf(IR,&integer); inew1=(int)integer; inew2=(int)integer+1; /*Loop for all columns*/ for(j=0;j=0) { fractionj=modf(JR,&integer); jnew1=(int)integer; jnew2=(int)integer+1; Ik[u][i][j]=Ik[u][i][j]+(D/nk)*(err[w][inew1][jnew1]*(1-fractioni)*(1-fractionj)+ err[w][inew1][jnew2]*(1-fractioni)*(fractionj)+ err[w][inew2][jnew1]*(fractioni)*(1-fractionj)+ err[w][inew2][jnew2]*(fractioni)*(fractionj)); }}}}}

/*Forward projection function*/

142 void forward() { double JR,IR,H,H1,K,LK,DELTAH,REALI,REALJ; double SA=sin(alpha); double SB=sin(beta); double CB=cos(beta); double CO=cos(omega[w]); double SO=sin(omega[w]); double SAO=sin(alpha+omega[w]); double CAO=cos(alpha+omega[w]); double B=L*d*SA*CAO/(b*SAO-L*SA); double A=(1-L*SA/b*SAO);

/*Loop for all rows*/ for(i=0;i=0) { fractioni=modf(IR,&integer); inew1=(int)integer; inew2=(int)integer+1; /*Loop for all columns*/ for(j=0;j

143 REALJ=(H1*d*SAO)/(d-H1*CAO); JR=ndet*(REALJ/det_size+0.5); if(JR<=(ndet-1) && JR>=0) { fractionj=modf(JR,&integer); jnew1=(int)integer; jnew2=(int)integer+1; p[i][j]=p[i][j]+(Ik[u][inew1][jnew1]*(1-fractioni)*(1-fractionj)+ Ik[u][inew1][jnew2]*(1-fractioni)*(fractionj)+ Ik[u][inew2][jnew1]*(fractioni)*(1-fractionj)+ Ik[u][inew2][jnew2]*(fractioni)*(fractionj)); }}}}}

144 Abstract

Digital tomosynthesis is an imaging technique to produce a tomographic image

from a series of angular digital images in a manner similar to conventional focal plane

tomography. Unlike film focal plane tomography, the acquisition of the data in a C–arm

geometry causes the image receptor to be positioned at various angles to the

reconstruction tomogram. The digital nature of the data, however, allows for input

images to be combined or mapped into the desired plane with the flexibility of generating

tomograms of many separate planes and of generating oblique planes from a single set of

input data.

Angular data sets were obtained of phantoms utilizing a Lorad stereotactic biopsy

unit with a coupled source and digital detector in a C-arm configuration. Tomographic images were reconstructed using iterative subtraction techniques with removal of out-of- focus planes, and an iterative reconstruction technique similar to Algebraic

Reconstruction (ART). These were compared with standard single view digital radiographs. Tested indicators of quality are the line spread function, Modulation

Transfer Function (MTF), Signal to Noise Ratio (SNR), Contrast to Noise Ratio (CNR),

Signal to Background Ratio (SBR). The method's effectiveness at removing the blurred

images of out-of-plane structures from the target plane was quantified in terms of the

SNR, CNR and SBR. Imaging and reconstruction was also performed on a cadaver

breast specimen.

The technology proved effective at partially removing out of focus structures and

enhancing SNR, CNR and SBR. However, spatial resolution was slightly degraded. The

145 metrics SNR, CNR and SBR all improved as the number of projections was increased.

The ART technique created images with greater clarity, and less blur, than did the iterative subtraction technique.

146