Objectives to understand:

1. Formation of tomographic sections using geometrical blurring. Dependence of section thickness on angle of rotation.

2. Acquisition of CT data as projections. Mathematical description of a projection.

3. Iterative reconstruction.

4. The Projection Theorem and associated mathematics.

5. Filtered Back Projection and associated mathematics.

6. CT exposure requirements.

7. Definition of CT image values.

8. Spiral CT and CT angiography.

Tomography objectives (continued):

In conventional radiographic projection imaging, the attenuation effects from material situated throughout the x-ray path are superimposed at the detector plane, as shown below.

•Source

Detector

This can lead to considerable ambiguity, for example when a low contrast object is superimposed on a dense or anatomically complex object.

The object of tomography is to obtain images corresponding to the attenuation distribution within a planar section through the body.

Another problem with conventional radiographic projection imaging is the presence of x-ray scatter, which makes it difficult to make quantitative assessment of attenuation values. The first of these problems was addressed by conventional film tomography sometimes called blurring tomography. Eventually, both were addressed by computed tomography (CT).

First let us consider the blurring approach which was introduced circa 1930 by Ziedses Des Plantes in Holland (in the same PhD thesis which introduced film subtraction angiography--try to beat that when you choose your thesis topic!)

The basic geometry for blurring tomography is shown in below.

••

Detector motion

Source motion

AB

A B AA B

A and B are objects at two different depths within the patient. By coordinating the motion of the detector and the source, objects such as A in the plane of rotation remain stationary on the detector. Objects such as B, which are outside the plane of rotation, are smeared out as the source and detector are rotated. The degree of blurring increases with the angle of rotation. Therefore the thickness of the in-focus plane decreases with increased rotation.

Several different types of blurring may be used. For example, linear, circular or hypocycloidal motions are used and result in different characteristic blurring artifacts. The choice of motion is made by considering which type of artifact is less likely to mimic the anatomy which may be under study.

Figure 3A below shows a frontal conventional radiograph of the head.

In Figure 3B, blurring tomography has been used to select a plane through the maxillary sinuses. The arrows point to a mass in one of the maxillary sinuses.

A B

In computed tomography, x-rays pass in a direction approximately transverse to the longitudinal axis of the body and pass only through the slice of interest, thus overcoming the problem of overlapping information from adjacent slices. The goal is to obtain an image representing the two dimensional distribution of attenuation coefficients at each picture element in the image. Various mathematical schemes may be used to reconstruct the image from the projection information.

An advantage of the slice geometry, analogous to that of the line scanned x-ray systems discussed in the section on scatter, is that scatter is greatly reduced. Additionally, in CT, electronic detectors capable of quantitative recording of the transmitted information are used. These two factors were essential in providing the revolutionary contrast resolution made available by CT in the mid 1970’s.

Figure 4 illustrates the concept of projection data. At several angles around the edge of the slice several ray paths, collectively called a projection, are passed through the slice of interest. Labeled in Figure 4 are the first projection P 1 and three of its rays and the ith projection P i and its jth ray pij.

In Sir Godfrey Hounsfield’s original scanner, there were eighty rays per projection and the image was represented by an 80 x 80 matrix. One might think this was the result of some advance mathematical optimization procedure. The fact is he was advancing his source and detector using a screw mechanism which had eighty turns.

Figure 4

Hounsfield’s scanner, for which he earned the Nobel prize in Medicine was called a rotate-translate scanner. Following rotation to a new projection angle, a linear ray scan was done to collect all the ray data for the projection.

Modern scanners use a continuously rotating gantry and arrays of detectors so that at each projection angle, all ray transmission values are simultaneously collected. There was an evolution of source and detector configurations leading to several generations of scanners.

Figure 5 below shows a third generation scanner, in which an array of several hundred individual detectors are rotated along with the x-ray tube in order to acquire the projection data.

Also shown is a fourth generation scanner in which only the x-ray tube undergoes circular motion and the radiation is detected by a stationary ring of detectors.

Finally, Figure 5 shows a more recently developed electron beam scanner configuration in which an electron beam is magnetically deflected along the surface of a cone and impinges on a circular target. X-rays are transmitted through the patient and detected by a slightly offset detector array. This type of scanner is particularly well suited to cardiac applications where scan times on the order of tens of milliseconds are desired.

Reference detector

IR

I0Iij

r

In each case the x-ray beam is collimated to define the desired slice and reference detectors just outside the slice are used to monitor beam intensity fluctuations.

Use of the reference detector to normalize the projection data is illustrated below in Figure 6.

The detected intensity along one ray in the projection is given by

(1)

where r is measured along the ray. The overall attenuation is due to all attenuation elements along the ray.

The reference detector provides a signal proportional to the incident intensity according to

The basic objective of CT is to solve for the attenuation values uij

at each point in the slice given the complete set of line integrals Pij.

We can get an approximate estimate of how many projections are required by referring to Figure 7 below which shows rays incident on a subject of radius D which is to be represented as an image with pixel size d.

Dd

The number n of pixels per diameter, which is equal to the numberof required rays per projection is given by

(5)

The total number of pixels, which is equal to the number of unknown attenuation coefficients in the image is then given by,

Therefore the number of projections with n rays per projection is given by

This analysis ignores the fact that pixels far from the axis of rotation are probed by a smaller number of rays than those near the center. When a more careful analysis is done in terms of the sampling of spatial frequencies, the required number is twice that calculated.

It should also be noted that in present fan beam scanners the rays are divergent and algorithms are required to obtain the information provided by an equivalent set of parallel rays which lend themselves more easily to mathematical analysis.

Figure 8, from Krestel-Imaging Systems for Medical Diagnosis, shows two scans of a CT phantom, one obtained with 180 projections (A) and the other (B) with 720 projections. Clearly the 180 projection scan is inadequate while the 720 projection scan is significantly better.

willi angles movie

Kalender et al

Kalender et al

Iterative Reconstruction

The early CT scanners used iterative image reconstruction schemes. These are time consuming, especially with the 512 x 512 image matrices used on modern scanners. Nevertheless, they are somewhat interesting. The main idea will be illustrated below for the simple case of a 2 x 2 matrix. Suppose the attenuation coefficient image is as shown in Figure 9a.

5 7

6 2

12

5 7

6 2

8

11 9

6 6

4 4

1st estimate

13 7

6 6

4 4

10 10

6.5 5.5

4.5 3.5

2nd estimate

6.5 5.5

4.5 3.5

10 10

Even in the original Hounsfield scanner there were 6400 pixels. One can imagine that this process would be time consuming even for a reasonably fast computer using modern matrices of over 250,000 pixels.

We will now examine the reconstruction of the attenuation coefficient image using two dimensional Fourier transform methods.

A plane wave is a two dimensional function of the form

(8)

characterized by sinusoidal variation in the direction of the propagation vector , where

This is illustrated in Figure 10 for the case of a wave

We have already seen that a one dimensional intensity distribution can be represented as a Fourier integral representing a continuous summation over sinusoidal functions of various frequencies. The extension of this idea is that a two dimensional function, such as an attenuation coefficient distribution (x,y) can be represented as a sum of a continuous distribution of two dimensional sinusoidal plane wave functions as

where is the two dimensional Fourier transform of u(x,y) representing the weightings of the plane waves having spatial frequency coordinate (kx,ky)

If it were possible by means of x-ray projection data to obtain the weighting coefficients , the attenuation distribution

could be found from equation 11. This is made possible by a mathematical relationship called the Projection Theorem.

Consider a coordinate system with the x’ direction perpendicular to the direction of the x-ray projection as shown in Figure 11.

For a given projection anglejthe detector array measures a projection consisting of detected rays

(13)

where the y’ integral is over all attenuation contributions in the patient.

Suppose that we now take the one dimensional Fourier transform of these detected values, obtaining

where the integrals cover the entire area of the slice.

If we compare equation 14 with equation 12, we obtain,

(15)

Therefore by performing the projection measurement and transforming the detected values, we obtain the Fourier expansion coefficients along the kx’ axis in frequency space with the kx’ axis making an angle relative to the kx axis.

The number of samples in k space will be equal to the number of elements in the detector array. However it should be realized that each data point in k space is computed from an integral over all of the detected ray values.

By changing the projection angle, the expansion coefficient values throughout k-space can be acquired. These are arranged in radial fashion as shown in Figure 12.

The Nyquist Theorem requires that the sampling intervals

kand kr be equal. Therefore

Where Np is the number of required projections and Nr is the number of samples along with each projection. This exceeds our previous simple estimate by a factor of two.

k = kmax */Np

kr = 2*kmax/Nr

Np = Nr/2

Once k-space has been filled up the data can be interpolated into a rectangular grid so that the attenuation image can be obtained from equation 11.

(11)

One problem with the use of the projection theorem to obtain the data needed for the calculation of the image from equation 11 is that all of the projection data must be obtained prior to image reconstruction.

This means that there is a delay in the appearance of the image following the scan. We will now discuss a technique which permits image reconstruction to occur simultaneously with data acquisition.

First we will discuss simple back projection, and then add a refinement required to remove image artifacts. Consider an object in the patient slice and the attenuation it produces in various projections as shown in Figure 13A.

If it is assumed that the attenuation associated with each detected projection is due to a uniform distribution of attenuation along the projection direction, this attenuation value may be projected back along the projection direction as shown in Figure 13B.

The mathematical representation of the back projected image B(x,y) is given by

(16)where

(17)

and lies along the x’ projection axis as shown in Figure 14.

This back projected information will reinforce at the location which produced the attenuation signal recorded in each projection.

However this simple scheme leads to the so called “Star Artifact” caused by the residual back-projected intensity outside the object.

We can do better by considering how information from an object is propagated to remote pixels in the image, in other words by considering the point spread function of the back projection operation.

Figure 15 shows the process by which an attenuation contribution from pixel A is communicated to a distant pixel B.

B ∆xB

A

RAB

The total fraction of the attenuation value of pixel A erroneously communicated to pixel B is proportional to the fraction F of back projected rays through pixel A which intercept pixel B. This is given by

The contribution of one pixel to another falls off like 1/ R where R is the distance between a pixel at r and a pixel at r’ as shown in Figure 16.

y

x

r

r’

r-r’ = R

This spreading of information from one point to another is equivalent to a convolution of the true image with a 1/R point spread function. The back projected image B(x,y) is related to the true image (x,y) by

The solution for (x,y) knowing B(x,y) and the point spread function is called Filtered Back Projection.

Recall that the convolution theorem states that if

That the Fourier transform of 1 / R is 1 / k is presented without proof. However, Figure 17, which represents results obtained with an image processing program called Alice (Hayden Image Processing) shows the 1 / R point spread function (A) and its Fourier transform (B).

A B

The profile through A, shown in C indicates the 1 / R behavior, while the profile through B, shown in D, indicates the 1 / k behavior. Similar behavior is found for all rays through these radially symmetric distributions.

C D

Equation 21 states that the k space representation of the back projected image is wrong by a factor of 1/k.

(21)

One solution to obtain the correct attenuation distribution would be to;

1 Back project to obtain B(x,y)

2 Take the Fourier transform to get

3 Multiply by k to get

4 Take Fourier transform to get (x,y)

Although this would work, a more efficient way which allows image reconstruction to proceed before all of the data is collected is to correct the projections P (x’,) prior to back projecting in order to remove the effects of the point spread function.

In order to do this we note from equation 21 that the k-space representation of the back projected image is wrong by a factor of 1 / k. The procedure usually used is to multiply this image by a filter function designed to remove the 1 / k mistake, i.e.,

Although the temptation is to just use , the filter function

is usually truncated at the maximum k value which can be adequately sampled by the pixel matrix.

The filter function and its Fourier transform are shown in Figure 18 along single lines in each space. The width and shape of R(x’) depend on the details of the pixel size and the chosen cutoff frequency.

Figure 18

The image space version of equation 22 is, according to the convolution theorem,

(23)

Substituting for the back projected image from equation 16, we have

Although the convolution integral in equation 24 is nominally a two dimensional integral, the projection data exists only at y’ =0, and the operation amounts to a linear convolution of each projection with the radial component of the filter function, which itself has only radial dependence.

The filtered back projection procedure then amounts to;

1 Measuring projections P(x’,)

2 Convolving with R(x’)

3 Back projecting according to equation 24.

The effect of the filtering is shown in Figure 19 where the projection data adds constructively at the location of a point object at point A, but tends to cancel at a point B just away from the actual point.

An estimate of the exposure required to see an attenuation difference we can make a couple of assumption and apply the signal to noise equations previously derived, namely

(25)

Where N0 is the required input fluence (assuming unit detector efficiency),S/n is the required signal to noise ratio, C is the contrast, A is the projected area, and T is the object transmission.

To get an approximate estimate for CT we assume that the required exposure is the same as would be required if all of the exposure were made in one projection and that the reconstruction process does not add significant noise to the image(which is usually the case). Two volume elements (voxels) to be distinguished are shown below.

The pixel size is d on each side, the slice thickness is t. The area is given by

The contrast is given by

Equation 25 then becomes,

(26)

Note the strong dependence on pixel size d. This is due to the fact that it enters into the projected area and the contrast in this geometry.

Suppose we wish to see a 0.6% difference in attenuation with signal to noise ratio of one. Assume that the average tissue coefficient is 0.19 cm- 1, t= 1cm, d = 1.5 mm and a patient thickness of 30 cm. This gives

and a required input fluence N of

Converting to exposure E in Roentgens we get,

An example of an abdominal CT image (Medicamundi 34/3) is shown in Figure 22.

The liver, spine and kidneys are clearly evident. The bone is brightest due to its high attenuation.

CT image values are usually stated in relative attenuation units rel

which are a scaled percentage of the attenuation difference relative to water as

The relative attenuation coefficients for k = 1000 are shown in Figure 23 taken from Krestel-Imaging Systems for Medical Diagnostics

CT images are usually displayed by selecting a range (window) of attenuation values centered at a chosen attenuation value (window level). Values within this window are digitally enhanced to fill the dynamic range of the CT video display. This permits detailed viewing of structures which are finely separated in attenuation.

Level

window

In conventional computed tomography the slices are obtained by incrementally moving the patient couch and repeating the scan. In this approach the spatial resolution in the slice direction is determined by detector size and collimation.

Another approach, which was proposed and developed by a former UW Medical Physics graduate, Professor Willi Kalender, now at the University of Erlangen, is to acquire data as the patient is continuously moved through the scanner.

The data required to reconstruct a single slice can be chosen from any portion of the continuous data set providing a continuous distribution of average slice position.

Interpolation of data is used to produce a consistent set of data corresponding to a single perpendicular slice through the patient.

Lieber Chuck:

• Spiral acquisition is great!

• The future is MRI!

• But some ignorants will continue on multislice and cone-beam spiral CT ....

• Thank you for your support!

Dein WilliSpiral Galaxy M88

The net result of this process is an improvement in resolution in the slice direction and more rapid acquisition of volume data. The scanning geometry is shown in Figure 24.

One of the most promising applications of this approach is the generation of angiographic images following the intravenous administration of iodinated contrast material.

Contrast is injected through an arm vein or through a catheter placed in the superior vena cava. Following a time delay to allow the contrast to go through the heart, the spiral scan is begun.

Rubin et al, Dx Imaging Jan. 1993

CTA

XRA

Arterial images are generated from the 3D data set using a projection algorithm such as the Maximum Intensity Pixel (MIP) algorithm. Shaded surface displays, such as in the image below are also used. The presence of calcium can be a source of image degradation.

The spiral CT technique has a distinct advantage over previously tried intravenous injection angiographic techniques in that it produces a full 3 dimensional data set which can be used to reconstruct views from arbitrary directions. Previously tried techniques such as intravenous DSA required additional injections in order to obtain new views.

A disadvantage which the spiral CT approach shares with intravenous DSA (discussed in the next section) is the uncertainty in the timing of the imaging and the arrival of the contrast material.

Technical development & clinical results

Cardiac CT Imaging

Drs J.L. Sablayrolles, F.Besse, C.Jardin, J.C. Roy, Q.SénéchalCentre Cardiologique du Nord - Saint Denis - FranceP.Giat, C.Coric GE Medical Systems

Cine-CT of prosthetic mitral valve in axial view

Cine-CT of prosthetic aortic valve in coronal view

Electron Beam CT Coronaries

Moshage et al.Radiology 1995196:707-714

Since CT acquires a three dimensional data set, the data can be viewed in easily interpretable surface and volume rendered displays, especially when spiral CT is used to improve resolution in the slice direction. Such displays have been very useful in planning surgical approaches. An example of such a display is shown in Figure 26 which shows a surface display of the bones in the region of the hip.

The 3D data can be used to digitally dissect the body enabling the surgeon to look inside various cavities and ascertain with precision the relationships between anatomical objects.

Computed Rotational Angiography

R Fahrig, AJ Fox, S Lownie and DW HoldsworthAJNR 18:1507-1514 Sept 1997

CRA 2D

Micro CT

Holdsworth and Thornton Microfil MV-122 contrast agent

Z

Time

SignalArtery Vein

Conventional Whole Body CTA Geometry

Limitations of Current Contrast-Enhanced CT Angiography

45 sec 59 sec 81 sec

Time-resolved MR Angiography

Asymmetric enhancement: A-V transit timeT. Carroll, J. Du et al

Z

Time

SignalArtery Vein

Whole Body Time-Resolved Acquisition

BestCase

WorstCase

Current Multiple Rotation Coronary CTA

Z-Scan CT

Z

Direction of conventional table motion

Conventional gantryRotation angle.

Linear X-ray sourceproduced by • scanned electron beam or• discrete pulsed source distribution

Rotating detector array + focused grid

Patient table

Z

Cone Beam CT

Cone Beam Detector