MEDICAL IMAGING 2nd Part Computed Tomography

Transcription

1 MEDICAL IMAGING 2nd Part Computed Tomography

2 Introduction 2 In the last 30 years X-ray Computed Tomography development produced a great change in the role of diagnostic imaging in medicine. In convetional projection radiograpgy the image represents the 2D projection of a 3D structure. To obtain information on the real structure the number of projections have to be increased A tomogram is an image of a slice or a plane within the body The basics of CT is to take a series of conventional cross-sectional x-rays while the patient is rotated slightly around an axis between each exposure. A series of projection data is obtained, thes data are used to reconstruct cross-sectional images.

3 Introduction : The austrian mathematician Radon develop a way to reconstruct the density distribution of an object if the line integrals (for every direction) are available : The physician Cormack develops a lot of theoretical work on x- rays creating the basis of CT scanning (without knowing Radon work) 1971: The first CT system has been created by Godfrey N. Hounsfield, he received the Nobel prize in 1979 (together with Cromack) 1972: The first clinical use of a CT system in London, the acquisition time of a couple of tomogram was 5 minutes

4 Introduction : The first II Generation CT system total-body scanner has been realised in the U.S., the acquisition time for 1 tomogram was 1 minute : III and IV Generations CT with acquisition time for 1 tomogram lower than 10 secs 1983: First electron beam CT, very expensive (scarce diffusion) 1989: First helical CT, very low acquisition time (less than 1 sec) and able to explore a large body volume 1992: Dual-slice CT scanner, able to acquire a large number of layers in a very low time 2000: Multislice CT scanner, multiple arrays of detectors; continuous development.

5 Computed Tomography Generations First-generation scanner Second-generation scanner 5 Single source collimated in a pencil beam and single detector Linear scan Arbitrary number of rays and angular projections can be measured, scattered radiation is mostly undetected Partial fan-beam and detectors array. Linear scan Able to complete a full scan in less time than I Generation scanners

6 Computed Tomography Generations Third-generation scanner Fourth-generation scanner 6 Fan-beam and detectos array. Source and detectors rotate in synchrony Dramatic decrease of scan time Fan-beam and large ring of stationary detectos array. Single rotating source Collimators cannot be used in detectors (they must receive energy from a mobile source) high detection efficiency and scattering can cause problems

7 Computed Tomography Generations 7 Main characteristics of the first four CT scanners generations I II III IV Scan Time (sec) Number of detectors (for each layer) Movement Rotation + Transl. Rotation + Transl. Rotation Rotation X-ray tube rotation Data amount (each tomogram) 57.6 kb 57.6 kb 1 MB 2-42 MB Matrix 80x80 80x80 256x x512

8 Computed Tomography Generations Fifth-generation scanner: electron beam computed tomography (EBCT); no moving parts, a electronically steered electron beam is used, this beam hit one of the four tugsten anode strip encircling the patient. This produces x-rays that are collimated in a fan-beam and revealed by a stationary ring of detectors. A full set of data are acquired in 50 milliseconds. Very expensive Sixth-generation scanner: helical CT addresses the need for rapid volumetric data acquisition. Conventional arrangement of source and detctors (III and IV generations) which can continuously rotate. During tube rotation patient table is slightly moved through the sourcedetectors plane. The image acquisition is not sequential (as it is for previous CT generations) but volumetric 8 A) sequential scanning B) helical scanning

9 Computed Tomography Generations Seventh-generation scanner: multislice CT scanners, a thick fan-beam is used and multiple detectors array are used to collect the x-rays. This technology is in continuous development, recently (and 256) multiple detectors array has been developed. 9 With the 256-slice system the entire heart can be covered in a single rotation Helical and multislice CT has made the requirements for new development in data processing techniques even more critical Future development: multislice scanners able to interact with a cone-beam Due to the large volume that can be explored it will be possible to stop using mobile patient table, this will decrease the complexity in image reconstruction

10 CT Instrumentation 10 Most of the commercial medical CT scanners use just one x-ray source. X-ray tubes usually wear out in less than one year and need frequent status control and calibration. X-rays generated by the tube require collimation and filtration. Typically fan-beam geometry (having degress of fan angle) is obtained through two pieces of lead that form a slit between them. Slit width is adjustable by the operator in order to obtain different fanbeam thickness, typically between 0.5 and 10 mm for single slice systems and 20 to 30 mm for multislice CT systems. Fan-angle Thickness mm X-rays fan-beam

11 CT Instrumentation 11 Different types of CT detectors: Solid state detectors contain a scintillation crystal able to interact with x-rays and produce photoelectrons that are captured by a photo-diode and converted to electric current Xenon gas detectors long thin tubes containing compressed xenon gas, when ionized the gas generates a current between anode and cathode. Less efficient than solid state detectors but highly directional Slip ring solves the mechanical problem of continuous electrical contact with rotating gantry; the gantry holds the x-ray tube (that needs high voltage to work properly) and the detectors that generate hundreds of signals to be transmitted out

12 Data Acquisition If a thin beam of monoenergetic x-rays, having an intensity of, propagates for a distance d through an omogenic material, the intensity of the attenuated beam is: I 0 12 I = Ie µd 0 I0 µ I Being µ the linear attenuation coefficient µ is a function of material atomic number and electronic density d If the beam propagates through different omogenic materials the intensity of the attenuated beam is: d µ i ( µ 1+ µ µ n) d i= I = I e = I e n I0 x µ 1 µ 2 µ n d I

13 Data Acquisition 13 If the material has a continuous variable attenuation coefficient the intensity of the attenuated beam is: I = I e 0 0 D dx Given a measurement of and knowledge of, the projection measurement is defined as: p I 0 I0 I = ln( ) = I 0 D µ dx Therefore the basic measurement of a CT scanner is a line integral of the linear attenuation coefficient x I 0 x D I

14 Data Acquisition A CT scanner has to be calibrated in order to obtain the reference intensity I 0 Multiple projections (along parallel lines) create a so-called attenuation profile 14 CT detectors are able to measure the line integrals of bidimensional distribution of linear attenuation coefficients Through a suitable number of attenuation profiles it is possible to identify the bidimensional distribution of attenuation coefficients and it is possible to reconstruct the bidimensional image. Attenuation profile

15 CT Numbers 15 Different CT have different x-rays tubes, so the same object will produce differenc values of µ on different scanners (the value of µ is a function of: x-ray energy, tissue density and atomic number) In order to compare data from different scanners is not possible to calculate absolute values of linear attenuation coefficents but at each pixel is associated a numerical value called CT number A CT number is calculated in Hounsfield units (UH) and is defined as: µ w µ t µ w h = 1000 µ Being the water attenuation coefficient and the tissue attenuation coefficient A CT number is 10 times the percentual difference between µ and t µ w divided by µ w w µ t

16 CT Numbers 16 As an example, a tissue having a attenuation coefficient 4% bigger than water has a CT number: h t (1.04 µ w µ w) = 1000 = 40 µ w Clearly h=0 for water, that is the reference value. Since µ = 0 for air, it is possible to find that h=-1000 for air. The largest CT number found naturally in the body are for bones, 1000 HU for average bone. CT numbers can surpass 3000 HU for metal and contrast agents A typical HU range is: / 3071 (4096 levels, 12 bits each pixel)

17 CT Numbers Examples of typical HU values 17

18 CT Numbers Windowing 18

19 CT Numbers Conventional CT scan without contrast showing possible tumor in the liver Conventional CT scan of the same patient using contrast

20 Image Formation 20 CT images are reconstructed as a numerical bidimensional matrix. Each element of the matrix is called pixel The number of pixels is variable between 256x256 and 1024x1024 For each couple of x and y, the pixel represent a micro bidimensional area and f(x,y) is the functional value of this area f(x,y) is a grey level value, positive integer power of 2 NxN matrix As a matter of fact this is a bidimensional representation of a tridimensional tissue volume f(x,y)

21 Image Representation 21 Each pixel of the bidimensional matrix represents the properties of a tissue volume called voxel Voxel dimension is a function of the field of view (FOV) In general the voxel cross-sectional plane dimensions are lower than the one of z axis (along-ray), voxels are parallelepiped Spatial resolution along z axis is lower than the one along x and y axes Anisotropy pixel y voxel x z

22 Parallel Ray Reconstruction 22 The basic CT measurements is a line integral of the effective linear attenuation coefficient within a cross-section µ We desire a picture of or a CT number over the entire cross-section. Can we reconstruct a picture of given a collection of its line integrals? µ The answer is yes! A line in the (x, y) plane is given by: { } L(, l θ) = ( x, y) xcosθ + ysinθ = l The unit normal to the line is oriented at an angle relative to x axis and the line is at distance l from the origin in the direction of the unit normal θ

23 Parallel Ray Reconstruction 23 The line integral of function f(x,y) is defined as: gl (, θ) = f( xy, ) δ( xcosθ + ysin θ ldxdy ) Being the line impulse defined as: δl ( xy, ) = δ( xcosθ + ycos θ l) The inpulse shifting property causes the integrand to be zero everywhere except on the line For a fixed θ glθ (, ), is called a projection θ l glθ (, ) For all and, is called the 2D Radon Transform of f( x, y)

24 Parallel Ray Reconstruction 24 If we make the following identifications f( x, y) = µ ( x, y) Unknown function to be reconstructed by CT I gl (, θ ) = ln( ) I CT Measurements It is possible to see that this mathematical abstraction characterizes the CT measurement situation It has to be noticed that this definition of projection corresponds to a collection of line integrals of parallel lines. This correspond to the geometry of 1G scanners only. However this formalism and methods can be applied also to reconstruct fan-beam projections. 0

25 Sinogram 25 glθ (, ) An image of with l and θ as rectilinear coordinates is called a SINOGRAM A sinogram is a pictorial representation of the Radon transform of and represents the data that are needed to reconstruct it There is no need to calculate the sinogram for angles higher than π since these projections are redundant f( x, y)

26 Backprojection glθ (, ) f( x, y) Let be the 2D Radon transform of. Consider the projection at θ = θ 0. There are an infinite number of functions that could give rise to this projection; it is not possible to determine uniquely form a single projection f( x, y) glθ (, ) l = l f( x, y) 0 Ll (, θ ) If takes on a large value at, then must be large 0 over the line (or somewhere over the line) 0 0 One way to create an image exploiting this property is to simply assign every point on the value of Ll (, θ ) gl ( 0, θ0) 0 0 l If we repeat this for every, the resulting function is called backprjection image and is given by bθ ( x, y) = g( xcosθ + ysin θθ, ) The backprojection image at a angle is consistent with the projection at θ 0 but its values are assigned with no prior information about the distribution of the image intensities θ 0 26

27 Backprojection It is possible to add-up backprojection images for every angle, obtaining the backprojection summation image, also called laminogram (, ) π f (, ) b x y = bθ x y dθ 0 27 Not the correct way to reconstruct f( x, y) Backprojection for θ = 30 Laminogram (blurriness)

28 d µ r 0 I Ie µ r = d 0 Backprojection I Ie µ r = d Backprojection assumption x x x x x I = I0e µ µ µ µ µ µ = µ /5 x ( d + d + d + d + d) r Object density is distributed along the whole x-ray path Star-like Artifact Backprojection (2 projections) Backprojection (4 projections)

29 Projection-Slice Theorem 29 Lets take the 1D Fourier transform of a projection with respect to l πρl G( ρθ, ) F gl (, θ) gl (, θ) e dl = = 1D Being { } j2 ρ the spatial frequency glθ (, ) If we substitute the analitical expression of it is possible to obtain j2πρl G( ρθ, ) = f ( x, y) δ( x cosθ+ y sin θ l) e dxdydl j2 l f( x, y) πρ δ( xcosθ ysin θ l) e dldxdy = + = f ( x, y) e j2 πρ ( xcosθ + ysin θ ) dxdy

30 Projection-Slice Theorem 30 The last expression is reminescent of the 2D Fourier transform of j2 ( xu yv) F( u, v) = π + f ( x, y) e dxdy Being u and v the frequency variables in the x and y directions u = ρ cosθ v = ρ sinθ G( ρ, θ) = F( ρcos θ, ρsin θ) f( x, y) Making the following identifications: and leads to a very important equivalence, called the projection-slice theorem The 1D Fourier transform of a projection is a slice of the 2D Fourier transform of the object. The 1D Fourier transform of the projection, calculated along a direction, equals the Fourier transform of the object calculated along the same direction passing through the origin

31 Projection-Slice Theorem Graphical representation of the theorem 31 ρ θ and can be interpretated as the polar coordinates of the 2D Fourier transform

32 The Fourier Method 32 A very simple image reconstruction method follows immediately from the projection-slice theorem Take the 1D Fourier transform of each projection, insert it with the corresponding correct angular orientation into the correct slice of the 2D Fourier plane, and take the inverse 2D Fourier transform f( x, y) = F G( ρ, θ ) 2D 1 { } This method is not widely used in CT due to some practical problems: Interpolating polar data onto a Cartesian grid 2D inverse Fourier transform is time-consuming

33 Filtered Backprojection 33 The inverse Fourier transform of F( uv, ) can be written in polar coordinates 2π 2 ( cos sin ) (, ) ( cos, sin ) j πρ f x y F e x θ + y θ = d d 0 0 ρ θ ρ θ ρ ρ θ Using the projection-slice theorem π 2 ( cos sin ) (, ) (, ) j πρ f x y G e x θ + y θ d d 0 = ρ θ ρ ρ θ The term xcosθ y being gl (, θ ) = g( l, θ + π) θ + sin is constant considering integration over ρ f x y ρ G ρθ e dρ dθ π 2 (, ) (, ) j πρ = l 0 l= xcosθ+ ysinθ

34 Filtered Backprojection 34 The inner integral of the previous formula is an inverse 1D Fourier transform ρ The term makes it a filtering equation. The Fourier transform of a projection is multiplied by a frequency filter and inverse-transformed After the inverse-transform, the filtered projection is backprojected (this is done replacing l = xcosθ + ysinθ ), and then is performed a summation of all filtered projection ρ This reconstruction approach is called filtered backprojection and is faster and more flexible with respect to the Fourier method The term ρ is called ramp filter

35 Convolution Backprojection 35 From the convolution theorem of Fourier transforms follows π 1 { } 0 1D l= xcosθ+ ysinθ f( x, y) = F ρ g( l, θ ) d θ Defining cl () = F 1 1D { ρ } π 0 [ θ ] = cosθ+ sin f ( x, y ) = c ( l ) g ( l, ) d l x y θ θ π = gl (, θ ) cx ( cosθ + ysin θ ldld ) θ 0 Substitution of the convolution integral to obtain the equation for convolution backprojection

36 Convolution Backprojection 36 Performing a convolution rather than a filtering operation is generally more efficient if the inpulse response is narrow (it is so in this case) cl () does not exists since ρ is not integrable In practice ρ is windowed with suitable windowing functions Convolution backprojection algorithms use an approximate impulse response cl () = F D W 1 1 { ρ ( ρ )}

37 Reconstruction 37 Filtered backprojection uses three basic steps to reconstruct an image from a sinogram: filtering, backprojection and summation Convolution backprojection uses the same steps except that filtering is performed using a convolution Filtering step: each row of the sinogram is filtered through a ramp filter to obtain the filtered sinogram Ramp filter is essentially a high-pass filter and its value for f=0 is 0 High-frequency details are accentuated

38 Reconstruction 38 Backprojection is applied to each filtered projection, rather than to the raw data Last setp is the summation of filtered backprojections This step is performed using an accumulator concept No need to store hundreds of backprojection images

39 Reconstruction 39

40 CT Image Quality 40 CT image quality can be expressed as a function of: Spatial resolution in the transversal plane xy Longitudinal resolution and slice sensitivity profile Noise Some indicators of CT image quality will be introduced Contrast Artifacts

41 CT Image Quality 41 Spatial resolution is a function of the smalles details that can be identified in the image; it is stricly dependant on the detectors (and, in general, on the intruments used for data acquisition and reconstruction). Spatial resolution is measure of how closely lines can be resolved in an image, it is measured in mm or LP/cm (line pairs/cm). The clarity of the image is not the number of pixels in an image, in fact spatial resolution refers to the number of independent pixel values per unit length. Detectable lines Undetectable lines

42 CT Image Quality 42 Pixel dimensions can be considered the upper limit of the spatial resolution It is not possible to identify details having dimensions lower than pixel size Details having dimensions comparable to the pixel ones provides contributions to adjacent pixels

43 CT Image Quality 43 Slice sensitivity profile (SSP) is the impulse response of the CT system along the z axis (patient long axis). Thick slice Thin slice

44 CT Image Quality Image width affects contrast and noise of object Optimised slice width: imaged slice object size 44 Object 5mm 5mm Low contrast Low noise 1mm Better contrast More noise

45 CT Image Quality Noise is the random variation of color (grayscale), some regions of the image that should be uniform show coarseness 45 Noise is produced by the sensors and circuitry of the scanner Noise intensity can be evaluated considering a region of the image; the mean pixels intensity of the region is calcualted as well as standard deviation standard deviation Noise = % mean intensity Noise can be reduced through reconstruction filters Noise is in inverse proportion to layer thickness and radiation dose root square

46 CT Image Quality 46 Different kind of filters can be used in backprojection to optimize spatial resolution against noise: smooth, standard, detail, bone, etc. Smooth Filtering Low contrast Low noise Sharp Filtering Better contrast More noise

47 Artifacts 47 Artifacts can appear in CT images, due to different reasons Aliasing: if projections are undersampled higher frequency information will appear as lower frequency artifacts. An high number of detectors is needed, it is needed to sample at numerous angles Beam hardening: when travelling through te body the x-rays beam spectrum component with low energy is almost completely absorbed and the mean energy increases (this artifacs affect the bones) Other: electornic or system drift (miscalibration), x-ray scatter (can be prevented using collimators at expense of lower efficiency), motion (typically a scan require 1 to 10 secs) Beam herdening artifact