Kedge subtraction vs. Aspace processing for xray imaging of contrast agents: SNR
Abstract
Purpose: To compare two methods that use xray spectral information to image externally administered contrast agents: Kedge subtraction and basisfunction decomposition (the Aspace method),
Methods: The Kedge method uses narrow band xray spectra with energies infinitesimally below and above the contrast material Kedge energy. The Aspace method uses a broad spectrum xray tube source and measures the transmitted spectrum with photon counting detectors with pulse height analysis. The methods are compared by their signal to noise ratio (SNR) divided by the patient dose for an imaging task to decide whether contrast material is present in a soft tissue background. The performance with iodine or gadolinium containing contrast material is evaluated as a function of object thickness and the xray tube voltage of the Aspace method.
Results: For a tube voltages above 60 kV and soft tissue thicknesses from 5 to 25 g/cm^2, the Aspace method has a larger SNR per dose than the Kedge subtraction method for either iodine or gadolinium containing contrast agent.
Conclusion: Even with the unrealistic spectra assumed for the Kedge method, the Aspace method has a substantially larger SNR per patient dose.
Key Words: spectral xray, Kedge subtraction, basis decomposition, contrast agent, photon counting,
1 Introduction
Kedge subtraction was one of the first methods to use xray spectral information to improve the visibility of contrast agents injected into the body^{2, 3, 4}. An alternative method utilizing spectral information is the basis decomposition method^{5} (the Aspace method). Alvarez^{6} showed this method can be used to provide near optimal signal to noise ratio (SNR) ^{7} with low energyresolution measurements. Although both methods use spectral information, they are quite different and an interesting question is which one provides a better SNR per dose for detecting an externally adminstered contrast agent in a soft tissue background?
This paper examines that question for idealized spectra and detectors. For Kedge subtraction, monoenergetic spectra with energies just below and above the Kedge energy and a quantum noiselimited, negligible pileup photon counting detector are used. For the Aspace method, a broad spectrum xray tube source is used with an ideal photon counting detector with pulse height analysis (PHA).
The imaging task is to detect contrast material embedded in soft tissue. The signal to noise ratios of the two methods are compared for equal dose, which is approximated as the absorbed energy. Since, in general, the square of the SNR is proportional to dose, the parameter compared is . This parameter is computed as a function of softtissue object thickness and the xray tube voltage for contrast agents containing iodine or gadolinium.
The imaging task does not measure the full capability of either method. The Aspace method extracts a great deal more information about the object than simply the presence of the contrast agent^{8, 9, 10}. The Kedge subtraction method can discriminate against body materials with different compositions so long as their attenuation coefficient is continuous at the Kedge energy. Nevertheless the imaging task used allows us to directly compare the fundamental performance of the two methods for an important clinical application. More general comparisons including the presence of other body materials and practical limitations on the xray source spectra for the Kedge subtraction method are under research.
Recent examples of work in this area include Roessl and Proksa^{11}, who applied the Aspace method to image contrast agents as did Zimmerman and Schmidt^{12}. Dilmanian et al.^{13} used a synchrotron radiation source at a nuclear physics laboratory to image contrast agents with Kedge subtraction. Shikhaliev^{14} prefiltered a broad spectrum source with high atomic number materials to provide a bimodal spectrum. The filtered transmitted spectrum was measured with a photon counting detector with PHA and the SNR of the contrast material thickness was computed. None of these papers compared the Kedge subtraction method to the Aspace method directly.
2 Methods
In this section, the imaging task for the signal to noise ratio definition is described. Then expressions for the SNR and absorbed energy with the Kedge subtraction and Aspace methods are derived. Finally, the SNR per absorbed energy is computed as a function of the object thickness and the tube voltage for contrast agents containing iodine or gadolinium.
2.a The imaging task
The imaging task assumes the object shown in Fig. 2.1. The task decides whether a contrast material is present from measurements of the transmitted xray energy spectrum.
Assuming normally distributed noise, the probability of error depends only on the signal to noise ratio^{15}, where the signal is the square of the difference in the expected values of the measurements between the soft tissue only and the soft tissue plus contrast materials and the noise is the variance of the measurements. The contrast material thickness will be assumed to be sufficiently small that the variance is essentially the same in the soft tissueonly and the contrast regions.
2.b Ideal Kedge subtraction
Figure 2.2 shows the attenuation coefficients of iodine and gadolinium as well as soft tissue. Notice the sharp discontinuities of the coefficients of the contrast materials. The absorption edge energies are unique to each element. The energies and the attenuation coefficients just above and below the discontinuities are shown in the table in the figure. Soft tissue and other biological materials have attenuation coefficients that are continuous throughout the diagnostic energy range.





iodine  33.17  10.18  31.59  
gadolinium  50.24  4.88  14.55 
For the ideal Kedge method, we use delta function xray spectra with energies just below and just above the Kedge energy, and , and compute the difference of the logarithm of the number of transmitted photons. Referring to Fig. 2.1, the expected values of the transmitted photon counts with the two spectra are and
(2.1) 
In this equation, is the sum of the incident photons of both spectra, is the softtissue attenuation coefficient at xray energy , is the soft tissue material thickness, is the contrast material attenuation coefficient, and its thickness. The incident photons were divided equally between the two spectra.
The Kedge signal is the difference of the logarithms of the photon counts
(2.2) 
Using Equations 2.1, this signal is
(2.3)  
In the background region, the contrast material thickness is zero, , and the background material thickness is so the signal is
(2.4) 
Since the soft tissue attenuation coefficient function is continuous, the Kedge subtraction background region signal can be made arbitrarily small by using measurement energies sufficiently close to the Kedge energy
(2.5) 
In the contrast material region, the background material thickness is and the contrast thickness is so the Kedge signal from Eq. 2.3 is
(2.6)  
For the ideal energies specified in Eq. 2.5, the first term is essentially zero so the signal in the contrast region is
(2.7) 
The probability distributions of the photon counts in Equations 2.1 can be modeled as independent Poisson since they are measured with different spectra. The variance of the logarithm of a Poisson random variable with expected value is ^{6}. Therefore, the variance of the Kedge signal, which is the difference of their logarithms, Eq. 2.3, is the sum of the variances of the logarithms of the individual counts
(2.8) 
The expected values are sufficiently large that we can use the normal approximation to the Poisson^{16}.
2.c Aspace processing
The Aspace method^{5} approximates the attenuation coefficient at points within the object as a linear combination of basis functions of energy multiplied by coefficients that depend on the position within the object.
(2.9) 
To apply this method to the imaging task in Section 2.A, we use the attenuation coefficients of the soft tissue and contrast material, and , as the basis functions^{17, 6}. With this basis set, the vectors of the basis set coefficients of the soft tissue and contrast materials are and . The superscript denotes a transpose.
In general, we need a three function basis set to approximate the attenuation coefficients of biological materials and a high atomic number contrast agent accurately^{16}. However, the two function set is sufficient to represent the materials in the object, which is assumed to be composed only of two materials, and it facilitates the comparison with the Kedge method.
The line integral of the attenuation coefficient along a line from the source to a detector pixel is
(2.10) 
where and the superscript denotes a matrix transpose. We summarize the line integrals as a vector the Avector.
The Aspace method estimates the Avector^{5, 18, 19} from measurements of the transmitted spectra. A photon counting detector with pulse height analysis is used so the counts in each bin are a different spectrum measurement. Neglecting scatter and pulse pileup, the expected value of the count in PHA bin is
(2.11) 
where is the incident spectrum and is the idealized bin response for bin , equal to 1 inside the bin and 0 elsewhere. The measurements can be summarized as a vector with components
where is the expected values of the bin count with zero object thickness. The estimator inverts to compute the best estimate of the Avector, , given .
Since the measurements are random quantities, the Avector estimates will also be random. If is their covariance, the signal to noise ratio for the imaging task is
(2.12) 
where is the difference of the Avectors in the regions with and without contrast material and the superscript denotes the matrix inverse. The optimal SNR is computed using the CramèrRao lower bound (CRLB), , the minimum covariance for any unbiased estimator^{20}. For the number of photons required with material selective imaging, the CRLB is^{16}
(2.13) 
In this equation, and is the covariance of the measurements. With the assumptions of no pileup and a quantum noise limited detector, is a diagonal matrix with elements and the matrix has elements^{6}
That is, each element is the effective value of basis function in the normalized spectrum
Because we use the attenuation coefficients of the object materials as the basis functions, the Avector is
(2.14) 
where and are the thicknesses of the soft tissue and contrast materials. In the soft tissue only region,
(2.15) 
and in the region with contrast agent
(2.16) 
2.d Absorbed energy
The spectra for the Kedge subtraction and Aspace processing are different so in order to compare the two methods on an equal basis we need a way to normalize their SNR. The method used is to divide the SNR by the xray energy absorbed in the object, which is used as a proxy for the xray dose. The absorbed energy is computed from the energy absorption coefficient of the soft tissue material, , which measures the energy absorbed by the object from the incident xray photons^{21}. Assuming the contrast material is so thin that it does not absorb significantly, the absorbed energy is
(2.17) 
In this equation, the incident energy spectrum is , where is the photon number spectrum incident on the object. The energy spectrum gives the sum of energies of the photons from to . The normalized energy spectrum is
where the denominator is the total energy of the incident photons
For the Aspace method, the absorbed energy is given by Eq. 2.17 with the incident xray tube spectrum calculated with the TASMIP algorithm^{22}.
For the idealized Kedge method described in Section 2.B, the two delta function spectra are assumed to be arbitrarily close to the Kedge energy, , with a total number of photons for both spectra equal to . With these assumptions, the absorbed energy is
(2.18) 
2.e SNR per absorbed energy as a function of object thickness
To compare the methods, the divided by the absorbed energy was computed as a function of the softtissue material thickness from 5 to 25 . Contrast agents with iodine or gadolinium as the high atomic number element were used. The contrast material thickness was fixed at . The absorbed energies with the two methods was computed as described in Section 2.A.
For the Kedge method, delta function spectra with energies infinitesimally below and above the contrast material’s Kedge energy were assumed as described in Section 2.B. The total incident photons were equally distributed between the two spectra.
For the Aspace method, a 100 kV xray tube spectrum computed with the TASMIP algorithm^{22} was used with photon counting detectors with five PHA bins. The PHA bins were adjusted to give an equal number of photons per bin for the spectrum transmitted through of soft tissue. An efficient estimator with low bias and with Avector noise covariance approximately equal to the CRLB was assumed.
2.f SNR per absorbed energy as a function of tube voltage
The spectrum used with the Aspace processing is controlled by the xray tube voltage so the SNR per dose was computed as a function of voltages from 40 to 100 kV. The object thickness was 20 . For comparison, the SNR per dose of the Kedge method with this object thickness was also plotted.
3 Results
3.a SNR vs. object thickness
Figure 3.1 shows the SNR per dose as a function of the soft tissue thickness. Panel (a) is for an iodine containing contrast agent and Panel (b) for gadolinium containing contrast agent . In the panels, the SNR of the two methods are on the left and their ratio on the right. The tube voltage for the Aspace method was 100 kV.
3.b SNR vs. tube voltage
Figure 3.2 shows the SNR per absorbed energy as a function of the xray tube voltage. The left graph is for an iodine contrast agent and the right for gadolinium contrast. The Kedge method SNR per dose is also plotted. Since it does not depend on the tube spectrum, it is a constant. The object thickness was fixed at 20 .
4 Discussion
Figure 3.1 shows that for soft tissue thickness from 5 to 25 the Aspace method’s SNR per dose is larger than that of the Kedge subtraction method. The ratio of the SNR values with the two methods depends on the contrast agent. For iodine containing agents, the ratio becomes larger as the object thickness increases approaching a value of 9 at 25 . The gadolinium ratio is approximately constant with a value approximately 3. The Kedge energy of iodine, 33.2 keV, is substantially lower than that of gadolinium, 50.2 keV, so the iodine Kedge signal is attenuated much more strongly as the object thickness increases.
Figure 3.2 shows that for tube voltages greater than 60 kV, the Aspace method SNR is larger than the Kedge subtraction method SNR for both iodine and gadolinium. For iodine, the Aspace method SNR is approximately 6 times larger than the Kedge subtraction value. For gadolinium, the Aspace SNR is close to the Kedge subtraction value at low tube voltages but increases to approximately 3.5 times larger at 100 kV. The change may be due to the fact that for low voltages the peak of the tube spectrum is close to the Kedge energy while for larger voltages the signal includes the contributions of the difference in attenuation of soft tissue and gadolinium above the Kedge energy.
The larger SNR of the Aspace method may seem surprising since the sharp discontinuity of the contrast material attenuation coefficient at the Kedge is markedly different from the background material attenuation coefficient. The superior SNR of the Aspace method is due to the fact that the contrast material attenuation is different from the soft tissue not only at the Kedge energy but also throughout the range of energies in the xray tube spectrum. The Aspace method measures this difference throughout the energy region leading to larger signal. Also, the tube spectrum above the Kedge is attenuated less than at the Kedge leading to a lower noise measurement.
The Kedge method implementation in this study used idealized measurement spectra with high flux and energies tunable infinitesimally close to the Kedge energy. No sources with these characteristics that can be deployed to clinical institutions currently exist. Early work^{2} used a broad spectrum xray tube source with a crystal monochromator but this did not provide a practical photon flux suitable for a clinical system. Other early work^{4} used an xray tube filtered by appropriately chosen materials. With this approach, there is a tradeoff between the tube loading and the width of the measurement spectra. For practical tube loading, the filtered spectra widths are not sufficiently small to give a large signal across the Kedge discontinuity. Another possibility^{13} is a synchrotron radiation source but currently this requires a large nuclear physics accelerator laboratory. There is research in “tabletop” synchrotron radiation sources but these have not proved practical at this time. As shown by the results in this paper, even with an ideal monoenergetic source, the Aspace method provides a better signal to noise ratio per dose.
The Aspace method used in this study, although idealized, may be implementable in a clinical environment. Xray tubes are widely used in diagnostic imaging. The photon counting detector assumed negligible pileup and perfect PHA bins but realistic pileup and overlap between PHA bin responses may not substantially reduce the Aspace method performance^{23, 24}. The effect of photon counting detector imperfections on the SNR is a subject of current research.
5 Conclusion
The Kedge subtraction and the Aspace method for imaging contrast agents containing iodine or gadolinium in a soft tissue background material are compared for their signal to noise ratio per unit dose. The Aspace method has a better for soft tissue object thicknesses from 10 to 25 and for tube voltages above 60 kV.
6 Supplementary material
Matlab language code to reproduce the figures of this paper is available online^{1}.
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