2.1

Forward model

All values in the projection domain are obtained using the Beer-Lambert law:

where μ_E is the linear attenuation coefficient at energy E, I₀ is the signal before attenuation, and L is the intersection length of the ray with the pixel of interest. The values of I₀ and L were set to 1 for all experiments.

2.2

Solution of linear systems

Limiting the estimation of the linear attenuation coefficient to a single pixel allows us to solve the linear system Ax = b using two analytical methods. The first is non-negative least squares, which is applied when the linear system has either full rank or is overdetermined.³ We solve underdetermined systems of equations using the conjugate gradient method,³ an iterative method that requires a good initial guess. It can be shown that for underdetemined systems of equations, conjugate gradient methods applied to A^TAx = A^Tb, such as CGLS and LSQR, will converge to the minimum norm solution A^T(AA^T)^‒1b, making their application feasible.³

2.3

Models of attenuation

We consider three different models for the energy dependent attenuation. The first model consists of a decomposition into base materials. In breast imaging, decomposition into adipose, fibro-glandular, and iodine components is the most evident choice. The system of equations for the single pixel problem in dual energy systems then takes the following form:

The subscripts L and H refer to low and high energies. The last row puts a constraint on the otherwise underdetermined system of equations by enforcing conservation of volume, possibly causing undesirable behavior when the selected base materials are suboptimal to represent all expected tissues. When more than two spectra are used to acquire the images, additional rows can be added, making this system overdetermined.

The second model makes use of the physical processes underlying the attenuation of x rays. It decomposes the attenuation profile into contributions due to photoelectric effect and incoherent scatter. The energy dependency of the photoelectric effect is usually modeled using the power law

where E is the x-ray energy, whereas the Klein-Nishina equation is employed to describe incoherent (Compton) scatter

where α = E/511 keV. However, this parameterization cannot model K-edges. Thus, in order to use this model to estimate the concentration of contrast agents, the attenuation characteristics of iodine must be included in the model. Consequently, the linear system of equations becomes:

meaning that measurements at three different energies are required for the linear system to have full rank.

This model can be used in dual-energy setups following the method that was proposed by Depypere et al.^{4, 5} If one can write ϕ and Θ as piece-wise linear functions of some parameter, e.g., the attenuation coefficient at a given energy, only two different measurements are needed. Hence, two parameters — the attenuation coefficient at a fixed energy and the iodine volume fraction — can be reconstructed using dual-energy setups. The difficulty of applying this method for breast imaging is that it is not possible to write the attenuation coefficient as a piece-wise linear function, as indicated in figure 1a. Blood, skin, adipose and glandular tissues have very similar properties. Since any possible combination of materials in the triangle adipose-glandular-blood is theoretically possible, a systematic error is introduced as soon as the background tissue does not lie on the chosen line.

Figure 1:

(a) Fitted contribution due to incoherent scatter as function of linear attenuation coefficient at 30 keV for materials present in the breast. (b) The S-parameters were calculated using compositions of materials found in the breast.

The third parameterization we included in our experiment was proposed by Midgley.⁶ It decomposes the linear attenuation coefficient into energy-dependent S-parameters that are weighted by composition-dependent a-parameters:

The number of S-parameters is not fixed and more parameters will lead to more accurate models. However, five S-parameters are sufficient to keep the maximum error below 2% in the energy range up to 50 keV.⁶ Moreover, following that a_k+1 ≥ a_k, μ_E monotonically decreases with energy. The S-parameters for biological tissues in breast imaging are shown in figure 1b. They were calculated by taking into account that a-parameters are functions of atomic number and electron density. Thus, the S-parameters are obtained by solving a least-squares problem,⁷ in our instance for elements that are found in breast tissues, specifically H, B, C, N, O, Na, Al, Si, P, S, Cl, Ar, K, Ca and Fe. As with the previous model, this parameterization scheme cannot account for K-edges. Thus, we extended this model to include the attenuation coefficient of iodine

Assuming that the iodine volume fraction f is very small, the resulting system of equations in a dual-energy setup can be written as

2.4

Experiments

We examine the effect of the attenuation model on the accuracy of skin and iodine reconstruction in dual-and triple-energy acquisitions. We use skin as a good example of a background material that needs to be modeled correctly in order to avoid systematic errors that will otherwise propagate into the estimated iodine concentration in model-based reconstruction. Since model-based reconstruction tries to minimize the mismatch in the projection domain, incorrect linear attenuation coefficients in background material must be compensated by adjusting iodine concentration, which leads to lower accuracy.

The tissue background for the experiment including iodine contrast consists of a mixture of 50% adipose tissue, 50% fibro-glandular tissue; the volume fraction of blood is 30%. Varying amounts of iodine (0–20 mg/mL) were then added to the blood fraction. Our contrast agent is modeled as pure iodine. We assume that the attenuation of the contrast agent suspension medium is equal to that of blood. All materials were modeled with corresponding tissue substitutes.^8–10 and the attenuation profiles of elements were obtained from XrayDB.¹¹

The energies of the low and high energy spectra in the dual-energy setup were set to 30 keV and 34 keV, and the third energy for the triple-energy setup was set to 38 keV. Water was used as initialization for Midgley’s parameterization. No initialization was needed for the other models since the resulting systems of equations were not underdetermined. The decomposition into photoelectric effect and incoherent scatter was considered only in the triple-energy setting.

3.1

Reconstruction of skin

Figure 2a shows the relative error of the reconstructed attenuation coefficient of skin with three methods. The largest deviation (8.9% at 10 keV and 1.44-10^‒4 volume fraction of iodine) results from the model that decomposes the background into adipose and glandular tissues. The most accurate result is achieved with Midgley’s model in a triple-energy setup (0.1% at 10 keV and ‒2.27·10^‒6 volume fraction of iodine). Complete results are shown in table 1.

Figure 2:

(a) Proportional error between reconstructed and true linear attenuation coefficient of skin when using different models of attenuation and number of spectra. (b) Error in estimated iodine concentration as function of the difference between the linear attenuation of the mixture below and above the K-edge of iodine and concentration in blood. The negative region indicates rise in the attenuation due to K-edge. The background tissue was 50% adipose and 50% fibro-glandular tissues; volume fraction of blood was 30%.

Table 1:

Mean and maximum absolute relative errors on attenuation between 10 keV and 49 keV, and reconstructed iodine volume fraction for skin.

3.2

Estimation of iodine concentration

Figure 2b shows the error in iodine concentration estimation as a function of the difference between linear attenuation at 30 keV and 34 keV, resulting from the iodine K-edge when reconstructed using Midgley’s model in dual- and triple-energy setups. The corresponding concentration of iodine is shown on top. The error is 62% and 3.39% at 0.97 mg I per mL of blood with double-energy and triple-energy setups, respectively. An even lower error (0.72%) is achieved with the decomposition into photoelectric effect and incoherent scatter with monochromatic radiation.