The x-ray attenuation property of a material within the diagnostic x-ray energy range is determined by a linear combination of photoelectric effect and incoherent Compton scattering, both of which are dependent on the incident x-ray photon energies [33]. Mathematically, we can express the attenuation coefficient of a unit mass of material in terms of these x-ray interactions:
where μ is the linear attenuation coefficient of the material at energy E and ρ is the material’s density which is energy independent; fp and fc are the functions describing the photoelectric effect and Compton scattering with α_p and α_c the coefficients representing their contributions. Alternatively, the physical interactions in Eq. 4.1 can be represented by two basis materials that are sufficiently different in their atomic numbers so as their x-ray interaction properties:
where σ_A and σ_B are coefficients describing the contributions of material A and B respectively. A common choice for the basis material pair is water and iodine. It can be shown that [34]:
and
where α_A,p and α_A,c are the contributions of photoelectric and Compton effects for material A respectively. Similarly, α_B,p and α_B,c and the contributions of the same physical phenomena for material B respectively. Equation 4.2 can be rewritten as follow:
Here, σ_Aρ and σ_Bρ are the equivalent densities (in gram per cm³) of the basis materials A and B respectively. The line integral over the linear attenuation coefficient in ray path can be expressed as:
D_A and D_B are the line integrals of the equivalent densities (also called the area densities) of materials A and B respectively in gram per cm². In dual-energy acquisition, projection at each view is measured at two different energy spectra yielding two non-linear equations per ray path:
and
the subscripts L and H represent the low and high energy spectra, P is the measured projection, I and I_o are the transmitted and incident x-ray intensities, and Ω (E) is the incident x-ray energy spectrum, respectively. The area under Ω(E) equals unity and hence the denominators on the right-hand side of the above equations equals one. While the mass attenuation coefficients of the basis materials A (water) and B (iodine) at energy E are readily available in standard NIST (National Institute of Standards and Technology) table [35], the unknown quantities in Eq. 4.6 are the two equivalent densities of the basis materials which can be approximately estimated using polynomial expansions:
and
where β, γ, δ, ε, ψ are the polynomial coefficients. The non-linear terms in Eq. 4.7 are essential to account for the beam hardening effects. Equation 4.6 essentially transforms a pair of attenuation measurements into two pairs of density projections with correction of beam hardening [23] (Fig. 4.3). The reconstructed images represent the equivalent densities of the basis materials in g/cm³ instead of the conventional linear attenuation coefficients in cm⁻¹. It should be emphasized that material decomposition does not identify the actual compositions in an image voxel but merely estimates the amount of each selected basis material required to produce the measured low and high kV attenuations in that voxel. If we select water and iodine as the material bases, any material other than water and iodine will be represented in both the ‘water-equivalent’ and ‘iodine-equivalent’ images (Fig. 4.4).

In addition to material density images, virtual monochromatic images can also be generated from the same set of dual-energy scan data. Virtual monochromatic images depicts how the scanned object will appear if a monochromatic x-ray beam (in kiloelectron volt, keV) is used for scanning. In monochromatic imaging at a specific x-ray energy E_o, integration over the entire energy spectrum in Eq. 4.6 is not required and the equation can be simplified as:
The above equation indicates that we can derive a set of monochromatic projections from the equivalent density projections and mass attenuation coefficients of the basis materials, from which virtual monochromatic images can be reconstructed with a standard tomographic reconstruction algorithm. Alternatively, virtual monochromatic images can be generated from a simple linear operation of the material density images that are generated from the material decomposition process discussed above. In this process, the attenuation value of any material is normalized to that of water as in standard CT images:
If water and iodine are selected as the material bases, then μ_A(E) = μ_W(E) and the above equation is simplified to:
where σ_Wρ and σ_Iρ are the ‘water-equivalent’ and ‘iodine-equivalent’ images [36]. Given the known linear attenuation coefficient values of water and iodine over the diagnostic energy range (~40 to 140 keV), virtual monochromatic images (displayed in Hounsfield Unit) can be readily generated with proprietary software (e.g. GE Healthcare’s GSI viewer) (Figs. 4.4 and 4.5). Virtual monochromatic imaging has several useful clinical applications, especially for beam hardening correction in CT myocardial perfusion imaging. When polychromatic x-rays transverse through the scanned object, the x-ray photons with lower energies are preferentially removed first due to photoelectric absorption and the effective energy of the x-ray beam increases (the beam is ‘hardened’) [37]. In myocardial perfusion imaging, the presence of iodinated contrast agent in the heart chambers leads to inconsistent x-ray attenuation (hence x-ray effective energies) among different ray paths which results in errors in image reconstruction. Virtual monochromatic images of the heart with minimal beam hardening can therefore facilitate a more accurate qualitative and quantitative assessment of regional myocardial perfusion with CT [20, 38–40] (Fig. 4.6).