The Alvarez-Macovski Model and Quantitative X-ray Computed Tomography

Abstract

X-ray computed tomography (XCT) is a standard technique of investigating the structure of both people and objects. There is a demand for extracting more quantitative information from XCT, such as density and atomic number. In order to generate physically sensible values for the tomogram voxels, XCT reconstruction algorithms need to be more physically accurate.

Poly-energetic (or polychromatic) x-rays creates problems in terms of x-ray interaction. Most x-rays sources are polychromatic, and x-ray attenuation changes with energy. This results in a complicated pattern of attenuation through materials, which is not correctly modelled in most reconstruction schemes.

For quantitative XCT reconstruction, an attenuation model is required. This links the x-ray attenuation coefficients in the voxel grey values to density and atomic number. X-ray attenuation models also give the energy dependence of the attenuation coefficient. We choose one of the most widely used two component model: the Alvarez-Macovski (AM) model. The AM model divides x-ray attenuation into the two major contributing components of photoelectric effect and Compton scattering, and relates the strength of each to the atomic number and density of the material.

In the simplest way, the AM model is used to directly recover density (or atomic number) from an x-ray tomogram. Most existing methods of density extraction from XCT tomograms assume a linear relationship between the density and reconstructed attenuation coefficients. We use a simplified Alvarez-Macovski model, which introduces a more realistic curvilinear relationship between density and attenuation coefficient. The calibration can be carried out in the same way as the usual linear method, and only requires one parameter. We carried out simulation studies of samples under different spectra and noise characteristics and found the AM model provided a robust method of recovery of density extraction on conventionally reconstructed XCT tomograms.

The simplified AM model can also be integrated directly into iterative reconstruction algorithms for XCT. This is done by modifying the forward projection step of the reconstruction algorithm so that a polychromatic projection weighed by the x-ray spectrum is used to better approximate what is recorded at the detector. For single spectrum XCT, simplifications are needed to reduce the unknowns in the AM model. We provide five such simplifications which can be selected depending on the circumstance, and evaluated their performances both in simulation and real world experimental data. We show that such a simple modification is indeed able to greatly reduce the effects of beam hardening and provide quantitative and accurate reconstructed voxel values for a variety of materials.

Finally, the AM model is also essential in dual spectrum/dual energy XCT (DECT). Usually, this requires special equipment or increased acquisition time. Using a recently discovered property of the statistics of the x-rays at the detector, we were able to carry out DECT using only a single spectrum x-ray on conventional x-ray detectors. This is possible because when the detectors are set to record each measurement multiple times and generate both the variance and mean readings, the two sets of data are independent. We can then use the variance and mean sinograms as the two inputs in a usual DECT algorithm and generate artefact free, quantitative XCT tomograms without simplifying the attenuation model itself. Furthermore, the variance and the mean data can also be used to generate a beam-hardening correction curve that can be used to qualitatively improve a conventionally reconstructed tomogram. We demonstrate this in various simulation studies and explore future practical applications. In the conclusion, we evaluate all of the work in the study, and speculate on future directions. Show more