A sub-matrix method for extracting x-ray coherent scattering form factors from image plate data

The coherent scattering form factor defines a material's small angle x-ray scattering properties. These scattering properties can provide useful medical diagnostic information if properly utilized. Measurement of the coherent scattering form factor is difficult, requiring expensive equipment and long measurement times. We show that it is possible to measure the coherent scattering form factor using standard clinical equipment through a matrix equation. The matrix elements are constructed from knowledge of the input x-ray spectrum. In the ideal case, the form factor can be extracted from this equation by inverting the matrix. For typical x-ray spectra and form factors, however, the matrix tends to be poorly conditioned, leading to large errors upon inversion. We have developed a sub-matrix method that constructs a series of smaller, well-conditioned matrices that can be accurately inverted to give the required form factor. We show through numerical simulations that the sub-matrix method can accurately measure the form factor of common tissue materials. Root mean square deviations of 0.0502 and 0.0804 were calculated for the form factors of water and fat with 90 kV spectra and 0.2 mm of tungsten filtration. Over the measurement range, the form factors vary between approximately 0.5 and 2.5. We show that the optimal spectral shape when using the sub-matrix method is one that is strongly peaked at high energies and that using an improperly chosen spectrum can result in a significant loss of accuracy. We also demonstrate that the sub-matrix method is not readily applicable for measurements of strongly ordered materials.