The difference spectrum [Fig. 1], at each spatial location is obtained by subtracting the [Fig. 1] from the [Fig. 1]:Display Formula
3Here, the symbol indicates differences between the basis component concentrations at different spatial locations. In calculating STC, the spectra are referenced to a patient-specific baseline of normal tissue. In other words, we correct the tumor spectra for any intrasubject variation to reveal only differences between tumor and normal tissue. Solving Eq. 3 for the STC spectrum leaves:Display Formula
4In Eq. 4, the last term is the fit of to the standard basis spectra. By fitting the difference spectrum as opposed to , the variation due to differences between the and basis spectra are minimized; thus the STC component can be recovered with relatively high precision. If we had fitted Eq. 1 instead of Eq. 3 using the standard basis spectra, the coefficients of the fit would have been large and the differences between and (the patient-specific basis spectra and the standard basis spectra) would overwhelm the subtle differences due to the STC component. Thus the single differential residual mainly reflects intersubject differences, thereby masking the subtle differences between tumor and normal tissue for a given patient. In the double-differential method, we subtract out the common residual for tumor and normal from fitting to the standard basis spectra; thus the selection of the basis spectra has little impact in revealing spectral differences due only to tumors.