Six previously extracted teeth were irradiated with 0, 2, 5, 10, 15, and 30 Gy. These teeth we placed in a gap in the dentition in the mouth of a volunteer and sets of EPR spectra were acquired on three days. All measurements were done with a clinical 1.2 GHz spectrometer at the EPR center at Dartmouth (Swartz et al, 2006
) with 1024 points recorded for each EPR spectrum. On each day, spectra for all six teeth were acquired using the same acquisition parameters. Typically these were: scan range 25 Gauss (10G = 1 mT), scan time 3 s, 30 to 90 scans that were averaged (more scans were used for low dose spectra), modulation amplitude 4 Gauss, modulation frequency 24.5 kHz, incident radio frequency (RF) power 100 mW. The N
= 6 × 3 = 18 EPR spectra are shown in . The spectra were adjusted for tooth-size, by measurements of the surface of the teeth in two orthogonal directions (D1
) and each original signal was divided by (D1D2
)/100. This provides the amplitude of the EPR signal per 100 square millimeters of the surface of the tooth.
Figure 1 Eighteen tooth-size adjusted EPR spectra in vivo with six teeth that received 0, 2, 5, 10, 15 and 30 Gy radiation. The irradiated teeth were inserted into a gap in the dentition of a volunteer. Measurements were repeated on three days with identical instrumental (more ...)
The reconstructed dose is derived from a calibration curve that is traditionally obtained as a linear regression of the peak-to-peak amplitude (P2P) on the radiation dose given. The method of abscissa prediction using regression analysis is called an inverse regression
(Draper and Smith, 1998
). To compare various methods of the dose reconstruction a precision measure should be used. We used an intuitively appealing standard error of prediction (SEP
) as the measure of precision computed as
is the number of spectra, Di
is the dose given and
is the predicted dose in the i
th measurement. This parameter of merit for dose reconstruction has been useful for linear and nonlinear calibration curves and corresponds to the formula of the standard error widely accepted in statistics (Snedecor and Cochran, 1989
). Geometrically, the standard error of the dose prediction is an average of horizontal
distances of the data points to the calibrations curve. SEP
gives rise to a quick computation of the confidence interval: the probability that dose is within interval
is 0.95 (the rule of two sigmas). If
is an estimated standard error of the linear regression in amplitude on dose given (the error of the amplitude), SEP
is approximately equal to
divided by the slope of the regression. In other words,
is an error on the y
-axis and SEP
is the error on the x
-axis computed from the former by scaling with respect to the slope of the calibration curve. Consequently, with the same error of the amplitude, SEP
is smaller for a steeper calibration curve.
We employed seven different computational methods and obtained the figure of merit for each method using all eighteen spectra from . The nomenclature of the methods considered in the present paper are shown in . Most of the methods have a modification parameter, such as the magnetic field range over which the fitting is carried out, and/or the value of the fixed linewidth. These parameters are noted in the abbreviations and the parameters are separated by the character “/”.
Methods nomenclature for in vivo EPR radiation dose prediction
With the empirical peak-to-peak method (EMP2P), the amplitude of the EPR signal is computed as the difference between the maximum and minimum value of the spectrum. For a small dose, this amplitude may be overwhelmed by noise at the ends of the spectra, so it can be useful to restrict the field interval that is utilized in the spectral fitting calculation. We specify the value of the modification parameter after the name. For example, the abbreviation EMP2P/F5 means that the empirical amplitude is computed for the 2×5 = 10 gauss interval corresponding to ±5 gauss of the expected position of the center of the radiation-induced spectrum.
The SMP2P method is the same as EMP2P/F but the amplitude estimation was performed after the signal was smoothed. We use a simple running-window
smoothing technique (also known as moving average) with the width W. Precisely, if W is an even integer and yi
is the original spectrum, the smoothed signal is computed as
In the next family of methods, a Lorentzian curve is used to fit the spectrum and estimate the amplitude of a radiation induced signal. As above, to eliminate noisy ends we may use the signal in the range ±F. To a reasonable approximation, the linewidth (LW) of the Lorentzian curve should be the same for all spectra and equal to that of highly irradiated teeth in which the radiation-induced signal dominates the spectrum. L/F/LWC method fits Lorentzian curve to all spectra with a common linewidth. We optimize fitting for fixed LW value and field range to obtain the minimal SEP.
The three Gaussian component model is similar to models used in in vitro studies mentioned in the Introduction. One component reflects the native signal and two components describe the radiation induced signals (RIS); the maximal amplitude of the RIS is correlated with the radiation dose through the calibration curve. This model also contains a linear slope to account for a possible baseline shift.
Finally, to improve the reconstruction at low doses one may carry out additional measurements with a strong standard signal and use these measurements in fitting tooth data. We used a lithium phthalocyanine (LiPc) crystal, sealed under vacuum, that was reproducibly placed in the resonator and measured its spectrum prior to the measurement of each tooth with the same instrumental settings. Under these conditions the modulation amplitude will affect the observed linewidth of the LiPc, so a spectral model that incorporates these effects was applied (Robinson et al., 1999
). Several parameters from the LiPc spectrum may be used for fitting the tooth data. In the present work, we use the center field parameter, so the tooth spectra were fitted with a fixed center, as determined using the LiPc. Two versions were tested: with LW varying from tooth to tooth (LLiPc) and the LW fixed (LLiPc/LW).