IOP PUBLISHING PHYSICS IN MEDICINE AND BIOLOGY
Phys. Med. Biol. 53 (2008) 7107–7124 doi:10.1088/0031-9155/53/24/007
Magnetization transfer proportion: a simplified
measure of dose response for polymer gel dosimetry
Heather M Whitney1,2, Daniel F Gochberg1,2,3 and John C Gore1,2,3,4
1 Vanderbilt University Institute of Imaging Science, Nashville, TN 37232-2675, USA
2 Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37232-2675, USA
3 Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville,
TN 37232-2675, USA
4 Department of Biomedical Engineering, Vanderbilt University, Nashville, TN 37232-2675,
USA
E-mail: heather.whitney@vanderbilt.edu
Received 14 August 2008, in final form 20 October 2008
Published 26 November 2008
Online at stacks.iop.org/PMB/53/7107
Abstract
The response to radiation of polymer gel dosimeters has most often been
described by measuring the nuclear magnetic resonance transverse relaxation
rate as a function of dose. This approach is highly dependent upon the choice
of experimental parameters, such as the echo spacing time for Carr-Purcell-
Meiboom-Gill-type pulse sequences, and is difficult to optimize in imaging
applications where a range of doses are applied to a single gel, as is typical for
practical uses of polymer gel dosimetry. Moreover, errors in computing dose
can arise when there are substantial variations in the radiofrequency (B1) field
or resonant frequency, as may occur for large samples. Here we consider
the advantages of using magnetization transfer imaging as an alternative
approach and propose the use of a simplified quantity, the magnetization
transfer proportion (MTP), to assess doses. This measure can be estimated
through two simple acquisitions and is more robust in the presence of some
sources of system imperfections. It also has a dependence upon experimental
parameters that is independent of dose, allowing simultaneous optimization at
all dose levels. The MTP is shown to be less susceptible to B1 errors than
are CPMG measurements of R2. The dose response can be optimized through
appropriate choices of the power and offset frequency of the pulses used in
magnetization transfer imaging.
1. Introduction
Polymer gel dosimeters comprise an aqueous matrix in which various monomers are uniformly
distributed. The monomers convert to polymers upon irradiation, so that various local
0031-9155/08/247107+18$30.00 © 2008 Institute of Physics and Engineering in Medicine Printed in the UK 7107

7108 H M Whitney et al
properties are modified in a dose-dependent fashion. The resulting spatial pattern of dose
deposition can be read out using one of several methods, including magnetic resonance imaging
(MRI) (Maryanski et al 1993) and optical (Gore et al 1996) and X-ray computed tomography
methods (Hilts et al 2000). Most practical applications to date have used MRI, and the dose
response has usually been described in terms of the transverse relaxation rate, R2 (Maryanski
et al 1994). Alternative imaging metrics have also been investigated, including magnetization
transfer (MT) imaging (Lepage et al 2002). MT imaging is sensitive to the exchange of
magnetization between proton pools in a sample that results from chemical exchange or
through-space dipolar interactions (Wolff and Balaban 1989, Henkelman et al 1993). In a
simple model that may be appropriate for polymer gels, two distinct proton populations are
considered to be coupled together. One pool corresponds to the mobile solvent protons, and a
second pool corresponds to hydrogen nuclei that are relatively immobile, associated with the
polymer in some way, and have different relaxation times or different resonant frequencies
because of chemical-shift effects. The integrated signal from both pools is measured after
the application of an appropriate saturating RF pulse at a frequency that is off-resonance to
the free water. The magnetization transfer ratio (MTR) has often been used as an index of the
degree of MT. It is defined as
MTR =
M0 − Msat
M0
, (1)
where M0 is the signal of the sample acquired without off-resonance saturation and Msat is
the signal acquired with saturation. Gel dosimetry measurements of MT have used slightly
different assessments of the magnetization measured with and without saturation. For example,
De Deene et al (2006b) proposed the ‘true magnetization transfer ratio’ which incorporates
the direct effect of the saturating irradiation on the free water. This ratio is given by
MT =
MH2O − Msat
M0
= MTR −
Mdir
M0
, (2)
where MH2O is the experimentally measured magnetization of water after a saturation pulse has
been applied and Mdir is described by the authors of the work to be the ‘direct effect contribution
which is due to the saturation of the water proton pool’. Other studies have published values
for the specific parameters that contribute to MT for different types of polymer gel dosimeters
as a function of dose (Gochberg et al 2001, 2003).
The aim of this paper is to provide a theoretical and experimental basis for using a slightly
different ratio of MT measurements for gel dosimetry that results in a linear response of
this quantity with dose in the range of 0–20 Gy. This approach will be compared to the
more traditional transverse relaxation rate measurement for dosimetry gels. In particular,
the relative sensitivities of the MT and transverse relaxation dose responses to the effects of
imperfections in imaging, including errors in the amplitudes of the radiofrequency pulses used,
are evaluated.
2. Theory
2.1. Dose response
Traditional transverse relaxation rate measurements of polymer gel dosimeters are performed
using spin–echo imaging. Assuming that the system is not saturated (i.e. TR  T1), the signal
at echo time TE is given by
S = M0 e
−R2TE. (3)

Magnetization transfer proportion 7109
In a region of linear dose response,
R2 = R2,0 + αD, (4)
where α and R2,0 are the slope and intercept, respectively. The signal difference generated by
any small dose increment D is then
S = −α · TE · S · D. (5)
This has a maximum value when TE = 1/R2, when the signal change is
S = −0.37
α
R2,0 + αD
D · M0, (6)
and decreases with dose D. For the most sensitive PAG and MAGIC gels, α is 0.19 and
0.503, and R2,0 is 0.9 and 7.653, respectively (De Deene et al 2006a, Luci et al 2007). Thus,
compared to the signal of an unirradiated dosimeter, a dose of 1 Gy will decrease the gel
MRI signal by 0.064M0 and 0.151M0 initially for PAG and MAGIC gels, respectively, but this
decreases to only 0.011M0 and 0.015M0, respectively, when comparing doses at 20 Gy, even
at the optimal echo spacing for that dose. The situation is further compromised because R2
may vary throughout the gel so no single choice of TE is then optimal for all regions. Thus,
even without considering errors inherent in calculating values of R2 from multiple echo data,
the signal changes available for discriminating regions of similar doses are small. A further
complication occurs in multi-echo sequences, which can produce maps of R2 from a single
acquisition but are sensitive to RF and static field inhomogeneities (Majumdar et al 1986a,
1986b), and the precise value of T2 may depend on the echo spacing (Baldock et al 2001).
A simple method using only two images, and from which a quantity that varies
approximately linearly with dose can be easily extracted, would be attractive. A simple
MT-sensitive imaging sequence is one that incorporates an off-resonance saturating pulse
immediately prior to imaging. The off-resonance pulse can be designed to saturate the
immobile or chemically shifted protons and to not affect the free water resonance directly.
MT effects alone then cause the signal from the mobile protons to decrease. This approach is
the conventional MT imaging option available on commercial MRI systems.
The effect of the off-resonance saturating irradiation is to reduce the signal of the mobile
water from M0 to Msat. The residual MRI signal is given by (Henkelman et al 1993)
Msat
M0
=
R1,mkf m + R1,f Rrf,m + R1,f R1,m + R1,f kmf
(R1,f + Rrf,f + kf m)(R1,m + Rrf,m + kmf ) − kmf kf m
. (7)
In this model, suppose the subscript ‘f’ stands for the free water pool and ‘m’ for a second
proton pool which has more efficient relaxation and a broader resonance or is chemically
shifted. This pool represents the polymerized product of irradiation and increases in direct
proportion to the degree of polymerization and dose, a linear approximation to an exponential
change (Lepage et al 2001). R1 is the longitudinal magnetization rate constant, kmf and kf m
are the rates of MT from the polymer pool to the free pool and vice versa, respectively, and
Rrf,i is the rate of loss of longitudinal magnetization in either the free or other pool. If we take
Rrf,f to be zero and Rrf,m to be much greater than all other rates, as is the case in an ideal MT
experiment, equation (7) becomes
Msat
M0
=
R1,f
R1,f + kf m
. (8)
For the rest of the discussion, R1,f and kf m will be referred to as R1 and k, respectively, to
simplify notation. Msat and M0 can be rearranged to give
M0 − Msat
Msat
=
k
R1
. (9)

7110 H M Whitney et al
We will call the ratio M0−MsatMsat the magnetization transfer proportion (MTP). The MTR is related
to the MTP:
MTR =
M0 − Msat
M0
=
k
R1 + k
; MTP =
MTR
1 − MTR
. (10)
The rate constant k will increase with dose in proportion to the amount of polymer produced,
as has been shown to be the case for methacrylic-based dosimeter gels in the dose range of
0–20 Gy (Gochberg et al 2003). We therefore may write
k = k0 + βD. (11)
We then have
MTP = k0T1 + βT1D. (12)
To calculate the MTP, two images (M0, Msat) must be measured, which then can be combined
to produce a quantity that should vary linearly with dose. The slope of this dose response is
βT1 =
β
R1
. (13)
The ratio of the slope to the intercept of the dose response is a useful indicator of sensitivity
when considering the detectibility of small changes in dose (Fong et al 2001); for the MT
measure this is βk0 , compared with
α
R2,0
for methods measuring transverse relaxation.
Equation (9) may also be derived by considering the relationship of Msat to M0 as given
by (Wolff and Balaban 1989)
Msat = M0(1 − kT1 sat), (14)
where T1 sat is the spin lattice relaxation time of the mobile water in the presence of the RF
irradiation and is given by
1
T1 sat
=
1
T1
+ k. (15)
A manipulation of the quantities Msat and M0 results in the same relationship found in
equation (9).
Equations (9) and (10) illustrate an advantage in calculating the effect of MT in terms
of the MTP: if k is linear in dose (as given in equation (11)), so is MTP, but not MTR, even
though fundamentally it contains the same information.
The interpretation of the MT–dose relationship can be further understood by considering
a two-pool model of MT in more detail, such as that suggested by the results of measurements
by Gochberg et al (2003). We can then write the second population as
pm = p
0
m + γD, (16)
where pm is the size of the relevant polymer proton pool after irradiation, p0m is the size of
this second pool in the unirradiated dosimeter, D is the absorbed dose and γ is the slope of the
pool size versus dose relationship. The MT rate constant, in the nomenclature introduced by
Gochberg et al (2003), is then
k = kmf
pm
pf
= kmf
p0m + γD
pf
, (17)
ignoring the (insignificant) changes in pf . Thus the slope of MTP versus D, as given in
equation (9) and incorporating equation (17), is kmf γpf R1 . We may also note that for a two-pool
exchange model of transverse relaxation
R2 = R2f + kmf
pm
pf
. (18)

Magnetization transfer proportion 7111
Thus for this model, α, the slope of the R2 versus dose line, is kmf γpf ; thus the ratio
α
R2,0
=
kmf γ
pf R2,0
.
We see that mapping R2 and the MTP are fundamentally related, both reflecting increases in
the contribution of the fraction of polymer gel. However, the fractional increase in R2 per dose
is smaller than the increase in MTP by the ratio R2R1 , which can be as large as 10. Moreover,
the slope-to-intercept ratio for the MT-based approach is γp0m whereas for the conventional R2
method it is γ
p0m+
pf R2f
kmf
, which is clearly smaller.
2.2. Effects of B1 inhomogeneities
Both quantitative R2 imaging and MTP imaging are vulnerable to variations and inaccuracies
in the RF (B1) field. Significant variations of the B1 amplitude occur within large samples,
especially at higher fields, so the flip angle experienced by any part of the sample may be
in error from the ideal intended value. The sensitivity of multi-echo measurements of R2
to such errors has been well documented (Majumdar et al 1986b, Sled and Pike 2000a).
When the refocusing pulses in a CPMG sequence are not precisely 180◦, the percent deviation
in estimating T2 is a function of T2 itself; i.e. samples with higher T2 value (and lower R2)
experience higher fractional deviations in T2 (and thus, dose extracted from a calibration curve)
for a given error in B1 than samples of lower T2. These inaccuracies in transverse relaxation
time estimates can result in an apparent nonlinearity of R2 versus dose and miscalibration of
the dose response curve.
The susceptibility of MT measurements to variations in B1 has not been as thoroughly
explored. In practice, the RF pre-pulse may produce incomplete saturation of the broad
component and/or partial saturation of the narrow water resonance. An apparent linear
dependence of the MTR on variations in B1 has previously been reported (Samson et al 2006).
We therefore have evaluated the effects of B1 errors on estimates of dose in polymer gels for
both the MTP and R2 approaches.
3. Methods
3.1. Gel preparation and irradiation
MAGIC (9% methacrylic acid) and MAGIC-2 gel dosimeters were produced as previously
described (Fong et al 2001, Luci et al 2007). The components of the two formulations are
the same but in different proportions, as the MAGIC-2 formulation has been optimized for
measurement of the transverse relaxation rate in the 0–20 Gy dose range. The manufacturing
process was as follows: a Pyrex beaker containing the water for the desired volume of gel was
placed in a water bath, along with a magnetic stirrer, on a hot plate. The desired quantity of
gelatin was added and allowed to bloom. The temperature of the system was brought to 45 ◦C,
at which point the water–gelatin solution was in liquid form. The monomer, ascorbic acid and
cupric sulfate were added and the system was stirred for approximately 3 min. The solution
was poured into 14 mL Pyrex screw-top test tubes and placed in a refrigerator until irradiation.
Samples were brought to room temperature, immersed in a water bath and irradiated with
6 MV photons with a dose rate of 2.84 Gy min−1. Total dose increments of 2 Gy were applied
in a parallel-opposed fashion in sub-increments of 1 Gy each, in a range of 0–20 Gy. A single
gel dosimeter from each formulation was removed after each application of 2 Gy of irradiation
to create the set of gels in the desired range and dose separation. Samples were returned to
refrigeration after irradiation. The MAGIC gels were manufactured 48 h before irradiation,
and the MAGIC-2 gels were manufactured 24 h before irradiation.

7112 H M Whitney et al
3.2. Imaging measurements
After a refrigeration period of 9 days following irradiation, samples were allowed to come to
room temperature and imaged at 4.7 T using a 31 cm bore Varian Inova (Varian Inc., Palo Alto,
CA, USA) spectrometer using a 63 mm quadrature coil. The imaging matrix was 64 × 64, the
field of view was 70 × 70 mm and the slice thickness was 4 mm. All samples from a given
formulation were imaged together. MT imaging was performed with a MT-prepared spoiled
gradient echo sequence (Sled and Pike 2000b), as this method is relatively fast and does not
have the potential disadvantage of heating of the samples. To prevent stimulated echoes,
spoiler gradients and rf spoiling were used to disperse transverse magnetization, and complete
spoiling was assumed. A 10 ms Gaussian MT pulse was applied at 30 offset frequencies
logarithmically distributed between 100 and 200 000 Hz for three different nominal MT pulse
angles (283◦, 566◦ and 849◦), as well as at power levels of ±5 and 10% from each of these
three powers. These powers are equivalent to 4.44, 8.88 and 13.32 μT, respectively (Tozer
et al 2003, Tofts et al 2005). These powers were chosen to be representative of the strength
of MT pulses available on clinical scanners. The signal was acquired via a 7◦ excitation sinc
pulse. The TR was 25 ms and TE was 4 ms, and the data from two acquisitions were averaged
together for each measurement.
Transverse relaxation measurements were made with a 32 echo CPMG-type pulse
sequence with TR of 15 s and TE of 10 ms. The transverse relaxation time was estimated
using a pixel-by-pixel fit (using Matlab’s ‘robustfit’ function) of the log of the data to a linear
model (see equation (3)). Transverse relaxation rate maps were measured as a negative slope
of the linear fit to the data for each pixel. One acquisition was taken for each measurement.
Reported values for MTP and R2 were taken as the average value of pixels from a circular
region (radius of 3 pixels) of interest for each sample.
4. Results
4.1. Magnetization transfer proportion measurements
Figure 1 displays an example MTP image of the data acquired from the MT experiment
for the MAGIC-2 dosimeter. The image was created by applying equation (10) to images
acquired for power of 8.88 μT at 200 000 Hz off-resonance (Msat) and 1375 Hz off-
resonance (M0).
Figures 2 and 3 plot the measurement of the MTP for both types of MAGIC gel over a
range of dose values, at four different offset frequencies and different MT powers. Each data
point is the mean of a region of interest for each dosimeter, and the standard deviation is the
standard deviation of those pixels in the region of interest.
There are a few important features to note from figures 2 and 3. Sensitivity, defined as the
slope of the MTP versus dose, varies with the offset frequency and power of the MT pulse. At
low powers and larger offsets the polymer pool is not fully saturated so the effects of MT are
reduced. Additionally, there is a strong dependence of the intercept on the offset frequency
and power chosen due to the direct effect of saturation on water, since it increases at small
offsets, where the direct effect is strongest. This issue is addressed more fully in section 5.
Given this lack of full saturation of the immobilized pool, we do not expect equation (9) or
equation (14) to be valid at all offsets and powers. Indeed, in section 5, we consider how the
theoretical expression for MTP should be modified under these circumstances. Nonetheless,
our experimental data do in fact show linear responses, though with a slope and offset that
depends upon the offset and power (in disagreement with equation (12), but as predicted

Magnetization transfer proportion 7113
Figure 1. Example MTP image for the MAGIC-2 dosimeter, acquired at 8.88 μT calculated from
images acquired at 200 000 and 1375 Hz off-resonance. The dose values are, beginning at the top
left and reading left to right, (row 1) 2, 10, 4 Gy; (row 2) 6, 12, 9, 14 Gy; (row 3) 18, 8, 16 Gy and
(row 4) 20 Gy. The dark space to the left of the 20 Gy dosimeter is a small vial of water used for
location reference purposes.
Figure 2. MTP versus dose for the MAGIC dosimeter.

7114 H M Whitney et al
Figure 3. MTP versus dose for the MAGIC-2 dosimeter.
Table 1. R2 for the measurement of the linearity of MTP versus dose, for the MAGIC gel dosimeter.
Offset MT power = MT power =
frequency 4.44 μT 8.88 μT
1375 0.969 0.989
1787 0.979 0.988
2322 0.961 0.990
3018 0.962 0.985
below). In order to maximize dose sensitivity for MT measurements, the appropriate offset
frequency and MT power should be chosen, as will be elaborated in section 5.
Tables 1 and 2 show values of the linear correlation coefficients R2, a measure of how well
these data are represented by a linear relationship, as a function of both offset frequency and
power of the B1 pulse, for the least-squares fit of MTP versus dose for MAGIC and MAGIC-2
dosimeters.
The slope and intercept of the dose response relationships can be combined to provide
an assessment of dose response sensitivity (Fong et al 2001). Experimental values for these
parameters for MTP are reported below. To aid in assessment of the performance of the MTP
in the face of B1 errors, the results acquired with the known B1 error are also reported.
Tables 3 and 4 display the slope and intercept values calculated from a linear fit of the data
acquired at different powers and a range of ±10% from the nominal powers, for the MAGIC-2
dosimeter. Values for the MAGIC dosimeter vary similarly.

Magnetization transfer proportion 7115
Table 2. R2 for the measurement of the linearity of MTP versus dose, for the MAGIC-2 gel
dosimeter.
Offset MT power = MT power = MT power =
frequency 4.44 μT 8.88 μT 13.32 μT
1375 0.894 0.976 0.982
1787 0.925 0.975 0.983
2322 0.896 0.966 0.982
3018 0.923 0.981 0.976
Table 3. Measured slope of the MTP versus dose line for a variety of MT powers and offset
frequencies for the MAGIC-2 dosimeter.
Error in B1 pulse angle
Offset frequency −10% −5% No error +5% +10%
MT power = 4.44 μT
1375 0.0070 0.0071 0.0081 0.0089 0.0116
1787 0.0056 0.0054 0.0069 0.0070 0.0093
2322 0.0043 0.0044 0.0058 0.0055 0.0072
3018 0.0032 0.0034 0.0047 0.0042 0.0061
MT power = 8.88 μT
1375 0.0254 0.0291 0.0308 0.0335 0.0361
1787 0.0205 0.0228 0.0247 0.0279 0.0299
2322 0.0170 0.0186 0.0209 0.0237 0.0248
3018 0.0140 0.0153 0.0178 0.0185 0.0207
MT power = 13.32 μT
1375 0.0521 0.0537 0.0580 0.0621 0.0677
1787 0.0425 0.0461 0.0480 0.0520 0.0578
2322 0.0341 0.0382 0.0399 0.0443 0.0467
3018 0.0294 0.0314 0.0338 0.0377 0.0409
Tables 5 and 6 display the calculated change in the slope-to-intercept ratio for the measured
MTP of the MAGIC and MAGIC-2 dosimeter. With the exception of the MTP acquired for
the MAGIC-2 dosimeter at the lowest power, the slope-to-intercept ratio of the MTP has a
range of approximately ±15% for a range of −10 to +10% error in B1 power.
4.2. Transverse relaxation simulation and measurement
The effect of B1 errors on measurements of transverse relaxation has been well characterized
in the literature. The observed value of R2 due to a given error in B1 power can be related as
(Sled and Pike 2000a)
R2,obs = R2 −
ln f
τ
, (19)
where f is an attenuation factor and τ is the echo spacing of the CPMG-type experiment. The
attenuation factor f incorporates the effect of errors due to imperfect B1 pulses, and for a hard
pulse sequence, assuming no error in B0, is given by
f =
cos(δ)
2
− cos(δ) +
1
2
, (20)
where δ is the degree of the refocusing B1 pulse. Note that this derivation assumes that
complete spoiling occurs of any transverse magnetization produced by incomplete nutations

7116 H M Whitney et al
Figure 4. Expected variation in R2 for a given error in B1 angle for the MAGIC dosimeter.
Table 4. Measured intercept of the MTP versus dose line for a variety of MT powers and offset
frequencies for the MAGIC-2 dosimeter.
Error in B1 pulse angle
Offset frequency −10% −5% No error +5% +10%
MT power = 4.44 μT
1375 0.1467 0.1639 0.1790 0.1944 0.1776
1787 0.1273 0.1448 0.1505 0.1704 0.1635
2322 0.1233 0.1270 0.1323 0.1549 0.1408
3018 0.1086 0.1130 0.1221 0.1427 0.1273
MT power = 8.88 μT
1375 0.3656 0.3825 0.4207 0.4388 0.4621
1787 0.3252 0.3530 0.3730 0.3904 0.4173
2322 0.2875 0.3254 0.3367 0.3546 0.3825
3018 0.2668 0.2986 0.3051 0.3285 0.3428
MT power = 13.32 μT
1375 0.5549 0.6145 0.6343 0.6519 0.6899
1787 0.5121 0.5400 0.5681 0.5757 0.6068
2322 0.4718 0.4948 0.5196 0.5256 0.5704
3018 0.4247 0.4572 0.4718 0.4783 0.5083
and stimulated echoes. Using experimental values for R2 for the MAGIC and MAGIC-2 gel
dosimeters and the above relationship, the expected deviation in R2 due to a given error in B1
can be calculated and the effect on the dose response can be estimated. The values used for
the dosimeter from the experiment described above were 5.00 and 14.40 s−1 for the MAGIC
dosimeter at 0 and 20 Gy, respectively, and 6.29 and 17.69 s−1 for the MAGIC-2 dosimeter.
These values were calculated from a linear fit of the R2 data acquired in the above experiment
to simulate the kind of values that would be taken from a calibration curve. Figure 4 displays
the expected variation in R2,obs for a given error in B1 power for the MAGIC dosimeter as
measured via a CPMG-type experiment using hard pulses.

Magnetization transfer proportion 7117
Table 5. Measured percent change in the slope-to-intercept ratio of the MTP versus dose line for
a variety of MT powers and offset frequencies for the MAGIC dosimeter. For most powers and
offset frequencies, the slope-to-intercept ratio does not vary more than 10% for even a 10% error
in B1 pulse angle.
Error in B1 pulse angle
Offset frequency −10% −5% No error +5% +10%
MT power = 4.44 μT
1375 −0.61% −1.73% 0.0941 −1.39% 10.05%
1787 −12.68% −11.43% 0.0941 −0.92% 5.96%
2322 −3.80% 0.87% 0.0815 6.87% 18.92%
3018 −11.07% −6.75% 0.0766 12.06% 13.41%
MT power = 8.88 μT
1375 −6.17% 0.42% 0.1151 5.51% 3.11%
1787 6.16% 5.15% 0.1058 10.53% 14.42%
2322 −5.31% 4.14% 0.1087 0.45% 12.76%
3018 −3.03% −4.00% 0.1051 8.16% 8.78%
Table 6. Measured percent change in the slope-to-intercept ratio of the MTP versus dose line for
a variety of MT powers and offset frequencies for the MAGIC-2 dosimeter. With the exception of
the values acquired at the lowest power, the slope-to-intercept ratio does not vary more than 10%
for even a 10% error in B1 pulse angle.
Error in B1 pulse angle
Offset frequency −10% −5% No error +5% +10%
MT power = 4.44 μT
1375 5.93% −4.48% 0.0452 1.38% 44.15%
1787 −3.49% −19.47% 0.0459 −10.51% 23.44%
2322 −19.66% −20.44% 0.0436 −18.69% 17.91%
3018 −24.73% −22.81% 0.0386 −23.05% 24.66%
MT power = 8.88 μT
1375 −5.16% 4.06% 0.0732 4.20% 6.86%
1787 −5.21% −2.57% 0.0663 7.60% 7.87%
2322 −4.26% −7.91% 0.0619 7.88% 4.58%
3018 −9.82% −12.34% 0.0584 −3.33% 3.40%
MT power = 13.32 μT
1375 2.64% −4.54% 0.0915 4.04% 7.18%
1787 −1.66% 0.97% 0.0845 6.83% 12.74%
2322 −5.67% 0.57% 0.0767 10.00% 6.81%
3018 −3.25% −3.92% 0.0715 10.06% 12.34%
The simulation shows that while the slope of the R2 versus dose line remains the same
(as indicated by the B1-independent difference in R2 between 20 and 0 Gy), the intercept (the
value of R2 at 0 Gy) will change significantly over a range of errors in B1. Table 7 lists the
expected percent change in the slope, intercept and slope–intercept ratio of R2 versus dose for
hypothetical errors in B1 power.
4.3. Comparison of the effect of B1 errors on dose estimates
Finally, a useful measure of the effect an error in B1 power has on a measurement is to consider
the apparent dose that would be calculated in the event of a specific B1 variation. When

7118 H M Whitney et al
Table 7. Expected change in the slope, intercept and slope–intercept ratio of R2 versus dose for
inclusion of B1 angle error. Negative percent error in B1 angle is not included as the result is
symmetrical about zero percent error in B1.
Percent error in B1 angle
0% 5% 10%
MAGIC
Slope (s−1 Gy−1) 0.470 0.470 0.470
Intercept (s−1) 5.000 5.618 7.4778
Slope–intercept ratio 0.094 0.0834 0.063
Percent change in the – −10.99% −33.13%
slope–intercept ratio
MAGIC-2
Slope (s−1 Gy−1) 0.570 0.570 0.570
Intercept (s−1) 6.290 6.908 8.768
Slope–intercept ratio 0.091 0.083 0.066
Percent change in the – −8.94% −28.26%
slope–intercept ratio
polymer gel dosimeters are used for radiation field assessment, a calibration curve is created
from dosimeters to which known doses are applied. The form of this line is
X = aD + b, (21)
where X is the measured value, such as R2 or MTP; a is the slope of the line and b is the
intercept. Once the desired measurement is made of the dosimeter to be assessed for dose, the
following equation is used to back-calculate to the apparent dose Dapp:
Dapp =
X − b
a
. (22)
If the parameter, be it MTP or R2, has been mis-measured due to B1 errors, the apparent
dose will be different than its true dose. This effect was investigated for the MTP and R2
measurements in the MAGIC and MAGIC-2 dosimeters. The slope and intercept of the
calibration curves for MTP data acquired with no B1 error (the nominal B1 powers referenced
above) were applied to MTP values measured with a manually adjusted power with known
error (±5 and 10% of the nominal B1 power) using equation (22). Additionally, the slope and
intercept calculated for the R2 values were applied to simulated R2 values acquired with a 5
and 10% B1 error. The percent error in apparent dose was calculated as
Dapp − D
D
× 100. (23)
Figure 5 displays representative data of the percent error in apparent dose acquired in both MTP
and R2 measurements, as a function of applied dose. The data show that at lower dose levels,
on the order of that which most radiation therapy dose fractionations deliver, measurements
of MTP with a 10% error in B1 show much less error in apparent dose than measurements
of R2.
5. Discussion
5.1. Saturation of the two pools
MT measurements of the dose response of gel dosimeters are potentially less dependent upon
errors in B1 power than traditional multi-echo measurements of the transverse relaxation rate.

Magnetization transfer proportion 7119
Figure 5. Percent error in the apparent dose for MTP measurements at ±10% B1 error and R2
measurements at 10% error in the MAGIC gel dosimeter. The MTP data were acquired at MT
power 7.92 (−10% error) and 9.68 (+10% error) μT and offset frequency 1375 Hz. The R2 data
were calculated by simulation using data from figure 4. Corresponding data for the MAGIC-2
dosimeter show an expected percent error in apparent dose for R2 at 2 Gy to be approximately
217%, while the percent error for the MTP was an average of 96%. At 4 Gy, the percent error in
R2 was approximately 109%, while the percent error for the MTP was an average of 54%. With
a few exceptions, the data followed this trend for all offset frequencies investigated (1375, 1787,
2322 and 3018 Hz off-resonance).
In principle, once sufficient B1 power is applied off-resonance, complete saturation of the
broad resonance can be achieved, and using larger powers will have little effect as long as
direct saturation of the narrow water line is avoided.
At the powers and offsets shown in this work, complete saturation of the bound pool
was not achieved, which may be more typical of conditions when using clinical scanners.
The effect of incomplete saturation of the bound pool in the experiment can be estimated by
adding a correction factor to equations (8) and (9), which were derived by taking Rrf,m to be
much greater than k, kmf , R1 and R1,m. A first-order correction for incomplete macromolecular
saturation can be derived by instead assuming that Rrf,m and kmf are much greater than k, R1
and R1,m. This gives
Msat
M0
=
R1
R1 + k
(
1 − kmfRrf,b+kmf
)
. (24)
The quantity 1 − kmfRrf,b+kmf can be interpreted as a measure of the effect incomplete saturation
has on measurements of MT. For the MTP, the inclusion of this error results in
MTP =
k
R1
(
1 −
kmf
Rrf,b + kmf
)
. (25)
Equation (25) is strictly linear in dose only if kmf is independent of dose, which is true for
BANG gels (Gochberg et al 2001) but not MAGIC gels (Gochberg et al 2003). Hence,
achieving complete saturation of the macromolecular pool would be ideal. Using published
values for R1, k and kmf for the MAGIC gel dosimeter, the level of saturation (Msat/M0)
achieved in our experiments can be estimated via equation (24). These estimated values for

7120 H M Whitney et al
(a)
(c)
(b)
Figure 6. (a) Rrf,f for a 20 Gy MAGIC gel dosimeter, (b) Rrf,m and (c) region for which the two
criteria (Rrf,f much less than all other rates, Rrf,m much more than all other rates) overlap.
dosimeters irradiated to 0 Gy are on the order of 84%, 65% and 53% saturation for the three
powers (4.44, 8.88 and 13.32 μT, respectively) of the MT pulse in the experiment, while
values for 20 Gy dosimeters are on the order of 76%, 49% and 33% for the three powers.
The experiment was designed to perform the MT experiment at power levels similar to those
available on clinical scanners, but the significant dependence of Msat/M0 on the rf power is
due to incomplete saturation of the macromolecular pool.
It would be ideal to use known values for R1, k and kmf to estimate the appropriate MT
power strength and offset frequency to use in MTP experiments. Such prior knowledge could
aid in ensuring that the assumptions behind equations (9) and (11) are fully met. For example,
prior work (Gochberg et al 2003) has estimated that for the MAGIC gel dosimeter, R1, k and
kmf for dosimeters at 20 Gy are approximately 1 Hz, 7.2 Hz and 122 Hz, respectively. To
estimate the range of powers and offsets for which Rrf,f is much less than and Rrf,m is much
greater than any other rate, these values can be calculated using lineshape assumptions for the
system in question and compared for the regime in which equations (9) and (11) are valid.
Figure 6 displays calculations of the saturation of the free and bound pools and an assessment

Magnetization transfer proportion 7121
Figure 7. Simulation of the direct effect on water for a range of offset frequencies and MT powers.
for whether or not the assumptions behind equations (9) and (11) were met. Values for Rrf,f
were calculated by assuming a Lorentzian lineshape for the free pool and using the relationship
Rrf,f =
ω21T2,f
1 + (2πT2,f )2
, (26)
where T2,f is the transverse relaxation time of the water pool,  is the offset frequency of the
saturation and ω1 is the strength of the MT pulse in rad s−1. Values for Rrf,m were calculated
by assuming that the bound pool could be characterized by a Gaussian lineshape and using
the relationship
Rrf,m = ω
2
1T2,m
√
π
2
T2,m e
−
(2πT2,m)2
2 , (27)
where T2,m is the transverse relaxation time of the bound pool. For this value, an estimated
value of 10 μs was used.
In figure 6(a), Rrf,,f is displayed as a function of both MT pulse power and frequency.
Values greater than 1, approximately R1 (the lowest rate of concern), are not plotted to
preserve the scale of interest. Figure 6(b) displays Rrf,m. Figure 6(c) is displayed by evaluating
at each offset frequency and MT power whether or not the criteria of Rrf,f  all other rates
and Rrf,m  all other rates are valid, and the shaded area indicates the values for which the
criteria are met.
A similar analysis can be used to appreciate the direct effect on the free water line.
Assuming steady state conditions, the solution for the uncoupled Bloch equations for the
transverse magnetization of the water is given by
Msat
M0
=
1 + (T2,f )2
1 + (T2,f )2 +
( ω1
2π
)2
T1,f T2,f
. (28)
Using values of T2,f of 1/14.4 Hz and assuming that R1 is on the order of 1 Hz for polymer
gel dosimeters at 4.7 T, the direct effect on the free water line can be estimated. Results for a
range of frequency offsets and MT powers are displayed in figure 7.

7122 H M Whitney et al
It can be estimated from figure 7 that at the offset frequencies and powers in this work,
there was some level of direct effect on the water (Msat/M0 having a range of approximately
40–70%), and this is likely another reason why our measurements of the MTP dose response
intercept vary at the different offset frequencies.
These simulations suggest that while an appropriate range of offset frequencies was used
for the measurement of MTP as displayed in figures 2 and 3, the MT power strength was less
than ideal for a dosimeter with these specific relaxation properties. However, our results show
that for clinically feasible implementation of MT experiments, where the power of the MT
pulse may not be able to be increased by much or the user may only be able to set a nominal
maximum power level, a linear dose response can still be attained. These results suggest
that complete saturation of the macromolecular pool and a priori knowledge of the sample
parameters are not necessary to achieve linearity in dose.
5.2. Comparison of dose measurement in the presence of B1 errors
The change in the slope-to-intercept ratio for the measurement of the MTP in MAGIC-type
gel dosimeters was approximately ±15%, while the value of the slope-to-intercept ratio for
R2 could be expected to deviate by as much as 33% for 10% error in B1 power.
The estimated (apparent) dose at dose levels used in most fractionation schemes varies
with B1, but for the range of changes considered here the MTP appears less susceptible to
errors in apparent dose than are measurements of R2.
While the measurement of the MTP is linear in dose for a wide range of MT pulse powers
and offset frequencies, it is important to note that the sensitivity can be maximized by the
appropriate choice of these two variables. A priori knowledge of the MT rates for a particular
type of dosimeter can be used to estimate the appropriate values for MT power and offset
frequency, but as has been shown is not necessary to achieve a linear dose response of MTP.
The MTP approach has not been tested in polyacrylamide-type (PAG) polymer gel
dosimeters. Some studies (Gochberg et al 2001, 2003) have suggested that MT may
behave differently in PAG versus methacrylic acid-based dosimeters, which may affect the
measurement of the MTP versus dose in PAG. Note also that the MTP is proportional to k
(equation (9)) and inversely proportional to pf , the size of the free pool (equation (17)), so
efforts to change these may result in greater sensitivity. Sensitivity should also differ between
dosimeters with different amounts of MT present due to formulation, as kmf is dependent upon
the amount of monomer. We have assumed, based on expectations and some experimental
data, that kmf is not dose dependent. In some conditions, this may no longer be valid (Gochberg
et al 2003), such that kmf = kmf,0 + mD. Then, the MTP has both a linear and a quadratic term
in terms of D. However, it will still appear linear until mD  kmf ,0.
The MTP has been described before in the imaging literature, using the name ‘equivalent
cross-relaxation rate (ECR)’ (Sogami et al 2001), for imaging studies for breast cancer.
However, the term ‘cross-relaxation’ has usually been reserved in nuclear magnetic resonance
to denote dipolar cross-relaxation, whereas MT includes chemical exchange and other effects.
We therefore feel it is not appropriate to use the ECR name and propose MTP for this quantity.
5.3. Other concerns
Image resolution should not significantly affect these types of measurements unless regions of
interest are drawn such that partial volume effects are present in the measurement of the MTP.
Creating regions of interest that are well within the boundaries of the samples will ensure that
this error is avoided.

Magnetization transfer proportion 7123
6. Conclusion
This work has shown that the measurement of the MTP in methacrylic acid-type gel dosimeters
is linear with dose and shows promise as a measure of dose response in polymer gel dosimeters.
The method has been validated at powers and offset frequencies similar to those used in clinical
applications, and prior knowledge of magnetization transfer quantities, while desirable to
maximize contrast, is not necessary to acquire useful dose response data. The method is less
susceptible to calibration errors than transverse relaxation rate measurements in the presence
of B1 inhomogeneities.
Acknowledgments
The authors thank George Ding and Charles Coffey for assistance with dosimeter irradiation
and Xiawei Ou for assistance with data acquisition. This work was supported by NIH grants
CA090844, EB000214 and EB001452.
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