Quantitative Magnetization Transfer Imaging Using
M. Gloor,*K. Scheffler, and O. Bieri
It is generally accepted that signal formation in balanced
steady-state free precession (bSSFP) is a simple function of
relaxation times and flip angle only. This can be confirmed for
fluids, but for more complex substances, magnetization trans-
fer (MT) can lead to a considerable loss of steady-state signal.
Thus, especially in tissues, the analytical description of bSSFP
requires a revision to fully take observed effects into ac-
count. In the first part of this work, an extended bSSFP signal
equation is derived based on a binary spin-bath model. Based
on this new model of bSSFP signal formation, quantitative MT
parameters such as the fractional pool size, corresponding
magnetization exchange rates, and relaxation times can be
explored. In the second part of this work, model parameters
are derived in normal appearing human brain. Factors that
may influence the quality of the model, such as B1 field
inhomogeneities or off-resonances, are discussed. Overall,
good correspondence between parameters derived from
two-pool bSSFP and common quantitative MT models is ob-
served. Short repetition times in combination with high sig-
nal-to-noise ratios make bSSFP an ideal candidate for the
acquisition of high resolution isotropic quantitative MT maps,
as for the human brain, within clinically feasible acquisition
times. Magn Reson Med 60:691–700, 2008. © 2008 Wiley-
Key words: balanced SSFP; magnetization transfer; quantitative
Balanced steady-state free precession (bSSFP; also known
as TrueFISP, FIESTA, and balanced FFE) (1,2) has become
a valuable and indispensable tool in diagnostic imaging
over the last several years, especially for cardiac and car-
diovascular applications (3–5). It is generally accepted that
signal formation in bSSFP can be derived from the Free-
man-Hill formula (6), being proportional to the ?T2/T1for
repetition times TR ?? T1,T2(7). However, only recently
subtle signal deviations (8–10) from the Freeman-Hill for-
mula indicated further contrast mechanisms, especially
the steady-state of bSSFP in tissues may be reduced up to
a factor of two from magnetization transfer (MT) effects
(11). Here, short TRs in combination with large flip angles
(?) saturate the magnetization of protons associated with
macromolecules and membranes in biological tissues (re-
stricted pool protons). As a result, subsequent exchange of
these protons with mobile ones (liquid pool protons) con-
stituting the steady-state leads to an overall signal reduc-
tion, if compared to a situation in absence of exchange. For
bSSFP, on-resonant excitation thus not only generates the
steady-state but also acts directly as MT sensitizing radio-
frequency (RF) pulse. From this, a new method for MT
imaging with bSSFP was proposed (12), which in contrast
to common MT methods (13–15) circumvents the need for
additional MT sensitizing prepulses.
MT was first demonstrated by Wolff and Balaban in the
late 1980s (16). Not only has it become a standard tool for
suppression of background signals from tissues in MR
angiography (14), but it was also realized that MT shows
great promise in the field of multiple sclerosis offering
tissue characterization beyond conventional T1, T2, and
T2* (16). In its simplest and common form of quantifica-
tion, MT effects are often condensed within the so-called
magnetization transfer ratio (MTR) (17). MTR has become
popular not only for characterizing subtle diseases in the
brain (15) but also for the assessment of breast, knee and
cartilage (14). Although great effort has been undertaken to
ensure reproducibility in MTR measurements (18), the
phenomenological breakdown of a complex tissue system
to a single parameter may be inappropriate simply by its
virtue of oversimplification, and thus may overlook useful
diagnostic information. Indeed, there has been given evi-
dence that MTR has only limited pathological specificity
(19), making MTR results incomplete and controversial.
As a result, binary spin-bath models have been investi-
gated in detail over the last several years by many research
groups, capable of gaining intrinsic MT model parameters
(19–21). Quantitative MT imaging (qMTI) yields the frac-
tion (F) of restricted pool protons, the magnetization ex-
change rate (kf), as well as the pool relaxation properties
(T1, T2). Recent studies indicate that especially F may be of
great diagnostic potential, because it correlates with the
myelin content in brain white matter (22). Quantitative
MT, in contrast to MTR, has the potential to yield ex-
tended and unquestionable diagnostic information; how-
ever, measurement protocols lack clinically applicable ac-
So far, quantitative MT parameters are commonly de-
rived from associated MT dispersion curves. Sampling of
these curves is time consuming because different MT-
weighted spin-echo or gradient-echo measurements have
to be acquired based on a set of MT pulses that differ in
power and off-resonance frequency (19,20). In contrast, the
overall MT sensitivity of bSSFP in combination with its
excellent SNR and the short acquisition time makes it an
ideal candidate to meet the clinical requirements for fast
and reliable high resolution qMTI. In the first part of this
work, an extended bSSFP signal equation is derived based
on a binary spin-bath model to take MT exchange effects
into account. Assuming that relaxation and exchange can
be separated into two independent processes within any
Division of Radiological Physics, Department of Radiology, University Hospi-
tal Basel, Basel, Switzerland.
Grant sponsor: Schweizerischen Nationalfonds; Grant number: SNF PP0B-
68783; Grant sponsor: Novartis Pharma AG.
*Correspondence to: Monika Gloor, Division of Radiological Physics, Depart-
ment of Radiology, University Hospital Basel, Petersgraben 4, CH-4031
Basel, Switzerland. E-mail: firstname.lastname@example.org
Received 1 November 2007; revised 10 March 2008; accepted 25 April 2008.
Published online in Wiley InterScience (www.interscience.wiley.com).
Magnetic Resonance in Medicine 60:691–700 (2008)
© 2008 Wiley-Liss, Inc.
TR, partial integration of the coupled Bloch equations
yields an extended steady-state eigenvector equation for
bSSFP that can be solved analytically. Numerical simula-
tions confirm the validity of the derived two-pool bSSFP
signal equation and thereby justify the conceptual separa-
tion of exchange and relaxation processes. In the second
part of this work, the extended two-pool bSSFP signal
equation is used to derive qMTI parameters from human
brain, such as the fractional pool size and corresponding
exchange rates. It will be demonstrated that the fitted
parameter values are in good correspondence with litera-
ture values. Possible issues and differences with respect to
common qMTI methods are discussed and analyzed. Fi-
nally, it will be shown that qMTI with bSSFP offers great
potential for generating qualitative high resolution MT
parameter maps within clinically feasible acquisition
times. In summary, this work introduces a novel fast and
quantitative MTI method that is based on bSSFP.
Single-Pool bSSFP Signal Equation
For a centered echo (TE ? TR/2) between alternating ex-
citations (??), the steady-state of bSSFP is calculated from
the Bloch equations using partial integration, yielding:
1 ? E1
1 ? E1E2? ?E1? E2?cos?, 
where E1,2? exp(–TR/T1,2) and M0is the equilibrium mag-
netization (6,7). This formal description has been con-
firmed for simple probes consisting of a single aqueous
phase (11). However, considerable deviations (up to a fac-
tor of two) were detected in tissues, and the molecular
origin of this apparent signal reduction was found to be
MT (11). Thus, at least for tissues the ordinary description
of the steady-state according to Eq.  is inappropriate and
demands for an extended bSSFP signal equation including
MT effects. In the following, signal formation in bSSFP is
analyzed based on a standard binary spin-bath MT model
(19,23) similar to the two compartment model for blood
oxygen saturation (24).
Two-Pool bSSFP MT Model
The minimal model for MT based signal analysis is com-
posed of two pools: a liquid pool of “free” protons (sub-
script f) and a semisolid pool of protons that are restricted
in motion (subscript r). The formal description of this
binary spin-bath model leads to a set of coupled differen-
tial equations, as described in any detail elsewhere (19,23).
Saturation of restricted pool protons is typically achieved
by off-resonance irradiation (frequency offset ?), that ide-
ally leaves the magnetization of free pool protons unaf-
fected. The effect of pulsed irradiation (25) on the longi-
tudinal magnetization of the restricted pool protons can be
described by a time-dependent mean saturation rate:
where G(?) is the absorption line shape, and where ?W????
depends on the shape ?1(t) ? ? |B1(t)| and on the duration
TRFof the RF pulse.
On-resonance RF pulses are applied for MT bSSFP, that
is, ? 3 0 (11), and the system of differential equations (19)
? ? R2,fMx,f,[3a]
? ? R2,fMy,f? ?1?t?Mz,f,[3b]
? R1,f?M0,f? Mz,f? ? kfMz,f? krMz,r? ?1?t?My,f,
? R1,r?M0,r? Mz,r? ? kfMz,f? krMz,r
? W?? 3 0,t?Mz,r, [3d]
where the subscripts x, y, z denote the various spatial
components of the magnetization vector M, R1,f(R1,r) refers
to the longitudinal relaxation rate, and R2,f(R2,r) to the
transverse relaxation rate of the pools (R1,2? 1/T1,2). Mag-
netization exchange is given by the pseudo-first-order rate
constants kf? RM0,rand kr? RM0,f, where R is the funda-
mental rate constant between the two pools and M0,f(M0,r)
denotes the equilibrium magnetization of the free (re-
stricted) pool. The fractional size of the restricted pool
amounts to F ? M0,r/ M0,f, and by definition kr? kf/ F.
Idealized Two-Pool bSSFP Signal Equation
The restricted pool in an idealized MT bSSFP experiment
is fully saturated, that is, Mz,r? 0. This leads to a decou-
pling of Equations [3c] and [3d] similar to the idealized
MT spoiled GRE case (26). Equation [3c] can be rewritten
? ?R1,f? kf??
R1,f? kfM0,f? Mz,f?? ?1?t?My,f, 
being formally analogous to a single-pool situation, but
with modified longitudinal relaxation and equilibrium
magnetization. Therefore, Equation  can be used to de-
scribe the idealized two-pool bSSFP signal using the sub-
R1,f3 R1,f? kf and M0,f3
R1,f? kfM0,f. 
However, it can be readily deduced that especially in the
range of low to moderate flip angles the Mz,r? 0 condition
is inapplicable leading to a substantial overestimation of
MT effects at low saturation levels. Thus, although the
idealized bSSFP MT model is only of limited experimental
interest, it represents the theoretical upper bound in the
overall achievable MT effects.
692Gloor et al.
Two-Pool bSSFP Signal Equation
Because the idealized two-pool model may fail to yield
adequate MT parameter estimations due to practical limi-
tations in the specific absorption rate (SAR) to satisfy Mz,r
? 0, the two-pool model Equations ([3a]–[3d]) must be
solved. Although, in principle this system of coupled dif-
ferential equations may be solved, the solution is complex
and cumbersome. Therefore, a different approach based on
partially integrated Bloch equations is presented, which
methodologically closely follows the vector model de-
scription of bSSFP, as introduced by Carr in 1958 (1) and
applied by many others (2,6,7) to derive signal properties
of bSSFP (see Eq. ). Beside the normal separation of
excitation and relaxation processes, it is further assumed
that exchange processes decouple from relaxation pro-
cesses within the short repetition times commonly used
with bSSFP. This approach is valid as long as fractional
pool size modifications from longitudinal relaxation pro-
cesses are negligible within TR, which is quite similar to
the concept of instantaneous rotation that allows for the
decoupling of excitation and relaxation. Based on these
simplifications, an analytical solution to the two-pool
bSSFP model can be derived as follows: Without loss of
generality, RF pulses are played out along the x-axis. As a
result, only y-components of the transverse magnetization
contribute to the steady-state signal, and the system of
differential Equations ([3a]–[3d]) with magnetization M ?
(Mx,fMy,fMz,fMz,r) can be reduced to M ? (My,fMz,fMz,r).
Exchange and relaxation parts of Equations [3b]–[3d] yield
a solution of form M(t) ? A(t) ? M(0) and M(t) ? E(t) ? M(0)
with matrix representations:
F ? 1?
F ? 1
1 ? Fe??F?1?k,r
F ? Fe??F?1?k,r
1 ? e??F?1?k,r
F ? e??F?1?k,r?and
where E2,f ? exp(?R2,ft), E1,f ? exp(?R1,ft) and E1,r ?
exp(?R1,rt). Excitation is captured in a matrix Rxthat con-
tains a rotation part for the free pool and a saturation term
for the restricted pool, whereas RF phase alternation is
described by Rz?? ? 180?? according to:
and Rz?? ? 180?? ??
The steady state equation can now readily be derived
following standard methods (7). The magnetization di-
rectly after the nthRF pulse is given by Mn
nthRF pulse. The magnetization directly before the (n?1)st
pulse in case relaxation takes place before exchange thus
?describes the magnetization directly before the
?? M0?, where M0??
M0,f?1 ? E1,f?
M0,r?1 ? E1,r??.
The eigenvalue equation for the steady-state magnetiza-
tion directly after an RF pulse is of form Mn?1
(here Rztakes into account alternating RF pulses) to finally
M???? ? Rx?I ? RzAERx??1AM0
?1 ? E1,f?B ? C
A ? BE1,fE2,f? ?BE1,f? AE2,f?cos?, 
A ? 1 ? F ? fwE1,r?F ? fk?,
B ? 1 ? fk?F ? fwE1,r?F ? 1??,
C ? F?1 ? E1,r??1 ? fk?,
fk? exp[?(kf? kr)TR],
fw? exp? ? ?W?? 3 0??TRF?, M0,f? 1,
E2,f? exp(?R2,fTR), E1,f? exp(?R1,fTR)
and E1,r? exp(?R1,rTR).
It is interesting to note that Eq.  is of the same form
as the well-known single-pool bSSFP steady-state equa-
tion (Eq. ) but shows some additional terms comprising
MT related parameters, such as F, kfand the mean satura-
tion rate ?W?W??. It is thus apparent that Equation 
converges to Equation  in the limit of a vanishing re-
stricted pool, that is, ?W?W??, F, kf, kr, E1,r3 0, as can be
expected from theoretical considerations. The solution to
the two-pool bSSFP signal equation as given by Equations
 and  represents the case where relaxation takes
place before exchange. For reasons of completeness, it is
noteworthy that the case where relaxation takes place after
exchange yields an almost identical solution as Equations
 and . For the sake of simplicity the following will
only consider the former solution (relaxation before ex-
Experiments and calibrations were performed on a Sie-
mens 1.5 T Avanto system (Siemens Medical Solution,
Erlangen, Germany) and all numerical simulations, data
analysis and visualization were done in Matlab 2006a (The
MathWorks, Inc., Natick, MA).
Quantitative MTI Using bSSFP693
Numerical simulations of the full set of nonsimplified
ordinary differential equations (ODE) (see Eqs. [3a]–[3d])
were performed to verify Equation  to justify the sep-
aration of relaxation and exchange processes (see Eq. ).
Therefore, a standard ODE solver was used to simulate the
steady-state signal as a function of RF pulse characteristics
(flip angle ?, RF pulse duration TRF), relaxation processes
(T1and T2), fractional pool size (F), and exchange proper-
ties (kf). For excitation, sinc-shaped RF pulses of variable
duration having a time-bandwidth product of 2.7 and one
side lobe were used. The mean saturation rate (?W?) is
calculated as a function of ? and TRFaccording to Equation
 based on Super-Lorentzian line shapes G(?) being ap-
propriate for the description of tissues (19,21):
The on-resonance singularity is handled by extrapolating
G(?) from approximately 1 kHz to the asymptotic limit ?
3 0, yielding G(0) ? 1.4 s ˙ 10?5s?1(11) for T2,r? 12 ?s (19).
As a result of the uncertainty, no distinction was made
between G(0) for white matter and G(0) for gray matter.
Possible issues are analyzed and discussed in any detail
later in this work. In this framework, the general uncer-
tainty in R1,r(23) is expressed by the fact that R1,ris set
equal to R1,f. This is in contrast but not so different from
the common R1,r? 1 s?1assignment.
In Vivo Experiments
All experiments were performed in 3D with a sagittal
orientation based on a 144 ? 192 ? 192 matrix yielding
1.3 mm isotropic resolution. Different MT sensitivities in
bSSFP can be achieved from a variation of the flip angle (?)
or the RF pulse duration (TRF) in Equation  (Fig. 1).
However, the signal change is considerably larger with RF
pulse elongation as compared to an increase in TR only
(12). From the overall similarity of Equations  and ,
it is seen that the main two-pool bSSFP signal character-
istics rely on the combination T2/T1(7). As a result, reli-
able T1or T2parameter estimation from Equation  thus
requires an independent determination of either T1or T2.
In summary, the protocol used for quantitative MT param-
eter estimation consisted of:
1.Two spoiled gradient echo (SPGR) sequences (TR/
TE ? 9.8 ms/4.77 ms, bandwidth ? 140 Hz/Pixel) with flip
angles ? ? 4° and ? ? 15° for calculation of a T1,fmap
according to the DESPOT1 method (27,28).
2. Eight bSSFP sequences with ? ? 35° (bandwidth ?
790 Hz/Pixel) and varying RF pulse durations (TR1/TRF,1?
2.92 ms/0.23 ms, TR2/TRF,2? 2.99 ms/0.3 ms, TR3/TRF,3?
3.09 ms/0.4 ms, TR4/TRF,4? 3.26 ms/0.58 ms, TR5/TRF,5?
3.53 ms/0.84 ms, TR6/TRF,6? 3.88 ms/1.2 ms, TR7/TRF,7?
4.28 ms/1.6 ms, TR8/TRF,8 ? 4.78 ms/2.1 ms). Minimal
TR/TRFvalues were determined by a compromise between
peripheral nerve stimulation as a result of fast gradient
switching and upper boundaries in ? due to limitations in
the SAR. Maximal TR/TRFvalues were chosen to yield an
elongation factor (?) of the RF pulse of 8–10 (see Fig. 1).
Minimal TR (irrespective of the RF pulse elongation) max-
imizes MT (12) and keeps the overall acquisition time
3. Eight bSSFP sequences with TR/TRF ? 2.99 ms/
0.27 ms (bandwidth ? 790 Hz/Pixel) and varying flip
angles (?1? 5°, ?2? 10°, ?3? 15°, ?4? 20°, ?5? 25°, ?6?
30°, ?7? 35°, ?8? 40°), which were distributed up to the
4. A multislice (16 slices, 5 mm slice thickness) B1field
map sequence (64 ? 64 matrix, 4 mm in plane resolution)
using stimulated echoes in a multipulse sequence (?–?–?
analogous to the ?–2?–? scheme (29)) for the assessment
of flip angle deviations.
5. For anatomical reference, an MPRAGE sequence (TR/
TE ? 1760 ms/3.35 ms, inversion time ? 906 ms, ? ? 7°,
bandwidth ? 190 Hz/Pixel) completed the qMTI protocol.
Typically, frequency variations of less than 20 Hz were
achieved within the brain by manual shimming to reduce
off-resonance sensitivities. Overall qMTI data acquisition
was completed within 30 minutes (including DESPOT1:
6 min; bSSFP (varying ?): 8 min; bSSFP (varying TRF):
11 min; B1map: 2 min; MPRAGE: 3 min). Measurements
on healthy volunteers were approved by the local ethics
committee. Five acquisitions on the same subject were
performed with time-lags of 1 to 3 weeks to analyze the
reproducibility of the method.
The software packages FSL (30) and AFNI (31) were used
for the image registration, brain extraction and white (gray)
matter segmentation. The effective measured B1field is
expressed as percentage difference from the actual flip
angle as set by the system (protocol). From this, effective
flip angles were calculated on a pixel-by-pixel base after B1
image registration and data interpolation (FSL) before all
data evaluations. Longitudinal relaxation time, T1,f, was
calculated from DESPOT1. The spin-bath model parame-
FIG. 1. BSSFP sequence scheme for exploration of MT effects from
RF pulse variation. a: For short RF pulse durations and minimal
repetition time TRMTthe steady-state signal is strongly attenuated
from MT effects. b: Minimal MT effects can be achieved from a
considerable increase in the RF pulse duration (factor ? leading to
an increased repetition time TRnon-MT), yielding low saturation of the
restricted pool protons.
694Gloor et al.
ters F, kfand T2,fwere estimated from a fit of Equation 
to all 16 bSSFP image acquisitions (pixel-by-pixel), using a
global nonlinear least-squares fitting routine. Lastly, MTR
maps were calculated from TRFvariation (TMT? TRF,1?
0.23 ms and Tnon-MT? TRF,8? 2.1 ms) according to MTR ?
100 ? (S0– SMT) / S0[%], where S0and SMTcorrespond to
the signal amplitude measured with TMTand Tnon-MT, re-
spectively (17). As a result, 3D T1,f, F, kf, T2,fand MTR
parameter maps could be extracted.
Figure 2 displays a comparison of the derived two-pool
bSSFP steady-state equation (Eq. ) with numerical sim-
ulations of differential Equations [3a]–[3d] for white mat-
ter using parameter values from (19). Within the range of
experimentally applied flip angles (see the Methods sec-
tion), the analytical solution slightly underestimates sim-
ulated values by 1.1% in maximum (Fig. 2a). In addition,
the idealized two-pool bSSFP steady-state signal (Eq. )
is displayed, where full saturation of the restricted pool
protons occurs for all flip angles. As expected, the two-
pool bSSFP equation converges to the idealized case only
in the limit of large flip angles (? ? 100°).
In earlier work, it was demonstrated that bSSFP shows a
strong dependency on TR (11) and especially on the RF
pulse duration (12). The derived two-pool bSSFP signal as
a function of TRF (and TR) is displayed in Figure 2b.
Simulated RF pulse durations covered a range of 0 ? TRF?
2.1 ms at a fixed time-bandwidth product of 2.7, with
corresponding TR variation between 3 ms and 5.1 ms.
Similar to Figure 2a, the analytical two-pool model equa-
tion slightly underestimates values from full ODE simula-
tions and the discrepancy increases with increasing RF
pulse durations. At TRF ? 2.1 ms the simulated value
exceeds the one deduced from the signal equation by
4.3%. This is most likely a result of the assumed instan-
taneous action of the RF pulses; a criteria that becomes
more and more falsified with increasing TRF. For short TRF
durations, the situation of infinitely many RF pulses per
second is achieved and therefore full saturation occurs
??W? 3 ?? representing the idealized case.
In summary, good agreement between the analytical de-
scription of the two-pool bSSFP model (Eqs. [10, 11]) and
the numerical simulations based on the full, that is, non-
simplified, ODEs (Eqs. [3a]–[3d]) was found. The slight
underestimation of the steady-state signal by Equation 
as compared to the numerical simulations is most likely
due to neglected T2effects during excitation processes, as
indicated by the increasing discrepancy with increasing
RF pulse durations.
Quantitative Magnetization Transfer Imaging (qMTI)
Quantitative MT parameter evaluation based on Equations
 and  is exemplarily displayed in Figure 3. After
brain extraction and registration, Equation  was fitted
pixel-by-pixel to bSSFP signal intensities. Figure 3a,b dis-
plays the signal dependencies on RF pulse duration (and
corresponding increase in TR) and flip angle for white and
gray matter, respectively. The global fits (i.e., both flip
angle and TRFvaried data sets share the same MT model
parameters) yield parameter estimates for the fractional
pool size F, the exchange rate kfand the relaxation time of
the free pool T2,f. Because the fractional pool size is corre-
lated with myelin (22), F shows higher values in white as
compared to gray matter.
Figures 4 and 5 display the results from qMTI using the
two-pool bSSFP model in normal appearing human brain.
In Figure 4, besides the anatomical reference, axial, sagittal
and coronal slices were shown for T1,f (based on DES-
FIG. 2. Two-pool bSSFP model
analysis for white matter using pa-
rameters from literature (T1,f ?
585 ms, T2,f? 81 ms, F ? 0.157, kf
? 4.45 s?1, R1,r? 1 s?1, T2,r? 12
?s (19)). a: Transverse magnetiza-
tion (Mxy) with and without MT and
MTR as a function of flip angle (?).
b: Variation of ? (TR ? 2.92 ms,
TRF ? 230 ?s): Excellent corre-
simulations and the prediction ac-
cording to the analytical solution of
found. The two-pool bSSFP signal
equation converges to the ideal-
ized case roughly at ? ? 100°,
thereby indicating full saturation of
restricted pool protons. c: Variation
of TRF (? ? 35°): Simulation and
solution match for low TRFbut de-
viate with increasing RF pulse du-
Quantitative MTI Using bSSFP 695