Page 1

Quantitative Magnetization Transfer Imaging Using

Balanced SSFP

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-

Liss, Inc.

Key words: balanced SSFP; magnetization transfer; quantitative

imaging; MT

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-

quisition times.

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

691

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: monika.gloor@stud.unibas.ch

Received 1 November 2007; revised 10 March 2008; accepted 25 April 2008.

DOI 10.1002/mrm.21705

Published online in Wiley InterScience (www.interscience.wiley.com).

Magnetic Resonance in Medicine 60:691–700 (2008)

© 2008 Wiley-Liss, Inc.

Page 2

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.

THEORY

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:

My? M0sin??E2

1 ? E1

1 ? E1E2? ?E1? E2?cos?, [1]

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. [1] 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:

?W???? ?

?

TRF?

0

TRF

?1

2?t?dtG???, [2]

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)

reduces to:

dMx,f

dt

? ? R2,fMx,f,[3a]

dMy,f

dt

? ? R2,fMy,f? ?1?t?Mz,f,[3b]

dMz,f

dt

? R1,f?M0,f? Mz,f? ? kfMz,f? krMz,r? ?1?t?My,f,

[3c]

dMz,r

dt

? 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

as:

dMz,f

dt

? ?R1,f? kf??

R1,f

R1,f? kfM0,f? Mz,f?? ?1?t?My,f, [4]

being formally analogous to a single-pool situation, but

with modified longitudinal relaxation and equilibrium

magnetization. Therefore, Equation [1] can be used to de-

scribe the idealized two-pool bSSFP signal using the sub-

stitutions:

R1,f3 R1,f? kf and M0,f3

R1,f

R1,f? kfM0,f. [5]

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.

Page 3

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. [1]). 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:

A?t? ?

1

F ? 1?

F ? 1

0

0

00

1 ? Fe??F?1?k,r

F ? Fe??F?1?k,r

E?t? ??

1 ? e??F?1?k,r

F ? e??F?1?k,r?and

E2,f

0

0

00

0

E1,f

0E1,r?,[6]

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:

Rx??,t? ??

cos?

? sin?

0

sin?

cos?

0

0

0

e??W??30???t?

and Rz?? ? 180?? ??

? 1

0

0

0

1

0

0

0

1?.[7]

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

where Mn

nthRF pulse. The magnetization directly before the (n?1)st

pulse in case relaxation takes place before exchange thus

evolves to:

?? RxMn

?,

?describes the magnetization directly before the

Mn?1

?

? A?ERxMn

?? M0?, where M0??

0

M0,f?1 ? E1,f?

M0,r?1 ? E1,r??.

[8]

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

yield:

?

? RzMn

?

M???? ? Rx?I ? RzAERx??1AM0

[9]

with solution

My

?? M0,fsin?

?1 ? E1,f?B ? C

A ? BE1,fE2,f? ?BE1,f? AE2,f?cos?, [10]

where

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).[11]

It is interesting to note that Eq. [10] is of the same form

as the well-known single-pool bSSFP steady-state equa-

tion (Eq. [1]) 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 [10]

converges to Equation [1] 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

[10] and [11] 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

[10] and [11]. For the sake of simplicity the following will

only consider the former solution (relaxation before ex-

change).

METHODS

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

Page 4

Numerical Simulations

Numerical simulations of the full set of nonsimplified

ordinary differential equations (ODE) (see Eqs. [3a]–[3d])

were performed to verify Equation [10] to justify the sep-

aration of relaxation and exchange processes (see Eq. [6]).

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

[2] based on Super-Lorentzian line shapes G(?) being ap-

propriate for the description of tissues (19,21):

G??? ??

0

1

?

2

?

T2,r

?3u2? 1?exp??2?

2??T2,r

3u2? 1?

2?du [12]

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 [10] (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 [10] and [1],

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 [10] 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

minimal.

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

SAR limit.

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.

Data Analysis

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.

Page 5

ters F, kfand T2,fwere estimated from a fit of Equation [10]

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.

RESULTS

Validation

Figure 2 displays a comparison of the derived two-pool

bSSFP steady-state equation (Eq. [10]) 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. [5])

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 [10]

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

[10] and [11] is exemplarily displayed in Figure 3. After

brain extraction and registration, Equation [10] 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-

spondence between

simulations and the prediction ac-

cording to the analytical solution of

thetwo-pool bSSFP

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-

rations.

numerical

modelis

Quantitative MTI Using bSSFP 695