A comparison between equations describing in vivo MT : The effects of noise and sequence parameters Quantitative models of magnetization transfer -LRB- MT -RRB- allow the estimation of physical properties of tissue which are thought to reflect myelination , and are therefore likely to be useful for clinical application . Although a model describing a two-pool system under continuous wave-saturation has been available for two decades , generalizing such a model to pulsed MT , and therefore to in vivo applications , is not straightforward , and only recently have a range of equations predicting the outcome of pulsed MT experiments been proposed . These solutions of the 2-pool model are based on differing assumptions and involve differing degrees of complexity , so their individual advantages and limitations are not always obvious . This paper is concerned with the comparison of three differing signal equations . After reviewing the theory behind each of them , their accuracy and precision is investigated using numerical simulations under variable experimental conditions such as degree of T1-weighting of the acquisition sequence and SNR , and the consistency of numerical results is tested using in vivo data . We show that while in conditions of minimal T1-weighting , high SNR , and large duty cycle the solutions of the three equations are consistent , they have a different tolerance to deviations from the basic assumptions behind their development , which should be taken into account when designing a quantitative MT protocol . 1 Introduction The Magnetization Transfer -LRB- MT -RRB- effect is based on the exchange of magnetization occurring between groups of spins characterized by different molecular environments . In biological tissues , two or more `` pools '' of protons can be identified : those in free water -LRB- the free , or liquid , pool -RRB- and those bound to large molecules -LRB- referred to as restricted , semisolid , or macromolecular , pool -RRB- . The latter protons are characterized by a very short transverse relaxation time -LRB- T2 -RRB- and therefore do not directly contribute to signal intensity in conventional magnetic resonance -LRB- MR -RRB- images . Nevertheless , it is possible to sensitise an MR experiment to the magnetic resonance characteristics of macromolecular protons by exposing the sample to radio-frequency -LRB- RF -RRB- energy several kilohertz away from the Larmor frequency . Protons in free water are relatively insensitive to such irradiation , but it can cause saturation of protons in the semisolid pool which , due to their short T2 and correspondingly large line width , are responsive to irradiation at these frequencies . In these conditions , any exchange of magnetization between pools results in a decreased intensity of the observed MR signal . From a quantitative MT model based on the exchange between two pools Henkelman et al. -LSB- 1 -RSB- derived a signal equation for the continuous wave -LRB- CW -RRB- case , in which RF irradiation of particular -LRB- constant -RRB- amplitude and several seconds duration is used to saturate the macromolecular pool . The parameters characterizing the two pools in the model are potentially interesting to measure , and they can be estimated by fitting Henkelman 's equation to a set of MR measurements obtained in the presence of MT pulses with a suitable set of amplitudes ω1 and offset frequencies Δf . As CW irradiation is impractical and generally not available for in vivo imaging experiments , in vivo MT-weighted MRI is generally obtained using the so-called pulsed MT acquisition , in which the long period of saturation is replaced by a much shorter irradiation pulse -LRB- typically applied just before each excitation pulse -RRB- along with intervals without irradiation -LRB- during which data is collected -RRB- . For data from this type of acquisition , Henkelman 's equation must be modified to allow for the short duration of the saturation pulses relative to T1 -LSB- 2 -RSB- . A number of such modified signal equations for pulsed MT , all based on the same original two-pool model , have been developed -LSB- 3 -- 5 -RSB- . While the numerical results published so far suggest reasonable consistency across the solutions predicted by these equations , no direct comparison is available , and the differing conventions and symbols used mean that evaluating discrepancies and similarities between them is not straightforward . This paper is concerned with the comparison of three of these signal equations -- two derived by Sled and Pike -LSB- 3,6 -RSB- , plus that of Ramani et al. -LSB- 5 -RSB- . Ramani et al. used a CW power equivalent approximation -LRB- CWPE -RRB- -LSB- 5 -RSB- where the pulse is simply replaced by a CW irradiation with the mean square amplitude that would give the same power over the interval between MT pulses . By means of the CWPE approximation , Henkelman 's steady state model can be straightforwardly applied to the in vivo MRI case , neglecting the imaging elements of the pulse sequence . Due its steady state nature , the implicit assumption within the equation is that the relative signal intensity in data obtained with different MT-weightings only depends on the characteristics of the MT pulse , and that T1 and T2 relaxations equally affect all measurements . As Ramani 's equation does not explicitly model the effects of the excitation pulses and TR , its description of the MT-weighted signal is valid only when the degree of T1-weighting in the acquisition sequence is minimal . As it effectively assumes that the MT pulse is applied continuously , another parameter likely to affect the accuracy of Ramani 's equation is the duty cycle , i.e. the duration of the MT pulse relative to the repetition period , whose effect has never formally been investigated . Sled and Pike -LSB- 6 -RSB- propose an alternative equation which can be fitted directly to the measured signal . Their solution is derived by approximating the pulse sequence as a series of periods of free precession , CW irradiation and instantaneous saturation of the free pool . It has the advantage of incorporating the effect of the excitation RF pulses , and also makes it possible to account for saturation effects of the excitation . Together with this solution , the authors propose also a simpler variant which neglects free precession , thus assuming a succession of instantaneous saturation of the free pool and CW irradiation of the macromolecular pool for the total duration of the interval between pulses . Both equations presented by Sled and Pike for in vivo applications require the numerical evaluation of ordinary differential equations -LRB- when modeling the rate of saturation of the macromolecular pool with a super-Lorentzian , see next section -RRB- , at least for the estimation of the effect of the MT pulse on the free pool , and they are therefore computationally intensive . Ramani 's solution has the advantage of being simpler , at the price of its inability to account for the effects of the excitation pulses . A recent paper presented an evaluation of these signal equation , validated using animal data -LSB- 7 -RSB- . Here we first review the theory behind them and then use numerical simulations to extend the range of experimental conditions under which each can be tested -LRB- investigating how duty cycle , saturation effects of the excitation and noise affect the MT parameters fitted by each of them -RRB- . We also perform a statistical comparison between MT parameters estimated using each signal equation in healthy brain tissue from in vivo data . 2 Theory 2.1 Coupled Bloch equations Assuming that the MT effect can be modeled using a liquid pool -LRB- A -RRB- and a macromolecular pool -LRB- B -RRB- , the magnetization of either pool can be described by its longitudinal component and its transverse components . The exchange between pools associated with the transverse components of magnetization can be considered negligible due to the extremely short T2 associated with the macromolecular pool -LSB- 2,6 -RSB- . The coupled Bloch equations for the system can thus be written as follows : In Eqs . -LRB- 1 -RRB- -- -LRB- 4 -RRB- , represents the transverse relaxation time of the liquid pool , and are the fully relaxed values of magnetization associated with the two pools -LRB- assumed dimensionless -RRB- , RA and RB are their longitudinal relaxation rates , and R is the exchange rate constant . Δf represents the frequency offset of the pulse , while ω1 -LRB- t -RRB- is the time dependent amplitude of the pulse expressed in rad s − 1 -LRB- i.e. the angular frequency of precession induced by the pulse -RRB- . RRFB -LRB- Δf , ω1 -LRB- t -RRB- -RRB- is the rate of saturation of longitudinal magnetization in pool B due to the irradiation by the amplitude defined by Δf and ω1 -LRB- t -RRB- , and depends on the transverse relaxation time of the macromolecular pool , . Li et al. -LSB- 8 -RSB- show that , in brain tissue , the specra associated with macromolecular pool are better modeled by a super-Lorentzian , -LRB- with being the average power of the MT pulse -RRB- , yielding : and we adopt this model here . We remark here that this notation is not universal . Some authors label the A and B pools as ` F ' and ` R ' , respectively -LSB- 6 -RSB- , or ` f ' and ` m ' -LSB- 9,10 -RSB- and use the symbol W instead of RRFB -LSB- 6 -RSB- . The pseudo first-order exchange rates , and , are often referred to as kf -LRB- or simply k -RRB- and kr . 2.2 Sled and Pike 's RP signal equation Assuming that the pulse sequence consists of an MT pulse followed by an excitation pulse and by a period of recovery , Sled and Pike -LSB- 6 -RSB- decompose it into a series of periods where Eqs . -LRB- 1 -RRB- -- -LRB- 4 -RRB- have exact or approximate solutions . These solutions can then be concatenated by imposing the appropriate initial conditions , leading to an expression for the measured signal which is less expensive to compute than numerically integrating the full set of differential equations . The effect of an MT pulse on the macromolecular pool is modeled as a rectangular pulse whose width is equal to the full width at half maximum -LRB- τRP -RRB- of the curve obtained by squaring the instantaneous amplitude of the MT pulse throughout its duration , and whose amplitude is such that the pulses have equivalent average power -LRB- rectangular pulse , or RP , approximation -RRB- . The effect of the pulse on the liquid pool is modeled as an instantaneous fractional saturation of the longitudinal magnetization . Such fractional saturation -LRB- S1A -RRB- is estimated by solving -LRB- numerically -RRB- the system of Eqs . -LRB- 1 -RRB- , -LRB- 3 -RRB- and -LRB- 4 -RRB- when R and RA are set to 0 . In matrix form -LSB- 6 -RSB- , considering the longitudinal components of magnetization onlyInstantaneous saturation of the free pool , caused by both MT and excitation pulses , is simply described by multiplying Mz by the matrix S -LRB- where θ is the excitation flip angle -RRB- The state of the magnetization after a period t1 -LRB- assuming starting time = t0 -RRB- is given by the solution to the system of Eqs . -LRB- 1 -RRB- and -LRB- 2 -RRB- for either free precession -LSB- FP -RSB- or CW : with According to Sled and Pike 's RP approximation , over the time interval T between application of MT pulses -LRB- typically the time required to excite and collect data for a single k-space line of a single image slice -RRB- , Mz undergoes instantaneous saturation , CW irradiation for a period τRP/2 , FP for a period -LRB- T − τRP -RRB- , and CW for another τRP/2 . After including all thee steps , we can impose the equalityand solve for Mz yielding an equation for the longitudinal components of magnetization . Recalling that the signal observed at readout is -LRB- where the repetition time TR may be ⩾ T depending of the details of the image acquisition -RRB- it is thus possible to model the MT-weighed signal . The solution to Eq . -LRB- 10 -RRB- can be easily computed in matrix form , and we adopt this procedure for all the following experiments . 2.3 Sled and Pike 's CW signal equation A simpler expression is presented in the same paper -LSB- 6 -RSB- , where the effect of the MT pulse on the macromolecular pool is described by a CW irradiation of duration T . In this case , over the same period T , Mz undergoes instantaneous saturation , and CW irradiation -LRB- of the restricted pool only -RRB- for a period T. Following the same procedure as described above , a more manageable analytic expression can be derived in this case : Here is the steady state solution obtained by Henkelman et al. -LSB- 1 -RSB- for CW irradiation of the restricted pool of duration T : which is equivalent to the first element of the vector . In Eq . -LRB- 12 -RRB- , We note here that for the specific case of the 3D spoilt gradient echo acquisition described in this paper -LRB- see Section 3 -RRB- T = TR . Sled and Pike also introduce , as a useful index which is believed to correlate with myelin content , the relative size of the macromolecular pool -LSB- 6 -RSB- F , defined asF can be fitted directly , by substituting in Eq . -LRB- 13 -RRB- and previous . More details on these signal equations can be found in -LSB- 3,6,11 -RSB- . 2.4 Ramani 's signal equation Henkelman 's solution -LSB- 1 -RSB- for the CW case is obtained by solving Eqs . -LRB- 1 -RRB- -- -LRB- 4 -RRB- in the steady state , i.e. setting the derivatives on the left hand side to zero . Ramani et al. -LSB- 5 -RSB- adopt the same approach in the pulsed MT case , simply replacing the MT pulse with a CW irradiation with the same mean square amplitudewhere PSAT is the mean square saturating field . In order to ease the comparison between the three signal equations , we break with the terminology of the original paper -LSB- 5 -RSB- , where the macromolecular fraction f -LRB- with f = F / -LRB- F + 1 -RRB- -RRB- was used , and instead rewrite the Henkelman -- Ramani expression using F and to obtainwhere M0 is the signal with no MT-weighting -LRB- again assuming a constant of proportionality of c = 1 -RRB- . Note that while in the equations presented in the previous section , M0 is the equilibrium magnetization , in Eq . -LRB- 17 -RRB- it simply represents the partially recovered magnetization available prior to the application of an MT pulse . 2.5 Fitting All the equations are written in terms of seven parameters : and , but these can not be uniquely determined -LSB- 1,12 -RSB- . Constraints are imposed by measuring the observed longitudinal relaxation rate of the sample , RAobs -LRB- = 1/T1obs -RRB- independently , linked to RA by Henkelman et al. -LSB- 1 -RSB- A further issue is that dependence of S -LRB- ω1 , Δf -RRB- on RB is weak , making fitting of this parameter unstable . Since the estimates of the other parameters are largely insensitive to its value , RB is usually kept fixed at 1 s − 1 -LSB- 1,5,6 -RSB- . This reduces the number of free parameters to 5 , which can be estimated by fitting the equation to five or more measurements with different combinations of ω1 -LRB- t -RRB- and Δf . 3 Materials and methods 3.1 Numerical simulations In order to compare the three signal equations , and to highlight their shortcomings , we need to test their performance against data corresponding to a known set of parameters . The easiest way to obtain such data is to synthetically produce them , using numerical simulations . We consider here the case of an MT-weighted spoiled gradient echo acquisition , where off-resonance saturation is achieved using Gaussian pulses -LRB- of duration τSAT -RRB- applied once every TR -LRB- just prior to RF excitation -RRB- , while on-resonance excitation is obtained using short 5-lobe sinc pulses -LRB- in the presence of a ` slab selection ' gradient -RRB- . Eqs . -LRB- 1 -RRB- -- -LRB- 4 -RRB- can be solved numerically to predict the longitudinal magnetization at the end of the MT pulse -LRB- i.e. just before the excitation pulse -RRB- , . Since the measured signal intensity is proportional to the transverse magnetization at readout -LRB- if we neglect decay -RRB- : The MT pulse is characterized by its maximum amplitude , B1SAT , MAX , by its duration , τSAT , and by the standard deviation of the Gaussian envelope , σ . The excitation pulse is characterized by its maximum amplitude , B1EXC , MAX , by its duration , τEXC , and by its bandwidth , BW . Both pulses can be described by their equivalent on-resonance flip angle , given by the integral over pulse duration of ω1 -LRB- t -RRB- . Additionally , we assume the excitation pulse to have no effect on pool B . The measured signal intensity can be estimated by calculating the solution of Eq . -LRB- 19 -RRB- as Mz tends towards a steady state , i.e. after solving Eqs . -LRB- 1 -RRB- -- -LRB- 4 -RRB- for several TRs , until the difference between Mz -LRB- TRn + τSAT -RRB- and Mz -LRB- TRn +1 + τSAT -RRB- is less than 0.01 % of Mz -LRB- TRn + τSAT -RRB- . We simulate four experiments to probe the effects of different parameters on the fits . Firstly we aim to identify any systematic biases in the three signal equations with respect to variable experimental conditions , by fitting them to noise-free simulated data -LRB- Experiments 1 -- 3 -RRB- ; next we investigate their robustness in the presence of noise , by using a Monte Carlo approach -LRB- Experiment 4 -RRB- . The simulated signal is computed by using a Runge -- Kutta ordinary differential equation -LRB- ODE -RRB- integrator with adaptive step-size control -LSB- 13 -RSB- . Spoiling of the transverse magnetization is simulated by setting the transverse components of magnetization equal to zero before the occurrence of every MT pulse . All experiments use the same set of MT parameters , chosen to be similar to values measured previously in white matter -LSB- 14,15 -RSB- and shown in Table 1 as ` test set ' . All also use the same excitation pulse parameters -LRB- τEXC = 3.2 ms , and BW = 2.5 kHz -RRB- , but other values differ : Experiment 1 : In the first simulation , we investigate the accuracy of Ramani 's , Sled and Pike 's CW and Sled and Pike 's RP signal equations as a function of duty cycle , by simulating 5 MT experiments using an excitation flip angle of 5 ° , TR = 30 ms , and varying τSAT between 5 and 25 ms , in steps of 5 ms. The full width at half maximum -LRB- FWHM -RRB- of the pulses varies accordingly between 2.29 -LRB- σ = 0.97 -RRB- and 11.45 ms -LRB- σ = 4.86 ms -RRB- , in steps of 2.29 ms. Each simulated set consists of 60 points , generated using only two fixed values of ω1CWPE -LRB- 250.2 and 850.7 rad s − 1 -RRB- -LRB- and thus different MT flip angles for each value of τSAT -RRB- , following Sled and Pike -LSB- 6,15 -RSB- , and 30 values of Δf per flip angle . The offset frequency ranges from 400 to 30,000 Hz , sampled at regular interval on a logarithmic scale . The three equations are fitted to the synthetic datasets using the Levenberg -- Marquardt method , as implemented in Numerical Recipes -LSB- 13 -RSB- , to yield the estimated parameters . Numerical derivatives are computed where an analytical expression is unavailable . The test set of parameters -LRB- see Table 1 -RRB- are used to provide the initial parameter estimates , and RAobs is obtained by solving Eq . -LRB- 18 -RRB- with respect to this quantity . The same ODE integrator used to compute the simulations is used to estimate S1A at every step for Eqs . -LRB- 7 -RRB- , -LRB- 11 -RRB- and -LRB- 12 -RRB- . Experiment 2 : Secondly , we explore the effects of saturation of the excitation on the estimated parameters , simulating the outcome of six MT experiments using regularly spaced excitation flip angles ranging from 5 ° to 20 ° and TR = 30 ms. We fix τSAT = 15 ms and keep the other parameters as in the first experiment . With the exception of the excitation flip angle , the input parameters are identical for all six cases . In order to check that our results are not specific to the choice of the sampling scheme , we repeat Experiments 1 and 2 with an alternative sampling scheme generated using four fixed values of ω1CWPE -LRB- 250.2 , 450.4 , 650.5 and 850.7 rad s − 1 -RRB- and 15 values of Δf per flip angle . Experiment 3 : As Sled and Pike used two sequences with two different MT pulse durations and two TRs to constrain the estimate of RM0B , we create a synthetic dataset formed by two `` sequences '' -LRB- 40 MT points with TR = 30 ms , τSAT = 15 ms , θ = 5 ° , 2 MT flip angles equal to 250 ° and 850 ° , respectively ; and 20 with TR = 45 ms , τSAT = 20 ms , θ = 6 ° , 2 MT flip angles equal to 353 ° and 1202 ° , respectively -RRB- . We then compare the accuracy of the MT parameters estimated fitting the three signal equations to such a dataset and to the `` single sequence '' dataset obtained in Experiments 2 when fixing τSAT = 15 ms and θ = 5 ° . Experiment 4 : Finally , in order to investigate the sensitivity to noise , we add complex noise with zero mean Gaussian real and imaginary parts to the dataset obtained in Experiment 2 for τSAT = 15 ms and θ = 5 ° . We then take the modulus to obtain a noisy data sample . The standard deviation of the Gaussian noise is set to be M0/Σ , where Σ is the desired SNR in the unweighted image , which we vary over the interval -LSB- 20 , 300 -RSB- . -LRB- The SNR values typically observed in 3D spoiled gradient echo scans from our system , with acquisition parameters similar to those detailed below -LRB- see Section 3.2 -RRB- , typically range between 40 and 100 depending on the coil used , resolution , use of parallel imaging , etc. -RRB- For each level of noise , we generate 10,000 sets of noisy independent samples and fit the three equations to each set . Look-up tables are used for the super-Lorentzian lineshape , the fractional saturation S1A , and their derivatives , in order to speed up the computation . 3.2 In vivo data A single subject -LRB- female , 34 years old -RRB- was scanned twice on a 1.5 T system -LRB- SIGNA Horizon Echospeed , General Electrics , Milwaukee , WI , USA -RRB- using a 3D MT-weighted fast spoiled gradient recalled-echo -LRB- SPGR -RRB- sequence -LSB- 15 -RSB- -LRB- TR/TE = 28/5 .1 ms , Gaussian MT pulses , duration = 14.6 ms , standard deviation = 2.84 ms , bandwidth = 125 Hz , matrix = 256 × 96 × 32 , FOV = 240 × 180 × 160 mm3 , to reconstruct twenty-eight 256 by 256 voxel slices -RRB- . The excitation flip angle was 5 ° on the first session and 15 ° on the second one , the interval between scans was 7 days . On each occasion a dataset of 20 MT points was obtained , using 2 MT pulse flip angles -LRB- 220 ° and 820 ° , corresponding to a CWPE amplitude of 251.1 and 861.2 rad s − 1 -RRB- , and 10 values of Δf per flip angle . Δf ranged between 400 and 20,000 Hz , and was stepped using a constant logarithmic interval . In addition to the MT data , three 3D SPGRs -LRB- TR = 13.1 ms , TE = 4.2 ms , same FOV and resolution as the MT sequence -RRB- were also obtained on each occasion , with three different excitation flip angles -LRB- θ = 25 ° , 15 ° , 5 ° -RRB- in order to independently estimate the longitudinal relaxation rate of the system , RAobs . The body coil was used for signal transmission and the manufacturer 's eight-channel head coil was used for reception . The total scan time was about 45 min . The study was approved by the Joint Research Ethics Committee of The National Hospital for Neurology and Neurosurgery and the Institute of Neurology , UCL , and the subject gave written informed consent before taking part . 3.3 Image analysis The two datasets -LRB- one from each MRI session -RRB- were processed on a Unix workstation -LRB- Sun Microsystems , Mountain View , CA , USA -RRB- , as described elsewhere -LSB- 15 -RSB- . Briefly : the 20 MT-weighted volumes obtained with the MT-weighted SPGR sequence and the three volumes obtained with the non-MT-weighted SPGR sequence were co-registered to the first MT-weighted volume using a modified -LSB- 16 -RSB- version of Automated Image Registration -LRB- AIR , available at http://air.bmap.ucla.edu:16080/AIR -RRB- -LSB- 17 -RSB- . RAobs was estimated on a pixel-by-pixel basis by fitting the theoretical SPGR signal equation to the signal in the non-MT-weighted SPGR images , as a function of the flip angle -LSB- 18 -RSB- . The 3 MT signal equations were fitted to the remainder of the images -LRB- as described for the synthetic data -RRB- yielding estimates of the MT parameters . Six bilateral regions of interest -LRB- ROIs -RRB- , three located in white matter -LRB- frontal , temporal and internal capsule -RRB- and three in gray matter -LRB- thalamus , putamen , caudate nucleus -RRB- were outlined on the T1-weighted images obtained from the non-MT-weighted SPGR scan -LRB- flip angle = 25 ° -RRB- . The 12 ROI outlines were then superimposed on the MT parameter maps , yielding 6 -LRB- 3 signal equations times 2 flip angles -RRB- estimates for each of the following : , and . Paired sample T-tests were used to compare the mean estimated parameters between equation solutions , and two-sample T-tests were used to test inequalities between flip angles , considering statistically significant two-tailed p values lower than 0.01 . 4 Results To provide a qualitative description of the accuracy of the three signal equations , we show in Fig. 1 the MT spectra simulated using Eq . -LRB- 18 -RRB- and the test set in Table 1 , together with those simulated using each of the three equations and the same test set , for two of the cases explored in Experiment 1 -LRB- A : duty cycle = 50 % , flip angle = 5 ° ; B : duty cycle = 17 % , flip angle = 5 ° -RRB- . All curves are normalized to the maximum intensity . As expected , the largest deviations between signal equations are observed at high power , for small offset frequencies , with Ramani 's equation providing the least accurate description . The deviation between Ramani 's predictions and the others are more pronounced for lower duty cycles -LRB- Fig. 1B -RRB- . 4.1 Duty cycle effect The estimates of , and against duty cycle are shown in Fig. 2 . Results for are omitted as the estimates from all signal equations converged to the test set irrespective of duty cycle . Both of Sled and Pike 's equations are less sensitive to changes in the duty cycle than Ramani 's equation -LRB- with the exception of -RRB- , providing very consistent estimates , except for , for which the CW variant yields a value closer to that used to create the simulation . The dependence of Ramani 's on the duty cycle is non-linear , with estimated values tending towards a plateau for duty cycles ⩾ 50 % . For Ramani 's equation , and are the parameters most sensitive to duty cycle changes . Although more stable , the estimates obtained at larger duty cycles are not necessarily more accurate than those obtained at the lowest duty cycle simulated -LRB- 17 % -RRB- . The same experiment repeated with a different sampling scheme yielded almost identical results for duty cycles > 17 % . For duty cycle = 17 % , Sled and Pike 's CW equation converged to and , and the RP variant converged to and , while Ramani 's equation produced results very similar to those obtained with the two-power scheme . 4.2 Saturation effect of the excitation The MT parameter estimates obtained from noise-free simulated data at various flip angles are reported in Fig. 3 , again with the exception of which is accurately determined by fitting all signal equations irrespective of the excitation flip angle , and therefore is not shown . As the flip angle -LRB- and thus the amount of saturation -RRB- increases , the estimates of the MT parameters based on Ramani 's signal equation increasingly deviate from the true values . Additionally , unlike the other parameters , the dependency of the estimated on the amount of saturation does not appear to be monotonic . Sled and Pike 's RP approximation provides the most consistent estimates across flip angles . The CW variant behaves similarly , at least for flip angles lower or equal to 15 ° , with increasingly biased results at higher flip angles . We report in Table 1 the estimated MT parameters obtained by fitting the three equations to the simulated noise-free 60 point dataset with excitation flip angle of 5 ° -LRB- minimum saturation effect from the excitation pulse -RRB- and duty cycle ≈ 50 % -LRB- at which point duty cycle effects have reached a plateau , and the three signal equations seem to provide consistent estimates -RRB- . All equations give similar results -LRB- very close to the ` test set ' of parameters used in creating the simulation -LRB- see Table 1 -RRB- -RRB- . The two variants of Sled and Pike 's solution provide more accurate estimates than Ramani 's equation for all parameters , with the CW approximation yielding a more accurate value for , and RP approximation yielding a slightly more accurate value for F -LRB- which Ramani 's equation tends to underestimate -RRB- . For all the signal equations , the largest error is in the estimation of -LRB- 7.6 % of true value for Sled and Pike 's RP equation , 10.2 % of true value for Sled and Pike 's CW equation , 21.2 % of true value for Ramani 's equation -RRB- . The results of Experiment 3 -LRB- fitting the signal equations to a combined dataset obtained with differing `` sequences '' , i.e. different TR , τSAT , and flip angle -RRB- are also shown in Table 1 . The estimates of appear highly sensitive to the combination of acquisition parameters , while all other parameters are not . Interestingly , is underestimated using both of Sled and Pike 's equations when using 2 sequences -LRB- and the error becomes larger -RRB- , while it is generally overestimated when using a single one . When using the four-power scheme the main difference compared to the two-power scheme was in the estimates of , which were slightly lower for the two variants of Sled and Pike 's solution -LRB- between 2.98 and 3.1 for the CW and between 3.3 and 3.4 for the RP approximation -RRB- . 4.3 Sensitivity to noise Fig. 4 compares the estimates of the MT parameters using the three signal equations in the presence of noise . Overall , the estimates obtained from the 60 point noisy dataset using any of the three signal equations are characterized by similar precision , although Ramani 's solution seems to provide slightly more robust results than Sled and Pike 's at SNR lower than 120 for and . The two equations proposed by Sled and Pike 's provide very similar estimates -LRB- with similar precision at all SNR levels -RRB- for and . The CW variant provides more accurate values for . The parameter whose estimate deviates most from its true value is -LRB- for all signal equations -RRB- , with Ramani 's estimates deviating more than the others . The standard deviation associated with this parameter , however , is smaller when using Ramani 's equation , at low SNR . The opposite is true -LRB- with the two variants of Sled and Pike 's formulation giving very similar performances -RRB- at high SNR . 4.4 In vivo results The MT parametric maps obtained by fitting all three equations to the 5 ° dataset obtained in vivo are characterized by similar quality -LRB- examples of F maps are shown in Fig. 5 -RRB- . Fitting the two equations proposed by Sled and Pike to the 15 ° dataset was more problematic ; in some voxels -LRB- most commonly in gray matter , and at the boundary between tissues -RRB- the equations gave physically meaningless parameter estimates , particularly for and . It is possible that this is the result of T1-contrast between tissues and CSF increasing the sensitivity of partial volume effect to motion . Fitting of Eq . -LRB- 17 -RRB- appeared to be more robust . Table 2 shows the values -LRB- mean and standard deviation -RRB- obtained for each white and gray matter ROI using each signal equation and both flip angles . 4.4.1 5 ° data -LRB- between equations -RRB- Although the estimates of and obtained with Ramani 's equation were statistically different -LRB- p < 0.001 , providing higher estimates and lower estimates of F and -RRB- than those obtained with either variant of Sled and Pike 's equations , for all parameters the absolute difference between Ramani 's and Sled and Pike 's estimates was always lower or comparable to the between-voxels standard deviation -LRB- within each ROI -RRB- -LRB- see Table 2 -RRB- . When comparing the two solutions proposed by Sled and Pike , none of the variables differed -LRB- p values of 0.02 for , 0.6 for F , 0.32 for and 0.47 for -RRB- . 4.4.2 15 ° data -LRB- between equations -RRB- Conversely , when comparing the parameters obtained by fitting different equations to the 15 ° dataset , the mean values were all significantly different -LRB- p < 0.001 -RRB- , with the exception of obtained using the two variants of Sled and Pike 's solution -LRB- p = 0.09 -RRB- . The largest differences were in the estimates of F and obtained using Ramani 's equation with respect to both Sled and Pike 's equations . The absolute difference between estimates , in this case , were approximately one order of magnitude larger than the between-voxel standard deviation -LRB- within each ROI -RRB- of the same parameters . 4.4.3 15 ° data vs. 5 ° data The estimates of obtained fitting any of the three equations to the 5 ° dataset were significantly different from those obtained by fitting the same equation to the 15 ° dataset -LRB- p < 0.001 -RRB- . For Ramani 's equation , the estimates of F and obtained from the 5 ° dataset were also significantly different from those obtained from the 15 ° dataset -LRB- p < 0.001 -RRB- . 5 Discussion We have shown that the estimates of , and obtained by fitting different equations describing the behavior of the two-pool model under conditions of pulsed MT to proton density-weighted noise-free simulated data with an MT duty cycle of approximately 50 % -LRB- typical of most in vivo applications -RRB- are in good agreement and deviate only slightly from the values used to create the simulations . There are some systematic differences , however , with Sled and Pike 's equations typically providing more accurate results . Overall , these findings are in line with those reported by Portnoy and Stanisz -LSB- 7 -RSB- . is substantially overestimated when using all signal equations , albeit less so when using the Sled and Pike 's RP approximation . The discrepancy between the estimate of this parameter and the observed T2 of the system has been observed before -LSB- 11 -RSB- . As white matter is known to have multiple free water T2 components , one of the explanations provided by researchers is that the T2 measured by spin echo experiments and measured by MT experiments represent different weighted averages of multiple water components . This explanation , however , does not apply to analogous results obtained in gels -LSB- 1,11 -RSB- . Similarly , the results of our simulations suggest that such a difference could merely result from the inadequacy of the model to estimate this parameter . It is unsurprising that , for both simulated and in vivo data , the largest quantitative difference between the results obtained when fitting different equations to the same -LRB- low flip angle -RRB- data is in the estimation of this parameter . When the effects of saturation from the excitation pulse are minimized , in fact , the main difference between Ramani 's and Sled and Pike 's equations is in the way the MT pulse effect on the liquid pool is modeled . It is interesting to notice that others have often reported this parameter to be underestimated -LSB- 11 -RSB- , and our own in vivo results are consistent with such observations -LRB- Table 2 -RRB- . However , when fitting the MT signal equations to simulated data , we observed the opposite trend . A possible explanation for this inconsistency is the sensitivity of Levenberg -- Marquardt fitting to the values used as starting points : while for synthetic data we can use the real -LRB- test set -RRB- numbers , when fitting in vivo data we can only provide a ` best guess ' . In order to compensate for this effect we repeated Experiments 1 and 2 running the fitting procedure 10 times , each time perturbing the initial guesses -LRB- independently for each parameter -RRB- by a random factor from a Gaussian distribution with zero mean and standard deviation equal to 10 % of the test parameter -LSB- 19 -RSB- . We retained and compared the set of parameters which gave the best fit out of the 10 trials in each case . This did not affect the estimates obtained using Sled and Pike 's equations . obtained using Ramani 's equation was even larger in this case , with all other parameters almost unaffected -LRB- data not shown -RRB- . An alternative explanation is that we used a larger number of points for simulations than for the human brain data . The range and spacing of the sampling points is likely to affect the results of the fitting -LSB- 20 -RSB- , and it is interesting to observe that even in the simulated data , is underestimated when using two `` sequences '' -LRB- as recommended by Sled and Pike -LSB- 6,11 -RSB- -RRB- . Contrary to their observations , however , the estimates of were virtually unaffected by this experimental parameter , and therefore our simulation experiments do not support the need for this type of acquisition . It is not clear why our result in this case should differ from that of Sled and Pike , but it is possible that this is due to our use of a different fitting routine . Because of our simulation results , we chose not to adopt the two-sequence approach for our in vivo comparison , as this would have required all the MT-weighted acquisitions to be normalized to the same maximum value , and also might introduce variable degrees of T1-weightings , which would further violate the assumptions underlying Ramani 's equation . Although the reason why is so strongly affected by the use of a dual TR protocol is unclear , it has been reported by others that even when using more accurate formulations , the estimates of R and are highly sensitive to the choice of data points -LSB- 7 -RSB- , and this has been explained as a result of the poor sensitivity of the equations to these parameters , or to systematic errors between the model and the data , which can vary with the sampling points . The poor sensitivity of the two-pool model to changes in partially explains the robustness of the other parameters against its misestimation . As T2 can be determined by the use of alternative techniques , the remaining parameters are generally of greater interest . We would like to stress , however , that we are not suggesting the two-pool model is an exact representation of the distribution of macromolecular pools or of their exchange . We simply report that all of these relatively simple signal equations appear to be able to provide consistent and practical information which might be useful in a clinical/clinical research context . Although we did not directly compare the sensitivity of different signal equations to changes similar to those caused by pathology , we have attempted to provide useful guidelines to the choice of the most appropriate one for a particular experimental setting . Although suggesting optimal sampling schemes is beyond the scope of this paper , it is clear from Fig. 1 that low offset frequencies -LRB- < 1 kHz -RRB- should be avoided when using Ramani 's equation . This is consistent with the set of points used by Ramani et al. in their original paper -LSB- 5 -RSB- , and with the findings of Portnoy and Stanisz -LSB- 7 -RSB- , and confirms that in pulsed MT experiments it is difficult to model the behavior of magnetization at low off-resonance frequencies . Our results also confirm the observation of Portnoy and Stanisz -LSB- 7 -RSB- that Ramani 's model tends to underestimate the signal especially close to the Larmor frequency . This deviation at low frequency offsets is likely to be a consequence of the poor ability of the CWPE approximation to characterize the liquid pool . As noted by Portnoy and Stanisz -LSB- 7 -RSB- , the cut-off of 1 kHz they empirically determined might vary with field strength and MT pulse amplitude and bandwidth . We should also note that it is apparent from Figs. 2 -- 4 that F and RA are strongly inversely correlated . This is a trivial consequence of the typical values of F in the human brain -LRB- of the order of 10 − 1 -RRB- , for which the denominator on the right hand side of Eq . -LRB- 18 -RRB- is approximately equal to , effectively coupling RA and F . This corresponds to conditions of rapid exchange -LSB- 9 -RSB- . The use of numerical simulations allows the investigation of conditions for which it is impractical to acquire in vivo data , for example very narrow or very long MT pulses . As expected , and shown by others -LSB- 7 -RSB- , even in noise-free condition , the estimates of the MT parameters provided by Ramani 's signal equation depend on the duty cycle -LRB- in a non-linear fashion -RRB- , suggesting the need for pulse sequences with a duty cycle of at least 50 % . The estimates obtained using Sled and Pike 's equations , on the other hand are , with the exception of , less sensitive to this parameter . We also showed that the degree of T1-weighting -LRB- i.e. of saturation of the excitation -RRB- has a large effect on the MT parameters estimated by fitting Ramani 's equation . This is a direct consequence of the assumptions underlying it . We therefore strongly recommend avoiding the use of Eq . -LRB- 17 -RRB- to fit data which deviate substantially from those assumptions . The two equations proposed by Sled and Pike , on the other hand , by removing the `` steady state '' assumption , appear to be less sensitive to this problem . The RP approximation gives estimates that are substantially insensitive to changes in T1-weighting , while the CW approximation becomes slightly affected by it for excitation flip angles larger than 14 ° -LRB- assuming a TR = 30 ms -RRB- . In this respect , the results obtained from synthetic data are in keeping with in vivo measurements . Ramani 's equation 's estimates of F obtained from the more heavily T1-weighted data -LRB- Table 2 , flip angle = 15 ° -RRB- are substantially lower than those obtained with less heavily weighted data , while estimates of are larger . Conversely , when fitting Sled and Pike 's equations to the 15 ° dataset , the estimates of F are slightly larger compared to those from the 5 ° dataset , although the difference is not statistically significant . A surprising finding from in vivo data was that estimates of obtained from all the equations increased slightly with the excitation flip angle , while this was not observed with simulated data . This discrepancy between simulated and real data may be explained by the presence of noise , and by the use of a smaller number of points for the real data . Noise in the raw data critically affects the estimation of MT parameters , as shown by our fourth simulation -LRB- Fig. 3 -RRB- . Monte Carlo simulations also suggest that at high SNR -LRB- ⩾ 150 -RRB- Sled and Pike 's estimates -LRB- at least of some parameters -RRB- are at least as precise as -LRB- and generally more precise than -RRB- Ramani 's estimates , but that Ramani 's equation provides more precise answers at lower SNR . We also note that at low SNR the uncertainty associated with the estimated parameters is much larger than the systematic difference between the mean parameters estimated by each signal equation . For comparison of the in vivo results with simulations , an estimate of the in vivo SNR is needed . The typical SNR for our system , measured in white matter on 5 ° - excitation minimally MT-weighted 3D SPGR images acquired using a 8-channel head coil and the parameters described in the paper , ranges -LRB- due to non-uniformity of the receive coil -RRB- between 60 and 100 -LSB- 20 -RSB- ; other systems are likely to be similar . It would be interesting to compare the standard deviations obtained by Monte Carlo iterations to the variance lower bound predicted by Cramer -- Rao theory -LSB- 21 -RSB- , as well as to the standard deviation obtained from real data using the bootstrap method -LSB- 22 -RSB- . A further limitation to this analysis of sensitivity to noise is in the use of suboptimal schemes for both synthetic and real data . For all quantitative techniques based on model fitting , the precision and accuracy of the parameter estimates depend on the choice of the sampling points , and we have previously shown -LSB- 20 -RSB- that the error in parameter estimates can be reduced by factors around 2 or 3 by using optimized sampling schemes . The precision of the estimates obtained from all three equations could therefore dramatically improve by using a more suitable set of MT points ; we recommend such optimization for practical applications , but this was not possible here as we needed identical schemes to allow direct comparison of the three equations . Nevertheless , since the schemes here used were likely to be equally suboptimal for all signal equations -LRB- and are similar to those used by the authors of the original papers -RRB- , we believe that our analysis should not have been unduly `` unfair '' to any of the solutions tested . It should also be noted , though , that the acquisition protocol reported by Sled and Pike -LSB- 6,11 -RSB- is based on a 60-point scheme to sample a single 7-mm thick slice . Here , despite the consequences on both scan time and SNR , we use only 20 points -LRB- for in vivo experiments -RRB- and smaller voxels as , for most clinical applications , such whole brain coverage and thinner slices are likely to be essential . Despite reducing the number of points by two thirds , a scan time of 35 min -LRB- required by the protocol we used in vivo -RRB- is still too long to be feasible in a clinical setting . We also estimated five parameters -LRB- and M0 -RRB- directly from the fitting , while the original paper by Sled and Pike -LSB- 11 -RSB- recommended the use of an independent estimate of the relative proton density , which allows the MT curves to be normalized to one , reducing to four the number of parameters to be extracted . While this clearly would improve the precision in the estimated quantities , it also lengthens the scan time , making it again less attractive for clinical applications , where the number of acquisitions is typically restricted by time constraints . A better fit could be obtained also by iteratively repeating the fitting using the estimate of M0 obtained from the current iteration as the starting point for the next . Preliminary results suggest that two iterations should suffice . Furthermore , simulated data suggest that when high SNR and a large number of MT points are available , fitting Sled and Pike 's signal equations provides more accurate results , and therefore is preferable . Between the two variants , the main differences seems to be in sensitivity to the degree of T1-weighting in the acquisition sequence -LRB- as a consequence of saturation from the excitation pulse -RRB- , suggesting that , in the absence of this confounding factor , the CW variant can be used without major disadvantages , given its reduced complexity . It should be noted , however , that we used a single set of MT parameters to create the simulations , without any attempt to explore the consistency of these results for a different type of tissue -LRB- e.g. gray matter -RRB- . Our results , therefore , are limited to this specific case until confirmed by further experiments . Regarding the in vivo results , it should be noted that we made no attempt to correct in vivo data for B1 inhomogeneities -LRB- although the use of the body coil for transmission should provide a fairly uniform B1 distribution at 1.5 T -RRB- . Both the measurements of RAobs , and the MT fitting are affected by deviations from the nominal flip angle -LSB- 6,18 -RSB- . However , we expect the error introduced by B1 inhomogeneity to equally affect the three solutions , and therefore not to have major consequences on the conclusions drawn from our experiments . In the present work we have restricted our analysis to the comparison of three signal equations derived from the two-pool model to predict signal intensity in pulsed MT experiments , without any attempt to modify them , or compensate for their limitations . Several aspects of MT modeling such as the quantification of the effects of the excitation pulses on the macromolecular pool , which is typically considered negligible -LSB- 2,6 -RSB- , need to be addressed . It also would be interesting to investigate the dependency of these 3 equations on other sequence parameters such as TR , as this quantity controls the efficiency of magnetization transfer -LSB- 23 -RSB- . This analysis would be complementary to the investigation of duty cycle effects we performed , and may confirm whether the relative insensitivity of Sled and Pike 's CW approximation to duty cycle is maintained for all pulse sequences . Furthermore , it would be interesting to explore possible modifications of Ramani 's equation to account for the imaging parameters of the pulse sequence -LRB- for example by incorporating an additional contribution of -LSB- 1-cos -LRB- alpha -RRB- -RSB- / TR to the CW saturation rate acting on the free pool -RRB- . Finally , providing reliable information about the optimal number of sampling points and their distribution would yield an additional element towards the choice of the most appropriate equation for a given application . All these areas deserve further investigation , which we hope to pursue in the future . 6 Conclusion We have shown that -LRB- 1 -RRB- Sled and Pike 's CW signal equation provides the most accurate estimates of MT parameters in the absence of noise ; -LRB- 2 -RRB- Ramani 's signal equation is sensitive to changes in the MT duty cycle , although this effect becomes stable for duty cycles ⩾ 50 % ; -LRB- 3 -RRB- Ramani 's equation is -LRB- as expected -RRB- not suitable for fitting T1-weighted data , and doing so leads to underestimates F and when the amount of T1-weighting is high . An error , albeit of smaller magnitude , is also introduced into Sled and Pike 's CW approximation estimates when the amount of T1-weighting is high . The RP variant is , on the other hand , extremely robust to this effect ; -LRB- 4 -RRB- in data with SNR typical of in vivo protocols , although the estimates of , and obtained from the fit of all three equations show some differences , the magnitude of the difference is smaller than the between-voxels within-ROI standard deviation , provided that T1-weighting of the imaging sequence is minimal ; -LRB- 5 -RRB- Sled and Pike 's equations are slightly less robust than Ramani 's one in conditions of low SNR .

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

mri
magnetization transfer
relaxation
human brain