Synthetic quantitative MRI through relaxometry modelling

Abstract Quantitative MRI (qMRI) provides standardized measures of specific physical parameters that are sensitive to the underlying tissue microstructure and are a first step towards achieving maps of biologically relevant metrics through in vivo histology using MRI. Recently proposed models have described the interdependence of qMRI parameters. Combining such models with the concept of image synthesis points towards a novel approach to synthetic qMRI, in which maps of fundamentally different physical properties are constructed through the use of biophysical models. In this study, the utility of synthetic qMRI is investigated within the context of a recently proposed linear relaxometry model. Two neuroimaging applications are considered. In the first, artefact‐free quantitative maps are synthesized from motion‐corrupted data by exploiting the over‐determined nature of the relaxometry model and the fact that the artefact is inconsistent across the data. In the second application, a map of magnetization transfer (MT) saturation is synthesized without the need to acquire an MT‐weighted volume, which directly leads to a reduction in the specific absorption rate of the acquisition. This feature would be particularly important for ultra‐high field applications. The synthetic MT map is shown to provide improved segmentation of deep grey matter structures, relative to segmentation using T 1‐weighted images or R 1 maps. The proposed approach of synthetic qMRI shows promise for maximizing the extraction of high quality information related to tissue microstructure from qMRI protocols and furthering our understanding of the interrelation of these qMRI parameters.

fully quantitative imaging within a clinical environment, 7 and clinical utility has been demonstrated, e.g. in the visualization of tumours. 8 Going beyond physical models that describe the MRI signal as a function of scanner parameters, such as flip angle, repetition and echo times, biophysical models that describe the interdependence of MRI parameters, such as R 1 , proton density (PD) and macromolecular tissue volume fraction, have more recently been proposed. [9][10][11] Combining such models with the concept of synthesizing images points towards an alternative approach: synthetic qMRI. In this case quantitative maps of fundamentally different physical properties are constructed through the use of biophysical models as distinct from constructing simple weighted images as in the conventional approach to synthetic MRI.
Such modelling approaches may enhance the robustness of quantitative imaging protocols that aim to quantify multiple parameters.
For example, high resolution (finer than 1 mm isotropic) and whole brain coverage leads to extended MPM protocol durations (25 min or more) and therefore increased vulnerability to motion, which could render valuable data unusable. In addition, at ultra-high field (>3 T) acquiring an MT-weighted volume can be particularly challenging due to the supra-linear increase in specific absorption rate (SAR) with field strength. The absence of an MT-weighted acquisition, due to either motion or SAR limitations, is particularly problematic since it prohibits the construction of an MT map, yet these have been shown to facilitate improved segmentation of deep grey matter (GM) structures. 12 These segmentation benefits are of great clinical importance because changes in regions such as the basal ganglia are associated with a number of pathological conditions, 13 including Parkinson's and Huntington's diseases, both of which are associated with involuntary movement such that remaining still during data acquisition may be particularly difficult for these patient groups.
In this study, the utility of synthetic qMRI is investigated within the context of the recently proposed linear relaxometry model. 11 In this model, which stems from the fundamental principles of the fast exchange regime, 14 the components of the apparent longitudinal relaxation rate (R 1 ) are expressed as a weighted sum of other qMRI metrics.
Quantitative maps of MT and effective transverse relaxation rate (R 2 *) are used as surrogates for the macromolecular and paramagnetic contributions to R 1 respectively. This model can be constructed on a participant-specific basis by pooling over GM and white matter (WM). The coefficients of this general linear model, which are global scalars for the whole brain, exhibited remarkable stability across a large, heterogeneous cohort. 11 This stability indicates that the mean of the population-derived model coefficients could be used on newly acquired maps to achieve the goal of synthesizing a full set of quantitative parameter maps from just a sub-set of the MPM protocol. The utility of doing so is demonstrated with two applications in the neuroimaging domain. In the first application, artefact-free quantitative maps are synthesized from motion-corrupted data by exploiting the overdetermined nature of the relaxometry model and the fact that the artefact is inconsistent across the quantitative maps, and is instead captured by the residuals of the model. In the second application, an MT map is synthesized without the use of an MT-weighted volume, which directly leads to a reduction in the SAR of the protocol. Using the synthesized MT map, we assess whether the previously established improvement in segmenting deep GM structures, relative to segmentation using T 1 -weighted images, is maintained.

| THEORY: LINEAR RELAXOMETRY MODEL
In the absence of exogenous contrast agents, the measured R 1 is dominated by contributions from free water spins, bound water spins at macromolecular sites and a smaller, spatially varying contribution from iron sites. 15,16 Under conditions of fast exchange, the measured R 1 can be expressed as a weighted sum of the relaxivities of these compartments 14 : Here R 1f is the relaxation rate of free water; f M is the fraction of spins at macromolecular sites with relaxivity r 1M ; f FE is the fraction of spins at iron sites with relaxivity r 1FE ; the index j sums over any unspecified contributions. The relaxivity describes the increase in the relaxation rate relative to free water sites, e.g. r 1M = R 1M − R 1f , where R 1M is the relaxation rate at macromolecular sites. A model of the apparent R 1 purely based on imaging data can be constructed by replacing the known contributors to R 1 with voxel-wise surrogate imaging markers 11 : Here, R 1f is taken to be a constant, β 0 . The macromolecular term, Similarly, a synthetic MT map is calculated by rearranging Equation 3 and using the measured R 1 map: 3. The PD-weighted acquisition or both the T 1and MT-weighted volumes are corrupted by motion. In these cases, all maps will be corrupted by motion to some degree and the possibility of improving the quality of the maps is reduced. The correction achievable will depend on the extent of motion artefact across the constituent volumes.

| METHODS
All data were acquired on a 3 T whole body MR system (TIM Trio, Siemens Healthcare, Erlangen, Germany) equipped with an RF body coil for transmission and a 32 channel RF head coil for receiving. The studies were approved by the local ethics committee and informed written consent was obtained from all participants prior to scanning.

| Data acquisition
Two studies were performed to assess the utility of synthesizing signal from the eight PD-weighted echoes was used to calculate a map of R 2 *. The first six echoes for each contrast weighting were then averaged to increase the signal-to-noise ratio. 21 Quantitative maps of the apparent R 1 were calculated from the PD-and T 1 -weighted volumes using the rational approximation of the Ernst equation 22 incorporating correction for transmit field efficiency as described above. 18 The FLASH acquisitions use both RF (50°phase increment) and gradient spoiling to minimize unwanted magnetization coherence pathways. However, residual errors can remain. To address this, we simulated the FLASH acquisitions using Bloch-Torrey equations for a range of expected transmit field efficiency. The correction parameters describing the linear dependence of the actual T 1 value on the apparent T 1 were derived from these simulations as described in Reference 23 and used to correct for imperfect spoiling of transverse magnetization.
Semi-quantitative maps of the percentage loss of magnetization resulting from the pre-pulse in the MT-weighted acquisition were calculated as described by Helms et al. 17 accounting for spatially varying T 1 times and flip angle inhomogeneities. 1

| Motion artefact correction study
MT and R 1 maps were synthesized for 12 motion-affected datasets.
These datasets had been excluded from various neuroimaging studies on the basis of failing visual inspection because of excessive levels of artefact consistent with intra-scan motion in one or more of the constituent FLASH volumes. Equations 3 and 4 were used to generate synthetic maps of R 1 and MT respectively using the mean relaxometry model coefficients reported in reference 11 . The success of motion artefact removal was evaluated by expert raters (n = 5, experienced physicists from WTCN), given that such evaluation has been shown to be a robust means of assessing motion artefact correction. [24][25][26] Each rater was presented with the original map and the corresponding synthetic map and had to decide which image had the least motion artefact, i.e. performed a forced choice rating assessment. The evaluation was carried out by each rater in two blocks, one for MT maps and  To achieve optimal inter-subject registration, the tissue probability maps (TPMs) derived from the MT maps were used to spatially normalize the data using the non-linear diffeomorphic DARTEL algorithm 28

| Motion artefact correction
Results of the forced choice assessment are presented in Table 1. In 50% of cases, the perceived level of motion artefact was significantly reduced in one of the synthetic parameter maps. MT was improved in four cases. An example of an improved MT map is shown in Figure 1. R 1 was improved in two cases and an example is shown in Figure 2. As would be expected, in no case was there a significant improvement in both synthesized maps.

| GM segmentation
Improved segmentation performance, i.e. segmentation specificity, is expected to result in significantly higher GM probability in GM regions and/or significantly lower GM probability in WM regions.   Figure 3A). Table 3 lists significant differences between the segmentation of the synthetic MT and R 1 maps. The synthetic MT map had significantly higher GM probability than the R 1 map in the following GM regions:

| Synthetic MT maps compared with R 1 maps
left and right pallidum and focally within the right substantia nigra ( Figure 3B, red; Table 3). The GM probability was significantly lower in two GM regions: the left gyrus rectus, the right lingual gyrus and within the interhemispheric fissure ( Figure 3B, blue; Table 3). Table 4 Table 4). The GM probability of the synthetic MT map was significantly higher in one WM voxel ( Figure 3C, red).

| DISCUSSION
The MPM protocol produces maps of R 1 , R 2 * and MT. The interdependence of these maps is described, to a large extent, by the principled linear relaxometry model of R 1 . We have demonstrated how this model, together with population-derived model coefficients, can be leveraged to synthesize quantitative maps from a subset of the MPM maps. This raises the possibility of generating synthetic quantitative maps of MRI parameters without actually acquiring the data typically required and introduces speed and/or robustness to quantitative imaging that will be of great importance in translating such approaches to a clinical environment.
Motion leads to inconsistencies across the imaging data used in the linear relaxometry model, which are largely captured by the model residuals facilitating artefact correction. Although motion is captured in the residuals these cannot be used to quantify the performance of the proposed method, since they will be influenced not only by artefact but also by any unspecified contributions to the measured R 1 , as well as any systematic bias. Therefore, to quantify the performance of the method we have used expert image quality rating, which is model independent and thus allowed an independent assessment of the method. This evaluation found significant data quality improvement in 50% of the cases evaluated by expert raters skilled in the identification of motion artefact.
In each case, significant artefact reduction was only achieved in one or other of the synthetic maps since at least a subset of artefactfree maps are required to afford an improvement. Significant image quality improvements occurred most frequently for MT maps. Intrascan motion occurring only during the MT-weighted acquisition will only affect the MT map. This is the optimal scenario for the presented correction scheme, an example of which is shown in Figure 1. Improvement occurred less frequently for synthesized R 1 maps since both the PD-and T 1 -weighted FLASH volumes used to calculate the R 1 map will also contribute to the R 2 * and MT maps, and therefore if intra-scan  In addition to variable performance, bias can be expected in the synthetic quantitative maps. We have previously reported 11    Any relaxometry model, such as the one used in this work, must make simplifying assumptions about the nature of the interaction between water compartments. A central assumption concerns the timescale over which the interaction between the different water compartments of the tissue occurs, which may be short, intermediate or long. 14 In our case, we build upon the assumption of fast exchange whereby we assume that the rate of exchange between compartments is higher than the difference in the relaxation rates of these constituent compartments. 37 As a consequence, the signal we measure is a weighted sum of the constituent compartments visible via MRI. 11,14 There are multiple reports that support this assumption by showing that the longitudinal relaxation rate in the brain is well described by a mono-exponential form and that deviations from this behaviour are small. e.g. 15,38 Limitations in the validity of this assumption may be a source of bias. Models such as the one presented here are also a first step towards in vivo histology, 2 in which quantitative maps are combined using biophysical models in order to extract descriptors of the underlying tissue such as myelin and iron levels 3 or the degree of myelination of fibres. 48 Any biases present in the parameter maps derived from this model would propagate through to these biological descriptors. Further development of the model will allow us to test and refine the validity of the model assumptions, such as the exchange between water compartments within the brain. Future work will assess FIGURE 3 Regions showing either higher (red) or lower (blue) GM probability when using the synthetic quantitative MT map as input to the segmentation algorithm as compared with A, a T 1 -weighted image, B, a quantitative R 1 map, and C, a quantitative MT map. For display purposes only, the results are presented at a statistical threshold of p < 0.001 without correcting for multiple comparisons and are overlaid on the average normalized MT map of the cohort. Note that the statistical analysis was restricted to a sphere, with a radius of 4 cm, in the centre of the brain. The outline of the sphere is indicated in black. The dashed arrows in A indicate WM regions in which the GM probability was lower for the synthetic MT map TABLE 3 Clusters in which the GM probabilities derived from the synthetic MT maps and R 1 maps were significantly different, p < 0.05 after small volume and family-wise error correction. Clusters with fewer than 10 voxels were excluded Biophysical models can provide insights into morphometric studies that show differential GM volume estimation, which is dependent on the contrast of the data used as input to the segmentation routine used for the morphometry. 12,46 Understanding these effects is of critical importance for computational neuroanatomy studies, particularly when there is ambiguity over the origin of observed changes, e.g. due to co-localized and interacting effects of atrophy and MRI parameter changes, as occur during ageing. 46 It has also been shown that age-related atrophy and differences in tissue microstructure can be captured by the qMRI parameters used in this study. 30,46 To circumvent any atrophy-related confound being introduced here, the cohort used was restricted to a narrow age range (18-25 years).
The linear relaxometry model itself assists with the interpretation of the differential segmentation performance we have seen. Considering the pallidum for example, the effects of reduced myelin content and increased iron content counteract each other in the R 1 map whereas in the synthetic MT map the effect of reduced myelin content in the pallidum should dominate since the iron effect has largely been removed. For this reason, there is greater contrast between the pallidum and the surrounding WM in the synthetic MT map than in the R 1 map, resulting in improved delineation of GM and WM by the segmentation algorithm. These effects of iron and myelin also combine to increase contrast in the T 1 -weighted acquisition. The reduced myelin content of the pallidum means that the T 1 relaxation time is longer than in the surrounding WM, leading to reduced signal intensity on a T 1 -weighted acquisition. The inevitable T 2 * weighting that is also present, since a T E of 0 ms cannot be achieved, means that the higher iron content additionally reduces the signal intensity of the acquisition due to more rapid signal decay. Hence, rather good segmentation performance was achieved for this structure using the T 1 -weighted acquisition as input.
In keeping with previous studies comparing segmentation performance on T 1 -weighted MDEFT (modified driven equilibrium Fourier transform) 47 and MT maps, 12 we also find that the MT map is the optimal choice to drive the segmentation algorithm, since it supports an even better delineation of subcortical structures (Figure 3). Therefore, if the time is available to acquire the extra data required to calculate a map of MT saturation, and if it is not corrupted by motion artefact, then this quantitative map should be the first choice for TABLE 4 Clusters in which the GM probabilities derived from the synthetic MT maps and the original MT maps were significantly different, p < 0.05 after small volume and family-wise error correction. Clusters with fewer than 10 voxels were excluded morphological studies based on segmentation. It should be borne in mind that the segmentation performance will also depend on the algorithm and prior information, i.e. the TPMs, used. These were constant across all analyses presented here, which utilized the segmentation algorithm and default TPMs of SPM12.
While this demonstration has been specific to the MPM protocol, the approach could be extended to other protocols to generate synthetic quantitative maps of MRI parameters without acquiring the data typically required. This raises the possibilities of improving efficiency and/or robustness in quantitative imaging, which is of great importance in translating such approaches to a clinical environment.

| CONCLUSIONS
Modelling and exploiting the interdependence of quantitative parameter maps facilitates greater insights into the underlying tissue microstructure and the removal of artefactual inconsistencies, e.g. due to head motion. Robustness to motion is a key requirement for quantitative imaging, particularly for the study of non-compliant participants, such as patients suffering from movement disorders. Here we have shown improved robustness to motion by creating synthetic qMRI maps free of artefact by applying biophysical models to motioncorrupted data. In addition, synthetic MT maps have been used to demonstrate improved segmentation of deep GM structures in comparison to that achieved with conventional T 1 -weighted data or quantitative R 1 maps. The proposed synthetic qMRI approach shows promise for furthering our understanding of the inter-relation of MRI parameters and for maximizing the extraction from qMRI protocols of high quality in vivo histology information related to tissue microstructure.