Patent Application: US-99887709-A

Abstract:
in mri by excitation of nuclear spins and measurement of rf signals induced by these spins in the presence of spatially - varying encoding magnetic fields , signal localization is performed through recombination of measurements obtained in parallel by each coil in an encircling array of rf receiver coils . through the use of magnetic gradient fields that vary both as first - order and second - order z2 spherical harmonics with position , radially - symmetric magnetic encoding fields are created that are complementary to the spatial variation of the encircling receiver coils . the resultant hybrid encoding functions comprised of spatially - varying coil profiles and gradient fields permits unambiguous localization of signal contributed by spins . using hybrid encoding functions in which the gradient shapes are thusly tailored to the encircling array of coil profiles , images are acquired in less time than is achievable from a conventional acquisition employing only first - order gradient fields with an encircling coil array .

Description:
in the o - space approach of this invention , parallel imaging performance is optimized using linear combinations of multiple spherical harmonics to form gradient shapes tailored to the spatial information contained in the coil profiles ( 15 ). in principle , arbitrary gradient shapes can be chosen for each successive echo to obtain different projections of the object . in its most general form , one begins with an array of surface coil profiles , assesses their spatial encoding ability , and then designs a nonlinear gradient encoding scheme that is maximally complementary to the coil array . in a typical circumferential coil array , the profiles vary smoothly throughout the fov , but the regions of peak sensitivity are relatively localized in angular regions near each coil . a circumferential array therefore provides more encoding in the angular direction than in the radial direction , a fact that has not been exploited by gradient encoding schemes in the past . according to an embodiment of the invention , we used a combination of the z2 spherical harmonic and the x and y linear gradients to image the axial plane . this choice of gradients is motivated by two factors : ( 1 ) the ability of the z2 gradient to provide excellent spatial encoding along the radial direction , where circumferential coil arrays provide the least encoding ; and ( 2 ) the ready availability of coil designs for producing the z2 spherical harmonic ( 16 ). in this approach , conventional phase encoding is discarded and replaced by projection acquisitions with the center of the z2 function shifted off - center using the x and y gradients . with each acquired echo , the object is projected onto a set of frequency isocontour rings that are concentric about a different center placement ( cp ) in the fov , suggesting the term “ o - space imaging .” by shifting the z2 quadratic shape off - center , it is ensured that there are enough overlapping isocontours from different projections at the center of the fov to resolve features in this region . the fourier transform of an echo obtained in the presence of a radially - symmetric gradient yields a projection of the object onto a set of rings centered on the applied readout gradient . with radial localization provided by the gradients , the surface coils are ideally positioned to provide spatial localization in the angular direction . furthermore , as will be shown in the results section , since the readout gradient provides spatial encoding in two dimensions , rather than just one as in cartesian trajectories , additional encoding is provided by increasing the gradient strength and sampling the echo more densely with essentially no impact on the imaging time . in cartesian parallel imaging , densely sampling the echo increases resolution in the x - direction but does not reduce the amount of aliasing in the phase encode direction . this is the primary impediment to achieving high acceleration factors . to achieve radial frequency encoding we image in the axial plane and apply a z2 dephase / rephase readout gradient . the radially - symmetric z2 gradient is translated to the desired location in the fov ( center placement ) using linear gradients and a b0 offset to complete the square , as shown in fig1 . following slice selection , the x , y , and z2 gradients are used to dephase and rephase the spins , as in projection imaging . the signal equation for echo s ( t ) is s m , l ⁡ ( t ) = ∫ ⁢ ∫ ⁢ ρ ⁡ ( x , y ) ⁢ c l ⁡ ( x , y ) ⁢ ⅇ - j2π ⁢ ⁢ g z ⁢ ⁢ 2 ⁢ 1 2 ⁢ ( ( x - x m ) 2 + ( y - y m ) 2 ) ⁢ ⁢ ⅆ x ⁢ ⁢ ⅆ y = a m , l , t ⁢ ρ where ρ ( x , y ) is the object , c 1 ( x , y ) is the receive coil sensitivity , ( x m , y m ) specifies the cp , g z2 is the strength of the z2 spherical harmonic in hz / cm 2 . gradient strengths g x and g y are chosen such that g x =− g z2 x m and g y =− g z2 y m in hz / cm . echoes formed using different cps during successive trs comprise a dataset from which the image is reconstructed . in the discrete case , the integral kernel can be represented as a projection matrix a m , n , t at time point t for the l th coil and m th cp . the echoes and encoding functions from multiple cps and coils are stacked to produce a single matrix equation : if radius r m is defined relative to each cp , the integral may be recast in polar coordinates : the inverse fourier transform of the echo now yields s m , l ( u ), the projection of the object along the isocontours encircling the m th cp . the projection specifies the amount of energy in the coil profile - weighted object that is smeared around ring - like regions that decrease in width with increasing r m ( as in fig2 a ). because the encoding function is not in the form of a fourier integral kernel , the data do not reside in k - space . consequently , image reconstruction cannot be achieved using k - space density compensation and re - gridding approaches similar to those employed in non - cartesian imaging with linear gradients . because of this fact , image reconstruction is performed by directly solving the matrix equation s = aρ by one of two methods : a spatial domain algorithm based on the projections and a frequency domain algorithm based on the echoes . when projections are obtained using the discrete fourier transform , each point in the projection s m , l [ u ] corresponds to the sum of the object intensity at all voxels lying within a band that can be approximated as the u th isofrequency ring . if the point spread function of the quadratic gradient is approximated as a discrete band of frequencies , then for an n s - point fourier transform there exist n s isofrequency rings , where n s is the number of samples taken during readout . the radius of the outermost ring specified by r max =√{ square root over ( bw / g z2 )} where bw is the sampling bandwidth . the sum over all voxels lying within a given ring is weighted by the l th surface coil profile at each point in that ring , when the object and coil profiles are represented in vector form , the set of all ring - domain equations may be vertically concatenated to form a single matrix equation accounting for all n s rings , m center placements , and l coils : where · denotes the dot product and w is a sparse matrix whose u th row weights each voxel according to its contribution to the u th ring of a given cp . the simplest version of w contains ones for each voxel lying within the u th ring and zeroes elsewhere . for an n × n reconstruction , the encoding matrix e is of dimension [ n s × m × l , n × n ]. direct inversion of this matrix is challenging for practical matrix sizes , but the sparsity of the matrix can be exploited by a version of the conjugate gradient algorithm known as lsqr ( 17 ) that is available as a function call in m atlab ( mathworks , natick , mass .). lsqr was selected for its ability to quickly solve sparse matrix equations with complex - valued and non - square matrix operators . the spatial - domain solution amounts to back - projecting individual points in each projection onto the corresponding rings in the image . the difficulty with this approach is in accounting for the spatially - varying , complex point spread function ( psf ) of each applied gradient shape . the psf is not the same for all voxels lying between two frequency iscontours , particularly for the innermost rings where r approaches zero . the radial gibbs ringing of the psf due to convolution with the fourier transform of the acquisition window boxcar only further complicates matters . these obstacles can be surmounted by directly solving the integral equation s = aρ in the frequency ( echo ) domain using the kaczmarz iterative projection algorithm — also known as the algebraic reconstruction technique ( art )— a row - action method that has found application in computed tomography and cryo - electron microscopy ( 18 ). this algorithm compares each echo time point with the inner product of the appropriate row of the projection matrix , denoted a m , n , t , with the n th iterate of the image estimator . the difference between these scalars weights the amount of basis function a m , n , t which is added to the estimator going into the next iteration : the algorithm typically converges in only a few iterations . the entire projection matrix — spanning all time points , coils , and center placements — is too large to fit in memory , precluding the use of lsqr as a m atlab call . but with the kaczmarz approach , only one data point is treated at a time , permitting individual basis functions to be recomputed on the fly or loaded from the hard drive in suitably sized chunks . while regularized versions of the kaczmarz algorithm exist for dealing with inconsistent sets of equations ( 19 ), it has been shown that strongly underrelaxed kaczmarz iteration ( small values of λ ) approach the minimum - norm least - squares solution for { circumflex over ( ρ )} ( 20 ), equivalent to the pseudo - inverse . we chose to use underrelaxation to minimize the effects both of random noise in s and of systematic error in the gradient shapes described by a , at the expense of increasing the reconstruction time . it should be noted that the true shape of each “ ring ” can be obtained by fourier transforming each n s × n 2 block of the encoding matrix along the temporal ( column ) direction . these image - domain matrices can be assembled into an equation for the unknown image using the projection data ( i . e ., ft of the echoes ). this version of the encoding matrix can be sparsified by truncating all values falling below a certain threshold , leaving only the point spread function in the vicinity of each ring . with the sparse encoding matrix able to be stored in memory all at once , lsqr would present an attractive alternative to kaczmarz for reconstructing the image . simulations were used to address four items . first , lsqr / image - domain reconstructions were used to quickly explore a variety of center placement schemes to find one that provides efficient o - space encoding . second , once a cp scheme had been chosen , kaczmarz / frequency - domain reconstructions were used to compare o - space and sense reconstructions over a wide range of acceleration factors . third , kaczmarz reconstructions were used to investigate the degradation of o - space and sense reconstructions in the presence of increasing amounts of noise . fourth , kaczmarz simulations were used to investigate the effects of increased ring density on the resolution of o - space images . two phantoms were used for each simulation : a low - noise axial brain image ( fig3 a ) obtained by averaging multiple conventional acquisitions ; and a numerical phantom ( fig3 b ) designed to illustrate the spatially - varying resolution and contrast properties of o - space encoding gradients . the numerical phantom incorporates small features in the center of the fov and sharp edges in the x and y directions . lesion - like features at four different contrast levels are also included . simulated 128 × 128 reconstructions were performed to determine a highly - efficient center placement scheme within the fov for datasets consisting of 32 and 16 echoes ( 21 ). by analogy to cartesian parallel imaging , this corresponds to 4 - fold and 8 - fold undersampling , respectively . a variety of coil geometries were also considered , ranging from 8 to 32 circumferentially - distributed loop coils , for whose b - fields an exact analytical expression exists in the magnetostatic limit ( 22 ). reconstructions were performed in the image domain using the projections of the sample about each cp ( 1d ft of each echo ). for simplicity and computational efficiency , a ring approximation to the true psf was used in which each voxel lying between frequency isocontours was averaged evenly over all voxels enclosed between the two contours . this results in a highly sparse image - domain encoding matrix that is less memory - intensive than the frequency - domain encoding matrix . cp schemes were evaluated based on the minimum mean squared error using the lsqr reconstruction as compared with the phantom . once a highly - efficient cp scheme was chosen , the kaczmarz algorithm was used to perform more accurate reconstructions by directly solving the frequency - domain matrix representation of the signal equation for both simulated and acquired echoes . o - space reconstructions were compared to sense reconstructions for r = 4 , r = 8 , and r = 16 ; for the r = 4 case , the reconstructions were then compared in the presence of varying amounts of noise . noise amplification in the sense reconstructions was mitigated using tikhonov regularization via the truncated svd of the aliasing matrix ( 23 ). uncorrelated gaussian noise was added to the phantom prior to multiplication by each of the 8 coil profiles used in the simulations . the noise standard deviation was scaled relative to the mean intensity in the phantom . noise correlations between the coil channels were neglected for the purposes of this study and will be treated in future work . it is expected that these correlations will similarly impact both conventional cartesian sense reconstruction and the proposed o - space approach . to explore the effects of ring density on resolution , the gradient strength and the number of readout points ( n s ) were sequentially increased while extra channel noise was injected into the echoes to model the increased sampling bw . for proof of concept , experimental data were collected on a 4 . 7t bruker animal magnet ( billerica , mass .) operating at 4t ( total bore diameter = 310 mm ) with a bruker avance console . the system is equipped with dynamic shimming updating ( dsu ) on all first and second - order gradients that can be triggered from within a pulse sequence ( 24 ). to avoid radial aliasing in o - space , the sampling bandwidth and z2 gradient strength g z2 were chosen so that the outermost frequency ring did not fall within the object for a particular cp : in practice , even with the z2 coil set to full strength over a 50 ms echo , most of the sampled rings lay outside the phantom , leading to suboptimal encoding , specific to this gradient set but not to the approach . the center placement scheme yielding the minimum mean squared error reconstruction in lsqr simulations is displayed in fig2 b . the phantoms and coil profiles used in all of the simulations are shown in fig3 . for this coil array , the g - factor was computed ( 2 ) in a 3 cm fov for each acceleration factor considered in this study as well as for the case of r = 2 ( fig4 ). adjusting for the fov , the g - factor maps were commensurate with those typically obtained from 8 - channel arrays on human systems . in the r = 4 kaczmarz reconstructions with 5 % noise , o - space images displayed comparable resolution and slightly lower noise levels as compared with sense images ( fig5 ). at r = 8 , however , the sense reconstruction with 8 coils becomes extremely ill - conditioned and either entirely overwhelmed by noise ( as in the original voxel - wise pseudoinverse sense method ) or plagued by severe residual artifacts , depending on the regularization parameter selected . by contrast , the o - space reconstruction at r = 8 shows a comparatively mild increase in noise and loss of resolution . at r = 16 , an acceleration factor for which 1d sense reconstructions are impossible , o - space images show pronounced degradation , but still retain impressive resolution and noise levels for images based on only 8 echoes . the compatibility of o - space imaging with extremely high acceleration factors suggests uses in functional brain imaging , cardiac imaging , and other applications in which spatial resolution is traded for temporal resolution , but artifact levels must remain low . when the acceleration factor is held constant at r = 4 and the noise level is varied ( fig6 ), the noise in o - space reconstructions is both lower and more spatially uniform than the noise in sense reconstructions . in sense images , the fact that each point in the reduced fov unwraps onto r - 1 points in the full fov causes severe noise amplification in the voxels in the reduced fov that have the most overlapping layers of signal from the phantom . in o - space imaging , by contrast , each voxel is smeared out along rings according to the spatially - varying psf of each center placement , preventing noise from concentrating in any one region of the image . by performing kaczmarz reconstructions of point sources at different locations in the fov , we calculated the psf of our chosen cp scheme . to isolate the effects of the gradient encoding alone , a single coil with uniform sensitivity was used . as expected , for each cp the source blurs along a ring that has the cp at its origin ( fig7 a , 7 b ). the source is localized to the point where all 16 of the rings overlap , with additional localization provided when a surface coil array is used . to illustrate the encoding provided by the z2 gradient alone , the encoding matrix for the z2 gradient was fourier transformed along the temporal dimension ( row - wise ), yielding the ring - like shapes that correspond to each point in an o - space projection . magnitude plots of a horizontal profile through the center of the fov ( fig7 c , 7 d ) illustrate the radial variation in resolution and the effects of echo truncation . since the derivative of the z2 field shape is zero at the center of the fov , all encoding in this region comes from the application of the x and y gradients to shift the rings off - center . truncation of an echo with duration τ results in convolution of the rings by sinc ( u / τ ) in u - space , corresponding to sinc ( r 2 / τ ) in the image domain . this leads to widening of the sidelobes as r approaches zero . as expected , increased ring density within the phantom contributes to substantial improvements in resolution . the 128 - ring o - space reconstructions show extensive blurring and background non - uniformity at both r = 8 ( fig8 a - c ) and r = 16 ( fig8 d - f ), suggesting that 128 rings are inadequate for highly accelerated acquisitions . but these artifacts diminish in the 256 - ring and 512 - ring cases for both r = 8 and r = 16 . this shows that o - space imaging benefits from using increased gradient strength and sampling bw during readout to boost ring density . this stands in contrast to cartesian acquisitions , where oversampling provides no reduction in aliasing artifacts along the undersampled direction ( s ). although there is no simple rule for “ nyquist sampling ” of an o - space echo , the gradient strength or dwell time may be increased up to point where the outermost ring is drawn inside the phantom , causing the outermost spins to alias back onto a low frequency ring . this definition of nyquist sampling depends on the location of the cp within the object , suggesting that the ring density may be allowed to vary between echoes for optimal encoding efficiency . the achievable ring density in the sample is limited only by the diminishing snr provided by very thin rings and safety regulations on gradient switching rates . in principle , surface coils could be used to “ unwrap ” radial aliasing caused by undersampling of the echo , but it &# 39 ; s not clear how such undersampling would improve resolution or acquisition time . volume - 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