J . Phys. E: Sci. Instrum. 21 (1988) 820-822. Printed in the UK Rapid communication microscopy of dynamic displace ments : k-space a nd q-space imaging NMR P T Callaghan, C D Eccles? and Y Xia Department of Physics and Biophysics. Massey University, Palmerston North, New Zealand Signal -Sampling + T Received 28 January 1988 Abstract. The superposition of pulsed gradient spin-echo (PGSE) and N M R imaging experiments results in a spin density image which is phase and amplitude modulated according to the local self-correlation function for nuclear spin displacements over the time between the KSE gradient pulses. It is shown that such an experiment can image both static and dynamic spin displacements, the reciprocal space for these image spaces being termed k-space and q-space respectively. Simultaneous imaging of diffusion and flow at microscopic resolution is demonstrated and the Poiseuille velocity di5tribution agrees well ivith the velocity map obtained for the motion of water in a 0.7 mm capillary tube. 1. Introduction Nuclear magnetic resonance imaging relies on the spatial dependence of the Larmor frequency in the presence of a magnetic field gradient (Lauterbur 1973. Mansfield and Grannell 1973, 1975). T h e process of signal acquisition in the time domain in the presence of such a gradient is sometimes termed k-space mapping. k-space is conjugate to the image space via the Fourier transformation. S(k)= i’ t Contrast Selective excitation Figure 1. Combined PGSE and filtered back projection imaging sequence. The contrast period provides a probe of q-space as g is varied while sampling of the FID during the resolution period provides a probe of static k-space. contrast period of figure 1. yields an echo amplitude and phase dependent on the nuclear spin self-correlation function P,(r’ - r, A ) which denotes the conditional probability that a spin initially at r has migrated to r‘ over the time A . In the present context the difference (r’ - r ) is labelled the dynamic displacement in order to clearly distinguish it from the static displacement, r. In the narrow gradient pulse approximation S(g) = !-I p(r) P>(r’ - r. A ) exp(i7dg ( r ’ - r ) ) dr‘ d r . (2) T h e case of motion comprising both self-diffusion with diffusion tensor D and directed flow uith velocity U has been treated by Stejskal (1965) who obtained S(g) = S(0) exp( - y’d’g p(r) exp(i2xk r) d r (1) where k = (1/2n)yGt,with j , the gyromagnetic ratio and G the imaging field gradient. In equation (1) S ( k ) is the time domain signal and p ( r ) is the nuclear spin density. An RF and gradient pulse sequence applicable to two-dimensional k-space imaging is shown in the selective excitation and resolution periods of figure 1 . Various contrast mechanisms are available in magnetic resonance imaging (Mansfield and Morris 1982). The most commonly used are longitudinal (Ti) and transverse ( T ? ) relaxation contrast in which the signal acquisition is preceded by a preparation period in which the nuclear spin magnetisation is subjected to the desired relaxation process. A n alternative form of contrast results when the k-space mapping is performed on a spin echo which has been formed under a matched pair of magnetic field gradient pulses. These pulses, denoted g. induce phase and amplitude modulation in the echo signal according to the nuclear spin displacements occurring during the contrast period. T h e pulsed gradient spin-echo (PGSE) method was originally proposed for the measurement of molecular selfdiffusion. More generally the i w x : sequence, shown in the Present addreis: Institutc f u r Molektilarhiologic und Biophysik. ETH-Hbnggerberg, CH-8093 Ziirich. Sm itzcrland. ? 0022-3735/88/080820+03 $02.50 @ 1988 IOP Publishing Ltd u Resoiution - D gA - iydg * u A ) . (3) For finite pulses the diffusion time A appearing in the first term of the exponent is replaced by an effective time (A - k)) while the second term is unaltered. It is apparent that directed flow causes a net phase shift in the echo while random motion induces an incoherent distribution of phase shifts leading to attenuation. The influence of these effects in N M K imaging is well known and in medical imaging phase shifts have been used as a signature for blood flow (O’Donnell 1985. Ridgway and Smith 1986). In this ~. paper we report o n a more systematic application of I Y ~ S to imaging in which the dynamic displacement profile is obtained at each point in static image space and in which separate velocity and diffusion images are computed in addition to the static spin distribution, p(r). T h e method involves obtaining a sequence of images under differing I’<;SE: gradients as originally proposed by Redpath et a/ (1981) and in the steady gradient case by Taylor and Bushell (1985). In particular we demonstrate the method in the case of water undergoing laminar flow inside a cylindrical capillary. Unlike previous investigations of velocity profiles using SMK imaging (Cho er a1 1986, Kose et a1 1985). the present experiment was carried out at microscopic resolution. an extension of the recent development of x h 4 R microscopy (Aguayo et a/ 1‘186, Eccles and Callaghan 1986). Furthermore the use of independent PGSF: and imaging gradients distinguishes the present work Rcrpitl i ~ c i r ~ i r ~ i i r r i i i ~ t r ~ i c i t i from c;irlicr velocity profile mciisurcmcnts hasctl on spinecho methods ( I 1;iyw;ird til 1073. Girroway 1074). In order t o clucit1;itc the formalism it is instructive to recast equation (2) in terms of ;I wavevector q = (1/2i)yc)R and to regard the echo amplitude as ii Fourier transformation in q-space. This space is conjugate to the dynamic displacement, (r' - r ) . whereas k is conjugate t o r . It is then apparent that the combined rwdimaging pulse sequence as shown in figure 1 causes a modulation of the signal in both k-space and q-space according to ill (4) lieconstruction of f',(r' - r . A ) rcquircs the inverse transform to he pcrfornicd in k-space. iiorm;ilis;ition with respect to p ( r ) . ;incl then tr~iiisform;itioiii n q-sp;icc. Note that the usu;il symmetry relation V ( k )= S( - k ) n o longer applies when phase shifts itre present i n the im;igc. This nccessitiitcs the sampling of all four qu;idr;ints of k-spiicc. The intlcpcndcncc of r' ancl r implies that six im;iging tlimcnsions arc rcprcsentcd b y the nested contrast ;ind resolution Fourier intcgr;ils. In the present work we report ii four-ilimcnsion~il application in which q is directed one-tlimcnsioii~illyalong the symmetry.asis of ii cylindricnl ciipillary. In practice 9 may he directed at will ancl the only tlinicnsional constraint is the available imiiging time. The d y n a nii c tl ispl ii cc i n e n t inii 1' h ;is ;I part i cu I ii r I y si in1' I c result for particles untlcrgoing Brownian motion superposed on ;I velocity I J p:irdlel t o g. I f q-spacc is s;implctl b y varying g in intervals #q,,then the contrast reconstruction yields ;I gaussian peak of digital FWII M (2N/x)(In(2)fO'g~DA)"' at position (N/2x)(yOg,,Aa)where N is the number of digitisation points in q-space, D the self-diffusion coefficient and I ) the velocity magnitude. 2. Experiment The Microscopic 60 MI Iz proton imaging iipparatiis is basctl Figure 2. I<c;il ; i n d iiiiqinary images lor witcr Ilowing throu$i ii 0 . 7 mm 11) ciipilliiry ohtainctl a s ,q is successively incrc;iscd. I ? imagcs arc shown from ii complete set o f I X . Diffusion c:iuscs successive im;igcs t o I x ;ittcnu;itctl while the velocity gradient ciiuscs ;iltcrn;iting circular rings ari ng from differing magnetisation phase. (Note that the display pain increased in thc second set o f six.) on ii JEOL FXOO spectrometer incorporating ;I specially huilt pulse progr;immcr and lcvcl controller. IW modulator. prcamplilicr. gradient current switching system antl IW prolw. 0rthogon;iI qii;iclrupolar gr;idicnt coils proviclc the t ransvcrsc (G,, G,.) imaging gradients while the slice selection (G:) and i ~ i s i i(g:) gradients are provided by ii planar array. Data processing is performed on-line using an Hitachi M B 16000 X( )SS microcom pu tcr . Water (tlopcrl with CuSO, t o reduce TI) was p:isscrl through ii 0.7 nini 11) Tcllon cnpilliiry using ;I constant SO mni head to m;iint;iin ii steady Ilow of approxini;itcIy 3 mm s I. This Ilow rate was suflicicntly sm;ill to keep cscitctl spins inside the iw coil during the entire IW excitation and iicquisition scclucncc. A 6 nini section of the lcllon c:ipillary tube was surroundctl by ii close-wound 3. I mm diameter RI. coil. 2.0 nim slices were excited using sinc-motlul;itcd 111-pulses iis shown i n figure I . Note that er1u;ition (4) implies cluat1r;iturc signal processing i n both contrast ;incl resolution domains. I n this work the resolution transform was computctl using liltcrcd heck projection t o obtain both real ant1 im;igin;iry images. The imaging gradient is reoriented i n 0' steps cvcry 32 acquisitions covering the range from 0" t o 360" cvcry 13 min. Figurc 3 shows the lirst 12 successive real ;incl im;igin- arv images oht;iincd iis (1 is incrc;isctl h y stepping the I Y ~ S I . . gr;idicnt under software control i n IS intervals u p to ;I ni;ixiiiiiiiii of 0.090 7'in I. 0 and A arc hcltl fixed at 2 ins antl 5 ins respectively. Thcsc i magcs clc;i rl y cs t i i hi t ;I It crn;i t i ng phase rings which grow progressively more closely spaced ;is the ii(isi.. gr;idicnt is increased. Thcsc rings arise hccausc of the tlistrihution i n molccu1;ir velocity from zero at the capillary w;ill to ii maximum nt the centre. Signal attenuation increases iis ,q is increasctl. hceiiirsc o f diffusive motion. thus rcnclcring the ini;igc effectively zero ;it thc 1Sth point in 9-space. Limitation to IS real antl iniagin;iry q-spacc images permits the entire tI;it;i set t o hc stored on one 5:" diskette. thus cn;ihIing the cs pcri inen t t o he totally ii ti tom atctl . Thcsc clat i i arc suhseqiicntly zero-lillctl to 256 sets hcforc q-tr;insforni;ition to produce. at cacti point i n image space. ;I one-dimcnsion~il prolilc of the dyiamic tlisp1;iccmcnt. The diffusion cocflicicnts and velocities corresponding to thcsc profiles ;ire coniputcd iit ciicli point i n the image array ;ind the resulting velocity and diffusion m a p arc shown in figure 3. along with ii t ypicii I tl y n ;in1 ic tl is place nien t proli IC from within t he i magc . I t is inimct1i;itcly apparent that the diffusion image is 82 1 Rrrpitl cot I I 111 11t I i c ~ r i i oIt nieiisurement of velocity and diffusion is demonstrated in which the Brownian motion is shown to be independent of the local net moleculx translation. Finally. it is shown that q-space transformation provides ;I quantitative contrast mechanism i n which the inxiging dimensionnlitv is substantially increased. The application here t o Newtonian fluid motion produces results consistent with expectations. Application to shear-dependent fluids could, in principle, provide precise non-invasive rheological measurements at the microscopic level. References Aguayo J B. Bl;ickbancl S J. Schocniger J. Mattingly M and I linterman M 19x6 Nuclear magnetic resonance imaging in ii single cell Nrrtirri. 322 I90 Figure 3. Velocity and diffusion images ohtaincd from q-spacc transformation of the data iis shown in figure 2. The lowcr section of the diagram shows ii dynamic displacement prolilc sclectcd a t a specific location in the image. The hroadcning arises from selfdiffusion while the offset from zero arises from the water proton velocity. essentially uniform while the velocity image exhibits ii cylindrically symmetric variation consistent with Poiseuille How. Figure 4((1)shows stacked profile plots for the diffusion and velocity. The experimental and theoretical Poiscuille velocity distributions for orthogonal sections through the capillary centre arc shown superposed in figure 4(h). The agreement is excellent. 3. Conclusion The results presented here are novel in three respects. I t is shown that the p;ir;iholic Poiseuille velocity distribution is appliaiblc on the microscopic scale inside a 0.7 inm capillary and ;it ;I transverse spatial resolution o f 2 S p n . Siniultaneous ( U ) Stacked prolilc plots of the diffusion and velocity maps respectively. shown in ligurc 3. (h) Vertical and horizontal sections through the velocity plot showing the experimental velocity fitted using a Poiseuillc distribution. Figure4. 822 Cho Z 11. Oh C t l . Kim Y S. M u n C W. Nalcioglu 0. Lee S J and Chung M K 1986 A new nuclear magnetic reson;ince imaging technique for unambiguous unidirectional iiiciisurenient of flow velocity J. Appl. PIiys. 60 1256 Eccles C D and Callaghan P T 1986 fligh resolution imaging: the NMR microscope .I. Mtrgtr. l < ~ w ) t r68 . 393 Garroway A N 1974 Velocity meiisurements in flowing fluids by N M I ~ J. I’1iy.s. 1): Appl. P1ry.s. 7 LIS9 tl:iywiird R J. Packer K .I and Tomlinson D J 1972 Pulsed fieltl-gr;idient spin echo NMR studies o f flow in fluids Mol. Phys. 23 IO83 Kose K. Satoh K. Inouye T and Yasuoka I1 19x5 NMR flow im;iging J. I’hys. Soc. Jtrlirrti 54 X I Lautcrbur P C 1973 Image formation by induced local interactions: examples employing nuclear magnetic resonance Nrrirrre 242 I90 Mansfield P and Grannell P K 1973 NMK ‘diffraction’ in solids J. f1iy.s. C: Solid Sicrti~PI1y.s. 6 L422 Mansfield P and Grannell P K 197.5 ‘Diffraction’ and microscopy in solids and liquids by N M I ~ Phys. Rev. B 12 3618 Mansfield P and Morris P G 1982 N M R Imaging in Riotiiedicitie Atlr~trtici~.v iri Magtii4i. Revotitrt~ci~ ed. J S Waugh (New York: Academic) O’Donnell M 1985 NMR blood How imaging using multiecho phase contrast sequences Med. Pliys. 12 59 Redpath T W. Norris D G. Jones R A and Hutchison J M S I984 A new method of N M I ~flow imaging Plys. Metl. Biol. 29 891 Ridgway J P and Smith M A 1986 A technique for velocity imaging using magnetic resonance imaging Rr. J. H ~ d i o l 59 . 003 Stejskel E 0 1965 Use of spin echoes in a pulsed magnetic field gradient to study anisotropic restricted diffusion and flow J. Cliiwi. Pliys. 43 3507 Taylor D G and Bushell M C I985 The spatial mapping of translational diffusion coefficients by the NMK imaging technique P1iy.s. Mid. Riol. 30 345

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