Rapid communication

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
P T Callaghan, C D Eccles? and Y Xia
Department of Physics and Biophysics. Massey University,
Palmerston North, New Zealand
+ 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.
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) =
- 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
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
D gA - iydg * u A ) .
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
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
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
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.
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
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
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.
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