- Email: [email protected]
Time-independent point-spread function for NMR microscopy
Mqnelic Resonance Printed in the USA.
Imaging, Vol. IO, pp. 269-278, All rights reserved.
1992 Copyright
0
0730-725X/92 $5.00 + 03 1992 Pergamon Press Ltd.
l Original Contribution
TIME-INDEPENDENT
POINT-SPREAD
FUNCTION
FOR NMR MICROSCOPY
E. W. MCFARLAND Francis Bitter National Magnet Laboratory and the Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA The resolution of NMR microscopy is analyzed in terms of the point-spread function, PSF(r), and the equivalent k space modulation transfer function, MTF(k). The analysis is developed for NMR spin warp and projection reconstruction imaging experiments; however, the framework provided is quite general. Incoherent spin motion is analyzed to predict what limits, if any, on spatial resolution are imposed by diffusion. Previous estimates of diffusion limits at l-5 pm were developed for specific imaging techniques, typically using a mean displacement argument. Although qualitatively correct, the quantitative predictions represent practical rather than fundamental limits. It is shown that diffusion-dependent “blurring” can be made arbitrarily small and that the practical limits are less stringent than previously thought. A major point illustrated by the PSF-MTF formulation is that the irreversible loss of coherence by randomly diffusing spins occurs faster than the physical displacement, thereby reducing their effect considerably on the frequency or phase of the net detected signal. The irreversible loss of signal due to diffusive motion will contribute to and possibly dominate the signal-to-noise limit of resolution. The resolution as measured by the width of the PSF and MTF for diffusion is shown to be independent of the signal acquisition time, and their functional forms allow selection of microscopic imaging parameters. An example of a three-dimensional spin-warp image of a green algae cell is shown with resolution of approximately 16 Km x 13 pm x 10 pm.
Keywords: NMR microscopy;
Transfer function analysis; Point-spread
INTRODUCTION
function;
Diffusion.
uniquely mapped into the image domain and (b) be detected above background noise? For this analysis of spatial resolution, a noiseless system will be considered where the superposition or convolution integral can be used to describe the mapping of the spin density distribution p(r) to an image function g(r’):
Several groups have worked to push the limits of spatial resolution in NMR images to the cellular level. ‘” These efforts have led to the widely held notion that it is the diffusion distance of spins over the data acquisition time that will ultimately limit the spatial resolution of NMR microscopy to approximately l-5 We are working towards microscopy of pm. 1,3~10-13 single cell preparations where coherent motion and regional magnetic susceptibility differences can be neglected.14-‘6 In these systems, ultimate limits of resolution can be approached. In this report the spatial resolution in an NMR image is analyzed in terms of transfer and point-spread functions. In this formulation, factors influencing resolution are separable and their relative importance readily appreciated. The question of resolution limit is how small a spacefixed volume, A V, can be defined such that rf photons originating from spins in A V = ( Ar)3 at r can be (a)
g(r’)
=
s
p(r)h(r’
-
r)dr
.
(1)
Use of this integral form to analyze generalized imaging system performance is well known and provides a quantitative means of analyzing resolution. 17,18For an impulse input p (r) = 6(r - r,,) the image domain output g(r’) = h (r’) is known as the impulse response function. The impulse response function may in general be complex; however, the real valued magnitude is often used as the system point-spread function, PSF(r’). The width in a given spatial dimension of the Address correspondence to E.W. McFarland, Department
RECEIVED 6/11/91; ACCEPTED 9/l l/91. This work was supported in part by the Whitaker Health Sciences Fund and an NSF P.Y.I. award (DIR-9057151).
of Chemical and Nuclear Engineering, University of California, Santa Barbara, CA 93106, USA. email: mcfar@squid. ucsb.edu. 269
270
Magnetic
Resonance
Imaging
PSF can be used as a quantitative measure of resolution in that dimension. ” Alternatively, the Fourier transform of h(r), FT[h(r)] = H(k), is known as the optical transfer function OTF(k) and is a measure of the system’s attenuation of a sinusoidal input at a spatial frequency k. It should be kept in mind that k refers to a real space spatial frequency component from the spatial decomposition of the input function p(r). The real valued complex magnitude of the OTF is defined as the modulation transfer function, MTF(k). The width of the MTF in a specific spatial frequency dimension can also be used as a quantitative measure of resolution, indicating the spatial frequencies below, in which the system adequately transfers the object detail to the image, Analysis of the resolution of an imaging system in terms of the PSF and/or MTF is well known and often called transfer function analysis (TFA). “-I9 As will be shown, by examining the individual components of the PSF or MTF, a better understanding and physical intuition of the relative importance of the different factors determining spatial resolution is possible. As a quantitative measure of resolution, the full width at half maximum (FWHM) of the PSF and MTF will be used,
Ar = FWHM[PSF(r)]
(mm)
and
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ANALYSIS
In NMR imaging, the spin system in each volume element AV at c is spatially encoded such that the time domain magnetization signal s(r, t) from A V will allow unique mapping of the spin density, p(r), to a specific voxel location r’ in the image plane. Typically, this is accomplished by collecting the total system signal s(t) in the presence of a single gradient for variable values of the orthogonal gradient strength and/or times and taking the n-dimensional Fourier Transform, FT[ ), or by the use of a backprojection, BP{ ), scheme.21’22 It is standard practice to define the spatial frequency variable k for NMR imaging in the presence of magnetic field gradients, G = grad{ Hz ] as23
k(t)
(mm-‘)
.
(2)
These definitions will only give quantitatively equivalent results in the case of Gaussian forms of the PSF; nevertheless, as a means of comparing the relative contributions of individual factors contributing to resolution, they will be shown to be useful. The required conditions for TFA are linearity and shift invariance of the imaging system. These conditions refer to the transformation properties of the imaging system with respect to the object and image and not to the actual physical phenomena, which for NMR is clearly nonlinear. Though they are almost never wholly satisfied in any practical imaging system, nevertheless, TFA has been successfully applied to transmission and emission imaging systems utilizing ionizing radiation. 17-19Tropper applied TFA to the echo planar imaging method to show the dependence of the PSF width on the switched gradient shape.20 Using TFA the resolution of NMR microscopy in the presence of diffusion will be analyzed in a framework applicable to many switched gradient experiments.
=Y
f s
VH,(t’)
dt’ .
0
Each element in the set of time domain signals is proportional to the spin density, p(r), or s(r, t) = p (r)f( k (t) , r) . Though time is not a vector quantity, to account for the fact that the gradient durations and strengths will often be different in the three orthogonal directions and spin labelling may be accomplished over different times, the variable t will be used with the realization that t = t, is the true time domain. We can write schematically
g(r’) Ak = FWHM[MTF(k)]
2, 1992
= FTk(s(t))
or for backprojection
=
s
p(r)FT[f(k(t),r)l
dr
at angles C$
P(r)f(k4(t),r)
dr
.
The function FT( f(k( t ), r)) then is the PSF or convolution kernel in Eq. (1) containing the desirable spatial encoding and relaxation information, as well as the unintentional geometric distortions and blurring. There are several implicit dependencies of J First, there is a dependence on the excitation scheme used, E(r,t), which together with the appropriate Bloch equation solution will determine the spin system evolution K (E(r, t), r, t). Second, sampling and digitization are contained in the functions X(t) and T(t), respectively, and relaxation in R(r,t). Third, diffusion effects will be described by W( 0, c ). We can then separate effects in liquid systems of uncoupled spins with a single resonance as: _f(W,r)
= K(E(r,t),C,t)X(t)T(t)R(r,f)W(D,C)
.
Time-independent
point-spread function 0 E.W.
The total system PSF is then PSF(r) = FTk ~f(k(t),
---I+
r)l
271
MCFARLAND
-_i__-JpH!___,, TE
TE/2
0
= FTk(K(E(r,t),r,t)X(t)T(t)R(r,t)W(D,t)) _-i2”_,,
or
GI -_I---_-_
PSF(r) = FT(K(E(r,t),r,t)J * FT(X(t)T(t))
t:
pGI tl
-
-
-
>B
-
>c
5 1 G=-yN,G,
* FT[R(r,t)J ---I--------
* FT( W(D,t))
(3) >D
where * denotes convolution. The MTF is equivalently written as
Fig. 1. Typical NMR imaging sequences. Following the rf excitation either the FID and/or the spin echo may be encoded with spatial information using magnetic field gradients. (A) A combination of three orthogonal gradients G, = G, + GY + G, is used in projection reconstruction imaging. (B) A single switched frequency encoding gradient can be used with a variable phase encoding gradient, and (C) for spin warp imaging.
MTFlk(t),rl =f(k(t),r) = K(E(r,t),r,t)X(t)T(t)R(r,t)W(D,t)
.
(4)
From these forms, each term of the PSF or MTF will be expressed individually below to understand their independent effects on the overall PSF or MTF. A. Input Signal [p(r)K(E(r,t),r)]
There are an indefinite number of forms for excitation and evolution, E(r,t) and ~(r,t), depending on the particular solution of the transient response of the spin system to a particular rf pulse and gradient sequence. For the purpose of discussion, we consider the free induction decay (FID) detected in the presence of a static gradient following a short rf pulse. Such a sequence might be used in a 3-D projection reconstruction scheme (Fig. 1A). In the absence of relaxation, diffusion, or digital sampling, the idealized time domain signal in the rotating frame is
s(t) =
s
Here, the spin density is uniquely mapped to g(r’) by = FT[s(k)] = &(r’ - r,,), which is the PSF in Eq. (1). Similarly, to calculate MTFs, a sinusoidal input at a frequency kO is used, p(r) = exp(iker). The signal by Eq. (5) becomes g(r’)
s(k) - 6(k - k,)
Thus, a sinusoidal density input provides an impulse in k space with uniform amplitude for all k and infinite width. The two idealized inputs have ideal resolutions from Eq. (2), Arp = FWHM(PSF[p(r)])
dr
p(r)K(E(r,t)r,f)
.
- 0
(mm)
and where Ak, = FWHM(MTF[k(r)l)
K(E(r, t)r,t)
= exp[ -iyG,*r.t]
.
For an impulse density input, p(r) = 6(r - rO), and s(t) - exp[ -iyG,.r,.t]
or
s(k) - exp[ -ik.r,]
- 03 (mm-‘)
(6)
(5)
.
where Ak/2r will give the MTF in the standard units of lines/mm. With these as a starting point, it will be shown that the additional factors of Eqs. (3) and (4) will degrade resolution by broadening the impulse in-
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Magnetic
put PSF and narrowing input.
the MTF
B. Sampling and Digitization
Resonance
Imaging
of the sinusoidal
[X(t,)T(
= 6(k - nk,)
=
27r
___
=
rGNs ts
-
2?r
and
Ns ks
n = 1,2,. . . N,
AkX7 = Nsks
It is the total sampling time, r = t,N,, that a spatial cutoff frequency ksNs which will tering of details at higher spatial frequencies quency encoding dimension. In the phase dimension, this cut-off is yt, ( Gp)maxwhere the maximum phase encode gradient often N, GJ2.
2, 1992
T2 values typically
D. Diffusion Effects fD(r, t)/
r)]
where k, = yGt, (or k, = yG, t, for the phase encode direction in spin warp sequences). Digitization will limit the bandwidth which is discretized into N, elements. The FWHM resolutions from Eq. (2) are then
Arxr
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liquids it is usually not restrictive. fall in the range 30-300 ms.
It is convenient to digitally sample NMR signals at finite intervals, t,, and for finite times, r. The effects on resolution are well known. The sampling functions X(t,) and T(T) are conveniently combined and expressed in terms of the product of k space functions X( k,)T( k,N,) for each dimension given by
X(k,)T(k,N,)
0 Volume
. (7)
results in cause filin a freencoding ( Gp)max is equal to
The effects of diffusion can also be analyzed in terms of their PSF and MTF. Previous discussions of the “ultimate resolution” in NMR microscopy have focused on the distance travelled by an individual spin due to diffusion.3,‘0-‘2,24*2s In a voxel element, A I/ resolution is not degraded by the signal produced from spins diffusing out of AVduring the acquisition time to some new location measured by the diffusion length. The resolution loss results from a reduction in signal from AV itself by the absence of the diffusing spins. While it is true that individual spins carry a path-dependent phase error, the random nature of diffusion guarantees that the net phase carried by all spins moving into a new volume will be zero in a uniform threedimensional system. Diffusion effects enter the time domain signal as an attenuation which has been well described.26p27 Using the known attenuation factors, the specific effects on resolution can be calculated. In any unit volume, AI’, the frequency is constant as determined by the space-fixed gradient. However, a distribution of spin phases will exist due to spins moving into the volume following the initial excitation. The signal from Avis
ss co
s(t)
=A
m
--m -cc
C. Relaxation Irreversible interactions is the frequency other sequence MTFs are PSFa
[R (r, t)] loss of coherence characterized by T2 well known. The effect will be only in encoding (or readout) dimension if all times are kept constant. The PSF and
= FT(exp[
x exp( -i(yG,rt)
where (b(r,~) is the random phase evolution in time, with a probability distribution given by Hahn as:27
exp
-t/Tz])
= FT(exp[
-k/(yGT,)]
P(@) =
)
and
+ +(r, t)) da dr
/
-
\
hr
3& -’
\
+‘G*DT~ ) 2G2Dr3
W(D, t) in Eqs. (3) and (4) for the impulse
input
is
then MTFR = exp[ -k/(yGT,)] with corresponding
2
ArR = YGTZ
widths
and
(8) P(@)exp(-i+(r,t))d+
from Eq. (2) of
AkR = 2 ln(2)yGT2
.
(9)
Relaxation is truly a fundamental limitation in as much as it can be a measure of the lifetime of an NMR state giving rise to an uncertainty in the transition energy between states. Fortunately, however, in biological
Following a 180” pulse at time TE/2, an independent diffusion period begins and the attenuation becomes the product of the attenuation before and after the 180” pulse. 28,2g W( D, t ) for a single spin echo is then
point-spread function 0 E.W. MCFARLAND
Time-independent
W(D,t)
=
1
Ar (pm) =
P(@)exp(-i+(r,t))d@
J-03
= exp
x exp
-y2G2D((
t - TE)3 + (TE/2)3) 3
-y2G2D(TE/2) 3
3
1 (10)
I
or normalized and in terms of k = yG( t - TE)
= MTF,(k)
= FT-‘[PSF,(r)]
.
(11)
The random phase terms enter in much the same way as relaxation, however, with a third power time (or k) dependence. Only diffusion occurring after the 180” pulse will effect resolution; however, the total attenuation will determine the signaLto-noise. The analytic form of FT[ W] is cumbersome; nonetheless, the FWHM of the Fourier transform of the time domain function (Eq. (10)) is
.,=,piym. To resolve two A V regions separated by a distance Ar, the gradient-determined frequency difference, AU = yGAr, must be at least equal to the width of FT[ W] . Expressing the resolution for the readout dimension in a spin echo image as in Eq. (2), we obtain:
273
43 ?G (g/cm)
’
(13)
For 10 pm resolution in water, the required gradient is approximately 80 g/cm; for 1 pm resolution in water, the required gradient is increased by lo3 to approximately 80,000 g/cm. The prohibitively large gradients are impractical for medical systems. However, small volume systems may utilize these large values. Fortunately, the diffusion coefficient in biological systems and other complex environments is typically less than that of pure water, thus lessening the gradient requirement. Furthermore, many biological systems may be cooled by at least lO”C, providing additional means for reducing D. This approach to diffusion analysis yields a considerably more intuitive result than discussion based on diffusion distance. The resulting diffusion determined resolution is not a function of time directly and is under the control of the experimenter by the choice of the gradient strength. The calculated resolution applies directly to projection reconstruction methods where the tomographic resolution is directly related to the resolution of the projections (assuming adequate sampling and neglecting the PSF of the algorithm and filter). Also, as will be shown below, this is the diffusion determined resolution in the readout dimension for spinwarp imaging. An analogy can be drawn between the effects of diffusion on distinguishing spatial volumes and the effects of chemical exchange on distinguishing two species separated by chemical shifts, o, and tit,.30 An effective exchange time 7, = ( Ar)2/2D can be defined. Two individual resonances can be “resolved” in the chemicalIy exchanging system when the condition for motional narrowing r(ob - o,( > 1 iS meL3' The gradients determine Iwb - w,I = yGAr, thus
(14)
(12) The reduction in resolution from diffusion is manifested as a broadening in the image domain or a narrowing in k-space and is independent of diffusion time. The approximate relationship for resolution in a room temperature water based system (D = 2.5 x lo-’ cm*/s) becomes
W(D, t) = exp
-y2D(G:t;
which has the same dependence as Eq. (12). More generalized acquisitions can be analyzed similarly. Consider a three-dimensional spin-echo sequence where G = G(r), Figs. 1B and C. It will be useful to examine a single k space dimension. The sampling and relaxation factors are unchanged in form; however, they will typically be asymmetric about their axes. With switched gradients, W(D, t) will be modified for t, < t < 2TE - t, as
+ G:(( t - TE)3 + (TE - tc)3))
3
(15)
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Magnetic
Resonance
Imaging
where t, = tb - t, and the constraint G, = G2(TE Q/t, is satisfied for rephasing. For t, = 7, t, = 7, TE = 27 and G, = G2 = G, the familiar echo attenuation expression is obtained at t = 2r.27 Because the diffusion phase loss is independent in each gradient interval, the attenuations are simply multiplied to obtain the total attenuation. 26-29 Noting that the first two terms in the numerator are constants, the MTFw (k) can be written for switched gradients as
(16)
MTFD( k,) = exp
where k, = yG2 (t - -t,). This is identical to the constant gradient case, Eq. (12), which shows that no loss of resolution occurs with the longer gradient, only a loss in signal intensity. For a fixed TE, one can keep the same resolution and increase the signal by switching off the gradient to prevent irreversible dephasing from diffusion. The corresponding PSF is obtained from the Fourier transform of the MTF. In the phase encode dimensions, k,, kz, the MTFs are written in terms of the variable phase gradient, G,, at a constant phase encode time t,. In this case, W( D, t ) becomes
W(D,t)
= exp( -y2yG’t’)
,
(17)
assuming the same phase encode times t, in y and z. The MTF in ky and k, are identical:
MTFD(k) For the phase are:
Ar, = 4
(18)
= exp
encoding
dimensipns,
1n’2iDtp I--
or for protons
and
Ak,,,
the resolutions
=
at room temperature: Ar(pm)
= 3m
.
(20)
The phase dimension resolution in y and z will be deas long as pendent on t,, which may be minimized can be increased to satisfy 2r/( yt, (G,),,) < (G,),, Ar as described above. Note that since the phase encoding dimension might be made to have a higher resolution than the readout dimension (Eq. (12)), one could use a 4-D acquisition if time efficiency was unimportant. 31,32 Finally, molecular motion during the time between
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phase encoding and frequency encoded readout can give rise to a registration artifact. During t, - tb the spins move and interact, possibly in several environments other than that in which they were phase encoded. The spin density and relaxation weighting in the image will reflect effects integrated over the path with an rms length of (2D( t, - t,,))“2. Though no loss of spatial resolution is apparent from this motion, implications for image interpretation exist and are discussed below. To minimize this effect the phase encoding gradients for the pulse sequence in Fig. 1 should be located adjacent to the readout period as in Fig. 1D. SAMPLE
RESULTS
The MTFs are shown for the phase and frequency dimensions as functions of k/(2a) in Fig. 2A, using parameters typical for NMR microscopy of water protons (D = 2.5 x 10e5 cm2/s, t, = 4 ms, TE = 8 ms, G2 = 160 g/cm). The total MTF is calculated from the product of the individual functions. In Fig. 2B the total MTF for the frequency encoding dimension is shown for the same parameters and G = 400, 2000, 4000, 8000 g/cm. Equations (7, 9, 12, & 13) can be used to select the gradient values and acquisition parameters for microscopic imaging of single cell preparations. An example of a three-dimensional spin-echo image of the green algae Nitella is shown in Fig. 3. The Nitella is a common algae ranging in size from 100 to 600 pm in diameter. The cell was contained in a water filled 600 pm ID micropipette. A three-turn solenoid rf coil of 100 pm diameter copper wire was wound on the outside of the pipette. The HI image was obtained at 360 MHz using the standard pulse sequence of Figs. IC and D with TE = 8 ms, and t, = 4 ms. The frequency encoding gradient was 40 g/cm and the maximum phase encode gradient was 80 g/cm. A bandwidth of 10 kHz was used and 256 time domain points were acquired. They and z gradients were switched through 128 and 64 phase encode values, respectively. Sixteen signal averages were used with a TR of 400 ms. Figure 3 contains selected views from the complete three dimensional data set. The cell is in the process of dividing and sending out the characteristic filamentous extensions. A light micrograph is shown for comparison. Table 1 gives the resolutions as determined by the PSF widths from Eqs. (7, 9, 12, & 19). Using a conservative underestimate of D = 2.5 x 1O-5 cm2/s, the total theoretical resolution of the image is approximately 11 pm x 7 pm x 10 pm. DISCUSSION The major conclusion of the present work is that by formulating NMR spatial resolution in terms of the
Time-independent
point-spread
function
MTF
k/2x (l/Km) (A) 1 MTF 0.5
0
0’
O.‘l
012
6.3
0.4
0.5
k/2n ( l/pm) @I Fig. 2. The modulation transfer function for NMR imaging of water protons using a direct Fourier imaging method with TE = 8 ms, t, = 4 ms, D = 2.5 x 10 em-5/s. (A) The MTFs obtained in a 160 g/cm readout gradient for (a) relaxation (T2 = 100 ms), (b) diffusion in the frequency encoding direction, (c) diffusion in the phase encoding direction, and (d) the total MTF in the frequency encode direction. (B) The total frequency encoding direction MTF with the above parameters and G = 8000, 4000, 2000, and 400 g/cm.
width of the PSF or MTF, the relative importance of the several known factors contributing to spatial resolution can be analyzed and improved physical intuition developed. Taken together, Eqs. (6,7,9, 12, & 19) contain the important factors and the dependencies that determine resolution. Equation (12), valid for
215
MCFARLAND
projection reconstruction and the frequency encoding direction of direct Fourier techniques, shows that the ratio [D/(-yG)] 1’3 is the important length parameter determining broadening and is independent of acquisition time. By increasing G, the separation in frequency between two regions, yGAr, can be kept larger than the broadening due to diffusive dephasing, Eq. (10). The corresponding relationship for resolution in a phase encoding dimension, Eq. (19), has the length parameter [Dt,] I” which is a measure of the diffusive displacement over the time the phase encoding gradient is applied. For NMR imaging using variable gradients, simply increasing the gradient strength and reducing the phase encode time will, in principle, always provide a means for distinguishing spatial domains. The technical problems of producing rapidly switched, large gradients, as well as the cost in terms of decreased signal-to-noise ratios, are well known.. The arguments developed above show that the effects of diffusion on spatial resolution are not fundamentally limiting as previously thought. The unavoidable effect, however, will be the irreversible and significant reduction in signal amplitude.26-29 The exponential reduction in signal amplitude to the second power of the gradient strength and third power of the gradient duration will be prohibitive when gradients of order 1000 g/cm are used. With such restrictions, the use of projection reconstruction methods whereby the FID is collected beginning at t = 0 are increasingly attractive.‘s21 Additional signal recovery can be obtained by using multiple rapid x pulses.26 Though several assumptions were made that limit the strict validity of TFA, these assumptions do not limit the physical intuition gained. In particular, the neglect of noise will prevent the deconvolution of each PSF from the raw image to regain the undistorted image. Additionally, to include the effects, important in many systems, of chemical shift, magnetic susceptibility, and anisotropic or restricted diffusion, more complicated analysis is necessary. The essential points and dependencies will, however, remain unchanged and the more complex expressions will be less physically intuitive. An important issue arises regarding the interpreta-
Table 1. Calculated
Arfreq--x (ccm)
0 E.W.
resolution
A rphase--y (Pm)
Arphase-z (Pm)
Sampling (Eq. 7) Relaxation (Eq. 9, T2 = 100 ms) Diffusion (Eq. 12, 13)
2.3 1.2 12.5
3.6
7.2
6.0
6.0
Total
16.0
9.6
13.2
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Fig. 3. Microscopic images of the green algae Nitella. (top) Light micrograph, scale as shown. (bottom) Selected views from the 256 x 64 x 128 3-D NMR image. The NMR image was obtained with NA = 16, TE = 8 ms, and TR = 400 ms. The cell is in the process of dividing and is sending out filamentous extensions typical of chlorophyta cell division. The 3-D image was obtained from 64 and 128 y and z phase encoding steps, respectively.
Time-independent
point-spread function 0 E.W.
tion of images with high apparent spatial resolution in the presence of molecular motion. Consider the irreversible dephasing of spins diffusing out of a unit volume. Their contribution to the total signal will be lost by their random motion in the gradient. As the dimension of the unit volume is reduced, the time for this to occur is reduced. For example, in a 200 pm diameter spherical volume of water at 3O”C, approximately 1 s is required for 50% of the protons to diffuse out of the sphere. If the dimension is reduced to 2 pm, it will take approximately 1 ms. This inescapable physical fact has significant implications on the information content of the microscopic image. For example, it is possible to obtain a proton density image with 2 pm spatial resolution by using a projection reconstruction scheme (Fig. 1A) and by choosing the gradients to satisfy the resolution criteria specified above. If, however, T1 weighting is desired and TR is reduced to 200 ms, the degree of T,weighting will depend on the T,'s of the surrounding regions through which the spins in the 2 pm imaging volume diffuse during the 200 ms interpulse delay. The T,weighting of the resulting image does not reflect the relaxation within the 2 pm volume, rather, it represents a weighted average of the surrounding environments. Similarly, T,-weighted spin echo images may be acquired at high resolution using the frequency and phase gradient sequences shown in Figs. 1B and D. However, the intervoxel movement during TE will result in an apparent T2in the image reflecting the relaxation that occurred in several surrounding environments visited by the diffusing spin during TE. Clearly, obtaining region-specific relaxation data poses an enormous problem and interpretation of NMR micrographs will be perhaps more difficult than obtaining them. Acquisition sequences used for macroscopic imaging, optimized for data acquisition rate or contrast, will likely be insufficient and will lead to confusing results when applied to imaging of microscopic volumes. Acknowledgments-The author wishes to acknowledge and thank the scientific staff of the Comprehensive NMR Facility (Supported by NIH Grant RRO0995) at the Francis Bitter National Magnet Laboratory, Cambridge, MA, for their technical support. Also A. Mortara, D.J. Ruben, and J.W. Wrenn provided extensive technical support and many helpful discussions. D. Cory and L.J. Neuringer’s reviews of the manuscript were greatly appreciated. REFERENCES Mansfield, P.; Grannel,
P.K. “Diffraction” and microscopy in solids and liquids by NMR. Phys. Rev. B 12(9):
3618-3634; 1975. Hinshaw, W.S. Image formation by nuclear magnetic resonance: The sensitive-point method. J. Appl. Phys. 47(8):3709-3721; 1976. Hedges, L.K. Microscopic NMR imaging. Ph.D. Thesis, SUNY Stony Brook, Department of Physics; 1984.
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4. McFarland, E.W.; Kushmerick, M.J.; Neuringer, L.J. Chemical exchange magnetic resonance. Magn. Res. Irnaging 6:507-515; 1988. 5. Aguayo, J.B.; Blackband, S.J.; Schoeniger, J.; Mattingly, M.A.; Hinterman, M. Nuclear magnetic resonance of a single cell. Nature 322:190; 1987. 6. Meyer, R.A.; Brown, T.R. Diffusion measurements by microscopic NMR imaging. J. Magn. Reson. 76:393399; 1988. 7. Zhou, X.; Porter, C.S.; Lauterbur, P.C.; Voth, B. 3D microscopic NMR imaging with (6.37pm)3 isotropic resolution. Abstracts. Eighth Annual Meeting of Society of Magnetic Resonance in Medicine, Amsterdam; 1989:286. 8. Clover, P.; Bowtell, R.; Harvey, P.; McJury, M.; Brown, G.; Manfield, P. A high resolution NMR Microscope for use at 11.7T. Abstracts. Eighth Annual Meeting of Society of Magnetic Resonance in Medicine, Amsterdam; 1989:285. 9. Johnson, G.A.; Thompson, M.B.; Gewalt, S.L. ; Hayes, C.E. Nuclear magnetic resonance imaging at microscopic resolution. J. Magn. Reson. 68: 129-137; 1986. 10. Eccles, C.D.; Callaghan, P.T. High-resolution imaging. The NMR Microscope. J. Magn. Reson. 68:393-398; 1986. 11. Cho, Z.H.; Ahn, C.B.; Juh, S.C.; Lee, H.K. Nuclear magnetic resonance microscopy with 4-urn resolution: Theoretical study and experimental results. Med. Phys. 15(6):815-824; 1988. 12. Callaghan, P.T.; Eccles, C.D. Sensitivity and resolution in NMR imaging. J. Magn. Reson. 71:426-445; 1987. 13. Hennessy, M.J.; Parsons, D. What is the physical limit of resolution of magnetic resonance microscopy (MRM). Abstracts. Fourth Annual Meeting of Society of Magnetic Resonance in Medicine, London, 1985: 1422-1423. 14. McFarland, E.W.; Mortara, A. Subcellular NMR microscopic imaging. Abstracts. Ninth Annual Meeting of Society of Magnetic Resonance in Medicine, New York; 1990:1127. 15. McFarland, E.W.; Mortara, A. Three-dimensional NMR microscopy: Pushing the SNR limit with temperature and microcoils. Abstracts. St. Louis: Experimental Magnetic Resonance Conference; 1991. 16. McFarland, E.W.; Fricke, S. NMR diffusion measurements in microscopic volumes. Abstracts. Chicago: Society of Magnetic Resonance Imaging; 1991. 17. Metz, C.E.; Doi, K. Transfer function analysis of radiographic imaging systems. Phys. Med. Biol. 24(6): 10791106; 1979. 18. Macovski, A. Medical imaging systems. T. Kailath (Ed.), Englewood Cliffs, NJ: Prentice-Hall; 1983. 19. Budinger, T.F. Transfer function theory and image evaluation in biology: Applications in electron microscopy and nuclear medicine. Ph.D. Thesis, Department of Biophysics, University of California, Berkeley; 1971. for the NMR 20. Tropper, M.M. Image reconstruction echo-planar technique, and for a proposed adaptation to allow continuous data acquisition. J. Mag. Reson. 42: 193-202; 1981. 21. Lauterbur, P.C. Image formation by induced local in-
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24.
25.
26.
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teractions: Examples employing nuclear magnetic resonance. Nature (London) 242:190; 1973. Kumar, A; Welti, D.; Ernst, R.R. Imaging of macroscopic objects by NMR fourier zeugmatography. Naturwissenschaft 62:34; 1915. Twieg, D.B. The /r-trajectory formulation of the NMR imaging process with applications in analysis and synthesis of imaging methods. Med. Phys. 10(5):610-621; 1983. Ahn, C.B.; Cho, Z.H. A generalized formulation of diffusion effects in urn resolution nuclear magnetic resonance imaging. Med. Phys. 16(1):22-28; Jan/Feb 1989. Callaghan, P.T.; Eccles, CD. Diffusion-limited resolution in nuclear magnetic resonance microscopy. 1. Mugn. Reson. 781-8; 1988. Carr, H.Y.; Purcell, E.M. Effects of diffusion on free precession in nuclear magnetic resonance experiments. Phys. Rev. 63(3):630-638;1954.
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27. Hahn, E.L. Spin echoes. Phys. Rev. 80(4):580-594; 1950. 28. Slichter, C.P. Principles of magnetic resonance, 3rd edition, Appendix G. New York: Springer-Verlag; 1990. 29. Torrey, H.C. Bloch equations with diffusion terms. Phys. Rev. 104(3):563-565; 1956. 30. Gutowsky, H.S.; Saika, A. Dissociation, chemical exchange, and the proton magnetic resonance in some aqueous electrolytes. J. Chem. Phys. 21:1688; 1953. 31. Maudsley, A.A.; Hilal, S.K.; Perman, W.H.; Simon, H.E. Spatially resolved high resolution spectroscopy by “four dimensional” NMR. J. Mugn. Reson. 51(147): 147-152; 1983. 32. Tannus, A.; Panepucci, H. A new approach to reduce diffusion effects in microscopy. Abstracts. Ninth Annual Meeting of Society of Magnetic Resonance in Medicine, New York, 1990; 392.
Imaging, Vol. IO, pp. 269-278, All rights reserved.
1992 Copyright
0
0730-725X/92 $5.00 + 03 1992 Pergamon Press Ltd.
l Original Contribution
TIME-INDEPENDENT
POINT-SPREAD
FUNCTION
FOR NMR MICROSCOPY
E. W. MCFARLAND Francis Bitter National Magnet Laboratory and the Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA The resolution of NMR microscopy is analyzed in terms of the point-spread function, PSF(r), and the equivalent k space modulation transfer function, MTF(k). The analysis is developed for NMR spin warp and projection reconstruction imaging experiments; however, the framework provided is quite general. Incoherent spin motion is analyzed to predict what limits, if any, on spatial resolution are imposed by diffusion. Previous estimates of diffusion limits at l-5 pm were developed for specific imaging techniques, typically using a mean displacement argument. Although qualitatively correct, the quantitative predictions represent practical rather than fundamental limits. It is shown that diffusion-dependent “blurring” can be made arbitrarily small and that the practical limits are less stringent than previously thought. A major point illustrated by the PSF-MTF formulation is that the irreversible loss of coherence by randomly diffusing spins occurs faster than the physical displacement, thereby reducing their effect considerably on the frequency or phase of the net detected signal. The irreversible loss of signal due to diffusive motion will contribute to and possibly dominate the signal-to-noise limit of resolution. The resolution as measured by the width of the PSF and MTF for diffusion is shown to be independent of the signal acquisition time, and their functional forms allow selection of microscopic imaging parameters. An example of a three-dimensional spin-warp image of a green algae cell is shown with resolution of approximately 16 Km x 13 pm x 10 pm.
Keywords: NMR microscopy;
Transfer function analysis; Point-spread
INTRODUCTION
function;
Diffusion.
uniquely mapped into the image domain and (b) be detected above background noise? For this analysis of spatial resolution, a noiseless system will be considered where the superposition or convolution integral can be used to describe the mapping of the spin density distribution p(r) to an image function g(r’):
Several groups have worked to push the limits of spatial resolution in NMR images to the cellular level. ‘” These efforts have led to the widely held notion that it is the diffusion distance of spins over the data acquisition time that will ultimately limit the spatial resolution of NMR microscopy to approximately l-5 We are working towards microscopy of pm. 1,3~10-13 single cell preparations where coherent motion and regional magnetic susceptibility differences can be neglected.14-‘6 In these systems, ultimate limits of resolution can be approached. In this report the spatial resolution in an NMR image is analyzed in terms of transfer and point-spread functions. In this formulation, factors influencing resolution are separable and their relative importance readily appreciated. The question of resolution limit is how small a spacefixed volume, A V, can be defined such that rf photons originating from spins in A V = ( Ar)3 at r can be (a)
g(r’)
=
s
p(r)h(r’
-
r)dr
.
(1)
Use of this integral form to analyze generalized imaging system performance is well known and provides a quantitative means of analyzing resolution. 17,18For an impulse input p (r) = 6(r - r,,) the image domain output g(r’) = h (r’) is known as the impulse response function. The impulse response function may in general be complex; however, the real valued magnitude is often used as the system point-spread function, PSF(r’). The width in a given spatial dimension of the Address correspondence to E.W. McFarland, Department
RECEIVED 6/11/91; ACCEPTED 9/l l/91. This work was supported in part by the Whitaker Health Sciences Fund and an NSF P.Y.I. award (DIR-9057151).
of Chemical and Nuclear Engineering, University of California, Santa Barbara, CA 93106, USA. email: mcfar@squid. ucsb.edu. 269
270
Magnetic
Resonance
Imaging
PSF can be used as a quantitative measure of resolution in that dimension. ” Alternatively, the Fourier transform of h(r), FT[h(r)] = H(k), is known as the optical transfer function OTF(k) and is a measure of the system’s attenuation of a sinusoidal input at a spatial frequency k. It should be kept in mind that k refers to a real space spatial frequency component from the spatial decomposition of the input function p(r). The real valued complex magnitude of the OTF is defined as the modulation transfer function, MTF(k). The width of the MTF in a specific spatial frequency dimension can also be used as a quantitative measure of resolution, indicating the spatial frequencies below, in which the system adequately transfers the object detail to the image, Analysis of the resolution of an imaging system in terms of the PSF and/or MTF is well known and often called transfer function analysis (TFA). “-I9 As will be shown, by examining the individual components of the PSF or MTF, a better understanding and physical intuition of the relative importance of the different factors determining spatial resolution is possible. As a quantitative measure of resolution, the full width at half maximum (FWHM) of the PSF and MTF will be used,
Ar = FWHM[PSF(r)]
(mm)
and
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ANALYSIS
In NMR imaging, the spin system in each volume element AV at c is spatially encoded such that the time domain magnetization signal s(r, t) from A V will allow unique mapping of the spin density, p(r), to a specific voxel location r’ in the image plane. Typically, this is accomplished by collecting the total system signal s(t) in the presence of a single gradient for variable values of the orthogonal gradient strength and/or times and taking the n-dimensional Fourier Transform, FT[ ), or by the use of a backprojection, BP{ ), scheme.21’22 It is standard practice to define the spatial frequency variable k for NMR imaging in the presence of magnetic field gradients, G = grad{ Hz ] as23
k(t)
(mm-‘)
.
(2)
These definitions will only give quantitatively equivalent results in the case of Gaussian forms of the PSF; nevertheless, as a means of comparing the relative contributions of individual factors contributing to resolution, they will be shown to be useful. The required conditions for TFA are linearity and shift invariance of the imaging system. These conditions refer to the transformation properties of the imaging system with respect to the object and image and not to the actual physical phenomena, which for NMR is clearly nonlinear. Though they are almost never wholly satisfied in any practical imaging system, nevertheless, TFA has been successfully applied to transmission and emission imaging systems utilizing ionizing radiation. 17-19Tropper applied TFA to the echo planar imaging method to show the dependence of the PSF width on the switched gradient shape.20 Using TFA the resolution of NMR microscopy in the presence of diffusion will be analyzed in a framework applicable to many switched gradient experiments.
=Y
f s
VH,(t’)
dt’ .
0
Each element in the set of time domain signals is proportional to the spin density, p(r), or s(r, t) = p (r)f( k (t) , r) . Though time is not a vector quantity, to account for the fact that the gradient durations and strengths will often be different in the three orthogonal directions and spin labelling may be accomplished over different times, the variable t will be used with the realization that t = t, is the true time domain. We can write schematically
g(r’) Ak = FWHM[MTF(k)]
2, 1992
= FTk(s(t))
or for backprojection
=
s
p(r)FT[f(k(t),r)l
dr
at angles C$
P(r)f(k4(t),r)
dr
.
The function FT( f(k( t ), r)) then is the PSF or convolution kernel in Eq. (1) containing the desirable spatial encoding and relaxation information, as well as the unintentional geometric distortions and blurring. There are several implicit dependencies of J First, there is a dependence on the excitation scheme used, E(r,t), which together with the appropriate Bloch equation solution will determine the spin system evolution K (E(r, t), r, t). Second, sampling and digitization are contained in the functions X(t) and T(t), respectively, and relaxation in R(r,t). Third, diffusion effects will be described by W( 0, c ). We can then separate effects in liquid systems of uncoupled spins with a single resonance as: _f(W,r)
= K(E(r,t),C,t)X(t)T(t)R(r,f)W(D,C)
.
Time-independent
point-spread function 0 E.W.
The total system PSF is then PSF(r) = FTk ~f(k(t),
---I+
r)l
271
MCFARLAND
-_i__-JpH!___,, TE
TE/2
0
= FTk(K(E(r,t),r,t)X(t)T(t)R(r,t)W(D,t)) _-i2”_,,
or
GI -_I---_-_
PSF(r) = FT(K(E(r,t),r,t)J * FT(X(t)T(t))
t:
pGI tl
-
-
-
>B
-
>c
5 1 G=-yN,G,
* FT[R(r,t)J ---I--------
* FT( W(D,t))
(3) >D
where * denotes convolution. The MTF is equivalently written as
Fig. 1. Typical NMR imaging sequences. Following the rf excitation either the FID and/or the spin echo may be encoded with spatial information using magnetic field gradients. (A) A combination of three orthogonal gradients G, = G, + GY + G, is used in projection reconstruction imaging. (B) A single switched frequency encoding gradient can be used with a variable phase encoding gradient, and (C) for spin warp imaging.
MTFlk(t),rl =f(k(t),r) = K(E(r,t),r,t)X(t)T(t)R(r,t)W(D,t)
.
(4)
From these forms, each term of the PSF or MTF will be expressed individually below to understand their independent effects on the overall PSF or MTF. A. Input Signal [p(r)K(E(r,t),r)]
There are an indefinite number of forms for excitation and evolution, E(r,t) and ~(r,t), depending on the particular solution of the transient response of the spin system to a particular rf pulse and gradient sequence. For the purpose of discussion, we consider the free induction decay (FID) detected in the presence of a static gradient following a short rf pulse. Such a sequence might be used in a 3-D projection reconstruction scheme (Fig. 1A). In the absence of relaxation, diffusion, or digital sampling, the idealized time domain signal in the rotating frame is
s(t) =
s
Here, the spin density is uniquely mapped to g(r’) by = FT[s(k)] = &(r’ - r,,), which is the PSF in Eq. (1). Similarly, to calculate MTFs, a sinusoidal input at a frequency kO is used, p(r) = exp(iker). The signal by Eq. (5) becomes g(r’)
s(k) - 6(k - k,)
Thus, a sinusoidal density input provides an impulse in k space with uniform amplitude for all k and infinite width. The two idealized inputs have ideal resolutions from Eq. (2), Arp = FWHM(PSF[p(r)])
dr
p(r)K(E(r,t)r,f)
.
- 0
(mm)
and where Ak, = FWHM(MTF[k(r)l)
K(E(r, t)r,t)
= exp[ -iyG,*r.t]
.
For an impulse density input, p(r) = 6(r - rO), and s(t) - exp[ -iyG,.r,.t]
or
s(k) - exp[ -ik.r,]
- 03 (mm-‘)
(6)
(5)
.
where Ak/2r will give the MTF in the standard units of lines/mm. With these as a starting point, it will be shown that the additional factors of Eqs. (3) and (4) will degrade resolution by broadening the impulse in-
272
Magnetic
put PSF and narrowing input.
the MTF
B. Sampling and Digitization
Resonance
Imaging
of the sinusoidal
[X(t,)T(
= 6(k - nk,)
=
27r
___
=
rGNs ts
-
2?r
and
Ns ks
n = 1,2,. . . N,
AkX7 = Nsks
It is the total sampling time, r = t,N,, that a spatial cutoff frequency ksNs which will tering of details at higher spatial frequencies quency encoding dimension. In the phase dimension, this cut-off is yt, ( Gp)maxwhere the maximum phase encode gradient often N, GJ2.
2, 1992
T2 values typically
D. Diffusion Effects fD(r, t)/
r)]
where k, = yGt, (or k, = yG, t, for the phase encode direction in spin warp sequences). Digitization will limit the bandwidth which is discretized into N, elements. The FWHM resolutions from Eq. (2) are then
Arxr
10, Number
liquids it is usually not restrictive. fall in the range 30-300 ms.
It is convenient to digitally sample NMR signals at finite intervals, t,, and for finite times, r. The effects on resolution are well known. The sampling functions X(t,) and T(T) are conveniently combined and expressed in terms of the product of k space functions X( k,)T( k,N,) for each dimension given by
X(k,)T(k,N,)
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. (7)
results in cause filin a freencoding ( Gp)max is equal to
The effects of diffusion can also be analyzed in terms of their PSF and MTF. Previous discussions of the “ultimate resolution” in NMR microscopy have focused on the distance travelled by an individual spin due to diffusion.3,‘0-‘2,24*2s In a voxel element, A I/ resolution is not degraded by the signal produced from spins diffusing out of AVduring the acquisition time to some new location measured by the diffusion length. The resolution loss results from a reduction in signal from AV itself by the absence of the diffusing spins. While it is true that individual spins carry a path-dependent phase error, the random nature of diffusion guarantees that the net phase carried by all spins moving into a new volume will be zero in a uniform threedimensional system. Diffusion effects enter the time domain signal as an attenuation which has been well described.26p27 Using the known attenuation factors, the specific effects on resolution can be calculated. In any unit volume, AI’, the frequency is constant as determined by the space-fixed gradient. However, a distribution of spin phases will exist due to spins moving into the volume following the initial excitation. The signal from Avis
ss co
s(t)
=A
m
--m -cc
C. Relaxation Irreversible interactions is the frequency other sequence MTFs are PSFa
[R (r, t)] loss of coherence characterized by T2 well known. The effect will be only in encoding (or readout) dimension if all times are kept constant. The PSF and
= FT(exp[
x exp( -i(yG,rt)
where (b(r,~) is the random phase evolution in time, with a probability distribution given by Hahn as:27
exp
-t/Tz])
= FT(exp[
-k/(yGT,)]
P(@) =
)
and
+ +(r, t)) da dr
/
-
\
hr
3& -’
\
+‘G*DT~ ) 2G2Dr3
W(D, t) in Eqs. (3) and (4) for the impulse
input
is
then MTFR = exp[ -k/(yGT,)] with corresponding
2
ArR = YGTZ
widths
and
(8) P(@)exp(-i+(r,t))d+
from Eq. (2) of
AkR = 2 ln(2)yGT2
.
(9)
Relaxation is truly a fundamental limitation in as much as it can be a measure of the lifetime of an NMR state giving rise to an uncertainty in the transition energy between states. Fortunately, however, in biological
Following a 180” pulse at time TE/2, an independent diffusion period begins and the attenuation becomes the product of the attenuation before and after the 180” pulse. 28,2g W( D, t ) for a single spin echo is then
point-spread function 0 E.W. MCFARLAND
Time-independent
W(D,t)
=
1
Ar (pm) =
P(@)exp(-i+(r,t))d@
J-03
= exp
x exp
-y2G2D((
t - TE)3 + (TE/2)3) 3
-y2G2D(TE/2) 3
3
1 (10)
I
or normalized and in terms of k = yG( t - TE)
= MTF,(k)
= FT-‘[PSF,(r)]
.
(11)
The random phase terms enter in much the same way as relaxation, however, with a third power time (or k) dependence. Only diffusion occurring after the 180” pulse will effect resolution; however, the total attenuation will determine the signaLto-noise. The analytic form of FT[ W] is cumbersome; nonetheless, the FWHM of the Fourier transform of the time domain function (Eq. (10)) is
.,=,piym. To resolve two A V regions separated by a distance Ar, the gradient-determined frequency difference, AU = yGAr, must be at least equal to the width of FT[ W] . Expressing the resolution for the readout dimension in a spin echo image as in Eq. (2), we obtain:
273
43 ?G (g/cm)
’
(13)
For 10 pm resolution in water, the required gradient is approximately 80 g/cm; for 1 pm resolution in water, the required gradient is increased by lo3 to approximately 80,000 g/cm. The prohibitively large gradients are impractical for medical systems. However, small volume systems may utilize these large values. Fortunately, the diffusion coefficient in biological systems and other complex environments is typically less than that of pure water, thus lessening the gradient requirement. Furthermore, many biological systems may be cooled by at least lO”C, providing additional means for reducing D. This approach to diffusion analysis yields a considerably more intuitive result than discussion based on diffusion distance. The resulting diffusion determined resolution is not a function of time directly and is under the control of the experimenter by the choice of the gradient strength. The calculated resolution applies directly to projection reconstruction methods where the tomographic resolution is directly related to the resolution of the projections (assuming adequate sampling and neglecting the PSF of the algorithm and filter). Also, as will be shown below, this is the diffusion determined resolution in the readout dimension for spinwarp imaging. An analogy can be drawn between the effects of diffusion on distinguishing spatial volumes and the effects of chemical exchange on distinguishing two species separated by chemical shifts, o, and tit,.30 An effective exchange time 7, = ( Ar)2/2D can be defined. Two individual resonances can be “resolved” in the chemicalIy exchanging system when the condition for motional narrowing r(ob - o,( > 1 iS meL3' The gradients determine Iwb - w,I = yGAr, thus
(14)
(12) The reduction in resolution from diffusion is manifested as a broadening in the image domain or a narrowing in k-space and is independent of diffusion time. The approximate relationship for resolution in a room temperature water based system (D = 2.5 x lo-’ cm*/s) becomes
W(D, t) = exp
-y2D(G:t;
which has the same dependence as Eq. (12). More generalized acquisitions can be analyzed similarly. Consider a three-dimensional spin-echo sequence where G = G(r), Figs. 1B and C. It will be useful to examine a single k space dimension. The sampling and relaxation factors are unchanged in form; however, they will typically be asymmetric about their axes. With switched gradients, W(D, t) will be modified for t, < t < 2TE - t, as
+ G:(( t - TE)3 + (TE - tc)3))
3
(15)
274
Magnetic
Resonance
Imaging
where t, = tb - t, and the constraint G, = G2(TE Q/t, is satisfied for rephasing. For t, = 7, t, = 7, TE = 27 and G, = G2 = G, the familiar echo attenuation expression is obtained at t = 2r.27 Because the diffusion phase loss is independent in each gradient interval, the attenuations are simply multiplied to obtain the total attenuation. 26-29 Noting that the first two terms in the numerator are constants, the MTFw (k) can be written for switched gradients as
(16)
MTFD( k,) = exp
where k, = yG2 (t - -t,). This is identical to the constant gradient case, Eq. (12), which shows that no loss of resolution occurs with the longer gradient, only a loss in signal intensity. For a fixed TE, one can keep the same resolution and increase the signal by switching off the gradient to prevent irreversible dephasing from diffusion. The corresponding PSF is obtained from the Fourier transform of the MTF. In the phase encode dimensions, k,, kz, the MTFs are written in terms of the variable phase gradient, G,, at a constant phase encode time t,. In this case, W( D, t ) becomes
W(D,t)
= exp( -y2yG’t’)
,
(17)
assuming the same phase encode times t, in y and z. The MTF in ky and k, are identical:
MTFD(k) For the phase are:
Ar, = 4
(18)
= exp
encoding
dimensipns,
1n’2iDtp I--
or for protons
and
Ak,,,
the resolutions
=
at room temperature: Ar(pm)
= 3m
.
(20)
The phase dimension resolution in y and z will be deas long as pendent on t,, which may be minimized can be increased to satisfy 2r/( yt, (G,),,) < (G,),, Ar as described above. Note that since the phase encoding dimension might be made to have a higher resolution than the readout dimension (Eq. (12)), one could use a 4-D acquisition if time efficiency was unimportant. 31,32 Finally, molecular motion during the time between
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phase encoding and frequency encoded readout can give rise to a registration artifact. During t, - tb the spins move and interact, possibly in several environments other than that in which they were phase encoded. The spin density and relaxation weighting in the image will reflect effects integrated over the path with an rms length of (2D( t, - t,,))“2. Though no loss of spatial resolution is apparent from this motion, implications for image interpretation exist and are discussed below. To minimize this effect the phase encoding gradients for the pulse sequence in Fig. 1 should be located adjacent to the readout period as in Fig. 1D. SAMPLE
RESULTS
The MTFs are shown for the phase and frequency dimensions as functions of k/(2a) in Fig. 2A, using parameters typical for NMR microscopy of water protons (D = 2.5 x 10e5 cm2/s, t, = 4 ms, TE = 8 ms, G2 = 160 g/cm). The total MTF is calculated from the product of the individual functions. In Fig. 2B the total MTF for the frequency encoding dimension is shown for the same parameters and G = 400, 2000, 4000, 8000 g/cm. Equations (7, 9, 12, & 13) can be used to select the gradient values and acquisition parameters for microscopic imaging of single cell preparations. An example of a three-dimensional spin-echo image of the green algae Nitella is shown in Fig. 3. The Nitella is a common algae ranging in size from 100 to 600 pm in diameter. The cell was contained in a water filled 600 pm ID micropipette. A three-turn solenoid rf coil of 100 pm diameter copper wire was wound on the outside of the pipette. The HI image was obtained at 360 MHz using the standard pulse sequence of Figs. IC and D with TE = 8 ms, and t, = 4 ms. The frequency encoding gradient was 40 g/cm and the maximum phase encode gradient was 80 g/cm. A bandwidth of 10 kHz was used and 256 time domain points were acquired. They and z gradients were switched through 128 and 64 phase encode values, respectively. Sixteen signal averages were used with a TR of 400 ms. Figure 3 contains selected views from the complete three dimensional data set. The cell is in the process of dividing and sending out the characteristic filamentous extensions. A light micrograph is shown for comparison. Table 1 gives the resolutions as determined by the PSF widths from Eqs. (7, 9, 12, & 19). Using a conservative underestimate of D = 2.5 x 1O-5 cm2/s, the total theoretical resolution of the image is approximately 11 pm x 7 pm x 10 pm. DISCUSSION The major conclusion of the present work is that by formulating NMR spatial resolution in terms of the
Time-independent
point-spread
function
MTF
k/2x (l/Km) (A) 1 MTF 0.5
0
0’
O.‘l
012
6.3
0.4
0.5
k/2n ( l/pm) @I Fig. 2. The modulation transfer function for NMR imaging of water protons using a direct Fourier imaging method with TE = 8 ms, t, = 4 ms, D = 2.5 x 10 em-5/s. (A) The MTFs obtained in a 160 g/cm readout gradient for (a) relaxation (T2 = 100 ms), (b) diffusion in the frequency encoding direction, (c) diffusion in the phase encoding direction, and (d) the total MTF in the frequency encode direction. (B) The total frequency encoding direction MTF with the above parameters and G = 8000, 4000, 2000, and 400 g/cm.
width of the PSF or MTF, the relative importance of the several known factors contributing to spatial resolution can be analyzed and improved physical intuition developed. Taken together, Eqs. (6,7,9, 12, & 19) contain the important factors and the dependencies that determine resolution. Equation (12), valid for
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MCFARLAND
projection reconstruction and the frequency encoding direction of direct Fourier techniques, shows that the ratio [D/(-yG)] 1’3 is the important length parameter determining broadening and is independent of acquisition time. By increasing G, the separation in frequency between two regions, yGAr, can be kept larger than the broadening due to diffusive dephasing, Eq. (10). The corresponding relationship for resolution in a phase encoding dimension, Eq. (19), has the length parameter [Dt,] I” which is a measure of the diffusive displacement over the time the phase encoding gradient is applied. For NMR imaging using variable gradients, simply increasing the gradient strength and reducing the phase encode time will, in principle, always provide a means for distinguishing spatial domains. The technical problems of producing rapidly switched, large gradients, as well as the cost in terms of decreased signal-to-noise ratios, are well known.. The arguments developed above show that the effects of diffusion on spatial resolution are not fundamentally limiting as previously thought. The unavoidable effect, however, will be the irreversible and significant reduction in signal amplitude.26-29 The exponential reduction in signal amplitude to the second power of the gradient strength and third power of the gradient duration will be prohibitive when gradients of order 1000 g/cm are used. With such restrictions, the use of projection reconstruction methods whereby the FID is collected beginning at t = 0 are increasingly attractive.‘s21 Additional signal recovery can be obtained by using multiple rapid x pulses.26 Though several assumptions were made that limit the strict validity of TFA, these assumptions do not limit the physical intuition gained. In particular, the neglect of noise will prevent the deconvolution of each PSF from the raw image to regain the undistorted image. Additionally, to include the effects, important in many systems, of chemical shift, magnetic susceptibility, and anisotropic or restricted diffusion, more complicated analysis is necessary. The essential points and dependencies will, however, remain unchanged and the more complex expressions will be less physically intuitive. An important issue arises regarding the interpreta-
Table 1. Calculated
Arfreq--x (ccm)
0 E.W.
resolution
A rphase--y (Pm)
Arphase-z (Pm)
Sampling (Eq. 7) Relaxation (Eq. 9, T2 = 100 ms) Diffusion (Eq. 12, 13)
2.3 1.2 12.5
3.6
7.2
6.0
6.0
Total
16.0
9.6
13.2
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Fig. 3. Microscopic images of the green algae Nitella. (top) Light micrograph, scale as shown. (bottom) Selected views from the 256 x 64 x 128 3-D NMR image. The NMR image was obtained with NA = 16, TE = 8 ms, and TR = 400 ms. The cell is in the process of dividing and is sending out filamentous extensions typical of chlorophyta cell division. The 3-D image was obtained from 64 and 128 y and z phase encoding steps, respectively.
Time-independent
point-spread function 0 E.W.
tion of images with high apparent spatial resolution in the presence of molecular motion. Consider the irreversible dephasing of spins diffusing out of a unit volume. Their contribution to the total signal will be lost by their random motion in the gradient. As the dimension of the unit volume is reduced, the time for this to occur is reduced. For example, in a 200 pm diameter spherical volume of water at 3O”C, approximately 1 s is required for 50% of the protons to diffuse out of the sphere. If the dimension is reduced to 2 pm, it will take approximately 1 ms. This inescapable physical fact has significant implications on the information content of the microscopic image. For example, it is possible to obtain a proton density image with 2 pm spatial resolution by using a projection reconstruction scheme (Fig. 1A) and by choosing the gradients to satisfy the resolution criteria specified above. If, however, T1 weighting is desired and TR is reduced to 200 ms, the degree of T,weighting will depend on the T,'s of the surrounding regions through which the spins in the 2 pm imaging volume diffuse during the 200 ms interpulse delay. The T,weighting of the resulting image does not reflect the relaxation within the 2 pm volume, rather, it represents a weighted average of the surrounding environments. Similarly, T,-weighted spin echo images may be acquired at high resolution using the frequency and phase gradient sequences shown in Figs. 1B and D. However, the intervoxel movement during TE will result in an apparent T2in the image reflecting the relaxation that occurred in several surrounding environments visited by the diffusing spin during TE. Clearly, obtaining region-specific relaxation data poses an enormous problem and interpretation of NMR micrographs will be perhaps more difficult than obtaining them. Acquisition sequences used for macroscopic imaging, optimized for data acquisition rate or contrast, will likely be insufficient and will lead to confusing results when applied to imaging of microscopic volumes. Acknowledgments-The author wishes to acknowledge and thank the scientific staff of the Comprehensive NMR Facility (Supported by NIH Grant RRO0995) at the Francis Bitter National Magnet Laboratory, Cambridge, MA, for their technical support. Also A. Mortara, D.J. Ruben, and J.W. Wrenn provided extensive technical support and many helpful discussions. D. Cory and L.J. Neuringer’s reviews of the manuscript were greatly appreciated. REFERENCES Mansfield, P.; Grannel,
P.K. “Diffraction” and microscopy in solids and liquids by NMR. Phys. Rev. B 12(9):
3618-3634; 1975. Hinshaw, W.S. Image formation by nuclear magnetic resonance: The sensitive-point method. J. Appl. Phys. 47(8):3709-3721; 1976. Hedges, L.K. Microscopic NMR imaging. Ph.D. Thesis, SUNY Stony Brook, Department of Physics; 1984.
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4. McFarland, E.W.; Kushmerick, M.J.; Neuringer, L.J. Chemical exchange magnetic resonance. Magn. Res. Irnaging 6:507-515; 1988. 5. Aguayo, J.B.; Blackband, S.J.; Schoeniger, J.; Mattingly, M.A.; Hinterman, M. Nuclear magnetic resonance of a single cell. Nature 322:190; 1987. 6. Meyer, R.A.; Brown, T.R. Diffusion measurements by microscopic NMR imaging. J. Magn. Reson. 76:393399; 1988. 7. Zhou, X.; Porter, C.S.; Lauterbur, P.C.; Voth, B. 3D microscopic NMR imaging with (6.37pm)3 isotropic resolution. Abstracts. Eighth Annual Meeting of Society of Magnetic Resonance in Medicine, Amsterdam; 1989:286. 8. Clover, P.; Bowtell, R.; Harvey, P.; McJury, M.; Brown, G.; Manfield, P. A high resolution NMR Microscope for use at 11.7T. Abstracts. Eighth Annual Meeting of Society of Magnetic Resonance in Medicine, Amsterdam; 1989:285. 9. Johnson, G.A.; Thompson, M.B.; Gewalt, S.L. ; Hayes, C.E. Nuclear magnetic resonance imaging at microscopic resolution. J. Magn. Reson. 68: 129-137; 1986. 10. Eccles, C.D.; Callaghan, P.T. High-resolution imaging. The NMR Microscope. J. Magn. Reson. 68:393-398; 1986. 11. Cho, Z.H.; Ahn, C.B.; Juh, S.C.; Lee, H.K. Nuclear magnetic resonance microscopy with 4-urn resolution: Theoretical study and experimental results. Med. Phys. 15(6):815-824; 1988. 12. Callaghan, P.T.; Eccles, C.D. Sensitivity and resolution in NMR imaging. J. Magn. Reson. 71:426-445; 1987. 13. Hennessy, M.J.; Parsons, D. What is the physical limit of resolution of magnetic resonance microscopy (MRM). Abstracts. Fourth Annual Meeting of Society of Magnetic Resonance in Medicine, London, 1985: 1422-1423. 14. McFarland, E.W.; Mortara, A. Subcellular NMR microscopic imaging. Abstracts. Ninth Annual Meeting of Society of Magnetic Resonance in Medicine, New York; 1990:1127. 15. McFarland, E.W.; Mortara, A. Three-dimensional NMR microscopy: Pushing the SNR limit with temperature and microcoils. Abstracts. St. Louis: Experimental Magnetic Resonance Conference; 1991. 16. McFarland, E.W.; Fricke, S. NMR diffusion measurements in microscopic volumes. Abstracts. Chicago: Society of Magnetic Resonance Imaging; 1991. 17. Metz, C.E.; Doi, K. Transfer function analysis of radiographic imaging systems. Phys. Med. Biol. 24(6): 10791106; 1979. 18. Macovski, A. Medical imaging systems. T. Kailath (Ed.), Englewood Cliffs, NJ: Prentice-Hall; 1983. 19. Budinger, T.F. Transfer function theory and image evaluation in biology: Applications in electron microscopy and nuclear medicine. Ph.D. Thesis, Department of Biophysics, University of California, Berkeley; 1971. for the NMR 20. Tropper, M.M. Image reconstruction echo-planar technique, and for a proposed adaptation to allow continuous data acquisition. J. Mag. Reson. 42: 193-202; 1981. 21. Lauterbur, P.C. Image formation by induced local in-
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