WO2007061438A2 - Slant-slice imaging with inhomogeneous fields - Google Patents

Abstract

A method of magnetic resonance imaging an object in the presence of a permanent background gradient, G0, in a polarizing field, B0. The method includes the steps of selecting a 2- dimensional slice of the object for excitation that is not perpendicular to Go, exciting the 2- dimensional slice of the object by applying a selective RF-pulse in the presence of a slice excitation gradient Gss, where Gss is a linear combination of G0 and a transverse gradient Gssapp, generated as a linear combination of basic gradient fields G1, G2 provided by a magnetic resonance scanner, and applying a readout gradient Gre where Gre is a linear combination of G0 and a transverse gradient Greapp generated as a linear combination of G1, G2, where Gss and Gre are not parallel at any point in the selected excited slice. The selected excited slice of the object may then be reconstructed and displayed in a conventional manner. The selected excited slice may also be phase encoded with a gradient Gph generated as, a linear combination of G1, G2. The functions X = (G1, B0), Y= (G2, B0) and Z = || B0(x,y,z) || , where G0=ΔB0, may define local coordinates that map the field of view of the magnetic resonance scanner onto a region of space, whereby Gss = B0 + G1 and Gre =B0- G1. The step of applying the readout gradient Gremay also include the step of applying at least one refocusing pulse.

Description

SLANT-SLICE IMAGING WITH INHOMOGENEOUS FIELDS

GOVERNMENT SUPPORT

[0001] The present invention was partially supported by the National Science Foundation under Grant No. NSF DMS02-07123. The government may have certain rights in the invention.

FIELD OF THE INVENTION

[0002] The present invention relates to methods and apparatus for obtaining magnetic resonance images in inhomogeneous magnetic fields and, more particularly, to imaging with a background field B₀ that varies both in magnitude and direction.

BACKGROUND OF THE INVENTION

[0003] In the standard approach to magnetic resonance imaging one uses a strong background field that is as homogeneous as possible. Commercial MR imaging magnets are homogeneous, within the field of view, to about 1 ppm. In "open" MRI systems, the field homogeneity is somewhat less, but still in this general range. One can imagine a variety of situations where it might be useful to do magnetic resonance imaging with the object placed entirely outside the magnet's bore. As a consequence of Runge's Theorem, it is possible to design coils so that this external field is as homogeneous as one would like, in a given region of space. However, this requires a large expenditure of power and complicated, difficult to design arrangements of coils. On the other hand, with very simple arrangements of permanent magnets or electromagnets, one can produce a field, B₀, such that, in a given region of space, exterior to the magnets, or coils: (1) U u / . . ,

The field is strong; (2) The direction of B₀ varies in a small solid angle; (3) The level sets of | B_o| are smooth; and (4) The size of V | B₀ | is not too large.

[0004] Several groups have considered problems of this sort. Generally speaking, the prior art uses pulsed gradients for spatial encoding, and refocusing pulses to repeatedly refocus the accumulating phase in the direction of the permanent gradient. These ideas are described in US 4,656,425 to Bendel, as well as in US 5,023,554 to Cho and Wong. The idea is further developed by Crowley and Rose as described in US 5,304,930 and US 5,493,225. Pulsed gradients are also used in SPRITE, though for different reasons, as noted by Balcom et al., Single-point ramped imaging with T₁ enhancement (SPRITE), J. Mag. Res. A₅ Vol. 123 (1996), pp. 131-134.

[0005] Another group considering such problems is that of Dr. Alexander Pines at University of California at Berkeley. His work is described in the recent PNAS paper: "Three-dimensional phase-encoded chemical shift MRI in the presence of inhomogeneous fields" by Vasiliki Demas, Dimitris Sakellariou, Carlos A. Meriles, Songi Han, Jeffrey Reimer, and Alexander Pines. Their approach is somewhat different in that they try to match inhomogeneities in the B₁ -field with that in the B₀ -field in order to effectively "cancel" them out. Their efforts are more directed towards spectroscopy and they consider very small field gradients.

[0006] Still another group working on problems of this sort is that of Bernhard Blumich at RWTH Aachen in Aachen, Germany. This group's work is described in "The NMR-mouse: construction, excitation, and applications" by Blumich B, Blumler P, Eidmann G, Guthausen A, Haken R, Schmitz U, Saito K, and Zimmer G in Magn Reson Imaging. 1998, pgs 479-484. Their approach is again different from what is described herein in that it uses a stroboscopic acquisition technique. Though it is very good for spectroscopy of materials, it is too time consuming and SAR intensive for in vivo applications.

[0007] The present invention provides practical approaches to imaging with an inhomogeneous background field, building on the work described in U.S. Provisional Patent Application No. 60/663,937 filed by the present inventors on March 21, 2005, the subject matter of which is hereby incorporated by reference in its entirety. In that disclosure, the present inventors presented acquisition techniques that use one or two refocusing pulses per line in /c-space. The present invention focuses on refinements of this method for acquiring data that lead to a fairly standard 2d-reconstruction problem.

SUMARY OF THE INVENTION ,

[0008] The invention provides a method of magnetic resonance imaging an object in the presence of a permanent background gradient, G₀, in a polarizing field, B₀. The method includes the steps of selecting a 2-dimensional slice of the object for excitation that is not perpendicular to Go, exciting the 2-dimensional slice of the object by applying a selective RF-pulse in the presence of a slice excitation gradient G_ss, where G_ss is a linear combination of G₀ and a transverse gradient G_ss ^app, generated as a linear combination of basic gradient fields G₁, G₂ provided by a magnetic resonance scanner, and applying a readout gradient G_re where G_re is a linear combination of G₀ and a transverse gradient G_re ^app generated as a linear combination OfG₁, G₂, where G_ss and G_re are not parallel at any point in the selected excited slice. The selected excited slice of the object may then be reconstructed and displayed in a conventional manner. The selected excited slice may also be phase encoded with a gradient G_ph generated as a linear combination of G₁, G₂. The functions X = (G₁, B₀), Y= (G₂, B₀) and Z = || B₀(x,y,z) || , where G₀=ABo, may define local coordinates that map the field of view of the magnetic resonance scanner onto a region of space, whereby G_ss •= B₀ + G₁ and G_re = Bo - G₁. The step of applying the readout gradient G_remay also include the step of applying at least one refocusing pulse.

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] Figure 1 illustrates slant slice imaging with an adjustable gradient of equal strength to the permanent gradient.

[0010] Figure 2 illustrates a timing diagram for the slant slice imaging method shown in Figure

1.

[0011] Figure 3 illustrates an image created from measurements made with a permanent background field and the imaging sequence shown in Figures 1 and 2.

[0012] Figure 4 illustrates slant slice imaging with an adjustable gradient of smaller strength than the permanent gradient.

[0013] Figure 5 shows images made using the slant-slice protocol with various values of the σ ratio v = — . The geometric distortion due to the slant of the read-out gradient is not corrected

in these images.

[0014] Figure 6 shows images from Figure 5 with the corrections for the geometric distortion. [0015] Figure 7 illustrates that when the ratio between G₀ and G₁ equals 1, their strengths are simultaneously increased. . . « . , , , , .

[0016] Figure 8 illustrates level sets of GO + Gl (circular arcs) and GO - Gl (hyperbolas), where a slice is a region between two circular arcs and the slice averaging is along the hyperbolas.

[0017] Figure 9 illustrates a "one-sided" 3d MR-imaging system designed using the techniques of the invention wherein the sample (patient) would lie on a patient table to one side of the magnet.

DETAILED DESCRIPTION OF ILLUSTRATIVE EMBODIMENTS

[0018] First of all, one must understand what is meant by "imaging in an inhomogeneous background field" in accordance with the invention. Because the local resonance frequency is determined by the magnitude of the local field, small variations in | B_o| are of much greater importance than small variations in its direction, hi accordance with the invention, imaging is performed in an inhomogeneous background field if V | B₀ | has a "large," time independent component throughout the imaging experiment. The inventors are not considering the sorts of "random" or localized inhomogeneities that arise from the physical properties of the object being imaged, e.g. susceptibility artifacts. Rather, the inventors assume that the function | Bo| has no critical points in the field of view, the level sets of | B_o| are smooth, and fit together nicely. One can write:

B₀ = B₀₀ + G₀, (1) where Boo is a constant uniform field, and the spatial variation of Bo around the constant background is captured by G₀. The choice of B₀₀ is, a priori, rather arbitrary. As the method of the invention leads to a 2d-imaging protocol, one may usually select B₀₀ as the value of B₀ at a central point in the excited slice.

[0019] In the approach described below, the permanent gradient in B₀ is used along with an adjustable gradient-field G as a slice select gradient. As used herein, the variation in B₀ is not an imposed gradient, which could be reversed, but rather a permanent and irreversible gradient that arises from the basic design of the imaging system and the placement of the sample.

[0020] Let D denote the region occupied by the sample, or field-of-view (FOV) and let G_χ and G₂ denote the basic gradient fields. The fundamental assumptions underlying the imaging approach of the invention are that:

1. An apparatus is provided that is capable of generating fields ^₁G₁ + ^₂G₂ in D, where η_x, η₂ can independently assume any value in a specified range [-m_g, m_g].

2. The functions: X = B₀ * G₁,Y = B₀ * G₂,and Z = \B₀\ (2) define a smooth one-to-one, smoothly invertible mapping from D onto a region of 3- dimensional Euclidean space topologically equivalent to a cube.

These are essentially the same as the assumptions made in the afore-mentioned provisional patent application to the present inventors.

[0021] In the afore-mentioned provisional patent application, the present inventors used the permanent gradient in B₀ alone to define the slice select direction. In accordance with the present invention, the permanent gradient is employed, along with a field of the form G_ss - Tf₁G_x + η₂ ^sG₂ , to define the slice selection direction, ' and a field transverse to this direction,' Gre as the read out gradient. A third field of the form G_ph = λ{η[G_x + η₂G₂) is used as a phase encoding gradient.

To describe the invention, a simple case is first considered wherein all of the gradients are linearly varying magnetic fields. As usual, it is assumed that the background field B₀ is sufficiently strong throughout D that components of fields orthogonal to B₀ have a very small effect on the measurements and can safely be ignored.

[0022] The present inventors distinguish between gradient fields, which are quasi-static magnetic fields used to induce perturbations in the background field and field gradients. Strictly speaking, the field gradient G generated by the gradient field G is defined to be:

If G₀1 « B₀ and G₀ is not too rapidly varying, then the following approximate value may be used:

for the field gradient, in the computations.

Analysis in the linear case

[0023] Let us suppose that ,B₀₀ = (0, 0, b₀ ) and let: B₀ = (0,0A) + (*,*,&*) = ^oo + G₀

G₁ = (*, *, x) and G₂ = (*, *,y) (5) .

Here * is used to denote negligibly small field components orthogonal to (O₃ O₃ZJ₀). The permanent field gradient is of the form:

G =(OAg₈), (6) and one can generate fields η₁G₁+η₂G₂, with gradients of the form:

G_η=_ηi(l,0,0)+η₂(0,l,0)₃ (7)

where lη \<m . ¹ r s

Strong transverse gradients (G^G₁)

[0024] The simplest case arises when m εg . In this case, one uses for the slice select gradient the field

+ G₀. The phase encoding gradient field is then G _h = λ(*,*,y), where λ assumes values in the range [— m_g,m_g]. The slice select field gradient is then G Kg₂Ag₂X while the read-out field gradient is G_re = (-g_z,0,g₂). With these fields, the approach to imaging in accordance with the invention is the following:

1. Place the sample in the static field, B_Q, long enough to polarize the nuclear spins.

2. Turn on the gradient field g G. to attain a slice select field gradient G =(g Ag" )•

3. Apply a selective RF-pulse to flip spins lying in the region of space where:

Λ - Δ/≤ H ^ +g.GJ≤ Λ + Δ/, leaving the spins outside this region essentially in their equilibrium state. If f_o=yb_Q, then the region of space excited is the slanted slice given by:

{{x,y,z) : -Δ/ < γg_z(x + z) < Δ/}. (8)

If w(s) denotes a function that is 1 for -Af≤s≤Afand zero outside a slightly larger interval, and τ_Q denotes the rephasing time for the selective RF-pulse, at the conclusion of the RF- pulse the transverse magnetization has the form: m(x,y,z;O) = smap(x,y,z)w(rgXx + z))e^iτorsΛx+z) (9)

Here α is the flip angle and p is a spin density function, normalized to take account of the magnitude of the equilibrium magnetization, and τ_Q is the rephasing time for the RF-pulse.

4. At the conclusion of the RF-pulse, the adjustable gradient g G is switched off and the excited magnetization is allowed to precess under the influence of the permanent gradient G₀ for T₁ units of time. At the end of the free precession period, a 180° refocusing pulse is applied, this produces a transverse magnetization of the form: m(x,y,z;l) = sinap(x,y,z)w(_rg₂(x + φ-^^'^¹' (10)

5. The magnetization is now rewound in preparation for read-out: the gradient field g G is again turned on and the excited magnetization is allowed to freely precess in the field

g_zG_l+G_(j for T₀ + — T₁ units of time. To phase encode, the gradient field is turned on:

-2IT₁

G- .. =

2 T_n + T, gfi: 2 '

At the conclusion of this free precession period, the transverse magnetization takes the form:

m(x,y,z;2) = sin ap(x,y,z)w(γg_z(x + z))e '²^^{1 * * y} (11)

6. To read out the magnetization, one immediately turns on the field -g G . IfJ=O at the start of the acquisition, then the signal available for sampling is:

S(t) = I svnap(x,y, z)w(jg_z (x + z))e ² e ² dxdydz (12)

D

One may change variables in this integral, letting a—z+x and b=z-x, to obtain:

This can be interpreted as the 2-dimensional Fourier transform of the slice average:

/o +A/ p(y,b) = - J p(—r-,y,—-)w(g₂a)da. (14)

^Z /o-Δ/ ^{l l}

TSz

[0025] The slice averaging is illustrated in Figure 1. The signal is therefore:

Sit) = jp(y,b)e ² dydb. (15)

At time t the signal is p(k(t)), where:

This pulse sequence is illustrated in the timing diagram shown in Figure 2 and an image of a phantom using this approach is shown in Figure 3. [0026] There are several possible variations in this approach that lead to the same result. For example, one could refocus immediately following the selective excitation. After the refocusing pulse, the magnetization would be allowed to freely precess, under the influence of the permanent gradient alone for 2τ_Q time units. At the conclusion of this free precession period, the magnetization could again be refocused. The signal would then be read out with read-out gradient equal to -g G_χ, as before. At the start of the read out, the transverse magnetization would equal:

m(x,y,z;2') = swLap(x,y,z)w(γg_z(x + z))e * ' " (16)

Weak transverse gradients (G^G₁)

[0027] What makes the previous case especially simple is the assumption that the apparatus can produce adjustable gradients, with the strength of the field gradient at least equal to the strength of the permanent field gradient. This is by no means necessary for the method to succeed. The only modification is in the definition of the slice average p(y,b). This more general case is now described.

[0028] The following is an illustrative case, with many possible variations that will not be spelled out. Suppose that one can generate adjustable field gradients that are smaller than the magnitude of the permanent field gradient in B . As before it is assumed that:

G₀ = (*,*,&*), Gi = (*,*,*) and G₂ = (*,*,y) , and that one can generate ^₁G₁ + ^₂G₂ where |η₁|,|η₂|<m «g_z- The same step is followed as described above. For the slice select gradient, one can use G_ϋ+g_χG_λ, as before one can use ^g_xG₂ for phase encoding and G₀ - g_xG_x as the read out gradient. Supposing that a scheme is used with two refocusing pulses, the calculations above are not repeated but one skilled in the art will appreciate that the signal equation now reads:

S(t) = jsmap(x,y,z)w(γg_z(x + z))e ² e ² dxdydz (17)

D

This can also be interpreted as a 2d-Fourier transform of an average over the excited slice. The only difference between this case and the previous case is that the average is over lines that meet the slice at a fixed angle, not necessarily 90°. This is shown in Figure 4.

[0029] To see this analytically, the variables setting is changed: _{, Ω =} Mz≤^' _{b =} M±M. ₍₁₈₎ g where g = ■sjgl + g². This gives:

_e ^2m ° ^; _e ^{ir gi oy}dbdyda (19)

Changing variables in the δ-integral, setting: _a _ (g_z ² - g_x ^z)a + 2g_χgzb _{b ^} (g_x ² -gl)a + g²s g² 2g_xg₂ obtains:

S(t) = jp(a,y)e-^ir(g^-^t)a+Λτogy]dyda. (21) where:

Thus, ^σ is p averaged along lines with slope (g ,0,g ). These are lines orthogonal to the read- out direction (— g_x,0,g_z), which are not, in general, parallel to the slice select direction (g Ag )• In the case at hand, the lines over which p is averaged make an angle θ with the slice select direction where:

The length of the intersection of these lines with the selected slice is minimized when θ = 90°, hence this length is slowly varying for θ close to 90°.

The dependence of SNR on the permanent gradient strength

[0030] The signal-to-noise ratio attainable using the procedures described above is now considered. The analysis of the SNR is essentially the same as it would be in a standard 2- dimensional imaging system. The main effect of the permanent gradient is to reduce the allowable spacing, Δt, between the acquisition of successive samples, thereby increasing the receiver bandwidth. Suppose that one has a slice thickness d and a rectangular field-of-view of ₁ size Z. If Δx denotes the (isotropic) pixel length, At is the acquisition time and iV-samples are collected in each direction, then:

[0031] It is desired to understand how the strength of the permanent gradient and the ratio v = g_x/g_z affects the SNR. This brings the FOV into the equation, along with the gradient strength.

Suppose that: g

then, to avoid aliasing, one would need to take:

Using that L=NAx, combining (24) and (25) one obtains:

Thus, the SNR is not directly affected by the presence of a strong permanent background gradient. It is indirectly affected, because in order to get the desired resolution the maximum frequency sampled in A-space must satisfy:

k ~ ^{1 + Vλ max} 2J(I - V²) ^"

If v is close to zero (g_x = g_z), then a thin slice is needed to get high resolution. This has the effect of lowering the SNR and would necessitate using several averages of each line. Beyond this, a large gradient also may lead to a thin slice, if one is constrained in the amount of RF- power one may apply. This also diminishes the SNR. These effects are illustrated in Figures 5 and 6, which show images of a pomegranate obtained using different values for v . In Figure 5 the geometric distortion has not been corrected, whereas in Figure 6 it has been. The slice thickness in Hertz is kept fixed throughout these images, which in turn means that the slice thickness in mm decreases as the gradient strength increases. AU images have the same intrinsic resolution. The SNR decrease is caused by the decrease in signal due to the thinning of the slice.

[0032] On the other hand, as noted by Rose and Crowley, a strong permanent gradient causes &-space to be rapidly traversed and so one can, in principle, refocus the transverse magnetization and reread the same line several times. Indeed, the time to traverse a line in A-space is proportional to the strength of the read-out gradient. Hence, if v is not too small, then within a single repeat time, one could refocus the magnetization and reread the line a number of times equal to m = g/ \ G₁ 1, and thereby regain the lost SNR, without any increase in imaging time. Figure 7 shows images where the ratio between G₀ and G₁ equals 1, but their strengths are simultaneously increased. The slice thickness (in mm) is held constant. As shown, the SNR decreases moderately as the gradient strength is increased. All images have the same intrinsic resolution.

The resolution in the linear case

[0033] If one can generate an adjustable gradient of strength equal to that of the permanent gradient, then the pixels are rectangular and the resolution is determined by the usual heuristic formula:

Δx * ^-. (27)

If the maximum adjustable gradient IG₁I is smaller than the permanent gradient, \G_Q\, then there is additional averaging involved in signal acquisition. To quantify this effect one can make the following simplifying assumption: the spin density p is constant along lines parallel to the slice select direction. Indeed, this is also a "worst case" analysis, when comparing the slant slice protocol to a protocol with averaging parallel to the slice direction (i.e. |G |=|G |). This assumption is reasonable for thin slices and a slowly varying spin density. In this case, at least within the excited slice, one gets:

^z)_W/[^{"% +%}]. (28)

One uses g_x =| G₁ \,g_z =| G₀ 1, to simplify the notation. To simplify the analysis, one may ignore the third dimension, which would, in any case be obtained by phase encoding in a direction orthogonal to the plane spanned by G₀ and G₁.

[0034] Ignoring the third dimension, the signal equation becomes:

where 2πk - γ{τ_Q - t)g. Letting: g² - g²

obtains:

One final change of variables, gives this integral a very simple interpretation:

a— a (3D

²sβ; obtains:

[0035] The measurement is the Fourier transform, at frequency — γ*-k, of the convolution g of/ with:

Thus the slanted slice has three different effects on the resolution:

2,g g 1. It reduces the effective maximum frequency sampled by a factor of — γ^~ and scales the g sample spacing in fc-space by the same factor:

W_o

2. It causes blurring due to the convolution with W along the slanted line. As g_x approaches g_z the convolution approaches convolution with a scaled delta function.

3. If the angle θ is close to zero, so that the effect of the convolution with JF cannot be removed, then the effective field of view is the support of W *f .

The effect of the convolution can, in principle, be removed if the Fourier transform of W does not vanish in the interval \rγ— jk^,— — jk_maκ]. Suppose that w(s) = χ^_r<Jtrd](s), then

Hence, the effect of the slant slice convolution can be removed, without excessive amplification of the noise, for frequencies that satisfy:

" « «[=?)^{■ (34)} Recalling that k_max is also scaled, the resolution is effectively given by

Δx « ^-p-, (35)

provided:

1 1 + v²

(36) ^{max <} 2d l-v^{2 '}

Note that k_mwi = γNhtg if 2JV+1 samples are collected. This shows that the resolution in the readout direction is effectively determined by g_x :

Putting together the two formulae shows that this approach has an effective resolution limit:

^Wn » ^d = ^ld —^ (³⁷)

COS ^ ' where θ is the angle between the slice select direction and the direction along which the spin density is averaged. With this modality, it may be desirable to use thin slices, measured many times, hi principle, this would allow the recovery of any lost resolution, though at the cost of additional acquisition time.

Analysis in the non-linear case

[0036] This section briefly describes the needed modifications if the gradient fields G_Q,G_χ,G₂ are not linear but satisfy the conditions enumerated above. For simplicity, a 2-dimensional case is considered, and the approach using two refocusing pulses, hi this case, B₀ = B₀₀ + G₀, where B₀ is the uniform field B₀₀ = (0,b₀). If G denotes an adjustable gradient field, then the local Larmor frequency is determined by:

This equation shows that the validity of the assumption that, for the purposes of analyzing the MR-signal, the gradient fields can be replaced by their projections onto B_Q, is equivalent to the assumption that:

\G +G\«b₀. (39)

This assumption pertains throughout the calculations that follow. [0037] Modifying the notation in the linear case, then: _{1 n f}

The assumptions on the fietlds" G^σ ₀'₅G₁ implty that Vg₀₅Vg₁ are linearly independent at every point within the field of view.

[0038] As shown by Epstein in Magnetic Resonance Imaging in Inhomogeneous Fields, Inverse Problems, Vol. 20 (2004), pp. 753-780, provided the direction of B₀ does not vary too much within the field-of-view, the selective excitation step proceeds very much as in the linear case. After the sample becomes polarized in the background field B₀, one may turn on the field G₁ and expose the sample to a selective RF-pulse. If w(s) is the slice profile, then the magnetization at the conclusion of the α-RF-pulse is: m(O') = sinap(x,z)w(r(g₀ + gi))e*^roy(a)+a). (41)

The transverse component is non-zero in the non-linear region of space where:

Mr(go(x>z) + giM)) ≠ 0- (42)

After a refocusing pulse: jn(V) = sinap(x,z)w(r(g_Q + gi))e^~/lbr(ai+a). (43)

The field G is turned off and the magnetization is allowed to freely precess for 2τ time units and is once refocused giving: m(2') = sinap(x,z)w(r(g₀ + g₁))e^~ir'^rl8a~gl) . (44)

[0039] Finally, at P=Q, the field -G₁ is again turned on to obtain the measured signal:

S(t) = jsin ap{x, z)w(γ(g₀ + g_t ))e^Kt~τ° ^Ms° ^~gi ^dxdz. (45)

D

Now using the basic assumptions, which imply that: gA = g_o +g, gB = g_ϋ - g_λ (46) define coordinates throughout the region of space occupied by the object, D, and define a map onto a regionD' of R² topological!/ equivalent to a square. Let dxdz = g²J(A,B)dAdB, the coefficient j is used to normalize so that g²J » 1 near the "center" of the slice. The signal equation becomes:

S(t) = g² jsmap(x(A,B),z(A,B))w(rgA)e^{Kt~To )rgB}J(A,B)dAdB. (47)

D

For each fixed B, A a (x(A, B),z(A, B)) traces a smooth curve in the xz-plane which is, in some sense, transverse to the slice. This is, of course, just the curve:

Rewrite the signal as a 1 -dimensional Fourier transform of the slice averaged function:

P(B) = g² \p{x(A,B),z(A,B))w{γgA)J(A,B)dA (48) provides:

[0040] The measurements are then samples of the Fourier transform of p. Samples of p can be reconstructed as a function of B. To reconstruct p in the slice defined -Δ/ < g^~ιγ(g_Q + g_λ) ≤ Δ/, one only needs to invert the relations in equation (46) to solve for (x,z) as functions of (A,B). Using the computation of J(A,B) one can also rescale the data according to the density of the individual slices. These steps are possible, at least numerically, if one knows the functions

[0041] For concreteness the example is considered where g_Q — z²,g_λ = x². The functions A,B define coordinates in the half plane x>0: A = z² + x² ,g_λ = z² — x². The image of the positive quadrant is the region where _^4>|5|. The area forms are related by the equation:

Figure 8 shows level lines of A and B in this quadrant. As illustrated, near to x=z the pixels are nearly rectilinear, but are less so near the axes. A typical pixel is shaded. The function p is given by:

From examination of Figure 8, it is evident that the simple notions of pixel and resolution, which are used with linear gradients, are not especially meaningful in the strongly non-linear case. Indeed it is evident that resolution in the reconstructed image, when transformed back to physical coordinates, is unlikely to be either isotropic at most points in the image plane, or homogeneous across the image.

[0042] Adding a third dimension is straightforward, given that one can generate two adjustable gradients G₁, G₂ so that the projections in the B₀-direction, g_o,g_λ,g₂ , define a smooth invertible mapping from the field of view to a region in R³ topologically equivalent to a cube. A field of the form G +g G can be used for slice selection, multiples of G₂ can be used to phase encode, and , .

G₀ - gfi_λ can be used as a read-out gradient. As explained in the afore-mentioned Epstein article, the measurements obtained in this way can be interpreted, after a change of physical (x- space) coordinates, as the Fourier transform of non-linear averages, of non-linear 2-dimensional slices of p(x,y,z).

[0043] The principal advantage of the approach of the invention is that it allows the usage of a magnet with a substantial permanent gradient to be used as the main magnet in an MR-imaging device. This is accomplished without significantly sacrificing either resolution, acquisition time or SNR. By using slanted slices one recovers, in almost its entirety, the formalism used to describe imaging with a homogeneous background field. In particular, one can use a simple FFT to reconstruct the image, along with a post-processing step to remove geometric distortions due to either non-linear gradient fields or to the slant slice acquisition. The method of the invention produces high quality images, with acquisition times comparable to what would be used in a standard imaging device.

[0044] The inventors have quantified the inherent limitations of the method as regards SNR and resolution, and neither seems, in any way insuperable. As noted above, a large permanent gradient increases the noise in each acquired line in exactly inverse proportion to the time required to acquire the line. Hence, by refocusing and remeasuring these lines, one can recover all the lost SNR, without any increase in overall repeat time. If g_x «g_z, then, for a given measurement time, this approach does have an intrinsically lower resolution than a standard imaging method, with a homogeneous background field. The only real constraint on the applicability of this method is that v not be too small, which allows for an enormous increase in the latitude available to designers of practical, high resolution, time and SAR efficient MR- imaging systems.

[0045] This technique could be used to build a "one-sided" 3d MR-imaging system as shown generally in Figure 9, wherein the sample (patient) would lie on a patient table 100 to one side of the magnet 110. This could be used for open MR-systems, or specialized MR-systems, for example dental MR. In many of these applications, gradient and RF-coils 120 are situated near to or around the sample, which is itself placed to one side of the static field generating magnet 110. As illustrated in Figure 9, the gradient coils are controlled by gradient amplifier and controller 130, while the RF coils are connected to RF transmitter/receiver 140, which provides output to computer 150 for processing of the image data for display on display device 160. In this configuration, one would not need to work against nature to design magnets with a very homogeneous field "outside the bore," but can solve the easier problems associated with designing magnets that have fields with a moderate but smooth permanent gradient, which the approach of the invention uses to good advantage. One can even imagine how this technique could be applied to advantage in an application like well-logging. While direct inversion of the measurements leads to images with geometric distortion, this distortion is completely determined by the field gradients. For a given magnet and gradient set, the geometric transformations, needed to remove the distortion, could be computed once and stored.

Claims

. . ,What is Claimed:

1. A method of magnetic resonance imaging an object in the presence of a permanent background gradient, Go, in the polarizing field, B₀, comprising:

selecting a 2-dimensional slice of the object for excitation that is not perpendicular to G₀;

exciting the selected 2-dimensional slice of the object by applying a selective RF-pulse in the presence of a slice selection gradient G_ss, where G_ss is a linear combination of Go and a transverse gradient G_ss ^app, generated as a linear combination of basic gradient fields G₁, G₂ provided by a magnetic resonance scanner;

applying a readout gradient G_re where G_re is a linear combination of G₀ and a transverse gradient G_re ^app generated as a linear combination of G₁, G₂, where G_ss and G_re are not parallel at any point in the selected excited slice; and

reconstructing the selected excited slice of the object.

2. A method as in claim 1, further comprising phase encoding the selected excited slice with a gradient G_ph generated as a linear combination Of G₁, G₂.

3. A method as in claim 1, wherein the functions X = (G₁, B₀), Y= (G₂, B₀) and Z = ||Bo(x,y,z) II , where G₀=AB₀, define local coordinates that map the field of view of the magnetic resonance scanner onto a region of space.

4. A method as in claim 3, wherein G_ss = B₀ + G₁ and G_re = B₀ - G₁.

5. A method as in claim 1, wherein the step of applying the readout gradient G_re further comprises applying at least one refocusing pulse.

6. A magnetic resonance imaging device that images an object in the presence of a permanent background gradient, G₀, in the polarizing field, B₀, comprising:

a magnetic resonance scanner that provides basic gradient fields G₁, G₂; an RF generator and RF coils that excite a selected 2-dimensional slice of the object for excitation that is not perpendicular to G₀ by applying a selective RF-pulse in the presence of a slice selection gradient G_ss, where G_ss is a linear combination of G₀ and a transverse gradient G_ss ^app, generated as a linear combination of the basic gradient fields G₁, G₂ provided by the magnetic resonance scanner;

a gradient generator and gradient coils that apply a readout gradient G_re where G_re is a linear combination of G₀ and a transverse gradient G_re ^app generated as a linear combination OfG₁, G₂, where G_ss and G_re are not parallel at any point in the selected excited slice; and

a processor that reconstructs the selected excited slice of the object.

7. A device as in claim 6, further comprising means for phase encoding the selected excited slice with a gradient G_pll generated as a linear combination OfG₁, G₂.

8. A device as in claim 6, wherein the functions X = (G₁, B₀), Y= (G₂, B₀) and Z = Il B₀(x,y,z) Il , where Go=AB₀, define local coordinates that map the field of view of the magnetic resonance scanner onto a region of space.

9. A device as in claim 8, wherein G_ss = B₀ + G₁ and G_re = B₀ - G₁.

10. A device as in claim 6, wherein the gradient generator applies the readout gradient G_reby applying at least one refocusing pulse.