mri in cylindrical coordinates

Pergamon

l Original Contribution

Magnetic Resonance Imaging, Vol. 12, No. 4, pp. 613-620, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in the USA. All rights reserved

0730-725X194 $6.00 + .OO

0730-725X(93)EOO86-4

MRI IN CYLINDRICAL COORDINATES

D.H. LEE AND SOONCHIL LEE

Department of Physics, KAIST, 373-l Gusongdong, Yusongku, Daejon, Korea

We obtained the nuclear density distribution of a disk in 25 ms by using an one-dimensional (1D) imaging technique in the cylindrical coordinates. A magnetic field gradient coil implementing the cylindrical coordinates in MRI was devised by simply changing the direction of current flow at the central part of a solenoid for easy construction. The current distributions, which give the maximum field gradient in the radial direction and the ratio of the field gradient in the radial direction to that in the axial direction, were found numerically as a function of the solenoid length and the number of turns of reverse current at the center. The ratio of the radial gradient and the axial gradient of the designed coil is large enough to produce an undistorted 1D image in the radial direction of a slice perpendicular to the axis. The reverse current prevents the image from being shifted in a spectrum. The gradient coil gives images which agree with theoretical expectations within 5% error. Our result is applicable to the imaging of dynamic objects with cylindrical symmetry which do not change substantially during the order of milliseconds.

Keywords: Cylindrical coordinates; Fast imaging; One-dimensional imaging; Gradient coil; Solenoid.

INTRODUCTION

The application of MRI to dynamic objects is limited mainly by its relatively slow data acquisition. Much work has been done to reduce imaging time and several successful fast imaging methods have been devel- oped during the last decade. One of the most effective ways of reducing imaging time is to reduce imaging dimension. The data acquisition time required for the two-dimensional (2D) image of 256 x 256 pixels is 256 times shorter than that required for the three- dimensional (3D) image of 256 x 256 x 256 pixels. The data acquisition time required for the one-dimensional (1D) image of 256 pixels is even shorter than the time required for the 2D image of 256 x 256 pixels divided by 256 because the coding process is not included in the 1 D imaging. If an image phantom has the arbitrary distribution of NMR imaging parameters such as a nuclear spin density and physical and chemical properties, the slice selection technique is usually taken for the reduc- tion of imaging dimension by confining our interest to a special plane or line. However, if the phantom has some spatial symmetry in its distribution of NMR imaging parameters, we can use the 1D or 2D imaging methods

without losing any information by choosing suitable coordinates which reflect that symmetry. For example, a sphere has only one degree of freedom in its imaging parameters in the radial direction. Therefore, the data acquisition for the imaging of a sphere can be done in the order of the echo time provided that the distribution of the imaging parameters in the radial direction can be measured by applying the ID imaging technique, that is, the reading method.

The model phantoms used for engineering studies often have cylindrical symmetry, such as the velocity distribution of fluid flowing in a tube or pore distribution in a porous disk. Few samples have the symmetry of the Cartesian coordinates which is usually adopted for clinical MRI’s. Since the cylindrical sample has two degrees of freedom in the radial and axial direction, the fastest way of obtaining the distribution of imaging parameters contained in a cylindrical sample is to choose a plane perpendicular to the axis by using the slice selection technique and apply the ID imaging method. There are two methods available for the 1D imaging of a selected slice in the radial direction. One method is to get an image in the Cartesian coordinates and transform it to the cylindrical coordinates by

RECEIVED 7/29/93; ACCEPTED 12/15/93.

613

Address correspondence to D.H. Lee.

614 Magnetic Resonance Imaging 0 Volume 12, Number 4, 1994

the inverse Abel transform. The other method is to get an image directly in the magnetic field gradient which implements the cylindrical coordinates. The advantage of the first method is that hardware change is not necessary in conventional MRI equipments. However, the necessity of the large signal-to-noise ratio of 2000 or so’ prevents the first method from being practical. In this work, we devised the radial gradient coil implementing the cylindrical coordinates in MRI for the imaging of dynamic objects. In combination with the slice selection technique, the radial gradient field enables us to finish the data acquisition for the imaging of the cylindrical samples in the order of the echo time which can be made shorter than 1 ms.

Two gradient fields are necessary for the imaging in the cylindrical coordinates, one in the radial direction, and the other in the axial direction. For the axial gradient field the Maxwell pair coil can be used as in conventional MRIs. The radial gradient field must satisfy several conditions. First, the field direction should be parallel to the static field direction, and the magnitude must increase or decrease monotonically in the radial direction. For the radial gradient field to be parallel to the static field, the current distribution with cylindrical symmetry about the static field direction such as a ring or solenoid current should be used. A ring current has been used in the other work’ to select a small cylindrical volume for the localized MRI. Small perpendicular components are allowed because perpendicular components contribute negligibly to the total field. The positions of nuclei cannot be specified unless the magnitude of field increases or decreases monotonically.

Second, the field gradient in the radial direction should be large enough to give the desired resolution in an image of a slice perpendicular to the symmetry axis. If the average radial gradient is G,, the resolution of an image is given by Zr/(yG,At), where y and At represent the gyromagnetic ratio of a nucleus and a data acquisition time, respectively. In ‘H NMR, for example, if the data acquisition time is 10 ms, G, should be at least 0.23 G/cm to give the resolution of 1 mm. In the case of a simple solenoid, the average radial gradient is not a monotonic function of the solenoid length but shows a peak at l/a = 0.61 where 1 and a are the solenoid length and the radius, respectively, as shown by the numerical simulation in the next section.

Third, the ratio of the field gradient in the radial direction to that in the axial direction must be large enough to get an undistorted image. The completely homogeneous field in the axial direction can be con- structed by an elliptical current distribution3 or an in- finitely long solenoid. In that case, however, the field becomes homogeneous also in the radial direction. Any other current distributions giving field gradient in the

radial direction simultaneously causes the field to be inhomogeneous in the axial direction. Due to this fact, it is not easy to find the current distribution generating the field suitable for the MRI in the complete cylindrical coordinates. However, it is possible to find the current distribution practical for the 1D imaging in the radial direction of a slice orthogonal to the axis of a cylindrically symmetric object. If the field change through the selected slice in the axial direction is less than the difference of fields between two neighboring pixels in the radial direction, an image can be reconstructed without noticeable distortion. If the thickness of the selected slice is 5 mm and the resolution is 1 mm, as in the above example, the ratio should be greater than 5 and the change of field through 5 mm in the axial direction be less than 0.023 G(= 0.23 G/cm x 0.1 cm) to give an undistorted image.

Last, the volume of a gradient coil which satisfies the above conditions is limited because the inductance of a coil determines the risetime of a input pulse to the coil. Therefore, the volume of a coil should be small enough that the time constant (L/R) of a gradient coil is less than the risetime of a pulse allowed by each experiment. We searched numerically the optimal current distribution satisfying these conditions based on the solenoid of finite length for easy construction.

Since the direction of the axial component of the field is same everywhere inside a simple solenoid, the image acquired in this field shifts to the higher frequency side in a spectrum. To compensate this offset field, we added reverse currents at the central part of a simple solenoid. For the ease of construction, the magnitude of the added reverse current was set to be twice the current of the based simple solenoid. In other words, the designed coil is the simple solenoid at the central part of which current direction is reverse to and amount same with that of the remaining part. In order to construct the gradient field satisfying the above conditions, the magnitude of the radial gradient and the ratio of the radial gradient to the axial gradient were calculated first as a function of the length of a simple solenoid. For the solenoid with the reverse current, the magnitude of the radial gradient and the ratio were calculated as a function of both the length of the whole solenoid and the central reverse region. The designed coil was tested in the ID imaging experiment of a phantom of cylindrical symmetry.

SIMULATION

The field produced by a solenoid can be approxi- mated as the sum of the fields produced by the Nring currents which compose the solenoid provided that the radius of the used coil is negligible compared with the

MRI in cylindrical coordinates 0 D.H. LEE AND S. LEE 615

length of the solenoid. The field distribution by one ring current (Fig. 1) can be calculated via elliptic integraL4 As we see in the figure, the field changes abruptly near a coil. Since a finite length solenoid shows the similar behavior, the imaging region cannot be extended some length, say, 80% of the radius. The field generated by a ring current increases monotonically in the radial direction near z = 0. The average radial gradient decreases with increasing z until z = 0.35a, and the field decreases clearly monotonically beyond z = 0.4a. Therefore, it is expected that while adding more rings makes the field more homogeneous in the z direction, it also reduces the average radial gradient per ring and even the total radial gradient field when the length of the solenoid is greater than 0.7a.

The solenoid length which gives the maximum ratio of the radial gradient and the inhomogeneity in the z direction depends on the range of the imaging region of interest. We assume that the imgaing region has the range of 0 4 r I 0.8a and -d/2 I z I d/2 where d is the thickness of the selected slice. Then the following quantity can be used as a measure of the ratio of the

0.0 0.2 0.4 0.6 0.6 1.0

0.6

I

0.0 0.2 0.4

average radial gradient and the inhomogeneity in the z direction.

R _ G, x resolution

G, x d/2 ’ (1)

where G, and G, are the average gradients in the radial and axial directions respectively defined as

2 G, = - s d o

d’2 B(O.ga,z) - B(O,z) dz ,

0.8a (2)

and

G,= 1 s ‘.‘a 0.8a o

IB(r,O) - B(r,d/2)( dr

d/2 9 (3)

where B (r, z) is the z component of the field at position (r, z) . Other components of solenoid field perpendicular to the z axis are neglected because they contribute

I - 0.0 0.6 0.8 1.0

0.6

Fig. 1. The magnetic field distribution by the circular ring current of radius a = 6.5 cm and current 1 A. z, and r represent the coordinates in the axial and radial directions, respectively. The unit of field is G.


negligibly to the total field as remarked above. G, is defined as the average of the absolute difference because sometimes we can be misled, since the gradient in the axial direction is very small if G, changes sign within the integral range. When G, changed sign we stopped the calculation because it violates the monotonically increasing (or decreasing) condition. If R is greater than 1, the average field difference between two pixels in the radial direction is less than the maximum

field change across the slice in the z direction. There- fore, 1 5 R may be thought as the criteria for the undistorted image. In fact, larger R is more desirable because the radial gradient at center is smaller than the average value.

In Figs. 2A and 2B, G, and R are plotted as a function of the length normalized w.r.t. the radius of a simple solenoid. It is assumed that the radius of the solenoid is 6.5 cm, and the slice thickness is 5 mm. The

oq----_ -.. ____~.__ .._... _~ ._._..-.._-.. _ .._.... ___ .._.-..._ ---~-

(A)]

Yb_ / /

0 / 1

2 ~ , \

,___._. ; 2-0.2;

\ \

_..I: ccl

41

-‘ii--------- I I I 1

1 2 3 4 5 6 7 8 9 10

1

I

1; I I

\

I \

I I \

E 0;. i I \ \

/ I

‘\ I \ I \ I \

i

25 ‘1 11

-34 -

-51 1 I 0 1 2 3 4 5 6 7 8 9 1

I/a

Fig. 2. (A) G, as a function of I/a. The numbers written near each graph represent the number of reverse current turns at center. (B) Corresponding R graphs.


number of turns of coil is 10 turns/cm, and current is 1 A. The magnetic field is calculated with the interval of 1 mm for 0 I r I 0.8~ and 0 I z zz d/2. The average radial gradient G, quickly increases with increasing solenoid length until it reaches its maximum value 0.46 G/cm at l/u = 0.61, and then slowly decreases after that. This decrease is expected from Fig. 1 because the average radial gradient changes sign before z/u 5: 0.35. The ratio R of the simple solenoid decreases very slowly after passing the maximum value 4.32 at l/a = 2.49. At the condition when G, is maximum, R is 3.62. Since R is greater than 1 and G, is large enough to give the resolution less than 1 mm with 1 A current, the simple solenoid of this condition is suitable for the one dimensional imaging in the cylindrical coordinates except the fact that the available spectral range is reduced.

The z component of magnetic field which is positive (or negative) everywhere inside a solenoid reduces the available spectral range because images are shifted to one side of the spectral window. If we make the field at center be zero, half of the window is available. To use the whole spectral range, it is desirable to shift the field so that the field at center and that at the end of imaging range (in our case, O.&z) have the same magnitude but are antiparallel to each other. One easy way of generating such a field is to make the direction of current flow reverse at the central part of a solenoid to reverse the field direction at center. We have inves- tigated the magnetic field distributions of this current configuration by adding reverse current rings one by

one. The sample plot of the ratio G, and R of the so- lenoids with the turns of reverse current of 1, 11, 25, 41, and 7 1, respectively, are drawn together with those of a simple solenoid in Figs. 2A and 2B. The sign of G, and R represent whether the magnetic field increases or decreases with increasing r, and the dashed line represents that the gradient in the radial direction changes sign within r = O&z in that solenoid length range. The gradient becomes more negative with increasing number of reverse current rings as expected. When the number of reverse turns is not bigger than 11, the sign of R and G, changes as the solenoid length increases. When the number of reverse turns are bigger than 25, the direction of the radial field gradient is always opposite to that of a simple solenoid. While the magnitude of the ratio R for large [increases monotonically with increasing number of reverse turns, G, for a given length is maximum when the number of reverse turns is 41. The magnitude of G, and R increase with increasing length because the rings locating at z I 0.35~~ add negative radial gradient to the field at center as explained above. It is worthwhile to note that at some conditions, the absolute value of R or G, is bigger than that of a simple solenoid though our original purpose of reversing the current direction at center was to prevent an image from being shifted.

For a given number of reverse turns, one solenoid length is determined by the above requirement, that is, B(O,O) + B(0.8~0) = 0. In Fig. 3, G,, R, and the total solenoid length which satisfy this requirement are

7-------- _--_

-_I_.___

0

i j(

# of reverse turns

Fig. 3. G, (+) and R (X) as a function of reverse turns for the cases satisfying the requirement B(0) + B(0.8~) = 0. The normalized solenoid length I/u (Cl) which satisfies such a requirement for each reverse turns is also plotted.


plotted as a function of reverse turns. R is minimum at 11 and slowly increases with increasing reverse turns, whereas G, has the maximum value 0.87 G/cm at 55.

The solenoid length satisfying the requirement increases monotonically with increasing number of reverse turns. Since the required length increases abruptly beyond 50, only the cases where l/a is less than 10 were taken into account. When G, is maximum, the normalized solenoid length l/a satisfying the requirement B (r = 0) + B (r = 0.8a) = 0 is 6.12 and the expected ratio R is about 3.91 which is large enough to produce an undistorted image.

METHOD

Following the simulation result, a magnetic field gradient coil was made which gives the maximum gradient 0.87 G/cm in the radial direction with 1 A current and 6.5 cm radius. The coil consists of 55 reverse turns at the center of cylinder, and 172 turns each at both ends as shown in Fig. 4. Since the inductance and re- sistance of this gradient coil are 0.89 mH and 5.9 Q, respectively, the time constant is 0.15 ms. The magnetic field distribution generated by this coil is illustrated in Fig. 5 in the range of 0 I r I 0.8a and 0 I z I d/2 when the slice thickness d is 5 mm. From Fig. 5, it is clear that the magnetic field change in the axial direction is negligible compared with that in the radial direction. The field increases nearly proportionally to the square of the radius which is a desirable property for the imaging in the radial direction as discussed below.

The spectrum obtained in a linear gradient field is

itself a spatial density distribution, that is, an image. However, the spectrum obtained in a nonlinear gradient field is not an image because the resolution is not a constant but varies as a function of position. If the magnitude of field is a quadratic function of r, the spatial resolution Ar is inversely proportional to r because Ar- AB/(tIB/tb), where ABisthespectralresolution. The spectral density corresponding to a position is proportional to the number of spins belonging to that position. In the cylindrical coordinates, the number of spins belonging to a radius r is proportional to rAr. Therefore, if the field varies exactly as a quadratic function of r, the number of spins is independent of r and the signal height is constant for an homogeneous density distribution. If the field change is not quadratic, the signal sensitivity changes with position. Therefore, the quadratic gradient field is most desirable for the imaging in the cylindrical coordinates. Since our gradient field is not exactly a quadratic function of r, the following transform is necessary to get a nuclear spin density as a function of r from the spectrum. First, the real position of a point in the spectrum should be found by using Fig. 5. If the corresponding position is r, the image is reconstructed by dividing the signals at each points in the spectrum by rAr. Here, the spatial resolution Ar is obtained from the relation Ar - AB/ (aB/ar), where aB/ar can be estimated from the slope at r in Fig. 5. Finally, we rearrange the evenly distrib- uted spectral points according to the resolution of that position.

A Maxwell pair selection gradient coil is wound in the same cylindrical tube so that its center coincides pre-

Fig. 4, The reading and the selection gradient coil devised following the result of the simulation. Large arrows represent current direction.


Fig. 5. The magnetic field distribution generated by the man- ufactured gradient coil with 1 A current.

cisely with that of the reading gradient coil. This coil consists of two sets of coil of 10 turns each and 7.06 cm apart from each other, as shown in Fig. 4. The sample consists of two concentric cylindrical acryl tubes of dif-

ferent diameters. A tube of 9.0 cm outer diameter and 7.9 cm inner diameter contains a smaller tube of 6.0 cm outer diameter and 5 .O cm inner diameter. The inside of the sample is filled with an aqueous solution of para- magnetic salt. The magnetic field strength is 1 .O kG and the repetition time and echo time are 500 ms and 25 ms, respectively.

RESULTS

Figure 6 shows the spectrum of the sample obtained by the designed gradient coils. In the figure, the frequency increases as going right, and the left edge corresponds to the center of the sample. The figure shows that the spectrum is clearly divided by the inner tube. Though the nuclear density increases linearly with increasing r, the signal height decreases as going higher frequency because the radial gradient, and therefore the resolution increases a little bit faster than r.

The image obtained from the spectrum by the transform explained in the preceding section is drawn in Fig. 7. with the theoretical expectation. This 1D image contains complete information of the nuclear density distribution in a disk. The data acquisition time required to obtain this image is basically a little longer than the echo time 25 ms, though the pulse sequence should have been repeated 64 times for averaging due to the poor signal-to-noise ratio of our 1 kG MRI system. Since the echo time can be reduced even shorter than 1 ms if necessary, the complete information for

0.8

0.4

0.2

(arb. unit)

Fig. 6. The spectrum obtained by the devised gradient coils. Frequency increases as going from left to right. The left edge of the spectrum corresponds to the center of the cylindrical sample.


r(cm>

Fig. 7. The reconstructed image (-) and theoretic image (---) as a function of the radius.

the imaging parameters contained in the dynamic disk, the correlation time of whose motion is longer than 1 ms, can be acquired in a system giving enough signal- to-noise ratio. The resolution increases with increasing radius as shown in the figure. The image signal height agrees well quantitatively with the theoretical expectation within 5% error. The image shows fairly sharp skirts at boundaries, and the small blurring is caused by the fact that the centers of three tubes, two for the

sample and one for the gradient coil, are not adjusted to exactly coincide.

REFERENCES

1, Majors, P.D.; Caprihan, A. J. Magn. Reson. 94:225; 1991.

2. Lee, S.Y.; Cho, Z.H. Magn. Reson. Med. 12:56; 1989. 3. Antony, MS.; Zirnheld, J.P. J. Appl. Phys. 22:205; 1983. 4. Smythe, W.R. Static and Dynamic Electricity. New York:

McGraw Hill; 1939.

mri in cylindrical coordinates

Documents