Abstract:
A method and apparatus to increase the effectiveness of a Magnetic Resonance Imaging (MRI) device by increasing the signal-to-noise ratio, allowing thinner slice thicknesses, and allowing more contiguous slices. In an MRI device, a patient is subjected to a constant magnetic field, and the RF pulses are used to excite the atoms in the body of the patient. The atoms release a corresponding RF signal when the atoms relax, which can be measured and mapped into a visual display. The RF pulses used to excite the atoms in the body of the patient use a modified Bessel function. The Bessel function having an approximately rectangular waveform in the frequency domain increases the signal-to-noise ratio, allows thinner slice thicknesses, and allows more contiguous slices, resulting in a better MR image and a more efficient MRI apparatus.

Description:
TECHNICAL FIELD 
     The technical field generally relates to magnetic resonance imaging techniques and more specifically to a method for improving the efficiency of a magnetic resonance imaging apparatus through particular RF pulses. 
     BACKGROUND OF THE INVENTION 
     Magnetic resonance imaging (MRI), also called nuclear magnetic resonance (NMR) imaging, is a non-destructive method for the analysis of materials and is used extensively in medical imaging. It is completely non-invasive and does not involve ionizing radiation. In very general terms, nuclear magnetic moments are excited at specific spin precession frequencies which are proportional to the local magnetic field. The radio frequency (RF) signals resulting from the precession of these spins are received using pickup coils. By manipulating the magnetic fields, any array of signals is provided representing different regions of the volume. These are combined to produce a volumetric image of the nuclear spin density of the body. 
     In MRI, a body is subjected to a constant magnetic field. Another magnetic field, in the form of electromagnetic (RF) pulses, is applied orthogonally to the constant magnetic field. The RF pulses have a particular frequency that is chosen to affect particular atoms (typically hydrogen) in the body. The RF pulses excite the atoms, increasing the energy state of the atoms. After the pulse, the atoms relax and release RF emissions, corresponding to the RF pulses, which are measured and processed into images for display. 
     When hydrogen nuclei relax, the frequency that they transmit is positively correlated with the strength of the magnetic field surrounding them. A magnetic field gradient along the z-axis, called the “slice select gradient”, is set up when the RF pulse is applied, and is shut off when the RF pulse is turned off. This gradient causes the hydrogen nuclei at the high end of the gradient (where the magnetic field is strong) to process at a high frequency (e.g., 26 MHz), and those at the low end (weak field) to process at a lower frequency (e.g., 24 MHz). When the RF pulse, of a single frequency, is applied, only those nuclei which process at the frequency will be tilted, to later relax and emit a radio transmission (i.e., the nuclei “resonate” to that frequency). For example, if the magnetic gradient caused hydrogen nuclei to process at rates from 24 MHz at the low end of the gradient to 26 MHz at the high end, and the gradient were set up such that the high end was located at the patient&#39;s head and the bottom part of the patient&#39;s feet, then a 24 MHz RF pulse would excite the hydrogen nuclei in a slice near the feet, and a 26 MHz pulse would excite them in a slice near the head. Thus a single “slice” along the z-axis is selected; only the protons in this slice are excited to a higher energy level, to later relax to a lower energy level and emit a radio transmission. 
     The second dimension of the image is extracted with the help of a phase encoding gradient. Immediately after the RF pulse ceases, all of the nuclei in the activated slice are in phase. Left to their own devices, these vectors would relax. In MRI, however, the phase encoding gradient (in the y-dimension) is briefly applied, in order to cause the magnetic vectors of nuclei along different portions of the gradient to have a different phase advance. 
     After the RF pulse, slice select gradient, and phase encoding gradient have been turned off, the MRI instrument sets up a third magnetic field gradient, along the x-axis, called the “frequency gradient” or “read-out gradient”. This gradient causes the relaxing protons to differentially process, so that the nuclei near the low end of the gradient begin to process at a faster rate, and those at the high end pick up even more speed. When these nuclei relax again, the fastest ones (those which were at the high end of the gradient) will emit the highest frequency of radio waves. The frequency gradient is applied only when the signal is measured. 
     The second and third dimensions of the image are extracted by means of Fourier analysis. The entire procedure must be repeated multiple times in order to form an image with a good signal-to-noise ratio. 
     Finally, in spin-echo imaging, there is the problem that the inhomegeneity of the main magnetic field induces variations in the rate of precession of nuclei. To fix this problem a 180° RF pulse is inserted into the cycle, at a time point halfway between the 90° pulse and the measurement of the radio transmission signal given off by the relaxing nuclei. 
     When 90° and 180° RF pulses are used, as described above, the technique is called “spin-echo imaging,” and takes several minutes to create a single image. Other techniques which can be used in MRI include “gradient echo imaging,” in which spin-echo&#39;s 180° pulse is replaced by a reversal of magnetic field gradients, “echo planar imaging,” a variation of spin-echo (also called “fast spin-echo imaging”), and “spin-echo inversion recovery imaging”. 
     MRI Techniques 
     In spin-echo imaging, an image section is defined by two RF pulses that are turned on following each other with a specified time interval. An echo signal is acquired once the same time interval has elapsed after the second RF pulse. The time between the first RF pulse and the signal is called the echo time TE. The echo signal results only when the tissue is exposed to both RF pulses, as is the case for stationary tissue. Blood flowing through the section will see the first pulse but move out of the section before the second RF pulse is given. Thus blood will give no signal or lower signal and appear darker than the surrounding tissue on the image. The intensity for blood is a function of the time TE/2 between the two pulses, the section thickness and the angle at which the blood flows through the section. For example, when the time between the pulses is 12.5 msec and the section thickness is 5 mm, blood entering the section perpendicular and moving with a velocity of 40 cm/sec will move completely out of the section in the 12.5 msec between the RF pulses and give no signal. Slower moving blood will move out only partially and appear with low but non-zero signal. Blood moving at an angle through the section, will take a longer time to traverse the section and may yield a non-zero signal. 
     Unlike spin-echo images, blood appears bright in fast gradient echo images. In gradient echo images, a stream of RF pulses is turned on in rapid time intervals with repetition times TR of 20-50 msec. When stationary tissue in an image section is exposed to this stream of RF pulses an equilibrium state is reached and results in a signal that is medium low (e.g., lower than the signal in spin-echo imaging). Moving blood enters the image section and is not exposed to the stream of RF pulses in the way the surrounding stationary tissue is. Thus moving blood gives a high signal. As in spin-echo imaging, the intensity of the blood signal depends on the section thickness, the velocity of the blood, the angle at which the blood enters the section and on the sequence acquisition parameters TR, TE and flip angle. 
     Physics of MRI 
     The proton nuclei of the hydrogen atom (and other atoms) possess a small magnetic moment. When placed within a magnetic field, a torque will be exerted upon them, resulting in a slight energetic advantage of one orientation (parallel to the field) over another (the anti-parallel orientation). Over time, random atomic collisions and other perturbations allow the complete system to reach a magnetic and thermal equilibrium with an excess of protons aligned with the magnetic field. The combined alignment of all of these protons results in a net magnetic moment; a subject placed within a magnetic field thus becomes “magnetized”. In biological tissues, this magnetization is exceedingly small, and generally not observable. In addition to their magnetic moment, atomic nuclei possess angular momentum—a quantum property known as “spin”. Because of this angular momentum, rather than aligning simply with magnetic fields, the individual nuclei process about it, much as a spinning top or gyroscope might, when placed in the earth&#39;s gravitational field. The precessional rate, or frequency, is characteristic of the atomic nucleus (e.g., protons) and is proportional to the strength of the magnetic field, a property crucial to the process of image formation. With the magnetic field strengths in use for today&#39;s typical MRI machines, the precessional frequency is between 5 MHz and 200 MHz. 
     As shown in  FIG. 1 , the hydrogen proton  100  possesses the quantum property of “spin” or angular momentum  102 , and has a small magnetic dipole moment  104 . When placed in a magnetic field  106 , a torque is exerted on the particle, causing it to process about the applied field  108 . The precessional frequency of the protons is directly proportional to the magnetic field strength. Protons process at about 43 MHz/Tesla. 
       FIG. 2  shows that the proton magnetization can be decomposed into the sum of a stationary (longitudinal)  202  and a rotating (transverse) component  204 . Each proton nucleus within a magnetic field thus yields a tiny field  206  that rotates about that applied field. The rotating field  206  from individual nuclei is generally aligned at random with respect to other protons in the subject or sample. In macroscopic systems, the average rotating field will effectively be zero, since that arising from any individual nucleus is canceled by another, oppositely oriented, neighbor. 
     In MR imaging, a second magnetic field is applied, which is orthogonal to the static field, and which rotates about the static field at the precessional frequency of the atomic nuclei. When the rotating field is present, the nuclei will process about it, forcing the magnetization away from equilibrium, and causing the ensemble of protons to process together, or in phase. The combined rotating magnetic moment thus produced by the ensemble of protons is observable as a time-varying electromagnetic signal. The second rotating magnetic field is applied at radio frequencies in the form of an RF pulse. 
     Two fundamental temporal parameters are used to describe the MR signal. The longitudinal relaxation rate, T1, is the rate at which nuclei, once placed in a magnetic field, exponentially approach thermal equilibrium, so that the magnetization (M) is described by the formula:
 
 M=M   o (1 −e   (−t/T1) ,  (1)
 
where, M o  is the equilibrium magnetization. In biological tissues, the proton T1 is quite long: from tens of milliseconds to seconds. Differences in the T1&#39;s of tissues are one of the primary bases of contrast in clinical MRI. A second parameter time constant describes the rate at which the MR signal decays. Once an MR signal is formed, i.e., after an RF pulse, it fades quickly; small variations in the local magnetic field, e.g., those caused by neighboring magnetic nuclei, cause the protons to process at slightly different rates and therefore to become out of phase with one another. Interactions among the magnetized protons, and motion in inhomogeneous fields, due for example to diffusion, also results in signal dephasing. The observed signal decay rate, (T2*) generally ranges from a few milliseconds to tens of milliseconds and, to a reasonable approximation, also follows first order kinetics. The MR signal, S(t), signal decays according to the formula:
 
 S ( t )= S   o ( e   (−t/T2*) )  (2)
 
where S o  is the signal strength immediately following the RF excitation pulse. The observed T2* decay is the net effect of all the dephasing terms:
 
1 /T 2*=1 /T 2+1/ T 2 m+ 1/ T 2 D +other terms,  (3)
 
where T2m represents the dephasing due to magnetic field inhomogeneitics and T2D is the diffusion-related signal loss. Like T1, the T2 signal decay rates differ among body tissues. By waiting for a period TE following the RF excitation pulse, differences in the signal decay rate will become evident as differences in the MR signal intensity; tissues with longer T2&#39;s will have stronger signals than those with short T2&#39;s, whose signals decay more rapidly. Modifications to the pattern of RF excitation (the “pulse sequence”) can modulate the contributions of the various relaxation processes to the resulting MR signal. In particular a spin-echo pulse sequence can be used to nearly eliminate the T2m contribution, increasing the relative contributions of other terms, such as proton diffusion, to the image contrast.
 
     In each MRI technique, various parameters such as T1, T2 and T2* can be manipulated to increase or decrease the weight or value of the parameter in the MR image. However, each technique is affected by certain harmful physical characteristics, affecting signal-to-noise ratio, SNR. Each technique benefits from increasing SNR. 
     SUMMARY 
     The present invention provides a method and apparatus to increase the effectiveness of a Magnetic Resonance Imaging (MRI) device by increasing the signal-to-noise ratio, allowing thinner slice thickness, and allowing more contiguous slices. In an MRI device, a patient is subjected to a constant magnetic field, and then RF pulses are used to excite the atoms in the body of the patient. The atoms release a corresponding RF signal when the atoms relax, which can be measured and mapped into a visual display. The RF pulses used to excite the atoms in the body of the patient use a modified Bessel function. The Bessel function having an approximately rectangular waveform in the frequency domain increases the signal-to-noise ration, allows thinner slice thicknesses, and allows more contiguous slices, resulting in a better MR image and a more efficient MRI apparatus. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The features, aspects, and advantages of the present invention will become better understood with regard to the following description, appended claims, and accompanying drawings where: 
         FIG. 1  illustrates the magnetic precession of a hydrogen proton; 
         FIG. 2  illustrates the components of the magnetic dipole of the hydrogen proton; 
         FIG. 3  illustrates a flowchart showing the operation of an MR imaging apparatus; 
         FIG. 4  illustrates and MRI machine; 
         FIG. 5  illustrates a Bessel function waveform; 
         FIG. 6  illustrates a Bessel function waveform of  FIG. 5  with an added constant q, as used in the method and apparatus of the present invention; 
         FIG. 7   a  illustrates a graph of the frequency domain of the Bessel function and sine function pulse of 5 msec, TE=30 msec; 
         FIG. 7   b  illustrates the first derivatives of the waveforms of the graph of the frequency domain of  FIG. 7   a;    
         FIG. 7   c  illustrates a graph of the frequency domain of the Bessel function and gaussian function pulse of 5 msec, TE=30 msec; 
         FIG. 7   d  illustrates the first derivatives of the waveforms of the graph of the frequency domain of  FIG. 7   c;    
         FIG. 8   a  illustrates a graph of the frequency domain of the Bessel function and sine function pulse of 5 msec, TE=20 msec; 
         FIG. 8   b  illustrates the first derivatives of the waveforms of the graph of the frequency domain of  FIG. 8   a;    
         FIG. 8   c  illustrates a graph of the frequency domain of the Bessel function and gaussian function pulse of 5 msec, TE=20 msec; 
         FIG. 8   d  illustrates the first derivatives of the waveforms of the graph of the frequency domain of  FIG. 8   c;    
         FIG. 9  illustrates a plot of the 5 msec 90° Bessel function pulse; 
         FIG. 10  illustrates a plot of the 5 msec 180° Bessel function pulse; 
         FIG. 11   a  illustrates a graph of the frequency domain of the Bessel function and sine function pulse of 3 msec, TE=15 msec; and 
         FIG. 11   b  illustrates the first derivatives of the waveforms of the graph of the frequency domain of  FIG. 11   a.    
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description is presented to enable any person skilled in the art to make and use the invention. For purposes of explanation, specific nomenclature is set forth to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that these specific details are not required to practice the invention. Descriptions of specific applications are provided only as representative examples. Various modifications to the preferred embodiments will be readily apparent to one skilled in the art, and the general principles defined herein may be applied to other embodiments and applications without departing from the scope of the invention. The present invention is not intended to be limited to the embodiments shown, but is to be accorded the widest possible scope consistent with the principles and features disclosed herein. 
     As shown in the flowchart of  FIG. 3 , in MRI, a body is subjected to a constant magnetic field  301 . RF pulses  303  are applied to the constant magnetic field. The RF pulses excite the atoms, increasing the energy state of the atoms. After the pulse, the atoms relax and release a corresponding RF pulse  305  which is processed  307  and displayed  309 . 
     A physical apparatus is shown in  FIG. 4 . An MRI apparatus has a bed  401 , where the object being examined is placed. The object may be a physical culture, a person, an animal, or any other physical object. The bed  401  is surrounded by magnetic coils  403 , which generate the constant magnetic field  301  of  FIG. 3 . An RF signal generator  405  creates the RF pulses  303  transmitted by the antenna  407  that excite the atoms in the object and an RF coil  409  receives the relaxation RF signals from the atoms in the object. Typically, the atoms are hydrogen atoms, but may be any other atom. For example, the atoms may be  1 H,  3 He,  13 C,  15 N,  17 O,  19 F,  23 Na,  31 P,  31 K,  65 Cu, or  129 Xe. A processor  411  processes the received RF emission signals and displays the information on a monitor  413 . 
     In MRI it is necessary to use RF pulses such that the slice profile of the images is a close approximation to a rectangle. Ideally, the slice profile of interest has a very particular gradient and an approximately rectangular slice profile is preferred. However, because the slice profile is less-than-ideally rectangular, a more rectangular slice profile is preferred. Although a perfectly rectangular slice profile is very difficult to achieve, a method is presented in which Bessel functions can be used in the design of RF pulses that offer improved slice profiles compared to those of the widely used sine, cosine, and gaussian functions. The RF pulses constructed from Bessel functions improve the image slice profile without reducing SNR and without increasing RF or gradient peak amplitudes. The pulse also allow thinner slice thicknesses, and allow more contiguous slices, which reduces the inefficiencies of either gaps or overlaps between adjacent image slices. 
     The Bessel function of the First Kind of order n for integer values of n≧0 is 
                       J   n     ⁡     (   t   )       =       t   n     ⁢       ∑     m   =   0     ∞     ⁢         (     -   1     )     m     ⁢       t     2   ⁢   m       /     (       2       2   ⁢   m     +   n       ⁢     (     m   !     )     ⁢       (     n   +   m     )     !       )                     (   4   )               
where t is time and m is an integer. For n=0, the Bessel function of the First Kind of order zero is
 
     
       
         
           
             
               
                 
                   
                     
                       J 
                       o 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         m 
                       
                       ⁢ 
                       
                         
                           t 
                           
                             2 
                             ⁢ 
                             m 
                           
                         
                         / 
                         
                           ( 
                           
                             
                               2 
                               
                                 2 
                                 ⁢ 
                                 m 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 m 
                                 ! 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 ( 
                                 m 
                                 ) 
                               
                               ! 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     Expanding equation (5) for m=0 . . . 9 yields
 
 J   o ( t )=1−( t   2 /2 2 )+( t   4 /2 4 (2!) 2 )−( t   6 /2 6 (3!) 2 )+( t   8 /2 8 (4!) 2 )−( t   10 /2 10 (5!) 2 )+( t   12 /2 12 (6!) 2 )−( t   14 /2 14 (7!) 2 )+( t   16 /2 16 (8!) 2 )+( t   18 /2 18 (9!) 2 ).  (6)
 
The oscillatory character of this function is shown in  FIG. 5 . To control the zero crossings or equivalently to control the number of lobes, equation (6) is modified by introducing a constant, q, as
 
                       J   o     ⁡     (     t   ,   q     )       =     1   -       t   2         2   2     ⁢     (     -   q     )         +       t   4         2   4     ⁢       (     2   !     )     2     ⁢       (   q   )     2         -       t   6         2   6     ⁢       (     3   !     )     2     ⁢       (     -   q     )     3         +       t   8         2   8     ⁢       (     4   !     )     2     ⁢       (   q   )     4         -       t   10         2   10     ⁢       (     5   !     )     2     ⁢       (     -   q     )     5         +       t   12         2   12     ⁢       (     6   !     )     2     ⁢       (   q   )     6         -       t   14         2   14     ⁢       (     7   !     )     2     ⁢       (     -   q     )     7         +       t   16         2   16     ⁢       (     8   !     )     2     ⁢       (   q   )     8         -       t   18         2   18     ⁢       (     9   !     )     2     ⁢       (     -   q     )     9                   (   7   )               
The character of this function is shown in  FIG. 6 . To control the peak amplitudes of the lobes, equation (7) is multiplied by a cosine function of a certain frequency, ω. The equation C(t) is defined to be C(t)=cos (ωt). This has a modulating effect on the Bessel function, as shown in equation (8).
 
 K ( t )= G*{G   1   *J   o ( t,q )* G   2   *C ( t )}  (8)
 
G is a multiplication factor used to calibrate the amplitude of K(t). G 1  and G 2  are percent amplitudes of each component function. For some applications, the cosine function in equation (8) is replaced by another Bessel function J o (t,Q) of a different constant, Q.
 
 L   1 ( t )= G*{G   1   *J   o ( t,q )* G   2   *J   o ( t,Q )}  (9)
 
Furthermore, for other applications, a superposition of a Bessel function, J o (t,Q) on the composite function (8) works very well. As shown, the superposition is
 
 L   2 ( t )= G*{G   1   *J   o ( t,q )* C ( t )+ G   2   *J   o ( t,Q )}  (10)
 
     Using the currently preferred equation (10) to design excitation pulses of various flip angles (5° to 90°) and 180° inversion pulses of RF durations ranging from 1 msec to 8 msec produced various experimental results. Experimental results are shown for 90° excitation and 180° inversion pulses with spin-echo sequences. 
     The parameter values listed below were calculated empirically to be the values in the time domain that produce a good approximation of a square wave in the frequency domain for the particular values used. 
     For 5 msec 90° RF pulses the following values are examples of parameters used: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 q 
                 Q 
                 ω 
                 G 
                 G 1   
                 G 2   
               
               
                   
                   
               
             
             
               
                   
                 −0.24 * 10 −6   
                 0.0 
                 0.4 * 10 −3   
                 55.0 
                 1.0 
                 1.0 
               
               
                   
                   
               
             
          
         
       
     
     For 5 msec 180° RF pulses the following values are examples of parameters used: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 q 
                 Q 
                 ω 
                 G 
                 G 1   
                 G 2   
               
               
                   
                   
               
             
             
               
                   
                 −0.21 * 10 −6   
                 −0.09 * 10 -−6   
                 0.4 * 10 −3   
                 100.0 
                 1.2 
                 0.8 
               
               
                   
                   
               
             
          
         
       
     
     These 5 msec Bessel function RF pulses were used in spin-echo sequences of different echo times TE to obtain MR images of phantom slices containing water doped with NiCl. The slice profiles of those images was measured and compared to the slice profile of sine and gaussian RF pulses. 
     Shown in  FIG. 7   a  is the slice profile  701  obtained with a Bessel function of a spin echo TE=30 msec. The measured slice thickness is very close to the calculated (from sequence parameters) slice thickness, between 7 and 8 mm. The slice thickness is measured, as is known in the art, as the distance between the half-maximum points of the intensity curve. Also shown is the same slice profile  705  obtained with a sine function. The steeper slopes of the slice profile  701  show clearly that the profile  701  is more rectangular than the corresponding profile  705 . In this graph, the x-axis is distance and the y-axis is intensity. 
     Shown in  FIG. 7   b  is the first derivative  703  of the slice profile  701  obtained with a Bessel function and the first derivative  708  of the slice profile  705  obtained with a sine function. The graph of the first derivatives more clearly shows the differences in slope of the slice profiles  701  and  705  obtained with the Bessel function and sine function. The slope  703  of the slice profile  701  is clearly steeper than the slope  708  of the slice profile  705 , which is also an indication that the slice profile  701  is more rectangular than the slice profile  705 . 
     Shown in  FIG. 7   c  is the slice profile  711  obtained with a Bessel function of a spin echo TE=30 msec. Also shown is the same slice profile  715  obtained with a gaussian function. The steeper slopes of the slice profile  711  show clearly that the profile  711  is more rectangular than the corresponding profile  715 . In this graph, as in  FIG. 7   a , the x-axis is distance and the y-axis is intensity. 
     Shown in  FIG. 7   d  is the first derivative  713  of the slice profile  711  obtained with a Bessel function and the first derivative  718  of the slice profile  715  obtained with a gaussian function. The graph of the first derivatives more clearly shows the differences in slope of the slice profiles  711  and  715  obtained with the Bessel function and gaussian function. The slope  713  of the slice profile  711  is clearly steeper than the slope  718  of the slice profile  715 , which is also an indication that the slice profile  711  is more rectangular than the slice profile  715 . 
     Shown in  FIG. 8   a  is the slice profile  801  obtained with a Bessel function with a TE=20 msec. Also shown is the same slice profile  805  obtained with a sine function. The steeper slopes of the slice profile  801  show clearly that the profile  801  is more rectangular than the corresponding profile  805 . As in  FIG. 7   a , in this graph, the x-axis is distance and the y-axis is intensity. 
     Shown in  FIG. 8   b  is the first derivative  803  of the slice profile  801  obtained with a Bessel function and the first derivative  808  of the slice profile  805  obtained with a sine function. The graph of the first derivatives more clearly shows the differences in slope of the slice profiles  801  and  805  obtained with the Bessel function and sine function. The slope  803  of the slice profile  801  is clearly steeper than the slope  808  of the slice profile  805 , which is also an indication that the slice profile  801  is more rectangular than the slice profile  805 . 
     Shown in  FIG. 8   c  is the slice profile  811  obtained with a Bessel function with a TE=20 msec. Also shown is the same slice profile  815  obtained with a gaussian function. The steeper slopes of the slice profile  811  show clearly that the profile  811  is more rectangular than the corresponding profile  815 . As in  FIG. 8   a , in this graph the x-axis is distance and the y-axis is intensity. 
     Shown in  FIG. 8   d  is the first derivative  813  of the slice profile  811  obtained with a Bessel function and the first derivative  818  of the slice profile  815  obtained with a gaussian function. The graph of the first derivatives more clearly shows the differences in slope of the slice profiles  811  and  815  obtained with the Bessel function and gaussian function. The slope  813  of the slice profile  811  is clearly steeper than the slope  818  of the slice profile  815 , which is also an indication that the slice profile  811  is more rectangular than the slice profile  815 . 
     Shown in  FIGS. 9 and 10  are plots of the 5 msec pulses with the above parameters.  FIG. 9  shows a plot of the 90° 5 msec waveform, and  FIG. 10  shows a plot of the 180° 5 msec waveform. 
     The parameter values listed below were calculated empirically to be the values in the time domain that produce a good approximation of a square wave in the frequency domain. 
     For 3 msec 90° RF pulses the following values were used: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 q 
                 Q 
                 ω 
                 G 
                 G 1   
                 G 2   
               
               
                   
                   
               
             
             
               
                   
                 −0.23 * 10 −6   
                 0.0 
                 0.4 * 10 −3   
                 55.0 
                 1.0 
                 1.0 
               
               
                   
                   
               
             
          
         
       
     
     For 3 msec 180° RF pulses the following values are examples of parameters used: 
     
       
         
               
               
               
               
               
               
               
             
           
               
                   
                   
               
               
                   
                 q 
                 Q 
                 ω 
                 G 
                 G 1   
                 G 2   
               
               
                   
                   
               
             
             
               
                   
                 −0.21 * 10 −6   
                 −0.09 * 10 −6   
                 0.4 * 10 −3   
                 100.0 
                 1.2 
                 0.8 
               
               
                   
                   
               
             
          
         
       
     
     Shown in  FIG. 11   a  is the slice profile  1101  obtained with a Bessel function with a TE=15 msec. Also shown is the same slice profile  1105  obtained with a gaussian function. The steeper slopes of the slice profile  1101  show clearly that the profile  1101  is more rectangular than the corresponding profile  1105 . As in  FIGS. 7   a ,  7   c ,  8   a , and  8   c , in this graph the x-axis is distance and the y-axis is intensity. 
     Shown in  FIG. 11   b  is the first derivative  1103  of the slice profile  1101  obtained with a Bessel function and the first derivative  1108  of the slice profile  1105  obtained with a gaussian function. The graph of the first derivatives more clearly shows the difference in slope of the slice profiles  1101  and  1105  obtained with the Bessel function and gaussian function. The slope  1103  of the slice profile  1101  is clearly steeper than the slope  1108  of the slice profile  1105 , which is also an indication that the slice profile  1101  is more rectangular than the slice profile  1105 . 
     It should be understood that although the currently preferred embodiment of the invention employs a modified Bessel function of the First Kind of order zero, other Bessel functions may be employed in practicing the principles of the present invention. 
     The foregoing description of the present invention provides illustration and description, but is not intended to be exhaustive or to limit the invention to the precise one disclosed. Modifications and variations are possible consistent with the above teachings or may be acquired from practice of the invention. Thus, it is noted that the scope of the invention is defined by the claims and their equivalents.