Rapid calibration of nutation angles in MRI

Techniques for rapidly and accurately calibrating RF transmitter parameters in a Nuclear Magnetic Resonance (NMR) magnetic resonance imaging (MRI) system obtain an estimate of flip (nutation) angle by determining a ratio of plural echo responses to a plural (e.g., three) RF pulse sequence. The ratio may be selected to be independent of relaxation times T.sub.1 and T.sub.2 so no relaxation waiting time between successive iterations is required. Accurate RF transmitter level calibration can be performed within on the order of three to five seconds. The techniques are robust and can discriminate flip angles over a wide range.

FIELD OF THE INVENTION 
The present invention relates to nuclear magnetic resonance (NMR), and more 
particularly to techniques for generating images using magnetic resonance 
(MRI). Still more particularly, the present invention relates to method 
and apparatus for rapidly and efficiently calibrating the parameters of RF 
(radio frequency) excitation pulses in MRI systems to provide desired 
nutation and/or "flip" angles. 
BACKGROUND AND SUMMARY OF THE INVENTION 
MRI systems are now in common use for a variety of medical and scientific 
imaging applications. The following generally relevant commonly-assigned 
U.S. Patents (incorporated by reference herein) describe some exemplary 
conventional MRI systems: 
U.S. Pat. No. 4,297,637 to Crooks et al; 
U.S. Pat. No. 4,318,043 to Crooks et al; 
U.S. Pat. No. 4,471,305 to Crooks et al; 
U.S. Pat. No. 4,599,565 to Hoenninger et al; and 
U.S. Pat. No. 4,684,891 to Feinberg. 
The following commonly assigned copending patent application generally 
relates to calibration of MRI systems: 
Serial No. 181,440 of Ching filed Apr. 14, 1988 entitled "MRI Compensated 
For Spurious NMR Frequency/Phase". 
The following principles of NMR are well known. All nuclei with an odd 
number of protons or neutrons behave, in effect, like small magnets. When 
placed in a steady external magnetic field, the magnetic axes of such 
nuclei precess at an angle about the imposed field axis at the so-called 
Larmor frequency. The Larmor frequency is related to the magnetic field of 
the nucleus by the so-called gyromagnetic ratio characteristic of the 
particular type of nuclei. As is well known, the direction of the net 
angular momentum or "spin" of a group of nuclei (and thus their net 
magnetic axis) can be reoriented with respect to the external magnetic 
field by electromagnetic signals having a frequency equal to the Larmor 
frequency. The electromagnetic signal produces a stationary magnetic field 
in the rotating frame of reference to nutate (reorient) the net spin 
resonance (Larmor frequency) nuclei by an amount determined by the 
amplitude and duration of the electromagnetic signal. 
Over a period of time, after removal of the electromagnetic signal, many 
magnetic moments will realign parallel to the external magnetic field. As 
nuclear realignment occurs, the relative phases of the individual spins 
begin to diverge as some nuclei precess faster and some slower then the 
central Larmor frequency. Thus, there is a gradual "dephasing" of the 
individual nuclear spins and a consequential loss of phase coherence. In a 
perfectly uniform magnetic field, such dephasing results from natural 
processes which cause nuclei to exchange energy with each other. The 
length of time that such dephasing takes to occur is related to the 
"spin-spin", or transverse, relaxation time constant T.sub.2. During 
realignment, the nuclear moments also lose energy to their surroundings 
and thus relax, orienting parallel to the external magnetic field. The 
"spin-lattice", or longitudinal, relaxation time T.sub.1, is related to 
this time of relaxation. 
As is also well known, nuclear spins initially aligned with the external 
magnetic field and then reoriented transverse to the initial direction 
induce a characteristic RF signal in an appropriately oriented coil 
connected to an RF signal receiver. Initially upon reorientation, a 
relatively strong voltage is induced in the receiver coil which gradually 
decreases in amplitude due to field inhomogeneity and to energy exchange 
between spins. This signal is called the "free induction decay" (FID). As 
is also well known, a "spin echo" or subsequent representation of the FID 
can be generated by bringing the respective spins back into phase 
coherence through use of a so-called "pulse sequence." For example, if at 
a time .tau. after the nuclear spins are "flipped" or reoriented (for 
example, 90.degree. with respect to initial direction) by a first 
electromagnetic pulse of appropriate frequency magnitude and duration, and 
then another electromagnetic signal of appropriate frequency, magnitude 
and duration is applied to effect 180.degree. nutation of the nuclear 
spins (hereinafter referred to as "180.degree. pulse"), the accumulation 
of further phase deviation for individual nuclear spins cause all of the 
individual spins to, at time 2.tau., again come into phase coherence to 
produce a so-called "spin echo" of the FID. 
Because the RF pulses "flip" the nuclei rotation axis, the reorientations 
(nutations) in the nuclei precession angle they generate are commonly 
referred to as "flip angles." 
As is also well understood by those skilled in this art, calibration of the 
RF transmitter and associated coils and components is critical to 
providing desired results based upon the phenomenon discussed above and/or 
based on other MRI phenomenon. The amplitude and duration of an applied RF 
pulse determines the nutation angle imparted to the nuclei precession. 
Thus, to obtain desired nutation angles it is necessary to generate RF 
pulses having corresponding desired durations and amplitudes. However, 
such RF pulse excitation of an object (e.g., a human patient) to be imaged 
is typically provided by applying the RF pulse to an RF coil closely 
coupled to the object--so that RF coil loading (and thus resulting 
radiated RF amplitude) depends on the position, size and other parameters 
of the object. Consequently, it is typically necessary to recalibrate the 
RF transmitter for each image acquisition (i.e., "study" or set of scans). 
Moreover, if the patient is moved, the Quality Factor (Q) and loading of 
the coil changes and the RF amplitude within the particular area of 
interest within the patient's body thus also changes. This typically 
requires the RF transmitter output level to be recalibrated for each new 
patient and also each time the patient is moved with respect to the RF 
coil in order to ensure desired nutation angles are being obtained for 
given RF transmitter output levels. 
It is possible to reduce or eliminate the necessity for repeated RF 
transmitter output recalibration by "de-Qing" the RF coil (so that coil 
loading is less affected by the positioning and other parameters 
associated with the patient). Unfortunately, a low Q RF coil uses RF power 
less efficiently, and the high magnetic field intensities provided by most 
superconducting magnet type MRI systems therefore require high Q RF coils. 
In order to increase the patient throughput of an MRI system and for other 
reasons (e.g., to minimize the apprehension and anxiety some patients 
suffer because of long scanning times), it is important to perform the 
requisite RF tuning and calibration as rapidly as possible. The following 
is a somewhat representative listing of documents relating to decreasing 
the time required for RF calibration: 
van der Muelen, P. and van Yperen, G. H., Proceedings, Society of Magnetic 
Resonance in Medicine Fourth Annual Meeting 1129 (1985); 
Sattin W., "A Rapid, High Signal-to-Noise RF Calibration System", 
Proceedings, Society of Magnetic Resonance in Medicine Seventh Annual 
Meeting 1016 (1988); 
Perman, W. H., Bernstein, M. A. and Sandstrom, J. C. "A Method For 
Correctly Setting the Flip Angle", Magn. Reson. Med. 9 16 (1989); 
Woessner, D. E. "Effects of Diffusion in Nuclear Magnetic Resonance 
Spin-Echo Experiments", J. Chem. Phys. 34 2057 (1961); 
U.S. Pat. No. 4,788,501, Leroux et al (1988); 
U.S. Pat. No. 4,739,267, Leroux et al (1988). 
Generally, known techniques for setting RF transmitter level include: 
seeking a maximum spin echo signal in a sequence; 
evaluating a ratio of signals in a three or more pulse sequence as a 
measure of flip angle; and 
cancelling a signal from a sequence of one or more pulses. 
The first method is the simplest and generally takes the longest time to 
perform. Not only is a search for a maximum a time-consuming endeavor, but 
accuracy demands a delay of several T.sub.1 relaxation times between 
repetitions. Commonly, the search for a maximum is automatically performed 
by obtaining NMR responses (e.g., four "spin echo" responses) from several 
different RF excitation levels (waiting at least a T.sub.1 relaxation time 
between different excitation cycles) and then applying conventional 
maximum-determining algorithms (e.g., curve fitting) to the resulting 
data. The excitation/acquisition process must typically be repeated five 
or six times before the algorithm converges accurately, and thus typically 
requires at least 45 seconds to perform. The method must be repeated each 
time the patient is moved significantly with respect to the RF coil, and 
thus may introduce significant delay into complex studies requiring 
different patient orientations. 
The Muelen and Sattin documents cited above teach a calibration method 
employing the first Hahn echo and stimulated echo from a three RF pulse 
sequence to measure the flip angle. Based on the estimate of the flip 
angle, a new level is chosen and the search continues. This is generally a 
more rapid procedure due to predicted nature of the duration. Generally 
once such methods are within range, the results they provide converge in 
two or three iterations. Muelen teaches generating an intensity ratio from 
a combination of the stimulated echo and first spin echo of a three pulse 
sequence of identical pulses, this ratio exhibiting no T.sub.2 dependence 
and a weak T.sub.1 dependence. Muelen teaches calculating the nutation 
angle from this intensity ratio. However, the T.sub.1 dependence 
introduces error into the calculation. 
Techniques which adjust the flip angle by seeking a cancellation in signal 
include an older method of setting a 180.degree. pulse by minimizing the 
FID as well as the more recent Woessner publication cited above which sets 
a 90.degree. pulse by minimizing an echo of the three pulse sequence. 
In contrast to the methods described above, the present invention actually 
provides an expression by which flip angle can be calculated substantially 
independently of T.sub.1 and T.sub.2. The present invention provides a 
robust technique for calibrating RF transmitter parameters in an MRI 
system. This technique can discriminate flip angles over a wide range, and 
is more rapid than prior art calibration techniques while avoiding 
systematic errors due to relaxation during the pulse sequence. 
Briefly, the present invention provides a technique for adjusting RF 
transmitter levels based on flip angles actually calculated from received 
responses. In the preferred embodiment, a three pulse sequence (e.g., 
.theta.-.tau.-.theta.-3.tau.-.theta.) is transmitted to generate plural 
NMR responses (where each of the three pulses has the same amplitude and 
duration and thus provides the same nutation angle .theta.. For example, 
the plural responses can include a "stimulated echo" response S.sub.1 and 
"spin echo" responses E.sub.2, E.sub.13 and E.sub.23. 
It is possible to use various combinations of these echo responses to 
provide simplified expressions for the flip angle. In particular, we have 
discovered that certain ratios of the echoes are independent of both 
relaxation times T.sub.1 and T.sub.2, are not restricted with respect to 
the range of flip angles, and provide accurate results over the 0.degree. 
to 180.degree. flip angle range generally of interest. These ratios can be 
used to calculate (estimate) flip angle resulting from a particular RF 
transmitter output level and may thus be used to automatically adjust (or 
to assist an operator in manually adjusting) the transmitter level for 
particular desired flip angles. Iterations of the excitation/acquisition 
sequence can be performed without waiting for relaxation because the 
calculations are substantially independent of relaxation times--and may 
therefore, for example, provide accurate automatic RF level calibration 
within on the order of three to five seconds (as compared to forty-five 
seconds required by the typical prior art maximum seeking algorithm).

DETAILED DESCRIPTION OF THE DRAWINGS 
The novel calibration procedure utilized by this invention typically can be 
achieved by suitable alteration of stored controlling computer programs in 
an existing MRI type apparatus. The block diagram of FIG. 1 depicts the 
general architecture of an example of such a system. 
Typically, a human or animal subject (or any other object to be imaged) 110 
is placed within a static magnetic field. For example, the subject may lie 
along the z-axis of a static magnet 108 which establishes a substantially 
uniform magnetic field directed along the z-axis within the portion of the 
object 110 of interest. For example, contiguous parallel slice-volumes p,q 
. . . z may be located within the volume to be imaged. Gradients (e.g., a 
fixed weak z gradient) may be imposed within this z-axis directed magnetic 
field along mutually orthogonal x,y,z axes by a set of x,y,z gradient 
amplifiers and coils 114 to phase encode the resulting NMR response 
signals which are generally then read out with the gradients turned off. 
NMR RF signals are transmitted into the object 110 and NMR RF responses 
are received from the object via RF coils 116 connected by a conventional 
transmit/receive switch 118 to an RF transmitter 120 and RF receiver 122. 
As will be appreciated by those in the art, separate transmit and receive 
coils may be used in some installations in which case the T/R switch 118 
may not be needed. 
All of the prior mentioned elements may be controlled, for example, by a 
control computer 124 which communicates with a data acquisition and 
display computer 126. The latter computer 126 may also receive NMR 
responses via an analog-to-digital converter 128. A CRT display and 
keyboard unit 130 is typically also associated with the data acquisition 
and display computer 126. 
As will be apparent to those in the art, such an arrangement may be 
utilized so as to generate desired sequences of magnetic gradient pulses 
and NMR RF pulses and to measure the desired NMR RF responses in 
accordance with stored computer programs (see FIG. 2, which shows an 
exemplary "spin-echo" acquisition sequence used in the preferred 
embodiment to obtain calibration data). As depicted in FIG. 1, the NMR 
system of this invention will typically include RAM, ROM and/or other 
stored program media adapted (in accordance with the descriptions herein) 
to generate a particular plural excitation RF pulse sequence (preferably 
three pulses), receive one or more resulting "spin echo" NMR responses, 
actually calculate an estimate of nutation angle from a ratio of selected 
response intensities, adjusting a parameter of the RF excitation pulse 
generating apparatus, and possibly performing one or more re-iterations of 
this overall process to obtain calibration of RF transmit level with 
desired flip angle. 
Consider FIG. 2, which shows the following three-pulse RF pulse sequence: 
EQU .theta..sub.(t=0) -.theta..sub.(t=.tau.1) -.theta..sub.(t=.tau.2). 
As those skilled in the art understand, this pulse sequence comprises three 
RF pulses, all three pulses having the same level and duration (and thus 
all providing the same nutation angle .theta.) with the pulse timing as 
stated. Provided .tau..sub.2 &gt;2.tau..sub.1, the following echoes (i.e., 
NMR echo responses) are generated after the third pulse: 
(i) the stimulated echo S.sub.1, 
(ii) the second spin echo of the first two pulses, E.sub.2, 
(iii) the spin echo of the second and third pulse, E.sub.23 and 
(iv) the echo of the first and third pulse E.sub.13. These responses are 
also shown in FIG. 2. 
Note that an additional E.sub.1 response not used in the preferred 
embodiment is generated between the second and third RF pulses. 
The following expressions describe the four pulses: 
##EQU1## 
The expression for E.sub.23 uses the value of the residual magnetization 
just prior to the second RF pulse. If we assume .tau..sub.1, .tau..sub.2 
&lt;&lt;T.sub.1, then this expression becomes 
EQU E.sub.23 =M.sub.o cos.theta.sin.theta.sin.sup.2 
(.theta./2)e.sup.-(2.tau..sub.2 -2.tau..sub.2)/T.sub.2 
Notice this signal passes through zero at .theta.=90.degree.. This is the 
echo signal which was used in some prior art techniques for RF level 
settings. Notice also that only E.sub.23 is a bipolar signal; all other 
echoes are either always positive or always negative for positive angles 
less than 180 degrees. There are more general expressions (i.e., 
generalizations of the expressions described above for echo intensities 
with different flip angles defined by different pulses in the RF pulse 
sequence. 
We wish to look for combinations of S.sub.1, E.sub.2, and E.sub.13 that are 
linear in M.sub.o and have the same T.sub.2 dependence as E.sub.23, such 
that an appropriate ratio can be formed from which to estimate .theta.. 
Let 
EQU A.ident.S.sub.1.sup.a (-E.sub.2).sup.b E.sub.13.sup.1-a-b. 
Further, let .tau..sub.2 =f.tau..sub.1 leading to the following T.sub.2 
dependence of A: 
EQU .sub.e -[2a(1-f)-2b+2f].tau..sub.1 /T.sub.2. 
For this to be the same dependence as E.sub.23 then 
##EQU2## 
The trigonometric part of A is: 
##EQU3## 
We can now look at the ratio A/E.sub.23 and select values for the 
parameters f and b to yield the desired algorithms for finding .theta.. In 
general we have 
##EQU4## 
We consider several easily solvable cases: They are b=-1,0,1/2,1, and 2 for 
a sequence in which f=4 and the four spin echoes are equally spaced apart 
at 5.tau..sub.1, 6.tau..sub.1, 7.tau..sub.1, and 8.tau..sub.1. FIGS. 
3(A)-3(E) are plots of these ratios as a function of .theta.. 
Given a measurement of E.sub.23 /A, .theta. can be calculated using the 
formulae: 
##EQU5## 
where .lambda.=tan.sup.2 (.theta./2). These are not the only solvable 
cases; other cases can be solved in closed form but involve solving cubic 
or quartic equations. In principle, any value of b can be used if one 
allows for a numerical solution of .theta.. 
We can also consider one case which does not use the signal from E.sub.23. 
This case is the limit of b approaching infinity: 
##EQU6## 
(Actually, this expression is the b'th root of A/E.sub.23 in the limit; or 
it can be verified directly.) As will be appreciated, this expression 
simplifies to the following in the preferred embodiment with f=4: 
##EQU7## 
Since our expression for E.sub.23 assumed that T.sub.1 was much longer 
than the echo times, these cases eliminate the need for that assumption. 
This is the only quantity with no T.sub.2 dependence which we can form out 
of the three other echoes. FIG. 3(F) shows the behavior of this expression 
as a function of .theta.. 
While the preferred embodiment uses three identical RF pulses in the 
excitation sequence in order to reduce computational complexity, it is 
possible to use a different number of RF pulses and/or to provide RF 
pulses with different amplitudes (and/or durations)--thus inducing 
different flip angles with different pulses in the sequence. For example, 
it might be desirable to use a three-pulse sequence of 
.theta.-2.theta.-.theta., or to use a pulse sequence having more (or 
possibly less) than three pulses (although we believe at least two pulses 
are necessary). One can find ratios of amplitudes of the resulting NMR 
echo responses to these more generalized sequences which are also 
substantially independent of T.sub.1 and T.sub.2. Moreover, the values for 
the factors a and b described above provide substantial independence of 
the T.sub.1 and T.sub.2 relaxation times even for these more general 
cases. 
Certain cases of the possible methods show severe limitations. For example, 
the curves corresponding to b=-1 and 2 are not invertible over the whole 
range of 0.degree.&lt;.theta.&lt;180.degree.. For b=2 this limits the range of 
measurable angles to be less than 109.5.degree.. A more severe restriction 
occurs for b=-1; in this case the restriction on the angle is 
.theta.&lt;70.5.degree.. 
In order to determine which of the many possible algorithms is most 
accurate, we determine the relative sensitivity of each technique towards 
error in the measurements. There are at least two separate issues to 
consider for signals acquired with noise: One concern is the relative 
sensitivity of all algorithms to random errors; and the other concern is 
the robustness of the measurement. By assumption, noise in the echo signal 
is a Gaussian white noise which is equal in rms amplitude for all 
measurements and is uncorrelated among the various signals. 
Under these assumptions, the rms error in the calculated nutation angle, 
.delta..theta. in proportional to the rms error in the signals, .delta.S, 
and is: 
##EQU8## 
The errors in the calculated angles show singularities whenever the first 
derivative of the curves of FIGS. 3(A)-3(F) vanishes. Intuitively, when a 
small change in E.sub.23 /A corresponds to a large change in angle there 
should be a high sensitivity to noise. 
The behavior of the rms error in the angle is shown in FIGS. 4(A)-4(F). 
These graphs correspond to the same cases as shown in FIGS. 3(A)-3(F). 
FIGS. 4(A)--(F) show the logarithm of the error in arbitrary units. In 
FIGS. 4A-4F we have neglected the T.sub.2 dependence of the error 
estimate. This results in an underestimate of the error in all of the 
displayed curves. The absolute magnitude of the rms error depends on the 
unknown parameters M.sub.o (which is proportional to the spin density in 
the irradiated volume) and .delta.S (which is dependent on coil efficiency 
and receiver bandwidth). Therefore, FIGS. 4A-4F can only show the relative 
error among the various algorithms. 
The overall noise sensitivity is similar for all the algorithms. We can see 
singularities in the case of b=-1 and 2 at the expected location of the 
crossovers in the curves. For small angles, b=1/2 and b=1 are slightly 
less sensitive to noise compared with the other algorithms. For large 
angles, b=1 is twice as sensitive to noise as the other methods. The 
overall least sensitive algorithm corresponds to b=1/2. 
The second concern is in the robustness of the estimate. For example, the 
solution for b=1 is only invertible for E.sub.23 /A&gt;-1. If a noisy 
measurement results in a value outside this range, the algorithm must deal 
with this as a special case. We can see that only b=1/2 makes no 
restrictions on the range of the measured quantities. We noted previously 
that all echoes other than E.sub.23 have a fixed sign. Therefore, we may 
implement the algorithm of b.fwdarw..ltoreq. and only acquire magnitude of 
the three other echoes. If this is the case, then this technique is also 
robust. 
A plot of the measured intensity of the four echoes as a function of the 
transmitter amplitude is shown in FIG. 5 (the x axis in FIG. 5 is in 
increments of RF transmitter amplitude in arbitary units). These data were 
acquired on a commercial MRI system using a mineral oil phantom with an RF 
pulse length of 10 msec and .tau..sub.1 =20 msec. FIG. 5 shows only the 
magnitude of the echoes. 
Notice in FIG. 5 that the curve for E.sub.23 crosses zero at a slightly 
higher RF level than the maximum of E.sub.2 or E.sub.13. 
The curves of FIG. 5 ideally should be the integrated intensity of the 
received signals. However, we have found that we can reduce susceptibly to 
the inherent DC offset of the RF receiver if Fourier transformations are 
applied to the signals and the analysis is performed on the first bin away 
from DC. This technique is preferable to phase alternation with signal 
averaging in the interest of shortest performance time. 
Graphs of the calculated flip angle from the data of FIG. 5 are shown in 
FIGS. 6A-6F for all of the candidate algorithms. For these graphs, the 
data were manually edited to resolve ambiguities of two solution cases 
(such as b=-1 and b=2) and to extend the calculated angle past 
360.degree.. 
As we can see, certain of the algorithms provide accurate estimates over a 
wide range of flip angles. For very low flip angles, the calculated flip 
angle is uniformly poorly calculated. However, in an iterative updating 
scheme, the next RF amplitude will be higher and bring the amplitude into 
the range of an accurate calculation. 
The calculation is more reliable close to 180.degree.. The calculated flip 
angle in the region around 180.degree. is most accurately calculated for 
the method that does not employ E.sub.23. FIGS. 6(A) and 6(E) show regions 
of increased error in the regions of the turnover of the function. FIGS. 
6(C) (b=1/2) and 6(F) (b.fwdarw..infin.) show the most linear behavior 
over the entire range. In fact, using these two functions it is possible 
to measure angles from zero to 360 degrees (the angles shown in FIGS. 
6(A)-6(F) above 360 degrees not resulting from calculation but instead 
being added to preserve the trend of the curves). 
These methods only yield a range of flip angles in the range of 0.degree. 
to 180.degree. for a single iteration. However, when used as part of an 
iterative routine to set the RF level, the iteration is stable out to 
270.degree.. In the range of 180.degree. to 270.degree., the measurement 
is inaccurate but will correctly result in a decrease in transmitter 
field. We are able to distinguish .theta.&lt;180.degree. from 
.theta.&lt;180.degree. by keeping track of relative phases of the signals of 
S.sub.1. This allows a complete determination of flip angle, up to 360 
degrees. 
FIG. 7 is a flowchart of exemplary program control steps for performing 
actual RF calibration of system 10. A trial three pulse sequence (e.g., 
using RF levels shown in the past to a least approximately yield a desired 
nutation angle) is first generated (block 200), and the resulting four NMR 
echo responses described above are acquired using conventional data 
acquisition techniques (block 202). Block 202 calculates a suitable ratio 
using one of the expressions discussed above to yield an estimate of the 
flip angle. If desired, detection and correction for "overflip" (as will 
be discussed shortly) can be performed at block 206. The calculated 
estimate of flip angle is then used to control readjustment of the RF 
transmitter level (block 208). A test is then performed to determine 
whether the calibration process has converged (e.g., by comparing 
estimated flip angle calculated by the just previous iteration of block 
204 with a newly calculated value) (decision block 210) and further 
iterations of blocks 200-208 are performed if necessary. Typically, both 
of the preferred techniques (i.e., using b=1/2 and b.fwdarw..infin.) 
converge in 2 or 3 iterations. Since we do not require T.sub.1 decay 
between pulses, the repetition rate is determined only by the computer's 
ability to acquire the data and perform the calculation. 
In our hands the results were quite satisfactory using b=1/2. However, we 
did notice a systematic inaccuracy in short T.sub.1 samples. The source of 
the inaccuracy is due to the .tau..sub.1 &lt;&lt;T.sub.1 assumption. If we relax 
this constraint, then the echo term is 
EQU E.sub.23 =M.sub.o ((cos.theta.-1)e.sup.-.tau..sub.2 /T.sub.1 +1) 
sin.theta.sin.sup.2 (.theta./2)e.sup.-2(.tau..sub.2 -.tau..sub.1)T.sub.2 
This signal vanishes at 
EQU .theta.=cos.sup.-1 (1-e.sup..tau.2/T1)&gt;90.degree.. 
For a short T.sub.1 sample, this error may be appreciable. In our work with 
a 20 msec interpulse time, the error on a mineral oil phantom (T.sub.1 
approximately equal to 125 msec) is roughly 10%. Notice that the curve of 
E.sub.23 in FIG. 5 crosses zero at a 10% higher RF amplitude than the 
maximum of E.sub.2 or E.sub.13. 
While this error is not enormous, there may be applications where the error 
is noticeable. For example, subcutaneous fat contributes a large amount of 
signal in surface coil imaging of the spine and may dominate the RF level 
setting algorithm. This would also become significant in the event that a 
mineral oil (or other very short T.sub.1) phantom is used as a quality 
control standard. In such applications, any decrease in signal to noise 
will be noticed. 
This problem may be overcome by using the method that does not use 
E.sub.23. We have found it also to be a reliable technique and does not 
result in systematic errors. 
As briefly discussed above, possible problems may arise if the flip angle 
is initially set to be greater than 270 degrees. In this case, the update 
increases the RF level and results in an overflip. It may be desirable 
therefore, to add the necessary logic to detect this condition and recover 
the proper transmitter setting. For example, the program can check whether 
it has converged on 540.degree. (see FIG. 7, block 206). In addition, the 
data obtained at high flip angles is more likely to produce errors in 
estimates due to RF field inhomogeneities. These effects are visible in 
FIG. 5 since each successive local minimum and maximum on each of the echo 
signals is broadened and the maxima attenuated. 
We have found the methods using b=1/2 and b.fwdarw..infin. to be very rapid 
and robust methods of adjusting RF amplitude. The entire process requires 
only a few seconds of data acquisition and consistently yields results 
that are as accurate as those provided by other, slower techniques. 
While the invention has been described in connection with what is presently 
considered to be the most practical and preferred embodiment, it is to be 
understood that the invention is not to be limited to the disclosed 
embodiment, but on the contrary, is intended to cover various 
modifications and equivalent arrangements included within the spirit and 
scope of the appended claims.