CA2089764C - Analyzing heart wall motion using spatial modulation of magnetization - Google Patents

Abstract

A technique for generating "optimal" radio frequency pulses for use in creating SPAMM imaging stripes and for quantitatively de-termining the displacements of the imaging stripes during a pre-imag-ing interval. The imaging operator selects the desired stripe parame-ters (304) so that the resulting SPAMM imaging stripe will have the desired narrowness, sharpness, flatness and the like. These parame-ters are then used by an optimal pulse generating system (212) to gen-erate the input radio frequency pulse sequence which will produce the desired stripes. Tagging of the heart wall or some other soft tissue structure with a grid of planes of altered magnetization is then used to provide finite element analysis to quantify regional heart wall motion and the like. The intersections of the tagging stripes are used as a set of fiducial marks within the heart wall, and the motion of these inter-sections through the heart cycle is used to track the motion of the un-derlying tissue.

Description

BYBTEM AND METHOD FOR (;EI~IER1~TIN(i OPTIMAL PULSES AND FOR
ANALYSING HEART WALL MOTION DBINt~ SPATIAL MODOL7~1TION
OF MAGNETISATION
COPYRIGHT NOTICE AND AUTHORIZATION
5 A portion of the disclosure of this patent document (APPENDICES A and B) is available on microfiche and contains material which is subject to copyright protection. The copyright owner has na objection to facsimile reproduction by anyone of the patent document or the patent disclosure, as it to appears in the Patent and Trademark Office patent file or records, but otherwise reserves all copyright rights w whatsoever.

APPENDIZ 8 is a computer program listing which is available in microfiche upon request.
BACKGROUND OF THE INVENTION
Field of the Invention The present invention relates to a method for generating optimal pulses for magnetic resonance imaging using 10 spatial modulation of magnetization (SPAI~i), and more particularly, to a method for specifying the desired magnetization in certain frequency ranges and the desired smoothness for each range and then using the specified values to generate a pulse sequence which will optimally achieve the :15 desired magnetization and smoothness for imaging. A system and method is also disclosed which adapts the techniques of motion analysis from radiologic imaging of invasively implanted metallic myocardial markers to the techniques of forming tagged magnetic resonance images of heart wall motion 20 using optimal pulse sequences.
Description of the Prior Art Nuclear magnetic resonance (NI~t) has many applications in chemistry and biology. Recent advances have made it possible to use NI~t to image the human body. Because 25 much of the body is water, which gives an excellent Nl~t signal, NI~t can yield detailed images of the body and information about various disease processes.
The physical basis of NMR is that many nuclei, such as protons, have a magnetic spin. When an external field, Bo 30 (in the z axis direction), is applied to a system whose elements have a magnetic spin, the individual elements of the system precess around Bo according to the Larmor frequency.
The Larmor frequency, for typical values of the external field Bo and for typical nuclei being investigated, is in the radio frequency (rf) range.
Generally, in NMR the system is perturbed by an alternating signal having a frequency w at or near the Larmor frequency of the spins of interest, whose magnetic field (B1) - is normal to Bo. This field is called a pulse. Any spin with a Larmor frequency close to w will interact with the applied field and rotate away from the z axis towards the xy plane.
This interaction is called excitation of the spins. The precise axis of rotation and amount of rotation depend on the strength of the pulse, the duration of the pulse, and the difference between w and wo. If B1 is very strong in relation to the off-resonance effects (w-wo) , then the rotation will be essentially around an axis in the xy plane and is called a hard pulse. Generally, the stronger the B1 field, the broader range of frequencies that it will perturb. The longer the B1 field is on, and the stronger the B1 field is, the more the spins will be perturbed.
In a typical application, the spins of interest will have different wo values, either because they have a different environment, or because Bo varies with location (a gradient). Furthermore, one needs to have different types of excitation of the spins depending on the application. The three most commonly needed types of excitation are described as a ~/2 pulse, which rotates spins from the z axis to the xy plane, a ~r pulse, which rotates spins from pointing in the positive z direction towards the negative z direction, and a refocusing pulse, which rotates spins of interest 180° about an axis in the xy plane, the axis being the same for all spins of interest. However, other types of excitation profiles can be needed, and one may need to apply a different type of excitation to different frequencies.
NMR imaging schemes depend on applying magnetic field gradients so that different nuclei at different locations experience different magnetic fields and, therefore, have different frequencies. Accordingly, the location of the nucleus determines its frequency. One then applies a pulse, ~~Ti'l.~ ~ ., ~ ~ ~., __ ~ , ,~
~i ~. ~ ~ ~ ~.. ;J ~
iP~.~I~S % ' ~ '~ ~ i~92 r-I . . . i UPN-0743 - ~ - PATENT
which rotates, or excites, the nuclei, and therefore, the magnetic dipole moment. One tries to excite only nuclei which have frequencies corresponding to the slice of tissue which it is desired to image, and to excite those nuclei to the same degree.
Thus, it is necessary to design pulses which perturb a particular range of frequencies to the same extent but that do not perturb all other frequencies. This "ideal" pulse is physically unrealizable. However, it has been possible to generate pulses which will do this more or less perfectly.
As a result, much work in the NMR imaging art has gone into generating pulses which will give very "sharp" slices, for the sharper the slice, the better the resolution of the image.
Previously, so-called "hard" radio frequency pulses have been used to give sharp slices for imaging. As known by those skilled in the art, a "hard" pulse is a constant amplitude rectangular radio frequency pulse in which the strength of the applied alternating field is sufficiently large that the pulse can be assumed to affect all frequencies of interest equally. Selective excitation is thus achieved by applying several pulses in a row and varying pulse widths and inter-pulse delay times. Different frequencies precess for different amounts during the delays between the pulses and thus respond differently to the pulse sequence. Many pulse sequences using hard pulses have been devised for different applications in NMR imaging, as described by P.J. Hore in an article entitled "Solvent Suppression in Fourier Transform Nuclear Magnetic Resonance", Journal of Maqnetic Resonance, Vol. 55, pp. 283-300 (1983), for example, and by 8randes in U.S. Patent No. 4,695,798.
However, hard pulse sequences have several major limitations. First the frequency response of hard pulses have "side lobes" or harmonics around the desired frequency range.
The location of the side lobes depends on the delay between pulses. Second, if the frequency range of interest is large, as it is in NMR imaging, it may be difficult to create pulses that are strong enough to be considered "hard".
SUBSTITUTE SHEET

Furthermore, the frequency response of hard pulse sequences currently in use is far from ideal.
Despite these limitations, hard pulses have been used effectively in the technique called spatial modulation of magnetization (SPAMM), which utilizes the motion sensitivity of magnetic resonance signals to generate images representative of heart wall motion. One of the principal sources of such motion sensitivity is that when the local magnetization of a material is altered, it carries the altered magnetization with it when it moves (within the limits of the relaxation times). It has been known to measure blood flow by locally altering the magnetization of blood and then detecting the passage of this "tagged" blood downstream.
Analogous techniques have been proposed for measurement of blood flow with magnetic resonance imaging (MRI) by detecting the passage of blood with locally saturated or inverted magnetization through the region being imaged. Other techniques have been proposed by Zerhouni et al. in "Human Heart: Tagging With MRI Imaging - A Method For Non-Invasive Assessment of Myocardial Motion", Radiolocrv, Vol. 169, 1988, pp. 59-63, for imaging of myocardial motion within the plane of the image by using selective saturation pulses to magnetically tag selected bands within the heart wall, thereby producing stripes of altered magnetization that move with the tagged tissue. However, such selective saturation for producing tagging stripes is limited by the number of tags that can be practically produced. Thus, although regional intramural heart wall motion has been followed by using the ability of MRI to produce and track regions of altered magnetization, it has heretofore been impractical for clinical use.
Axel et al. describe the use of SPAMM for following heart wall motion in a clinical setting in articles entitled "MR Imaging of Motion With Spatial Modulation of Magnetization", RadioloQV, Vol. 171, June 1989, pp. 841-845 and "Heart Wall Motion: Improved Method of Spatial Modulation of Magnetization for MR Imaging", Radioloctv, Vol. 172, August '1 ,~ f -w. l.s ~ s1 i~r;~~a~S ~, A ~ r P 1992 1989, pp. 349-350. Heart wall motion may also be followed by producing signal phase shifts proportional to velocity as described by P. van Dijk in "Direct Cardiac NMR Imaging of Heart Wall and Blood Flow Velocity", Journal of Combuter Assisted Tomoq_raphy, Vol. 8, June 1984, pp. 429-436. Nayler et al . in "Blood Flow Imaging by Cine Magnetic Resonance" , Journal of Computer Assisted Tomography, Vol. 10, September/October 1986, pp. 715-722 and Wedeen in U.S. Patent No. 4,752,734 describe similar techniques for monitoring blood flow. Although the above-referenced publications have described methods for production of images that can display effects of regional heart wall motion, they have not considered the quantitative analysis of the resulting images.
Prior art imaging approaches for analyzing regional heart wall motion have used contrast angiography or radioactive labeling of red cells to study the changing shape of the endocardial surface or tomographic imaging with ultra sound, x-rays or magnetic resonance to study the changes in shape of the inner and outer surfaces of the heart wall or its local change in thickness. However, the lack of any significant landmarks within the heart wall prevents using these techniques to study intramural distributions of motion or any components of motion other than radial components.
Moreover, the motion of the curved heart wall through the fixed imaging plane often creates an artificial apparent motion within the imaging plane, thereby resulting in an artificial appearance of active thickening.
To overcome the limitations imposed by the restriction to imaging of the surfaces, rather than internal tissue, radiopaque markers have previously been implanted in the heart in order to permit the tracking of individual points within the myocardium during contraction of the heart. In particular, Meier et al. disclose in an article entitled "Kinematics of the Beating Heart", IEEE Transactions on Biomedical Enqineerinq, Vol. 27, No. 6, June 1980, pp. 319-329 and in an article entitled "Contractile Function in Canine Right Ventricle", American Journal of Ph~sioloqy, Vol. 239, SUBSTITUTE SHEET

_ _ 1980, pp. H794-H804, a technique whereby at least three small radiopaque markers are implanted in a living heart such that their movement can be ascertained. Meier et al. describe a theoretical framework of kinematics which accounts for the large time-varying displacements of the markers whereby, for arbitrary small segments of myocardium, displacements can be described completely by a rotation tensor and a stretch tensor. Similar results are disclosed for three-dimensional motion analysis of the myocardium by Fenton et al. in an article entitled "Transmural Myocardial Deformation in the Canine Left Ventricular Wall", American Journal of Physiology, Vol. 235, 1978, pp. H523-H530 and by Waldman et al. in an article entitled "Transmural Myocardial Deformation in the Canine Left Ventricle", Circulation Research, Vol. 57, 1985, pp. 152-163. However, the invasive nature of the implantation of the radiopaque markers in accordance with these techniques coupled with the difficulty of tracking a sufficient number of such markers to adequately define regional wall motion has largely restricted the techniques disclosed in the above-referenced articles to research applications. In other words, because of the invasiveness of these procedures, these procedures are presently unacceptable for clinical use.
SPAMM has been used to simultaneously create parallel sheets of altered magnetization that show up as stripes in subsequent images. Motion of the "tagged" tissue between the times of SPAMM application and imaging results in a corresponding displacement of the stripes, thereby permitting study of motions. It is desired to adapt the MR
imaging techniques for monitoring regional motion with the heart wall as described above for use with the analysis techniques developed by Meier and others for imaging of invasively implanted radiopaque markers. In particular, it is desired to adapt the analysis techniques developed for imaging of invasively implanted radiopaque markers to MR
studies of heart wall motion using the noninvasive MR imaging techniques of SPAMM whereby actual displacements within the myocardium of the heart may be measured.

WO 92/03089 ~ ~ g ~ ~ 6 PCT/US91/OSR69 It is also desired to find pulse sequences with the desired frequency response and amplitude such that the best possible SPAMM imaging stripes can be used for such analysis.
For example, in an article entitled "Heart Wall Motion:
Improved Method of Spatial Modulation of Magnetization for MR
Imaging", Radiolocrv, Vol. 172, August 1989, pp. 349-350, one of the present inventors implements SPAMM using a binomial series of radio frequency pulses which are separated by equal gradient pulses. In this sequence, during the radio frequency pulses all spins are on resonance, so the pulses are "hard", i.e., their effect on the spins is independent of the spatial location. The frequency encoding then occurs during the gradient pulses. The result is a sequence of hard pulses with free precession between the pulses. Because of the nice stripes generated when such binomial pulses were used, binomial pulses have been preferred for use with SPA1~B~I.
However, binomial pulse sequences provide limited flexibility because the relationships of the amplitudes of the binomial pulses are predefined and cannot be adjusted by the SPAMM
user.
Others have also expended considerable effort in devising pulses which have better frequency characteristics.
One of the best of these is the hyperbolic secant pulse described by M. S . Silver et al . in an article entitled "Highly Selective ~/2 and ~r Pulse Generation", Journal of Mactnetic Resonance, Vol. 59, pp, 347-351 (1984). Unfortunately, even the response to this pulse does not have an ideal frequency response. Moreover, implementation of this pulse requires a spectrometer which can simultaneously modulate the amplitude and phase of the applied external field, which many spectrometers cannot do. Indeed, patents have issued on the instrumentation for the delivery of certain shaped pulses to the transmitter, such as the afore-mentioned patent to Brandes. Further continuing efforts to devise improved pulses are described by, for example, H. Yan and J. Gore in an article entitled "Improved Selective 180° Radio Frequency Pulses for Magnetization Inversion and Phase Reversal", Journal of Magnetic Resonance, Vol. 71, pp. 116-131 (1987).
However, such other techniques of pulse generation for MR
imaging do not provide selective pulses that have periodic excitations needed for creating the selective stripes of SPAMM, and, in any event, such pulse sequences are generally too long to be used with SPAMM. Better pulses for use with SPAMM are thus not suggested in the prior art.
In addition, the frequency response of a general system such as SPAMM has been studied by one of the present inventors in a sequence of papers: S. M. Eleff et al., "The Synthesis of Pulse Sequences Yielding Arbitrary Symmetric Magnetization Vectors", Journal of Magnetic Resonance, Vol.
72, pp 298-306 (1987); M. Shinnar and J.S. Leigh, "Frequency Response of Soft Pulses", Journal of Magnetic Resonance, Vol.
75, pp. 502-505 (1987); M. Shinnar and J.S. Leigh, "The Application of Spinors to Pulse Synthesis and Analysis", Journal of Magnetic Resonance in Medicine, Vol. 12, pp. 93-98 (1989); M. Shinnar et al., "The Use Of Finite Impulse Response Filters in Pulse Design", Journal of Magnetic Resonance in Medicine, Vol. 12, pp. 81-87 (1989); M. Shinnar et al., "The Synthesis of Pulse Sequences Yielding Arbitrary Magnetization Vectors", in journal of Magnetic Resonance in Medicine, Vol.
12, pp. 74-80 (1989); and M. Shinnar et al., "The Synthesis of Soft Pulses With a Specified Frequency Response", Journal of Magnetic Resonance .in Medicine, Vol. 12, pp. 88-92 (1989).
It was shown in these articles that the frequency response of a series of pulses can be written as a Fourier series, whose coefficients are nonlinear functions of the pulse amplitudes.
A method was also described whereby, if one specified the desired MZ magnetizat.ion as a Fourier series, one could generate a pulse sequence which will actually yield that magnetization. Furthermore, the way of specifying the desired MZ magnetization as a Fourier series was realized to be identical to a solved problem in the design of finite impulse response filters. Gsing this technique, one could specify WO 92/03089 PCT/US91/058~9 zos97s4 _... _ 10 the desired magnetization in certain frequency ranges, the desired smoothness for each range, and then generate a pulse sequence which would optimally achieve that desired magnetization.
The present inventors have joined forces to investigate whether the afore-mentioned pulse generation technique could be applied to the design of a pulse sequence or sequences which would be better than the afore-mentioned binomial sequence for SPAMM imaging. Since the stripes laid down for the SPAMM technique correspond to setting the z magnetization MZ in certain frequency ranges, it seemed ideally suited for this pulse generation technique. As will be described herein, the present inventors have in fact found that improved images can be produced by using pulses generated in accordance with the afore-mentioned pulse generation technique in place of binomial pulses and that the analysis techniques developed for imaging of invasively implanted radiopaque markers to MR studies of heart wall motion using the noninvasive MR imaging techniques of SPAMM may be used to quantitatively measure movement within the myocardium of the heart.
SUMMARY OF THE INVENTION
In accordance with a first aspect of the present invention, methods are provided of constructing pulse sequences to selectively excite frequency bands in NMR imaging using the techniques of SPAMM. The characteristics of the pulses are preferably predetermined by the SPAMM user and used by the pulse generating system of the invention to generate the pulses which will yield the desired imaging stripes. Such "optimal" pulses have been found to produce improved NMR
images, for "optimal," as used herein, means that the pulse sequence will have the smallest possible error in approximating the desired pulse.
In accordance with this aspect of the invention, a sequence of N hard pulses, P1-PN, is selected which satisfies the following conditions:

_ _ (1) Each pulse is assumed to be instantaneous, or sufficiently short that its duration can be ignored.
(2) The same amount of time, t, occurs between any two pulses. During this time the magnetization precesses around the z axis with frequency w.

(3) The total duration of the pulse sequence, Nt, is sufficiently short that relaxation can be ignored.
Then, if it is assumed that the z magnetization, as a function of frequency, started out at equilibrium, with MZ(w)=1, then the resultant z magnetization, assuming the system started in equilibrium, can be written as an (N-1)th order Fourier series in wt, where w is the off-resonance frequency. Furthermore, if all pulses have the same phase, then the z magnetization is symmetric in frequency (MZ (w) -MZ(-w)), and can be written as an Nth order Fourier cosine series in wt. This Fourier series is not a Fourier transform of the hard pulse sequence, and the relationship between the pulses and the response is non-linear.
In preferred embodiments, the pulse generating method of the invention has the following characteristics:
(1) If one is given a Fourier series (in wt) description of the desired z magnetization of a hard pulse sequence of N pulses, then one can actually do an inversion of this nonlinear problem and determine a hard pulse sequence which will actually yield that desired response.
(2) Furthermore, if one is given a Fourier cosine series, one can generate a series of hard pulses around a fixed axis which gives the desired response.
(3) In general, one starts with a specification of the desired z magnetization not as a Fourier series in wt, but rather as having certain desired values over several frequency ranges. It was realized by one of the inventors that the techniques of digital filter design, specifically that of finite impulse response filters, not part of the art of NMR, could be applied to yield the desired Fourier series.
Because the theory of digital filter design is quite advanced, one can generate Fourier series which optimally give the WO 92/03089 PCT/US91/OSR~9 j .. ~p~9764 desired response. One thus can generate hard pulse sequences which have the optimal frequency response.
The method in accordance with the first aspect of the invention thus provides a technique of synthesizing a sequence of hard pulses for generating any desired realizable magnetization. The sequence of hard pulses is generated around a fixed axis which will generate any desired realizable magnetization which is symmetric in frequency. Preferably, the techniques of finite impulse response filters are used for specifying the desired magnetization. In accordance with a preferred embodiment of the invention, the generated hard pulse sequences are used in the pre-imaging SPAI~i sequence to develop improved image slices of a patient's heart.
The novel advantages, features and objects of the first aspect of the present invention are specifically realized by a method of detecting motion of an image slice of a body portion of a patient using a magnetic resonance imaging device, comprising the steps of:
inputting parameters specifying the desired visual characteristics of the image slice;
synthesizing a radio frequency pulse sequence which, when applied to the body portion prior to imaging by the magnetic resonance imaging device, will cause the creation of an image slice having the desired visual characteristics;
applying to the body portion an external magnetic field so as to produce a resultant magnetization:
applying a pre-imaging sequence to the body portion so as to spatially modulate transverse and longitudinal components of the resultant magnetization to thereby produce a spatial modulation pattern, said pre-imaging sequence comprising the synthesized radio frequency pulse sequence and magnetic field gradient pulses;
applying an imaging sequence to the body portion to make visible said spatial modulation pattern of intensity;
and detecting motion of the body portion during a time interval between application of the pre-imaging and imaging .... - 13 -sequences by examining displacements of the spatial modulation pattern of interest occurring during the time interval.
In accordance with a second aspect of the invention, the techniques developed for motion analysis from radiologic imaging of invasively implanted metallic myocardial markers are adapted to the analysis of tagged MR images of heart wall motion. The apparatus of the invention may be incorporated into existing lit imaging devices by way of a unit specially designed for this purpose, or a computer program may be implemented to provide motion analysis for 2-D or 3-D motion in accordance with this aspect of the invention so as to produce a visual display of the results. The applications include, but are not limited to, analysis of heart wall motion as described herein.
The novel advantages, features and objects of this aspect of the invention are realized by a noninvasive method of determining displacements of a body portion of a patient during a given time interval, comprising the steps of:
applying to the body portion an external magnetic field so as to produce a resultant magnetization;
applying a pre-imaging sequence to the body portion so as to divide the body portion into a grid of intersecting stripes having altered magnetization, the stripes intersecting at a plurality of intersecting points;
applying an imaging sequence to the body portion;
quantitatively measuring physical displacements of predetermined ones of the plurality of intersecting points during the given time interval; and providing an image representing the displacements.
In a preferred embodiment of the method of this aspect of the invention, the pre-imaging sequence applying step comprises the step of specifying a triangular grid of the body portion wherein the predetermined ones of the plurality of intersecting points are respective vertices of respective triangles of the triangular grid. Also, the displacements measuring step may comprise the step of determining which of the intersecting points in the triangular grid at the WO 92/03089 2 ~ 8 ~ '~ ~ PCT/US91 /OSR69 beginning of the given time interval correspond to the predetermined ones of the plurality of intersecting points in a deformed triangular grid at the end of the given time interval. The displacements measuring step may also comprise the step of comparing the triangular grid at the beginning of the given time interval with the deformed triangular grid at the end of the given time interval. Accordingly, the image representing the displacements may comprise a comparison of an image representing the triangular grid at the beginning of the given time interval with an image representing the deformed triangular grid at the end of the given time interval. Preferably the body portion is the patient's heart wall.
BRIEF DESCRIPTION OF THE DRAWINGS
The above and other objects and advantages of the invention will become more apparent and more readily appreciated from the following detailed description of the presently preferred exemplary embodiment of the invention taken in conjunction with the accompanying drawings, of which:
FIGS. 1 (a)-(d) are timing diagrams of a binomial SPAMM pulse sequence for a two-dimensional grid formation prior to imaging.
FIG. 2 is a schematic diagram of a magnetic resonance imaging system in accordance with an embodiment of the invention whereby optimal rf pulses are calculated for application during SPA1~I imaging.
FIG. 3 is a flow chart illustrating a preferred embodiment of the SPANIrt imaging technique of the invention.
FIG. 4 is a diagram illustrating the effect of choosing a narrower stripe on the smoothness of the interstripe region.
FIG. 5 is a diagram illustrating the stripe differences between a seven pulse binomial sequence and a seven pulse optimal pulse sequence generated in accordance with the techniques of the invention.

FIG. 6 is a schematic representation of a 2-D
transformation of a triplet of points (R1-R3) in the myocardium from an initial state (left) to a later state (right).
FIG. 7 is a schematic representation of a 3-D
transformation of a quadruplet of points (R1-R4) in the myocardium from an initial state (left) to a later state (right).
FIG. 8 illustrates an overall flow chart for a program for analyzing motion from "tagged" I~t images in accordance with the invention.
SUBSTITUTE SHEET

2.~89'~64 -DBTI~ILBD D88CRIPTION OF TH8 pREBENTLY pREF$RRBD ~BHBODIME1~1TB
A SPAIN imaging technique in accordance with presently preferred exemplary embodiments of the invention will be described below with reference to FIGS. 1-8. It will be appreciated by those of ordinary skill in the art that the description given herein with respect to those figures is for exemplary purposes only and is not intended in any way to limit the scope of the invention. All questions regarding the scope of the invention may be resolved by referring to the appended claims.
The SPAI~i technique may be used to image the heart by simply using a three pulse sequence consisting of two nonselective RF excitation pulses separated by a constant amplitude magnetic field gradient pulse. The sequence is then followed after a predetermined delay by a conventional imaging sequence. The first RF ulse P produces transverse magnetization which is initially phase synchronized. The first RF pulse is followed by a gradient pulse which produces a regular variation of the phase of the excited spins along the direction of the gradient, with a wavelength inversely proportional to the integral of the gradient pulse amplitude.
The second RF pulse turns some of the phase-modulated transverse magnetization back into longitudinal magnetization with a corresponding modulation. The modulation takes the form of a sinusoidal dependence of the longitudinal magnetization on position along the direction of the modulating gradient. The amplitude of the modulation depends on the strength of the RF pulses. For example, two 45° pulses will produce modulation between the extremes of saturation and full magnetization. However, longitudinal relaxation between the SPAI~i and imaging sequences reduces the amplitude of the modulation. Accordingly, the pulse angles of the applied radio frequency pulses may sum to more than 90 ° to account for the longitudinal relaxation.
In the subsequent imaging sequence, the modulated longitudinal magnetization results in a corresponding intensity modulation in the final image, appearing as SUBSTITUTE S~E~1 _ , ., .. , regularly spaced stripes perpendicular to the direction of the modulating gradient. Then, since the magnetization of any material moves as the material moves, any motion of the material between the time of the initial SPAMM sequence and the subsequent imaging sequence will be reflected in the corresponding displacement of the stripes in the final image.
SPAMM is analogous to some techniques used to produce selective excitation of chemical shifts in nuclear MR spectroscopy (e.g., for solvent suppression). In particular, the use of equal-strength nonselective RF pulses to produce selective saturation is roughly equivalent to the 1-1 case of binomial excitation pulses in spectroscopy as described by Hore in "Solvent Suppression in Fourier Transform Nuclear Magnetic Resonance", Journal of Magnetic Resonance, Vol. 55, 1983, pp. 283-300, with the modulating gradient pulse taking the place of the chemical shift.
However, improved images have been obtained when more than two radio frequency pulses separated by gradients are used to generate the SPAMM stripes. For example, the selective excitation binomial pulse sequence has been shown by Axel et al. in the article entitled "Heart Wall Motion:
Improved Method of Spatial Modulation of Magnetization for MR
Imaging", Radioloav, Vol. 172, 1989, pp. 349-350, to result in improved SPAMM heart wall images. In particular, Axel et al. therein disclose the use of a pulse sequence with a train of nonselective RF pulses, the relative amplitudes of which are distributed according to binomial coefficients such as 1-3-3-1 or 1-4-6-4-1, for example. The absolute amplitudes are adjusted for the total amplitude of modulation desired.
Separating each of the RF pulses is a gradient pulse oriented perpendicularly to the desired stripe orientation and with a strength and duration dependent upon the desired stripe spacing. For the gradient amplitude G, the stripe spacing will be equal to the reciprocal of 7 f Gdt, where 7 is the gyromagnetic ratio.
FIGS. 1 (a)-(d) illustrate an example of a binomial SPAMM pulse sequence for two-dimensional grid formation prior WO 92/03089 PCT/US91/OSf~~9 ~~'~~89764 to imaging, with RF pulse amplitudes distributed as 1-4-6-4-1. Preferably, there is a phase shift of 90° from RFI to RFQ to avoid artifacts, while gradients Gl and G2 are orthogonal so as to produce orthogonal sets of stripes.
Gradient pulses used for modulation have an amplitude and duration such that the area under the pulse shape is inversely proportional to the desired stripe spacing. A brief delay between gradient pulses and RF pulses allows the gradients to settle. Also, to avoid perturbing the stripe pattern with the effects of local field variations or regional chemical shift differences (e. g., fat vs. water), the time interval between RF pulses should be kept short.
Thus, a two-dimensional array of stripes may be used to form a tagging grid by following the initial SPAMM
sequence with a second one, with the gradient oriented in an appropriate new direction. To avoid producing artifacts from stimulated echo formation, the phase of the RF pulse is preferably shifted by at least 90° between the two sequences.
For effective composite flip angles other than 90 ° , the degree of saturation in the stripe intersections may differ from that in the stripes.
The SPAMM sequencing for forming a two-dimensional array of stripes which forms a tagging grid as described above may be implemented on a conventional MR imaging system such as the Signa 1.5-T MR imaging system from GE Medical Systems.
A simplified schematic diagram of such an MR imaging system is shown in FIG. 2. As shown, the MR imaging system has a main magnet 200 which provides a steady magnetic field Bo which polarizes the nuclei of the protons of the specimen or subject for which an image is desired. Generally, within magnet 200 there is a space in which the specimen or individual to be examined is placed.
The MR imaging system of FIG. 2 also includes a set of three orthogonal direct current coils 202, 204, and 206 for producing spatial linear field gradients Gx, Gy, and GZ. These coils are driven by gradient generator 208, which in turn is controlled by a controller 210 which communicates with a host _ lg computer 212. Host computer 212 generates the pulse sequences and controls application of the field gradients. Host computer 212 thus implements the pulse generation and displacement measurement processes of the invention to be described in more detail below.
The MR imaging system of FIG. 2 also includes a radio frequency (RF) coil 214 which generates a radio frequency field in the specimen being analyzed and also senses a free induction decay or spin echo signal which is generated after termination of the radio frequency pulse. RF pulse generator 216 excites RF coil 214 in response to host computer 212 to apply the desired pulse sequences. The signal processor 218 receives the small microvoltage level spin echo signals, converts them into digital form and inputs them into host computer 212 for formation of an image (typically using Fourier transform techniques). The resulting images are then manipulated for display and stored in storage device 220 for later retrieval or immediately displayed on display device 222.
Hence, the apparatus of FIG. 2 may be used to implement the pre-imaging SPANIrI sequence and to implement the imaging sequence for detecting heart wall movement. In a preferred embodiment, SPA1~I pulses for heart imaging may be integrated into a conventional cardiac-synchronized imaging sequence, where the SPA1~I sequence can be started with a trigger pulse derived from the electrocardiogram. A two-dimensional grid of stripes can then be produced with two 1-4-6-4-1 RF pulse sequences, for example, along with gradient pulses. Although binomial sequences produce excellent SPAIN
stripes, the present inventors have now found that "optimal"
pulses may be obtained by allowing the SPAI~i user to specify the desired stripe parameters and then generating the pulse sequence which will produce the desired stripes. Accordingly, this pulse generation technique will now be described in detail with respect to FIGS. 3-5.
FIG. 3 is a flow chart illustrating a preferred embodiment of the SPAMM imaging technique of the invention.

0~ 8 ~ ~_ ~ F p 1992 As shown, the pulse generating system of the invention may be implemented as a process which runs on host computer 212 (FIG.
2). A preferred embodiment of such a process is given in microfiche Appandis A and should be referred to in conjunction with the description of FIG. 3. However, before describing the pulse synthesis algorithm with respect to FIG. 3, it will first be explained how the units of the pulse synthesis algorithm correspond to the units of SPAMM imaging.
In particular, the algorithm to synthesize the pulse sequence specifies the desired magnetization in terms of frequency units and the duration of the portion of the pulse sequence during which the spins are freely processing. In MR
imaging, on the other hand, one is interested in the magnetization at a desired location. The relationship between frequency and location is adjustable, depending on the gradient strength. However, one can easily convert the units so that one can see the relationship. In other words, since the magnetization response of such a hard pulse sequence is periodic, with a period 1/t, where t is the time the spins freely precess between any two pulses (not the total pulse sequence duration), this periodicity may be used to lay down a series of stripes.
The pulse synthesis algorithm of the invention allows one to specify a series of frequency bands, with the desired magnetization and desired smoothness being specified in each band. The specification of the frequency bands is most naturally done in terms of the periodic frequency, 1/t.
For example, one may choose to have the z magnetizations of spins with frequencies between 0 and .1/t be 0 and those between .2/t and .8/t be 1. However, because the pulses are all around the same axis, one is guaranteed symmetry around zero frequency and therefore around .5/t. That is, Mz(m) -Mz(-~), where w is the frequency.
Thus, to convert the above to imaging, it becomes apparent that this approach automatically specifies the stripe thickness in terms of the interstripe distance. The interstripe distance is set by determining how long the SUBSTITUTE SHEET

WO 92/03089 ; ~ PCT/US91/05869 _ - 21 -gradient will be on between pulses, which determines the interstripe frequency difference, and then the gradient strength, which determines the conversion of frequency to distance. Accordingly, the interstripe distance is ultimately determined by the product of the average gradient strength (Hz/cm) and the time the gradient is on between pulses.
Increasing the gradient strength or the time will decrease the interstripe distance.
One skilled in the art will appreciate that the above discussion assumes the gradient is off during the application of the radio frequency pulses. This guarantees that, subject to the uniformity of the radio frequency field of the coil, the pulses are sufficiently hard to excite all spins equally. However, one may want to leave the gradient on during the pulses. This requires that enough radio frequency power be used that the pulses are sufficiently "hard" to equally excite all frequencies of interest. For example, if the radio frequency pulses are very narrow, they must have higher amplitudes for a given gradient strength.
Otherwise the procedure is unchanged from the approach where the gradient is off when the radio frequency pulses are applied.
As shown in FIG. 3, 1~t imaging starts at step 300 in accordance with the invention by setting the value of internal program constants (Appendix A lines 1-14) (step 302) and then requesting the user to input the desired pulse parameter data (Appendix A lines 15-57) (step 304). As noted above, this data generally includes the number of desired RF
pulses to be used in the pre-imaging sequence, the desired stripe thickness and the size of the transition zone between the striped and non-striped regions. The Fourier cosine series which optimally fits the input specifications is then calculated (Appendix A lines 58-108) (step 306) using an algorithm similar to that previously disclosed by T.W. Parks et al. in an article entitled "A Program for the Design of Linear Phase Finite Impulse Response Filters," IEEE
Transactions on Audio and Electroacoustics, Vol. AU-20, pp.

195-199 (1972). The mathematical theory behind application of this technique for pulse generation is specifically set forth below. Tr.e resulting Fourier cosine series can then be renormalized (Appendix A lines 109-130) (step 308) so that the absolute value of the Fourier ' cosine series is always less than one.
Once the Fourier cosine series is renormalized, it is then converted to a complex exponential Fourier series (Appendix A lines 135-142) (step 310). The resulting data is then formatted so that it can used by a pulse generating routine, or the data is fed directly into a pulse generating subroutine for synthesizing an optimal pulse sequence (Appendix A lines 131-143 and subroutine "PULS"). A Fourier series for the spinor components ( (1+MZ) /2 = ; P; Z and ( 1-MZ) /2 - ;Q;Z is then calculated from the Fourier series for M:
(Appendix A subroutine "PULS" lines 16-38) (step 312). P and Q are then calculated using cepstral deconvolution (Appendix A subroutine "PULS" lines 39-53) (step 314). For each set of 2U P and Q, the pulse sequence which yields the spinor with P and Q as its spinor components is then calculated (Appendix A
subroutine "PULS" lines 54-88) (step 316). This latter step is generally accomplished by attempting to reduce the Fourier series order one order at a time and then choosing which values of P arid Q to use. The result will be the optimal RF pulse sequence which can then be output to a file or other media for storage (step 318).
3r) Once the optimal pulse sequence has been generated, the pre-imaging and imaging sequences may be started. This is accomplished by applying the external magnetic field Bo to the specimen (step 320) and feeding the optimal pulse sequence to the RF coil for application with the gradient pulses as the SPAMM pre-imaging sequence (step 322). The imaging sequence is then applied to detect the stripe displacements (step 324) .

_ - 23 -Further details regarding the optimal pulse generating technique of the invention can be found upon closer inspection of the Fortran source code listing of Appendiu A.
The disclosed pulse generating technique of the present invention thus obtains the B1 field which will actually yield a desired excitation profile. If the desired excitation profile is physically realizable, the method of the invention will invert the nonlinear relationship between the input B1 field and excitation to obtain the desired B1 field.
Moreover, if one starts out with a desired excitation profile which is not physically realizable, the method of the invention gives a procedure for a physically realizable profile which is closest to the desired profile, and then finds a B1 which will yield the desired profile.
The following represents the mathematical formulation of a pulse sequence which will yield arbitrary symmetric magnetization vectors for use as RF pulses in SPAMM
imaging in accordance with the invention.
As described by the one of the present inventors in an article entitled "The Synthesis of Pulse Sequences Yielding Arbitrary Symmetric Magnetization Vectors", in Journal of Magnetic Resonance, Vol. 72, pp. 298-306, the first step of the pulse generation method of the invention is to start with a physically realizable z magnetization which may be achieved by a hard pulse sequence and to determine a hard pulse sequence which will actually yield that z magnetization.
In particular, an N hard pulse sequence to be applied around the x axis within a total duration of T has a magnetization response MZ which can be written (as a function of frequency) as a Fourier cosine series:

Mz (m) - E a~cos (jmT/ (N-1) ) j=0 Equation (1) with ~ MZ (w) ~ always __<1.
If a specification of a desired MZ(c~) is given, then we know, since magnetization is preserved by the pulses, that WO 92/A3089 .: ~ ~ ~ ~ ~ ~ ~ PCT/US91/058b9 MZx (w) + M2~ (w) + M2Z (w) - 1. The first step of the invention is thus to find an appropriate solution of Mx and My which will satisfy this equation. For this purpose, we write Mx,, (w) - MY (w) - iMx (w) . The above condition then becomes ~M,~y(w) ~2 + M2Z(w) - 1.
This reduces to a phase retrieval or deconvolution problem, well studied in signal analysis, although generally not familiar to those skilled in the NMR art. There are a number of approaches to solving this problem. For example, one can convert MZ(w) to a complex polynomial in s -exp(iwT/(N-1)) and then solve for the roots of the polynomial 1 - MZZ (w) , or equivalently for the two polynomials ( 1 - MZ (w) , 1 + MZ(w)). One then groups the roots of the polynomials by means of symmetry considerations and chooses half of them.
There are also techniques such as cepstral deconvolution which allow for the solution of M,ry(w) as an (N-1) degree polynomial in s = exp ( iwT/ (N-1) ) . This is a representation of M,n, (w) as an N-lth degree complex Fourier series. One can guarantee that since the polynomial for MZ had real coefficients, the polynomial for M,~, will also have real coefficients. The Fourier coefficients may be generated by viewing MZ as a function of tt, where ft - wT/ (N-1) , using a digital fast Fourier transform and looking at the cosine coefficients.
The following are the details of one possible approach to the resulting deconvolution problem:
Letting n=wT/ (N-1) , we first express ~ M,~, (f~) ~ Z as a polynomial in cos(tt). Namely, using Tschebyscheff polynomials, one can write cos(nn) - Tn(cos(n)) as a polynomial in cos(ft). Thus, one can write:
- E b3 (cos (n) ) ~=Q (cos (D) ) .
j=0 Equation (2) WO 92/03089 ~ ~ ~~~91/05869 .._ - 25 -We now assume that there is no loss of magnetization during the entire pulse sequence. That is, Nr«TZ for N pulses with a constant time spacing r. Then:
IMZcn) IZ+I~,cn) I~=IMoIZr Equation (3) where Mo is a constant which may be normalized to 1. We can therefore write M,~, (tt) - sqrt ( 1- I M~ (tt) I 2) e'b, where ~ is the phase relationship we are trying to construct. We shall use this relationship to write M,ry(n) as a complex Fourier series.
to since MZ (tt) is real , ~ MZ (f~) ~ 2 - (MZ (t~) ) 2. We can therefore write, for u=cos(n):

P(u)=i-Q(u)~1-IMZ(n) I~ E b~u~=IMn(n) (Z
j=0 Equation (4) Viewing P(u) as a polynomial in u, we now factor P(u) to solve for its roots:

P ( a ) - bu,II ( u-u~ ) j=1 Equation (5) We now define s=eia, where u=(s+s-1)/2. We can therefore define rational function R(s) and give its factorization as follows:

p((9+8 1)/2) - b~2-~s2~(s-s~) - 1-IM=(n) IZ
j=1 Equation (6) Next, we group the roots into conjugate pairs noting that s~ is a root of R(s) if and only if (s~+s~-1)/2=uk, for some k. Thus, if s~ is a root, s~-1 is a root. Furthermore, because P(u) has real coefficients, if u~ is a root, u~* is a root (with * denoting complex conjugate). So if s~ is a root, WO 92/0089 , PCT/US91 /05869 S3* is a root as well. We also note that ~s~=1 occurs only when a is real, and ~u~_l. Since P(u)=1-~MZ(n)~2>_0 for u=cos (err) , thus any root u~ with u~ real and ~ u~ ~ <1 must occur with even multiplicity. Therefore, roots s~ can be grouped as follows:
1) r~ real, ~ r~ ~ - 1 can be grouped into r~, r~-1, j=1,L.
2) f~ complex, ~ f~ ~ - 1. Then f~, f~-1, f~-1* are roots, j=1,M.
l0 3) g~ real, ~g~) - 1. Every g~ occurs with even multiplicity, 2*K~, j = 1, K.
) h~ complex, ~h~~ - 1. Again, h~ occurs with even multiplicity, 2*H~. Furthermore, if h~ is a root, so is h~* - h~-1. Thus, we can group roots into h~, h~-1, with multiplicity 2*H~, j - 1,H.
We next need to select a phase function for M,ry(ft).
Accordingly, for each set of roots, we choose a representative. Thus, for each set of real roots a~ we choose either r~ of r~-1. For each set of complex roots, f~, we choose a conjugate pair, either f~, f~*, of f~-1, f~-1*. This gives us 2~L~'~ choices. Each choice gives us a different phase function for M,n, (n) .
Using our choices in the previous step, we now define the following rational function to define M,~,(f~):
L
- Ag ~(n (s-r3)n (s-f~) t$-f3*) ) 7._1 7._1 R H
3 0 (II ( s-g~ ) R~ II ( s-h~ ) ( s-h3-1 ) .
7._1 7._1 Equation (7) ~~~U~91 /05869 L H R
with A2 = b~,2-~/(II(r~)II(f~f~=)II(h~)R~I
j=i j=i j=1 Equation (8) It can be shown that A is real , so G ( s ) has real coef f icients .
We now claim that R(s)=G(s)G(s-'), which can be easily verified by multiplying through. We define:
M,~(il)=G(s) , for s=e'~.
Equation (9) We claim that this M,~, (n) satisfies Equation (3 ) , and since G(s) has real coefficients, G(s*)=G(s)*. Therefore, for IsI=1, Il'~y (n) IZ- G($)G($)* - G(s)G(s=) _ G(g)G(g'1) _ g(g) _ 1- I Ms (il) I Z.
Equation (10) In defining G, we could choose either A or -A.
Thus, there are 2L'~'1 choices in defining G. MZ (tl) and M,ry (f1) are thus expressed as finite Fourier series and the xy magnetization after the jth pulse is defined, M,n,(n), for j=0 and j=N. The next step is to find a pulse sequence which will yield that Fourier series.
One can express the effect of a pulse of f degrees around the x axis on the magnetization (Mx, My, MZ) as a matrix P(f). The effect of free precession for a time (T/(N-1)) at frequency c~ can also be written as a matrix R(w). Thus, the effect on the magnetization of a free precession and a pulse is (P(f)~R), viewed as a matrix product.
We then apply R 1P 1 ( f ) to the given (Mx, MY, MZ) The resultant expression is still a Fourier series. We then solve for the f that will yield a Fourier series of order N-2. We then repeat the procedure until we reduce the magnetization to a constant MZ and constant M,~,. We then apply a P 1 ( f ) which will yield MZ=1 and M,~,=0 . The resultant series of pulses will yield, when applied in reverse to a system with f ~8 9'~ 6 4 - 28 -MZ=1 and ,=0, a system with magnetization a al to Qu the desired one.
That this part of the invention could be done was announced in an abstract published by one of the inventors entitled "An Exact Synthesis Procedure For Frequency Selective Pulses", in 5th Proceedings of the Magnetic Resonance in Med-, pp, 1452-53, August, 1986. However, the details of how to obtain the pulse sequence were not given in that abstract. The technique for obtaining the pulse sequence first appeared in the aforementioned articled entitled "The Synthesis of Pulse Sequences Yielding Arbitrary Symmetric Magnetization Vectors" by one of the present inventors. The relevant portions of the calculations set forth in that paper for obtaining the hard pulse sequence will be reproduced here.
Having generated M,~, (t~) , we have defined the xy magnetization after the jth pulse, M,ry(ft)~, for j=0 and j=N.
We will develop a simple recursion relationship to allow the calculation of M,~, (n) ~_1 from M,n, (ft) ~ . That is, we will work backwards. In addition, a formula for calculating the jth pulse from M,~,(n)~ will be derived. Having accomplished these two substeps, the pulse sequence will be completely defined.
The following matrix definitions will be used for the derivations.
_, with T denoting the transpose, and R - (cosft -sinfl 0~ P - ~1 0 0 ~sinsZ cosh o~ ~o cosh -sink ° ° iJ ~o sink cosh where M is the magnetization vector, P~M simulates a pulse and flips the spins around the x axis, and R~M rotates the spins in the xy plane. The sequence we will use applies a rotation and then a pulse. In matrix notation this is (P~R). The initial rotation with M=[0,0,1] has no effect, and the final pulse is not accompanied by a rotation since all spins are pre-rotated (that is, the operation is (P~R) not (R~P)).

_. - 29 -Spins are neither created nor destroyed during the application of the pulse sequence. Removing the effects of ( P ~ R) from MP wil l yield Mp_1.
This is accomplished by multiplying by (P~R) inverse, (P~R) 1:
( cos0 cos~psintZ sin~psinst ~
( p' R) p 1 - ' -slnil COS~pCOSn sin~pCOSiZ I
( o -sin~p COS~P
Hence, Mp_i = [ (p'R)p 1) Z'Sp = ~ (p'R)p 1) { [~rmymz]p=) Equation (11) The following derivations hinge on the fact that the only perturbations on the spin system are repeated applications of precession around the z-axis followed by a pulse around the x-axis. Rotation around the y-axis is not covered by the resulting formulae. Noting that M,~, - -My + iMX and substituting into Equation (il) yields:
[M=] p_1 = -sinøp [M~,] p + cos~p [Ms] p or [MZ] p_1 - ( 1/2 ) 8ln~p [M~ + M~*] p + [Ms] pCOS~p.
Equation (12) Similarly, [I'~,] p-1 = [-My + iMz] p_1 = [M=sinQ - l~icos~cosn -Mzsin~cosn + iM=cosn + iM~,cos~sinil + iMssiniZsintZ] p =
[M=] p ( icosn + siniZ) + [M"] pcos~ (-cosn + isinfZ) + [Ms] psin~ (-cosn 2 5 + isiniZ) _ [M=] p ( ie-'n) + (I~,] pcos~ (-e-'a) + [Ms] psin~ (-e'~) [M=] psin~ (-e-'~) _ [Ms] psin~ (-e-'~) + ( 1/2 ) e-'s( ( [M~] p +
[M~*] p) cosh + ( [M~] p [l~i~,*] p) ) . Thus, [~,] p-1 = ( 1/2 ) ( 1+cos~p) e-'n [M~] p - ( 1/2 ) ( 1-cos~p) e-'n [M~*] p Slll (~p) B ~[Mz] p.
Equation (13) Numerically, we have [MZ]N and [M,~,]N as Fourier coefficients. We therefore need to convert Equations (12) and (13) into equivalent form. This can be accomplished by _ 2~~J'~64 -equating terms of equal order. Let ar represent the rth Fourier coefficient for MZ which corresponds to ei~ and let fr correspond to the rth Fourier coefficient for M,ry. We note that since:
M= (n) - Ms (-Q) and ar = a_r, that:
[a=Jp-i = (1/2)sin~p([fr]p + [f_r]p) + cos~p[arJp Equation (14) and [frJp-i = (1/2) (1+cos~p) [fr+i]p -lo (1/2) (1-cos~p) [f_~r+mlp sfn~p[amlp.
Equation (15) For a sequence of N pulses, the Fourier coefficients range from -(N-1) to (N-1). That is, [fr]p = [ar]p = 0 for ~r~>_p.
We now set r = (p-i) in Equation (14) and get [a~l)~1 = o =
sin~p ( [ fuel] p + [ f_~~1)] p + cos~p [a~.l] p. Rearranging yields tamp = -[a~ilp/ ( [fp-1]p + [f_~~1)Jp) Equation (16) Setting r - -p in E
[ f-p] p_1 (1/2)(1+cos~ ) f ration (15) yields _ p [ -p+1J p ( 1/2 ) ( 1-COS~~ (f~#~~P Sln~p [ a_~1J p.
Since we know that a= a_r, it can be shown that [fuel]p and [fly]pare of opposite sign. Therefore, we can define:
COS~p = ( [f~lJp '~ [f-(p-1)J p) / ( [fp_1]p - [f_~~1)Jp~ ' Equation (17) Substituting into Equation (16) we obtain the last of the recursion values:
Slll~p ~ -[a~l~p~([fp-1]p - [f_~~1)Jp)' Equation (18) The synthesis of the pulse sequence is completed by using the MZ Fourier coefficients, [ar]N, where MZ(n) is generated using a digital fast Fourier transform and from the M,~, Fourier coefficients, and by using [fr]N from the ' - 31 -_ generation of M~, (t1) to calculate the leading flip angles cos~p and sin~n from Equations ( 17 ) and ( 18 ) with p = N. We then decrease p, the current pulse, which started at N, and calculate [a]p, [f]p, cos~p, and sin~p for p = N-1, ..., 2. We can then calculate the first pulse by calculating the net _ angle required at Ms (t1=0) and the sum of all angles calculated in the previous two steps, i. e. , if Ms (t~0) = 0, and the sum of the hp's for p=2,N was :r/3, then the first pulse angle should be -r/3.
The above procedure can also be generalized to cases where Ms was specified as a complex Fourier series. In such a case, the above procedure could be repeated. However, the phase retrieval problem is more complicated, and the coefficients of M~, will. in general be complex. One may then repeat the above inversion procedure. However, then a pulse is specified not only by the number of degrees it rotates, but also by the phase of the pulse in the xy plane. A solution of the inverse problem was published by one of the present inventors in an article entitled "The Synthesis of Pulse Sequences Yielding Arbitrary Magnetization Vectors", Journal of Magnetic Resonance in Medicine, Vol. 12, pp 74-80 (1989), Hence, the. present invention may be easily extended to cases where the magnetization does not correspond to a fixed axis.
As noted above with respect to FIG. 3, the preferred embodiment of the pulse generator of the invention uses spinors in the optimal pulse determination. Spinors allow the rotation operator to be monitored, as described, for example, by L.D. Landau and E.M. Lifschitz in Quantum Mechanics: Non Relativistic Theorv, pp. 188-196, Pergamon Press, London, 1965, and by I.V. Aleksandrov in The Theory of Nuclear Mactnetic Resonance, pp. 169-181, Academic Press, NY 1966, and are thus presently preferred. Spinors are useful when one wishes to specify not only the final z-magnetization, but also the rotation operator. For example, in imaging, one may wish to specify a pulse that rotates 180' about an axis in the xy plane, like an inversion pulse. Furthermore, one may wish to specify that all frequencies get rotated about the same axis. This type of pulse is a refocusing pulse. Therefore, looking at the pulse as an operator on the magnetization, one would like to specify the components of this operator. The present invention may be extended to this area as well using spinors. Such an extension using spinors has been described by~one of the present inventors in an abstract in August 1988 and later in more detail in a paper entitled "The Application of Spinors to Pulse Synthesis and Analysis", Journal of Maqnetic Resonance in Medicine, Vol. 12, pp. 93-98 (1989), The present invention has been described as a system which generates "optimal" RF pulses for use in generating SPAMM images. As noted above, the RF pulses applied to the RF coil if they are the RF pulses requested by the operator of the MR imaging device. Accordingly, some comments regarding the features of such pulses will be given here to help the operator determine what pulse features may be desirable.
1) Interstripe distance. One wants to have stripes that are sufficiently close so that one can track many different points. However, the stripes should be sufficiently far apart so they can be easily separated. The major import of this interstripe distance is that if one can give narrower stripes, one might be able to space them closer together.
2) Narrowness. The stripes laid down by the pulses should be narrow. However, they should be at least a pixel wide, maybe even wider so that they can be easily seen on the image. Since the pixel size may change with the imaging technique and the gradient strength applied, the optimal stripe thickness may vary from specimen to specimen. This condition is easily adjustable by using the pulse generating algorithm of the invention to specify the line width.
3) Sharpness. The transition from stripe to normal image should be as sharp as possible, so as little of the picture is disturbed as is necessary. This is again a parameter which can be specified with the pulse generating 'NO 92/03089 PCT/US91/05869 algorithm of the invention. This parameter can be specified as a fraction of the interstripe distance. Also, the observed narrowness of the stripes is actually a function of the sharpness. That is, the stripe is observable not only on the band where it is specified, but also over part of the transition zone, where the z magnetization is between the specified value and 1. As long as the z magnetization is sufficiently small, it will be observed as a dark line.
Thus, for the sharpest lines, one can specify an extremely narrow line and use the transition zone to make it sufficiently wide so as to be visible.
4) Flatness. Away from the stripes, the image intensity should be minimally disturbed, so as not to interfere with the interpretation of the image. This can be specified in the pulse generation algorithm of the invention.
Also, as shown in FIG. 4, there may be a tradeoff between flatness and sharpness, as increasingly sharp transitions lead to ripples. In other words, as shown in FIG. 4, the flatter pulse 400 is less sharp than the rippled pulse 402. The effect of a sharp transition can be modeled, so that one can choose an appropriate tradeoff.

5) Persistence. The stripes laid down should persist through the cardiac cycle, at least to the end of systole.
This may be accomplished by using a flip angle of more than 90° to generate the stripes. This will lead to the stripes initially having a white center, as the middle is inverted more than 90°. The white stripe gradually disappears in the cardiac cycle. For example, a flip angle of 120° after a short delay will provide the same stripe as a 90° flip angle with a lesser delay.

6) Determination of the center. One would like to be able to measure the center of the stripe with some precision.
The presence of a white line in the middle may help in this determination.

7) Duration of the pulse sequence. The pulse sequence should not be too long. The precise determination of what too long means can change, but one would like to devote not WO 92/03089 ~ ~ ~ ~ ~ ~ ~ PCT/US91/05869 more than roughly 30 msecs to generating the stripes. That way one can lay them down at end diastole, with minimal cardiac motion during the generation of the stripes. In order to apply the pulses, one has to give the pulse, then cycle the gradients on and off, which takes roughly 2 msecs. This means that given limitations of the equipment, and that one is generating two orthogonal sets of stripes, one cannot use more than a 7 pulse sequence. Sometimes, even that can be too long, and one has to use a shorter sequence. However, one skilled in the art will appreciate that an instrument which can be cycled faster could use even longer pulse sequences.
As shown in FIG. 4, if one could use more pulses, one could make even sharper stripes.
Sequences of 5 and 7 pulse trains were synthesized using the technique described herein for different frequency constraints. The pulse trains were then implemented on a GE
Signa Research magnet as the encoding pulse train of a 1 dimensional SPA1~I experiment, and applied to a copper sulfate phantom. The gradient pulses were applied between each pulse as in FIG. 1, and a dephasing gradient was applied at the end of the pulse train. An imaging sequence was then done. The images were photographed, their intensities measured, and the stripe width measured. The stripes resulting from one such pulse sequence are shown by way of example as curve 500 in FIG. 5. Curve 500 was generated as a seven pulse sequence having the following respective flip angles (summing to 120°) 12, 15.5, 20.7, 22.8, 20.8, 15.6 and 12.6. For comparison purposes, the stripes which resulted when a seven pulse binomial pulse sequence having the following respective flip angles (also summing to 120°): 1.9, 11.3, 28.1, 37.5, 28.1, 11.3 and 1.9, was applied to the same arrangement are shown in FIG. 5 as curve 502. From FIG. 5, it can be readily appreciated that the synthesized pulses were substantially better than the binomial pulses (i.e., sharp and flat).
Thus, the optimized pulses of the invention can significantly improve the SPAMM technique for cardiac imaging.
Also, because the pulse values can be adjusted, a large family ',70 92/03089 / S91 /05869 '~'~

of pulse sequences with fairly similar performance can be generated by simply adjusting the relative weightings of the narrowness, sharpness, and flatness criteria. The operator is also given the opportunity to specify how much ripple or how little sharpness is tolerable. Furthenaore, for any given z magnetization, one can synthesize many different pulse sequences (up to 2N-1, where N is the number of pulses) which yield the same z magnetization, depending on root selection in the phase retrieval part of the pulse generation algorithm.
This can also be done independent of the gradient strength.
The ability of the MR imaging techniques using optimal pulses in conjunction with periodic spatial modulation of magnetization (SPAI~t) to define regional motion within the heart wall opens up the possibility of adapting the aforementioned analysis techniques for imaging of invasively implanted radiopaque markers to the analysis of MRI studies of heart wall motion. In particular, the techniques of SPA1~I
may be used in accordance with the invention prior to imaging to "tag" the heart wall for motion analysis. For example, as will be described below with reference to FIGS. 6-16, acquisition of two sets of tagged images in orthogonal planes will permit definition of the full strain tensor throughout the imaged myocardium as a function of cardiac cycle phase, thereby allowing regional myocardial function to be quantitatively assessed.
As will be apparent from the following description, the two-dimensional tagging of the heart wall with a grid of planes of altered magnetization offers the possibility of using finite element analysis to quantify regional heart wall motion. This is accomplished in accordance with the present invention by using the intersections of the tagging stripes as a set of fiducial marks within the heart wall. The motion of these intersections through the heart cycle tracks the motion of the underlying tissue. This grid of reference points provides the basis for a triangular tiling of the heart wall between them. Then, by approximating the regional heart wall motion between each such triangular finite element as being locally homogeneous, the motion of the triangular boundary will completely define the motion within it. In particular, the motion of the region may be decomposed into its equivalent rigid body (translation and rotation) and deformation (strain) components as described by Meier et al.
< in the aforementioned articles entitled "Kinematics of the Beating Heart", IEEE Transactions on Biomedical Engineerin~c, Vol. 27, 1980, pp. 319-329, and "Contractile Function in Canine Right Ventricle", American Journal of Physiology, Vol.
239, 1980, pp. H794--H804. The strain tensor then can be decomposed, in turn, into its principal strains or eigenvectors. Such a decomposition of the motion has the useful property that it is independent of the particular orientation of the tagging grid from which it is derived.
Meier et al. describe that the motion of tissue between groups of markers (or, as described herein, the intersections of the tagging stripes) can be analyzed using the concepts of continuum mechanics. For example, for non-collinear triplets of markers or intersections in accordance with the invention, the assumption of locally homogeneous motion in the triangle between them penaits complete characterization of that motion from the time dependence of the coordinates of the markers. In other words, a hypothetical unit circle embedded in the initial state of the tissue will be transformed into an ellipse by the motion, and eigenvalues of the strain give the major and minor axes of that ellipse as shown in FIG. 6. Similarly, these concepts can be extended to three dimensions with groups of four markers or intersections defining a tetrahedron, if the motion is again considered locally homogeneous within the tetrahedron. In other words, as shown in FIG. 7, a hypothetical unit sphere embedded in the initial state would generally be transformed into an ellipsoid by the motion, and the e3genvalues of the transformation would give the axes of the ellipsoid.

:v0 92/03089 ~ ~ ~ ~/US91/05869 One method for analyzing and mapping the regional variation in wall motion (hence contractile strength), is to divide the motion of the heart wall into rigid body displacement and regional deformation. Rigid body motion can easily be visualized by a vector representing the displacement of the centroid of a region of the left ventricular wall.
Deformation cannot be visualized as easily since it has many components. Thus, one method of displaying information about the regional deformation in the heart wall is through the calculation of the eigenvalues and eigenvectors of deformation. This method is based on the principles of continuum mechanics and was first used to describe heart wall motion by Meier et al., the assumptions and methods of which will be summarized here.
Even though the deformation of the entire heart varies temporally throughout the heart cycle, the deformation of a small region of the heart can be represented as a linear transformation, provided that the vectors that it operates on are "small enough". In two dimensions, this linear mapping is obtained by calculating the change in lengths of both components of two vectors in the same region, where a vector is defined as:
X = / $i \, which is mapped into a vector X' _ / $1~\
\ $2 \ $2'/
by a tensor:
F = / F11 F12 \
\ F21 F22 /, where F is actually calculated from a tensor of column vectors, A and B, in the undeformed and deformed state, respectively, where:
A = / $1 $2 \
\ Y1 Y2 /
B = / $1' $2' \
\ Yl' Y2' /.

=489764 - 38 -Hence, F = Ainv * B.
By the Polar Decomposition Theorem, F can be separated into two operators, R and U, where R is an orthogonal rotation matrix and U is a positive symmetric stretch tensor which has real positive eigenvalues which are the principal strains of the transformation. The angle a of R is the rigid body rotation. Thus, R can be written as:
R = / cos (a) sin (a) ~-sin (a) cos (a) /, where a is calculated such that the off-diagonal elements of U are equal. The resulting formula for a is:
( F21 - F12) / (F11 + F22).
Equation (19) The resulting positive symmetric tensor U can be decomposed into its eigenvalues and eigenvectors by setting the determinant of U-I(a) - 0. This equation for two dimensions results in a quadratic equation for a, which can be solved leaving two eigenvalues for U. The eigenvectors are obtained by solving the following equation for the components of the eigenvector V1 and V2:
(Q) (V) - (a) (v) ~
Equation (20) Other types of displays may also give insight into the regional variation by providing scalar representations of the strain vectors, namely the strain invariants. The first invariant, calculated directly from U, equals the sum of the diagonal components of U, i.e., I1 - U11 + U22 in two dimensions. The second invariant equals to the determinant of the elements of U, i.e.:
I2 = {(U11)(U22) - (U12)(U21)} (in 2 dimensions).

In practice, the vectors connect two intersections of SPAMM stripes so that they range from 5 to lOmm in length.
More details of this analysis technique may be found by referring to the aforementioned articles by Meier et al., as incorporated by reference above.
In order to apply the above-mentioned analysis techniques to "tagged" MR images of the heart in accordance with the invention, an initial approximation for a two-dimensional motion analysis is made that the intersections of stripes in the image of the heart define unique tagged points in the myocardium. In a preferred embodiment, portions of a suitable interactive computer program such as that available on microfiche as Appendix B (which is a listing of an embodiment of an analysis program in accordance with the invention which is written in the programming language "C" for use on a Sun workstation using Sunview display tools) is used for determining the locations of the tagging stripe intersections within the heart wall. These intersections are identified and recorded for a series of images at a given location at different phases of the cardiac cycle. The sequential positions at later phases corresponding to each such identified point in the initial image of the series are then linked. Linked lists of positions can then be used to provide the basis for a triangular tiling of the heart wall by choosing appropriate groups of three neighboring points.
Generally, only triangles completely contained within the heart wall can be unambiguously used. The motion of the regions delineated by the resulting sets of serial triangular configurations can then be analyzed as described above for the motion analysis with respect to the serial locations of triplets of implanted metallic markers. This analysis yields a set of variables characterizing the rigid body motion and deformation of each triangular region defined for any pair of cardiac images at different phases. Preferably, for any pair of cardiac images at different phases the initial reference image is defined in such a pair so as to be of a well-defined state such as end diastole. Appendix B sets forth a preferred ~20~~9''~f4 -embodiment of a computer program which performs the above functions and is available on microfiche upon request.
Having measured the location of many such points and calculated the variables characterizing the motion of the corresponding triangular regions they define in accordance with the techniques taught by Meier et al. and set forth above, the resulting large data set must be presented in a readily analyzable form so that the diagnostician can properly analyze the imaged data. This is most readily done visually using computer graphics to either overlay suitable points, lines or shaded or colored areas on the original heart images from which they were derived, or simply to display the derived graphics by themselves as "functional" images of the regional heart wall motion. Preferably, the contours of the inner and outer surfaces of the heart wall are also displayed for reference, particularly when displaying the functional images alone. If desired, the actual computed values of the motion variables for any region in the functional image can still be interactively displayed. Many different variables can be chosen for such displays, but examples of point or line overlays would include the locations of the selected points, the sequential positions of these points (shown as displacement vectors), and the configurations of the triangles defined by the points. The triangles can be shaded or colored according to selected scalar motion variables, such as the magnitudes of the displacement of the centroids, the eigenvalues, or the angle of the eigenvectors. Higher order variables such as the strain tensor can be represented with appropriate symbols.
A generalized flow chart of the computer program of Appendiu B running on host computer 212 or a peripheral computer so as to perform the above-identified functions of the preferred embodiment is shown in FIG. 8. As shown, host computer 212 or the peripheral computer reads MR images from the clinical scanner's archive magnetic tapes (storage 220) at step 800. The images are then sorted by location and cardiac cycle phase and filed for subsequent display and analysis at step 802. The images are then flexibly displayed (e.g., with zooming) on display monitor 222 at step 804 so as to aid in picking the locations of the points of intersection of the tagging stripes in the heart wall at step 806. Once the desired stripe intersection points are selected at step 806, triangles are selected at step 808 in accordance with the techniques described above. The necessary steps for motion analysis in accordance with the techniques described by Meier et al. and set forth above, wherein eigenvalues and eigenvectors are calculated, are then performed at step 810.
The resulting motion is then displayed at step 812 using a graphics package or some other display device whereby the necessary information is relayed to the diagnostician. Of course, any display technique which relays the desired information may be used by one skilled in the art. For example, the display may compare the triangular files just before imaging to the deformed triangular tiles immediately after imaging.
The analysis in accordance with the present invention can be extended to three-dimensional motion in a variety of different ways. For example, two sets of tagged images can be acquired in different orientations, and the resulting sets of two-dimensional components of the motion may be combined to find the three-dimensional motion of the underlying overlapped volume. In other words, information on the through-plane motion into the MR images is incorporated using an orthogonal set of tagged images. Alternatively, one set of two-dimensional tagged images can be acquired to study in-plane motion, with the addition of phase shift sensitization to through-plane motion. Thus, one set of images can contain three-dimensional motion information, with in-plane motion stored as stripe displacements, thereby integrating the motion between the time of tagging and imaging, and through-plane motion stored as phase shifts proportional to velocity at the time of imaging. With a series of images made at different delays after tagging, the velocity can be integrated or the displacements differentiated WO 92/03089 r to give the three-dimensional motions on a common scale. With both approaches to three-dimensional motion, the images are recorded at fixed planes in space, yielding an Eulerian description of the motion. Such a description is convenient for transforming to an equivalent Lagrangian description of the motion in material coordinates relative to an initial configuration at the time of tagging.
Thus, the striped intersections in an image in accordance with the invention need not define unique points, thereby allowing the possibility of a component of motion through the image plane. A triangle of intersection points in one image plane and a suitable intersection point in an adjacent image plane may also define a tetrahedron.
Displacement of the vertices of the initial tetrahedron can be estimated for subsequent cardiac phases from suitable interpolation of the MR motion images. This data can then be analyzed as for the corresponding data from tetrahedral quadruplets of implanted metallic members in accordance with the techniques used by Fenton et al. in an article entitled "Transmural Myocardial Deformation in the Canine Left Ventricular Wall", American Journal of Phvsioloctv, Vol. 235, 1978, pp. H523-H530, and by Waldman et al. in an article entitled "Transmural Myocardial Deformation in the Canine Left Ventricle", Circulation Research, Vol. 57, 1985, pp. 152 163, for example.
Because of the higher dimensionality of three-dimensional data and the resulting variables characterizing the motion, it is more difficult to display such data in a readily analyzable form. Possibilities for such three-dimensional display include stereo or interactively rotatable displays such as wire frame displays of the tetrahedrons, shaded surface displays of the heart reflecting the underlying motion variables, and interactive "cutaway" displays of the heart wall. Of course, other display techniques in accordance with the invention will become apparent to those skilled in the art.

The analysis technique of the invention allows the diagnostician to interactively pick corresponding locations of tag intersections on consecutive images. The resulting set of points can be interactively used to specify a triangular tiling of the wall from which an eigenvalue analysis of the regional deformation may be carried out. Displays of the derived motions that may be found useful by those skilled in the art include graphic overlays on the initial images of an "optical flow" display of the serial displacements of each tagged point, a display of the initial and deformed grids defined by the tiling, and a set of crossed line segments superimposed on each triangle, whose lengths and orientations correspond to the regional principal strains. Of course, other display techniques may be used by those of ordinary skill in the art.
Thus, the approach to analysis of tagged MR images of heart wall motion in accordance with the invention can provide quantitative measures of regional heart wall motion.
The values calculated for the variables characterizing the action do not in principle depend on a specific orientation of the tagging grid used. Instead, these values can be used to generate functional images that directly display the regional distribution of wall motion. Moreover, the technique of the invention is also useful for studying the flows of blood or cerebrospinal fluid (CSF) and for distinguishing slow flowing blood from thrombus. In addition, by using the techniques of the invention, an infarct can be localized and identified and then displayed on a display monitor for analysis by a diagnostician. Furthermore, not only may the technique of the invention be used to separately image motion within the heart wall, but also the invention may be generally used to image elastic strain deformation in all soft tissue structures.
However, a limitation of the motion analysis from tagged images in accordance with the invention is that the spatial sampling is limited by the stripe spacing, which is, in turn, limited by the imaging resolution. The duration of stripe persistence is thus limited both by longitudinal relaxation of the tissue and blurring caused by respiratory motion. The spacing of the stripes can be easily adjusted by changing the strength and duration of the gradient pulse in the pre-imaging sequence in accordance with the techniques described above. By so adjusting the gradient pulse, stripes as fine as tens of microns apart have been created, thereby enabling very small motions involved in diffusion and perfusion to be monitored.

' y0 92/03089 _, Thus, the SPAMM imaging technique in accordance with the invention provides a simple and effective way to directly demonstrate cardiac motion. It is also useful for studying the flows of blood or CSF and for distinguishing slow flowing blood from thrombus. Adaptation of the methods described in accordance with this invention permit a wide range of other applications, including demonstrating the distribution of magnetic field and RF field inhomogeneities and gradient non-linearities, calculating gradients and demonstrating chemical shift differences. Moreover, the MR imaging technique of the invention is much more flexible than any previously contemplated for motion analysis.
Although an exemplary embodiment of this invention has been described in detail above, those skilled in the art will readily appreciate that many additional modifications are possible in the exemplary embodiment without materially departing from the novel teachings and advantages of this invention. For example, the radio frequency pulses need not be applied along the same axis. They can instead be incremented in value to create a net phase shift. Also, since the pulses herein described are designed to the specifications of the observer, minor variations in the amplitude and phase of the optimal pulses are believed to be within the scope of the invention as herein described and claimed. In addition, the imaging described herein can be applied to a patient's blood or the patient's cerebrospinal fluid as well as the patient's heart wall to determine motion in accordance with the techniques of the invention. Furthermore, since the most time-consuming part of the analysis process of the invention is the interactive picking of locations of stripe intersections, image processing approaches may be used to speed up or automate this step so as to greatly facilitate the motion analysis. Accordingly, all such modifications are intended to be included within the scope of this invention as defined in the following claims.

APPENDIX A
0001 IMPLICIT REAL*8(A-H,O-Z) PI2,AD,DEV,X,Y,GRID,DES,WT,ALPHA,IEXT,NFCNS,NGRID

0003 DIMENSION IEXT(512),AD(512),ALPHA(512),X(512),Y(512) 0004 DIMENSION H(512) 0005 DIMENSION DES(8196),GRID(8196),WT(8196) 0006 DIMENSION EDGE(20),FX(10),WTX(10),DEVIATE(10) 0007 DIMENSION XINT(8196) 0009 PI2=6.283185307179586 0010 PI=3.141592653589793 0011 NFMAX = 256 i MAXIMUM LENGTH OF EXPONENTIAL

SERIES

0012 NBANDS=2 i HAS 2 BANDS

0013 JB=2*NBANDS

0014 LGRID=6 ~ GRID DENSITY

0021 PRINT *,' PULSE SYNTHESIS PROGRAM ' 0022 PRINT *,~************************************~

0023 PRINT *,' PROGRAM WILL SYNTHESIZE A HARD PULSE

SEQUENCE YIELDING

0024 PRINT *,' ANY DESIRED FREQUENCY SPECIFICATIONS ' 0025 PRINT *,' NUMBER OF PULSES ? ' 0026 READ *,NFCNSC

0027 NFILT=2*NFCNS-1 0028 IF (NFILT .GT. NFMAX .OR. NFILT .LT. 3) CALL ERROR

THE END OF T

0034 C THE SECOND BAND (WHICH IS THE PART BETWEEN THE

STRIPES) RUNS FROM

MIDDLE OF THE

BANDS

AMPLITUDES IN THE

WIDER THAN ONE

0040 C RIPPLE (A FUNCTION OF THE NUMBER OF PULSES

AND THE WIDTH

0041 C OF THE STRIPE) 0042 PRINT *,' NOW ENTER DESIRED VALUES. ' 0043 PRINT *,' BAND EDGES UNITS ARE FRACTION OF

INTERSTRIPE DISTANCE' SUBSTITUTE SHEET

pGT/US91 /05869 ~O 92/03089 0044 EDGE(1)=0 AND # 1 STARTS AT 0 AND ENDS AT ' ' PRINT *, 0046 READ *,EDGE(2) ' 0047 PRINT *,' AND SHOULD HAVE VALUE ?

0048 READ *,FX(1) SHOULD HAVE WEIGHT ? ' ' PRINT *, 0050 READ *,WTX(1) 5 BUT STARTS AT??' ENDS AT
' 0051 .

PRINT *, 0052 READ *,EDGE(3) 0053 FX(2)=1 ' AND SHOULD HAVE WEIGHT ?' INT *

0054 , PR

0055 READ *,WTX(2) 0056 EDGE(4)=.5 ************************************

FOURIER SERIE

MAGNETIZATION

GRID (1)=EDGE(1) 0062 DELF=LGRID*NFCNS

0064 DELF=0.5D0/DELF

0065 J=1 0066 L=1 0067 LBAND=1 0068 140 FUP=EDGE(L+1) 0069 145 TEMP=GRID(J) CALCULATE THE DESIRED MAGNITUDE RESPONSE AND THE

C

DE5(J)=EFF(TEMP,FX,WTX,LBAND) 0074 WT(J)=WATE(TEMP,FX,WTX,LBAND) 0076 J=J+1 0077 GRID(J)=TEMP+DELF

0078 IF(GRID(J). GT. FUP)GO TO 150 GRID(J-1)=FUP

DES(J-1)=EFF(PUP,FX,WTX,LBAND) 0081 WT(J-1)=WATE(FUP,FX,WTX,LBAND) 0083 LBAND=LBAND+1 0084 L=L+2 IF(LBAND .GT. NBANDS) GO TO 160 0085 GRID(J)=EDGE(L) 0088 16 0 NGRID=J-1 UP A NEW APPROXIMATION PROBLEM WHICH IS

EQUIVALENT TO THE ORIGINAL PROBLEM

S FOR THE EXTERMAL FREQUENCIES--EQUALLY

=FLOAT(NGRID-1)/FLOAT(NFCNS) ~~.1 E3ST6'~"~Ta ~ S ~-; ~E'~' WO 92/03089 PCT/US91/058b9 0099 DO 210 J=1,NFCNS

0100 210 IEXT(J)=(J-1)*TEMP+1 0101 IEXT (NFCNS+1)=NGRID

0102 NHl=NFCNS-1 0103 NZ=NFCNS+1 APPROXIMATION PROBL

0107 CALL REMEZ(EDGE,NBANDS) 0109 DO J=1,2048 0110 XINT(J)=O. DO

0112 dev=dabs(dev) 0113 aconl=1/(1+dev) 0114 C PRINT *,ACON1 0115 DO J=1,NFCNS

0116 XINT(J)=ALPHA(J)*aconl 0118 PRINT *,XINT(1) 0120 CALL FFT(xint) 0121 amax=0.0 0122 do j=1,4096 0123 xl=dabs(XINT(j)/2.d0) 0124 if(xl .gt. amax) amax=xl 0125 enddo 0128 acon-aconl/amax 0129 PRINT *,' AMAX=',AMAX

0131 OPEN(UNIT=21, NAME='poly.DAT' TYPE='NEW') 0132 C , HERE ARE STORED COEEFICIENTS OF FOURIER SERIES

(COMPLEX EXPONENTI

0134 write(21,*) 2*nfcns-1 0135 do j=l,nfcns-1 0136 h(j)=alpha(nfcns+1-j)*acon/2.d0 0137 write(21,*)h(j) 0138 enddo 0139 write(21,*)alpha(1)*acon 0140 do j=l,nfcns-1 0141 write(21,*)alpha(j+1)*acon/2.d0 0142 enddo 0143 close(unit=21) 0145 CALL PULS(NBANDS) SU1C~~TITUTE SHEET, 4g 0002 REAL*8 FUNCTION EFF(TEMP,FX,WTX,LBAND) MAGNITUDE RESPONSE
0005 C AS A FUNCTION OF FREQUENCY, 0007 IMPLICIT REAL*8(A-H,O-Z) 0008 DIMENSION FX(5),WTX(5) 0009 EFF=FX(LBAND) 0003 REAL*8 FUNCTION WATE(TEMP,FX,WTX,LBAND) 0005 C OF FREQUENCY, 0007 IMPLICIT REAL*8(A-H,O-Z) 0008 DIMENSION FX(5),WTX(5) 0009 WATE=WTX(LBAND) 0004 1 FORMAT('********** ERROR IN INPUT DATA **********~) SUBSTITUTE SHEET

0003 SUBROUTINE REME2(EDGE,NBANDS) CONTINUOUS

0008 C FUNCTION WITH A SUM OF COSINES. INPUTS TO THE

0009 C ARE A DENSE GRID WHICH REPLACES THE FREQUENCY AXIS, THE

0010 C DESIRED FUNCTION ON THIS GRID, THE WEIGHT FUNCTION

ON THE

15 0011 C GRID, THE NUMBER OF COSINES, AND AN INITIAL GUESS OF

THE

0012 C EXTREMAL FREQUENCIES. THE PROGRAM MINIMIZES THE

CHEBYCHEV

0014 C FREQUENCIES (POINTS OF MAXIMUM ERROR) AND THEN

CALCULATES

0015 C THE COEFFICIENTS OF THE BEST APPROXIMATION.

25 0017 IMPLICIT REAL*8(A-H,O-Z) PI2,AD,DEV,X,Y,GRID,DES,WT,ALPHA,TEXT,NFCNS,NGRID

0019 DIMENSION EDGE(20) 0020 DIMENSION IEXT(512),AD(512),ALPHA(512),X(512),Y(512) 30 0021 DIMENSION DES(8196),GRID(8196),WT(8196) 0022 DIMENSION A(256),P(256),Q(256) 0026 ITRMAX=25 0027 DEVL=-1.0 0028 NZ=NFCNS+1 0029 NZZ=NFCNS+2 0032 IEXT(NZZ)=NGRID+1 0033 NITER=NITER+1 0034 IF(NITER .GT. ITRMAX) GO TO 400 45 0035 DO 110 J=1,NZ

0036 DTEMP=GRID(IEXT(J)) 0037 DTEMP=DCOS(DTEMP*PI2) 0038 110 X(J)=DTEMP

0039 JET=(NFCNS-1)/15+1 50 0040 DO 120 J=1,NZ

0041 120 AD(J)=D(J,NZ,JET) 0042 DNUM=0.0 0043 DDEN=0.0 0044 K=1 0045 DO 130 J=1,NZ

0046 L=IEXT(J) ~E~:L~~~'1TU i E ~N~~T

k. 51 0047 DTEMP=AD(J)*DES(L) 0048 DNUM=DNUM+DTEMP

0049 DTEMP=K*AD(J)/WT(L) 0050 DDEN=DDEN+DTEMP

0051 130 K=-K

0052 DEV=DNUM/DDEN

0053 NU=1 0054 IF(DEV .GT. 0.0) NU=-1 0055 DEV=-NU*DEV

0056 K=NU

0057 DO 140 J=1,NZ

0058 L=IEXT(J) 0059 DTEMP=K*DEV/WT(L) 0060 Y(J)=DES(L)+DTEMP

0061 140 K=-K

0062 IF(DEV .GE. DEVL) GO TO 150 0065 150 DEVL=DEV

0066 JCHNGE=0 0067 K1=IEXT(1) 0068 KNZ=IEXT(NZ) 0069 KLOW=O

0070 NUT=-NU

0071 J=1 0076 200 IF(J .EQ. NZZ) YNZ=COMP

0077 IF(J .GE. NZZ) GO TO 300 0078 KUP=IEXT(J+1) 0079 L=IEXT(J)+1 0080 NUT=-NUT

0081 IF(J EQ. 2) Y1=COMP

0082 COMP=DEV

0083 IF(L .GE. KUP) GO TO 220 0084 ERR=GEE(L,NZ) 0085 ERR=(ERR-DES(L))*WT(L) 0086 DTEMP=NUT*ERR-COMP

0087 IF(DTEMP .LE. 0.0) GO TO 220 0088 COMP=NUT*ERR

0089 210 L=L+1 0090 IF(L .GE. KUP) GO TO 215 0091 ERR=GEE(L,NZ) 0092 ERR=(ERR-DES(L))*WT(L)-0093 DTEMP=NUT*ERR-COMP

0094 IF(DTEMP .LE. O.O) GO TO 215 0095 COMP=NUT*ERR

0097 215 IEXT(J)=L-1 0098 J=J+1 0099 KLOW=L-1 0100 JCHNGE=JCHNGE+1 0102 220 L=L-1 ITIJTE SHEET, ~' -ST
cUg, 0103 225 L=L-1 0104 IF(L .LE. KLOW) GO TO 250 0105 ERR=GEE(L,NZ) 0106 ERR=(ERR-DES(L))*WT(L) 0107 DTEMP=NUT*ERR-COMP

0108 IF(DTEMP .GT. 0.0) GO TO 230 0109 IF(JCHNGE .LE. 0) GO TO 225 0111 230 COMP=NUT*ERR

0112 235 L=L-1 0113 IF(L .LE. KLOW) GO TO 240 0114 ERR=GEE(L,NZ) 0115 ERR=(ERR-DES(L))*WT(L) 0116 DTEMP=NUT*ERR-COMP

0117 IF(DTEMP .LE. 0.0) GO TO 240 0118 COMP=NUT*ERR

0120 240 KLOW=IEXT(J) 0121 IEXT(J)-L+1 0122 J=J+1 0123 JCHNGE=JCHNGE+1 0125 250 L=IEXT(J)+1 0126 IF(JCHNGE .GT. 0) GO TO 215 0127 255 L=L+1 0128 IF(L .GE. KUP) GO TO 260 0129 ERR=GEE(L,NZ) 0130 ERR=(ERR-DES(L))*WT(L) 0131 DTEMP=NUT*ERR-COMP

0132 IF(DTEMP .LE. 0.0) GO TO 255 0133 COMP=NUT*ERR

0135 260 KLOW=IEXT(J) 0136 J=J+1 0138 300 IF (J .GT. NZZ) GO TO 320 0139 IF (K1 .GT. IEXT(1)) K1=IEXT(1) 0140 IF(KNZ .LT. IEXT(NZ)) KNZ=IEXT(NZ) 0141 NUT1=NUT

0142 NUT=-NU

0143 L=0 0144 KUP=K1 0145 COMP=YNZ*(1.00001) 0147 LUCK=1 0148 310 L=L+1 0149 IF(L .GE. KUP) GO TO 315 0150 ERR=GEE(L,NZ) 0151 ERR=(ERR-DES(L))*WT(L) 0152 DTEMP=NUT*ERR-COMP

0153 IF(DTEMP .LE. 0.0) GO TO 310 0154 COMP=NUT*ERR

0155 J=NZZ

0157 315 LUCK=6 SU$STITUTE SKEET

_. 5 3 0159 320 IF(LUCK .GT. 9) GO TO 350 0160 IF(COMP .GT. Y1) Y1=COMP

0161 K1=IEXT(NZZ) 0162 325 IrNGRID+1 0163 KLOW=KNZ

0164 NUT=-NUT1 0165 COMP=Y1*(1.00001) 0166 330 L=L-1 0167 IF(L .LE. KLOW) GO TO 340 0168 ERR=GEE(L,NZ) 0169 ERR=(ERR-DES(L))*WT(L) 0170 DTEMP=NUT*ERR-COMP

0171 IF(DTEMP .LE. 0.0) GO TO 330 0172 J=NZZ

0173 COMP=NUT*ERR

0174 LUCK=LUCK+10 0176 340 IF(LUCK .EQ. 6) GO TO 370 0177 DO 345 J=1,NFCNS

0178 345 IEXT(NZZ-J)=IEXT(NZ-J) 0179 IEXT(1)=K1 0181 350 KN=IEXT(NZZ) 0182 DO 360 J-1,NFCNS

0183 360 IEXT(J)=IEXT(J+1) 0184 IEXT(NZ)=KN

0186 370 IF(JCHNGE .GT. 0) GO TO 100 APPROXIATION

0192 NM1=NFCNS-1 0193 FSH=1.OE-06 0194 GTEMP=GRID(1) 0195 X(NZZ)=-2.0 0196 CN=2*NFCNS-1 0197 DELF=1.0/CN

0198 Irl 0199 KKK=O

0200 IF(EDGE(1) .EQ. O.O .AND. EDGE(2*NBANDS) .EQ. 0.5) KKK=1 0201 IF(NFCNS .LE. 3) KKK=1 0202 IF(KKK .EQ.1) GO TO 405 0203 DTEMP=DCOS(PI2*GRIDS(1)) 0204 DNUM=DCOS(PI2*GRID(NGRID)) 0205 AA=2.0/DTEMP-DNUM) 0206 BB=-(DTEMP+DNUM)/(DTEMP-DNUM) 0208 DO 430 J=1,NFCNS

0209 FT=(J-1)*DELF

0210 XT=DCOS(PI2*FT) 0211 IF(KKK .EQ. 1) GO TO 410 0212 XT=(XT-BB)/AA

SUBSTITUTE SNcE'~

4~;~~;~,~ ~~~ 54 0213 FT=DACOS(XT)/PI2 0214 410 XE=X(L) 0215 TF(XT .GT. XE) GO TO 420 0216 IF((XE-XT) .T. FSH) GO TO 415 0217 L=L+1 0219 415 A(J)=Y(L) 0221 420 IF((XT-XE) .LT FSH) GO TO 415 0222 GRID(1)=FT

0223 A(J)=GEE(1,NZ) 0225 IF(L .GT. 1) L=L-1 0227 GRID(1)= GTEMP

0228 DDEN=PI2/CN

0229 DO 510 J=1,NFCNS

0230 DTEMP=0.0 0231 DNUM=(J-1)*DDEN

0232 IF(NM1 .LT. 1) GO TO 505 0233 DO 500 K=1,NM1 0234 500 DTEMP=DTEMP+A(K+1)*DC05(DNUM*K) 0235 505 DTEMP=2.0*DTEMP+A(1) 0236 510 ALPHA(J)=DTEMP

0237 DO 550 J+2,NFCNS

0238 550 ALPHA(J)=*ALPHA(J)/CN

0239 ALPHA(1)=ALPHA(1)/CN

0240 IF(KKK .EQ. 1) GO TO 545 0241 P(1)=2.0*ALPHA(NFCNS)*BB+ALPHA(NM1) 0242 P(2)=2.0*AA*ALPHA(NFCNS) 0243 Q(1)=ALPHA(NFCNS-2)-ALPHA(NFCNS) 0244 DO 540 J=2,NM1 0245 IF(J .LT. NM1) GO TO 515 0246 AA=0.5*AA

0247 BB=0.5*BB

0249 P(J+1)=0.0 0250 DO 520 K=1,J

0251 A (K) =P (K) 0252 520 P(K)=2.0*BB*A(K) 0253 P(2)=P(2)+A(1)*2.0*AA

0254 JM1=J-1 0255 DO 525 K=1,JM1 0256 525 P(K)=P(K)+Q(K)+AA*A(K+1) 0257 JP1=J+1 0258 DO 530 K=3,JP1 0259 530 P(K)=P(K)+AA*A(K-1) 0260 IF(J .EQ. NM1) GO TO 540 0261 DO 535 K=1,J

0262 535 Q(K)=-A(K) 0263 Q(1)=Q(1)+ALPHA(NFCNS-1-J) 0265 DO 543 J=1,NFCNS

0266 543 ALPHA(J)=P(J) 0268 IF(NFCNS .GT. 3) RETURN

SUBST1TUT~ SHEET.

55 zU8~7~~
02 69 ALPIiA ( NFCNS+1 ) =0 . 0 0270 ALPHA(NFCNS+2)=0.0 SUBSTITUTE SHEET

0003 REAL*8 FUNCTION D(K,N,M) 0008 INPLICIT REAL*8(A-H,O-Z) 0009 COMMON PI2,AD,DEV,X,Y,GRID,DES,WT,ALPHA, IEXT,NFCNS,NGRID

0010 DIMENSION IEXT(512),AD(512),ALPHA(512),X(512),Y(512) 0011 DIMENSION DES(8196),GRID(8196),WT(8196) 0013 D-1.0 0014 Q=X(K) 0015 DO 3 L=1,M

0016 DO 2 J=L,N,M

0017 IF (J-K) 1,2,1 0018 1 D=2.0*D*(Q-X(J)) 0021 D=1.0/D

0002 REAL*8 FUNCTION GEE(K,N) THE

FORM

0007 IMPLICIT REAL*8(A-H,O-Z) 0008 COMMON PI2,AD,DEV,X,Y,GRID,DES,WT,ALPHA,IEXT, NFCNS,NGRID

0009 DIMENSION IEXT(512),AD(512),ALPHA(512),X(512),Y(512) 0010 DIMENSION DES(8196),GRID(8196),WT(8196) 0012 P=0.0 0013 XF=GRID(K) 0014 XF=DCOS(PI2*XF) 0015 D=0.0 0016 DO 1 J=1,N

0017 C=XF-X(J) 0018 C=AD(J)/C

0019 D=D+C

0020 1 P=P+C*Y(J) 0021 GEE=P/D

0005 1 FORMAT(' ************ FAILURE TO CONVERGE

**********~/

0006 1'OPROBABLE CAUSE IS MACHINE ROUNDING ERROR'/

0007 2'OTHE IMPULSE RESPONSE MAY BE CORRECT'/

0008 3'OCHECK WITH A FREQUENT RESPONSE') ~.~'aTf'",~'("f'r'E ed-t~~-~- z~~~~s~

0002 SUBROUTINE PULS(NPOLY) ALGORITHM AND USE IT

0007 IMPLICIT REAL*8(A-H,O-Z) 0008 COMPLEX*16 COEFF(256) 0009 REAL*8 POLY(256),POLY1(256),POLY2(256) 0010 REAL*8 CEP1(256),CEP2(256),CEP3(256),CEP4(256) 0011 REAL*8 PULS1(256),PULS2(256),PULS3(256),PULS4(256) 0012 REAL*8 PULS5(256),PULS6(256),PULS7(256),PULS8(256) 0013 REAL*8 XINT1(256),PULSE(256) 0014 OPEN(UNIT=21, NAME='POLY.DAT' TYPE='OLD') 0015 , READ (21,*) NFILT

0016 DO J=1,NFILT

0017 READ(21,*)POLY(J) 0018 POLY(J)=POLY(J)*.999999 0019 POLY1(J)=POLY(J) 0020 POLY2(J)=-POLY(J) 0021 CEP1(J)=0.0 0022 CEP2(J)=0.0 0023 CEP3(J)=0.0 0024 CEP4(J)=0.0 0025 PULS1(J)=0.0 0026 PULS2(J)=0.0 0027 PULS3(J)=0.0 0028 PULS4(J)=0.0 0029 PULSS(J)=0.0 0030 PULS6(J)=0.0 0031 PULS7(J)=0.0 0032 PULS8(J)=0.0 0034 CLOSE(UNIT=21) 0035 NFCNS=NFILT/2+1 0036 POLY1(NFCNS)=POLY1(NFCNS)+1 0037 POLY2(NFCNS)=POLY2(NFCNS)+1 0039 DO J=1,NFILT

0040 COEFF(J)=POLY1(J)/2 0042 CALL CEPSTRUM(COEFF,CEP1,CEP2,NFILT) 0043 do j=l,nfcns 0044 cep2(j)=-cepl(j) 0045 enddo 0046 DO J=1,NFILT

0047 COEFF(J)=POLY2(J)/2 0049 CALL CEPSTRUM(COEFF,CEP3,CEP4,NFILT) 0050 do j=l,nfcns 0051 cep4(j)=-cep3(j) 0052 enddo 0054 CALL PULSR1(CEP1,CEP3,NFCNS,PULS1,DIFMAX1) 0055 DIFMAX=DIFMAX1 ~L~~~~'~ t"j -~ St"~~C~' 0056 DO J=l,NFCNS

0057 PULSE(J)=PULSE1(J) 0059 CALL PULSRl(CEP2,CEP3,NFCNS,PULS2,DIFMAX2) 0061 IF(DIFMAX2 .1T. DIFMAX) THEN

0062 DIFMAX=DIFMAX2 0063 DO J=l,NFCNS

0064 PULSE(J)=PULS2(J) 0068 DO J=1,NFCNS

0069 CEP1(J)=-CEP1(J) 0070 CEP2(J)=-CEP2(J) 0072 CALL PULSR1(CEP1,CEP3,NFCNS,PULS3,DIFMAX3) 0074 IF(DIFMAX3 .1T. DIFMAX) THEN

0075 DIFMAX=DIFMAX3 0076 DO J=1,NFCNS

0077 PULSE(J)=PULS3(J) 0081 CALL PULSR1(CEP2,CEP4,NFCNS,PULS4,DIFMAX4) 0083 IF(DIFMAX4 .LT. DIFMAX) THEN

0084 DIFMAX=DIFMAX4 0085 DO J=1,NFCNS

0086 PULSE (J) =PULS4 (J) 0090 OPEN(UNIT=21, TYPE='NEW' NAME='PULSE. OUT') 0091 , WRITE(21,*) NFCNS

0092 DO J=1,NFCNS

0093 WRITE(21,*) PULSE(J) 0095 CLOSE(UNIT=21) S-~iT'3TE SHEET.
S~

_ 60 0002 SUBROUTINE CEPSTRUM(COEFF,CEPMAX,CEPMIN,N) 0003 IMPLICIT COMPLEX*16(A-H,O-Z) 0004 COMPLEX*16 COEFF(1),CINT(4096),CINTl(4096) 0005 REAL*8 XINT1,XINT2,CEPMAX(N),CEPMIN(N) 0006 M=N/2+1 0007 DO J=1,4096 0008 CINT(J)=O ~ INITIALIZE

0010 DO J=1,N

0011 CINT(J)=COEFF(J) 0013 CALL FFT2C(CINT) 0014 DO J=1,4096 0015 XINT1=CDABS(CINT(J)) 0016 IF(XINT1 .LT. l.D-20) XINT1=1.D-20 0017 CINT(J)=DCMPLX(DLOG(XINT1)/2.) ~ computes logofsqrt 0019 c now do fft 0020 CALL FFT(CINT) 0022 do j=1,2047 0023 CINT1(J+1)=DCONJG(CINT(J+1)/2048) 0024 cint(j+1)=0 0025 CINT1(4097-J)=O

0026 cint(4097-j)=dconjg(cint(4097-j)/2048 0027 enddo 0028 cint(2049)=0 0029 cint(1)=dconjg(cint(1)/4096) 0030 cintl(2049)=0 0031 cintl(1)=dconjg(cint(1)/4096) 0033 c now have cepstrum; need to compute minimal phase 0034 call fft(cint) 0035 do j=1,4096 0036 cint(j)=dconjg(cdexp(cint(j))) 0037 enddo 0038 call fft(cint) 0039 cepMIN(1)=dREAL(cint(1)/4096) 0040 DO J=2,M

0041 CEPMIN(J)=DREAL(CINT(4098-j))/4096 0043 xintl=dsqrt(dabs(dreal(coeff(1))/(cepmin(1)*cepmin(m )))) 0044 do j=l,m 0045 cepmin(j)=cepmin(j)*xintl 0046 enddo 0048 call fft(cintl) 0049 do j=1,4096 0050 cintl(j)=dconjg(cdexp(cintl(j))) 0051 enddo 0052 call fft(cinti) 0053 DO J=1,M

SUBSTITUTE SHEET

"'~O 92/03089 PCT/US91 /05869 0054 CEPMAX(J)=DREAL(CINT1(j))/4096 0056 xintl=dsqrt(dabs(dreal(coeff(1))/(cepmax(1)*cepmax(m )))) 0057 do j=l,m 0058 cepmax(j)=cepmax(j)*xinti 0059 enddo ~!J~ T~TJ'~~ ~f~~~T

'~~~~ ~~~ '62~

0003 SUBROUTINE PULSR1(POLl,POL2,N,PULSES,DIFMAX) MAGNETIZATION AND OUTPUT

0007 c spinor is (poll,pol2) _ 0008 c poll is al + a2s +a2s 2+... as is pol2 0009 IMPLICIT REAL*8(A-H,O-Z) 0010 REAL*8 POL1(1),POL2(1),PULSES(1) 0011 REAL*8 XINTl(200),XINT2(200),XINT3(200),XINT4(200) 0012 C FIRST INITIALIZE XINT1,XINT2 0013 DO J=1,N

0014 XINT1(J)=POL1(J) 0015 XINT2(J)=POL2(J) 0017 DIFMAX=0 0018 DIFO=0 0019 DIFN=0 0021 NFl=(N-1)/2 0022 DO J=1,NF1 0023 J1=N+1-2*J

0024 FLIP=DATANZD(XINT2(1),XINTl(1)) 0025 IF((FLIP) .GT. 90) FLIP=FLIP-180 0026 IF((FLIP) .LT. -90) FLIP=FLIP+180 0027 Xl=DCOSD(FLIP) 0028 Y1=DSIND(FLIP) 0029 DIF1=2*FLIP

0030 PULSES(N+1-J)=DIF1 0031 DO K=1, J1 0032 XINT1(K)=X1*XINT1(K)+yl*XINT2(K) 0033 XINT2(K)=-Y1*XINT1(K+1)+X1*XINT2(K+1) 0035 DIF=ABS(DIFl-DIFO) 0036 IF(DIF .GT. DIFMAX) DIFMAX=DIF

0037 DIFO=DIFl 0039 FLIP=DATAN2D(XINT1(J1),-XINT2(J1)) 0040 IF((FLIP) .FT. 90) FLIP=FLIP-180 0041 IF(FLIP .LT. -90 FLIP=FLIP+180 0042 X1=DCOSD(FLIP) 0043 Y1=DSIND(FLIP) 0044 DIFl=2*FLIP

0045 PULSES(J)=DIF1 0046 DO K=l,Jl-1 0047 XINT2(K)=X1*XINT1(K)-Y1*XINT2(J1+1-K) 0048 XINT4(K)=-Y1*XINT1(Jl=1-K)+X1*XINT2(K) 0050 DO K=l,Jl-1 0051 XINT1(K)=XINT3(K) 0052 XINT2(K)=XINT4(K) 0054 DIF=ABS(DIF1-DIFON) 0055 IF(DIF .GT. DIFMAX) DIFMAX=DIF

SUBSTITUTE SHEET

WO 92/03089 ~ PCT/US91/05869 0056 DIFN=DIF1 0059 FLIP=DATAN2D(XINT2(1),XINT1(1)) 0060 IF(FLIP .GT. 90) FLIP=FLIP-180 0061 IF(FLIP .LT. -90) FLIP=FLIP+180 0062 DIF1-2*FLIP

0063 PULSES(1)*-DIF1 0064 DIF-ABS(DIF1-DIFO) 0065 IF(DIF .GT. DIFMAX) DIFMAX=DIF

0066 DIF=ABS(DIFl-DIFN) 0067 IF(DIF .GT. DIFMAX) DIFMAX=DIF

0068 PRINT *,DIFMAX

0069 RE~gN

SUBSTITUTE SHEET

Claims (20)

WE CLAIM:

1. A method of nonintrusively determining displacements of an image slice of a body portion of a patient during a given time interval using a magnetic resonance imaging device, comprising the steps of:
inputting parameters specifying the desired visual characteristics of said image slice;
synthesizing a radio frequency pulse sequence which, when applied to said body portion prior to imaging by said magnetic resonance imaging device, will cause the creation of an image slice having said desired visual characteristics;
applying to the body portion an external magnetic field so as to produce a resultant magnetization;
applying a pre-imaging sequence comprising said synthesized radio frequency pulse sequence and magnetic field gradient pulses to the body portion so as to divide said body portion into a grid of intersecting stripes having altered magnetization, said stripes intersecting at a plurality of intersecting points;
applying an imaging sequence to the body portion to make visible said grid of intersecting stripes;
quantitatively measuring physical displacements of predetermined ones of said plurality of intersecting points during said given time interval: and providing an image representing said displacements.

2. A method of detecting motion of an image slice of a body portion of a patient using a magnetic resonance imaging device, comprising the steps of:
inputting parameters specifying the desired visual characteristics of said image slice;
synthesizing a radio frequency pulse sequence which, when applied to said body portion prior to imaging by said magnetic resonance imaging device, will cause the creation of an image slice having said desired visual characteristics;
applying to the body portion an external magnetic field so as to produce a resultant magnetization;

applying a pre-imaging sequence to the body portion so as to spatially modulate transverse and longitudinal components of the resultant magnetization to thereby produce a spatial modulation pattern of intensity, said pre-imaging sequence comprising said synthesized radio frequency pulse sequence and magnetic field gradient pulses;
applying an imaging sequence to the body portion to make visible said spatial modulation pattern of intensity; and detecting motion of said body portion during a time interval between application of said pre-imaging and imaging sequences by examining displacements of the spatial modulation pattern of interest occurring during said time interval.

3. The method as in claim 2, wherein said parameters include stripe thickness in said spatial modulation pattern of intensity.

4. The method as in claim 2, wherein said parameters include weights for relating sharpness and flatness of respective stripes in said spatial modulation pattern of intensity.

5. The method of claim 2, wherein said parameters include a number of radio frequency pulses to be included in said radio frequency pulse sequence.

6. The method of claim 2, wherein said parameters include a size of a transition zone between a striped and a non-striped region in said spatial modulation pattern of intensity.

7. The method of claim 2, wherein said pre-imaging sequence comprises said synthesized radio frequency pulse sequence with said magnetic field gradient pulses interspersed between respective radio frequency pulses of said synthesized radio frequency pulse sequence.

8. The method of claim 2, wherein said pre-imaging sequence comprises said synthesized radio frequency pulse sequence and overlapping magnetic field gradient pulses.

9. The method of claim 2, wherein said synthesized radio frequency pulse sequence is applied to said body portion so as to cause a selective periodic excitation of said resultant magnetization and said synthesized radio frequency pulse sequence comprises respective radio frequency pulses having amplitudes which are not related in accordance with a binomial amplitude function.

10. The method of claim 2, wherein said radio frequency pulse sequence is synthesized in accordance with the following steps:
calculating a Fourier cosine series in M Z which optimally fits said parameters, where M Z is a magnetization response of said body portion to said radio frequency pulse sequence;
converting said Fourier cosine series into a complex exponential Fourier series;
generating a Fourier series for spinor components of said complex exponential Fourier series of the form:
(1+M Z)/2=P 2 and (1-M Z)/2=Q 2;
deconvoluting said Fourier series for spinor components to find P and Q; and calculating a radio frequency pulse sequence which yields a spinor with P and Q as its spinor components.

11. A method of nonintrusively determining displacements of a body portion of a patient during a given time interval, comprising the steps of:
applying to the body portion an external magnetic field so as to produce a resultant magnetization;
applying a pre-imaging sequence to the body portion so as to divide said body portion into a grid of intersecting stripes having altered magnetization, said stripes intersecting at a plurality of intersecting points;
applying an imaging sequence to the body portion;
quantitatively measuring physical displacements of predetermined ones of said plurality of intersecting points during said given time interval; and providing an image representing said displacements.

12. The method of claim 11, comprising the further step of specifying a triangular grid of said body portion wherein said predetermined ones of said plurality of intersecting points are respective vertices of respective triangles of said triangular grid.

13. The method of claim 12, wherein said displacements measuring step comprises the step of determining which of said intersecting points in said triangular grid at the beginning of said given time interval correspond to said predetermined ones of said plurality of intersecting points in a deformed triangular grid at the end of said given time interval.

14. The method of claim 13, wherein said displacements measuring step comprises the step of comparing said triangular grid at the beginning of said given time interval with said deformed triangular grid at the end of said given time interval.

15. The method of claim 14, wherein said image representing said displacements comprises a comparison of an image representing said triangular grid at the beginning of said given time interval with an image representing said deformed triangular grid at the end of said given time interval.

16. The method of claim 11, wherein said body portion is the patient's heart wall.

17. An apparatus for noninvasively determining displacements of a body portion of a patient during a given time interval, comprising:
means for applying to the body portion an external magnetic field so as to produce a resultant magnetization;
imaging means for applying a pre-imaging sequence to the body portion so as to divide said body portion into a grid of intersecting stripes having altered magnetization, said stripes intersecting at a plurality of intersecting points, and for applying an imaging sequence to the body portion;
means for quantitatively measuring physical displacements of predetermined ones of said plurality of intersecting points during said given time interval; and a display for displaying an image representing said displacements.

18. The apparatus of claim 17, further comprising means for specifying a triangular grid of said body portion wherein said predetermined ones of said plurality of intersecting points are respective vertices of respective triangles of said triangular grid.

19. The apparatus of claim 18, wherein said displacements measuring means comprises means for determining which of said intersecting points in said triangular grid at the beginning of said given time interval correspond to said predetermined ones of said plurality of intersecting points in a deformed triangular grid at the end of said given time interval.

20. The apparatus of claim 19, wherein said displacements measuring means comprises means for comparing said triangular grid at the beginning of said given time interval with said deformed triangular grid at the end of said given time interval.