Magnetic Resonance Printed in the USA.

Imaging, All rights

Vol.10,

pp. 623-626,

1992 Copyright

reserved.

0

0730-725X/92 $5.00 + .OO 1992 Pergamon PressLtd.

0 Original Contribution

A FAST Tl ALGORITHM JIAN GONG AND JOSEPH P. HORNAK Center for Imaging Science, Rochester Institute of Technology, Rochester, NY 14623USA Multispectral tissue classification using magnetic resonance I’, , T2, and p images may be useful in diagnosing and locating certain pathology. Techniques for generating the Tl images necessary for this classification scheme often require longer data collection and post processing times than are practical. As a consequence, further development of this classification scheme may be limited. This paper addresses an improvement in the post processing time required to generate Tl images. A nonlinear least-squares algorithm is described for rapidly generating spinlattice relaxation time images from variable repetition time magnetic resonance images. The algorithm generates a 256 x 256 pixel Tl image from nine variable repetition time images in approximately 60 set on a VAX-6510 computer. Keywords:

MRI; Spin-lattice relaxation times; Image processing; Postprocessing

The magnetic resonance (MR) signal in an imaging procedure is a complex function of intrinsic properties of the imaged tissue, such as the spin relaxation times of and spin exchange constants between each nuclear spin, and extrinsic properties or imaging parameters. The intrinsic portion of this complex relationship may for many applications be approximated by a single apparent relaxation time. The apparent spin-spin (7”) and spin-lattice (T, ) relaxation times are often informative in diagnosing pathology. For example, multispectral tissue classification, the classification of tissues based on their properties in multiple spectral regions, using T, and T2 images, has been proven useful by others in locating and diagnosing certain pathologies. ‘s2 T2 images may be rapidly obtained from variable echo time (TE) MR images using a linear least-squares algorithm. Unfortunately, both the acquisition and postprocessing of data to generate T, images is not as rapid. Tl data acquisition may take upwards of 1 hr using a saturation recovery sequence, although shorter sequences are available.3 Calculation of Tl images from magnetic resonance images is a computationally intensive procedure, requiring for a 256 x 256 pixel image that 65,636 Tl values be determined using a nonlinear least-squares algorithm. The postprocessing of MR images to generate at T, image may take several hours depending on the specific T, algorithm utilized, the RECEIVED

Address

12/2/91;

number of images used to calculate T, , and the computer hardware. Rapid non-least-squares techniques have been described in the literature,4*5 which generally sacrifice accuracy for speed. This communication describes a rapid nonlinear least-squares technique for calculating the apparent T, images in less than 1 min from saturation recovery type images. The fast Tl algorithm is derived in detail both to give the reader a clear understanding of the procedure and to stimulate development of similar algorithms for other routinely used pulse sequences. Under ideal conditions, that is, the absence of spin exchange, primarily one component filling an image voxel, and perfect radiofrequency Bi pulses, the magnetic resonance signal from a saturation recovery type imaging experiment may be approximated as S(TRi) = k(1 - e-TR”r’)

to Joseph P. Hornak,

(1)

where TR is the repetition time of the experiment and k is a proportionality constant dependent on such factors as the spin density and amplifier gains. In the case of the commonly used spin-echo sequence, k is also a function of the spin-spin relaxation time and TE. A T, value is typically calculated by fitting Eq. (1) to imaging data obtained at several values of TR. The imaging data is the actual MR signal Si at TRi. A leastImaging Science, Rochester Institute of Technology, Roch-

ACCEPTED 2/21/92.

correspondence

algorithms; Tl relaxation times.

ester, NY 14623.

Center for 623

624

Magnetic Resonance Imaging 0 Volume 10, Number 4, 1992

squares technique seeks to minimize n data points, where

~(T,,/c)

for the Z(T,)

IA

T,

\\

$(7’t,k)

= i

[S(TRi)

,

(2)

=0 .

(3)

- S;]’

i

Therefore

&)(T,,k)

=

2

6k +

$

AT, 1

6k nor AT,may be zero then

Since neither

3~0

2~0, aT

and

ak

(4)

Fig. 1. Plots of the family of functions Z( T1) defined by Eq. (9) where TR = 250, 500, 1000, 1500,2000,3000,4000, and 6000 msec.

From Eq. (4) one has

[$

[k(l

_

e-WT~)

_

$1

(1 -

e-TRl’Tl)

= 0

(5) x

[sit1

_

e-WTl)

-

~~(1

-

e-TRJTl)]

(9)

and

,g

.

[k(l - e--TR,/TI ) k explicitly

Expanding

&y(l

s;] TR~~-TR,/TI

_

=

0

.

(6)

yields

$

_e-TW’~)

$TR;

e

The function Z( T, ), shown in Fig. 1, goes to zero in the limit as Tl = 0 and 03. The single root of Eq. (9) is the T, value sought. The fast T, algorithm determines the root T,by a controlled iterative process depicted in Fig. 2. Two T, values, T,,and T,b,are used to seed the algorithm.

-TRi/Tl

i=l

i=l

,$

_e--TWT~)2

(1

=

_e-WT~).

&Rie-Wi(l i=l

Z(T,) (7)

It should be noted that Eq. (7) contains one unknown, T, . In order to find the root solution T,of Eq. (7), it should be rearranged by cross multiplying the numerators and denominators. After reordering the summations one obtains

k 2 i=l

si(

1 _

e-T&/T1

)TRje-TRj’Tl

(1

_

b

e-TR,‘Tl)

j=1

=

k 2 i=l

(1

_

Let the difference Z(T,).

e-TR,/T~)2SjTRj

e-TRj/T~

.

(8)

j=l

of the two sides be a function

Fig. 2. A plot of the function Z( Tr ) as defined by Eq. (9) depicting the controlled iterative process for finding Tt . T,, and Tr b are starting points for the fast T, algorithm. Tl c is the zero crossing of a line joining T,, and Tlb. See text for details.

A fast T, algorithm 0 J. GONG AND J.P. HORNAK

T,, is chosen to be less than the minimum T, , and T2,, greater than the maximum T,in the image. A straight line is drawn between Z(T,,)and Z(T,,). If the Z value at the zero-crossing, T,,, of this line, Z(T,,), is less than an arbitrarily chosen minimum E, then clc is the sought after root, otherwise T,,becomes T,,(or T,,)and the process is repeated. This process is repeated for each pixel in the image space to yield a T, image. Background noise located in those regions of the Tlimage devoid of signal may be eliminated by only calculating T,for those pixels where the average pixel intensity in the n images is greater than a noise level. The fast T, algorithm is inherently quicker than the general Levenberg-Marquardt6 nonlinear leastsquares method at finding the T, of data based on Eq. (1) because the fast T,algorithm lacks the computational overhead necessary to make the Marquardt method general. The accuracy of the fast T, algorithm needs to be tested. A Fortran version of the algorithm executed on a Digital Equipment Corporation (Maynard, MA) VAX-65 10 computer was used to test the accuracy of the algorithm at finding the Tl of data based on Eq. (1). Simulated ideal saturation re-

625

Table 1. T, Calculated from simulated saturation recovery curves Range of random noise added to signal T, (msec) 250 lg 1500 2000 3000

0

10

249(O) 249(2) 500(O) 1000(O) 1500(O) 2003(O) 3004(O)

500(3) lOOl(6) 1501(g) 2002(14) 3008(33)

30

loo

250(5)

250(17)

499(9) lOOO(18) 1504(28) 2005(50) 3011(106)

500(29) 1002(57) lSlO(98) 2005(144) 3060(379)

300 253(53) 502(85) 1016(169) 1569(327) 2060(454) 3140(1218)

T, values and (standard deviations) are the result of 256 simulations. TR = 0.25, 0.5, 0.75, 1, 1.5, 2, 3, 4, 6 set and k = 1000.

covery curves based on Eq. (1) with k = 1000, TR = 6000, 4000, 3000, 2000, 1500, 1000, 750, 500, 250 msec, and T, between 250 msec and 3 set were fit using the algorithm with E = 10P3. Varying amounts of random noise were added to the simulated curves before fitting. The results of these fits are tabulated in Table 1. The accuracy with which the algorithm calcuare lates the root T,and the speed of the computation

Fig. 3. A 256 x 256 pixel T,image of the human brain with three adjacent phantoms produced by the fast T,algorithm in 60 set from nine variable TR spin-echo images. The horizontal line across the bottom of the image is a flow artifact and the three black pixels in the brain are pixels set to zero by the noise level criterion described in the text.

1992

626

determined by n and the magnitude of E. An Evalue of lo-’ yields the root T1 to within a few milliseconds on noise free data. The ability of the algorithm to find T, from data with noise is governed by the signal-to-noise ratio (S/N) and the TR values.‘~’ The longest TR should be approximately three times the longest T,, while the shortest TR may be approximately equal to the shortest T,. Increasing the value of n improves the accuracy of the measured T, but increases the computation time. This algorithm was used to construct the representative T, image of the human head with three adjacent phantoms shown in Fig. 3. This image was constructed from nine 256 x 256 pixel spin-echo-images recorded on a General Electric (Milwaukee, WI) Signa imager using a quadrature bird cage head coil and operating at 1.5 T. The nine spin-echo images were recorded with one excitation, 192 phase-encoding steps, and had a TR/TE = 6000, 4000, 3000, 2000, 1500, 1000, 750, 500,250/15 msec. The 6-set TR image had S/N of 200 for cerebrospinal fluid. The image took approximately 60 s to calculate using an E = 10p3, T,, = 100 msec, and TIb = 8 sec. (For comparison purposes, the VAX6510 is 28 times the performance of a VAX 1l/780.) Proportionately shorter calculation times were required with fewer than nine variable TR images. The average phantom T, values were within 1% of the similar quantity obtained on the Signa imager using the two parameter TI calculation option for a region of interest. The fast T, algorithm rapidly and, with the appropriate data set, accurately calculates an apparent T,

image from n variable TR images. Further reductions in computation time will be possible with lower level computer languages. The remaining obstacle to routinely generating T, images, which this paper does not address, is the long imaging time. This time will only be reduced by changing the pulse sequences.

REFERENCES 1.

2. 3.

4. 5.

6.

7.

8.

Hornak, J.P.; Blaakman, A.; Rubens, D.; Totterman, S. Multispectral image segmentation of breast pathology. SPIE Image Processing 1445:523-533; 1991. Higer, H.P.; Bielke, G. Tissue Characterization in MR Imaging. New York: Springer-Verlag: 1990. Crawley, A.P.; Henkelman, R.M. A Comparison of OneShot and Recovery Methods in Tr Imaging. Magn. Reson. Med. 7~23-34; 1988. Sperber, G.O.; Ericsson, A.; Hemmingson, A. A fast method for rr fitting. Magn. Reson. Med. 9:113; 1989. Jesmanowicz, A.; Wong, E.C. ; Wu, J.C. ; Hyde, J.S. Fast algorithm for computation of Tr , T,, and diffusion coefficient images. San Francisco: Tenth Annual Meeting of the Society of Magnetic Resonance in Medicine; August 1991: Abstract #740. Press, W.H.; Flannery, B.P.; Teukolskv, S.A.; Vetterling, W.T. Numerical Recipes, The Art of Scientific Computing. Cambridge, UK: Cambridge University Press: 1988. Guggenhem, E.A. The determination of the velocity constant for a unimolecular reaction. Phil. Msg. [7]2:538543; 1926. Wolberg, J.R. Prediction Analysis. Princeton, NJ: D. Van Nostrand Co.: 1967.

A fast T1 algorithm.

Multispectral tissue classification using magnetic resonance T1, T2, and rho images may be useful in diagnosing and locating certain pathology. Techni...
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