The Elliptical Nature of the Cone of Uncertainty in DTI Tractography | Lecture notes Project Management

The Cone of Uncertainty is Elliptical: Implications for DTI Tractography

C. Koay1, L-C. Chang1, and P. J. Basser1

1NICHD, NIH, Bethesda, MD, United States

INTRODUCTION

The diffusi on tensor i s always m easured i n the prese nce of bac kground noi se. Withi n the fram ework of recently prop osed met hods of er ror propag ation

and of 1st-order matrix perturbation for DTI [1,2], one can easily characterize the noise induced variability of the estimated tensor itself and of tensor -

derived qu antities, sc alar or vec tor. Of par ticular inter est in tract ography i s the uncert ainty of the eigenvect ors or princi pal directions. With this end in

view, the c one of unc ertainty w as propos ed as a lo cal measur e of tract dispersi on by Bass er [3] and l ater used by Basser & Pajevic [4] and Jones [5] as

a tool for visualizing and quantifying uncertainty along the fiber tract. The cone of uncertainty is not the only measure for quantifying tr act disp ersion,

there is another approach that based on the perturbation method, which was proposed by Anderson [6] and used by Lazar [7,8]. None of these studi es

provided th e key geom etric object relating t o the local measure of tract disper sion, whi ch is the cov ariance mat rix of the m ajor eige nvector of th e

diffusion t ensor. Her e we present t hree diff erent method s based on t he error pr opagation f ramework [1], the 1st-order matrix perturbation method [2], and

the dyadic formalism [3,9], to show that the local variability in the purported fiber direction is, in general, not described by a right circ ular cone of

uncertainty but a right elliptical cone. Consequences of this finding are discussed.

METHODS

Error Propagation Method Based on t he framew ork presen ted in [1] , propaga tion of err ors amou nts to findi ng an app ropriate “ coordinat e” transf ormation

between cov ariance s tructures of inter est. In DTI, the diff usion tensor can be rep arametri zed by th e eigenval ues and th e Euler angles in order to

construct th e covarianc e matrix of the maj or eigenvec tor of the diffus ion tensor. T he eigenvec tors of this cov ariance matr ix whose eig envalues are

nonzero ar e then th e major an d minor ax es of the ellipse of the cone of uncert ainty.

1st-order Matrix Perturbation Method The pioneering work of Hext [ 10] provi ded a metho d to constr uct the cov ariance str ucture of the major ei genvecto r

based on the 1st-or der matrix p erturbati on, see Eq [4.17] of [1 0], but the i mportanc e of this for mula has not been poi nted out in p revious D TI studies.

Here, we suggest Hext’s approach as an alternative to the other two methods proposed here for constructing the elliptical cone of uncert ainty.

Dyadic Product Formalism This formalis m was propos ed by Bass er et al.[3] t o overcom e the probl em of bias i n computing sam ple mea n of the

eigenvect or. Here, we s how how th e cone of unc ertainty ca n be constru cted from the average d yadics. Let },,{ 111 N

be a sample of major

eigenvect ors and let

∑

Tqqqq 111

11 be th e average dyadic [3] . Further, l et the eig envalue dec ompositio n of the aver age dyadic s be

∑

=ψψλ= 3

11 i

iii

qq where 321 λ≥λ≥λ . Then, the c ovariance matrix of the mean major ei genvector can be shown to be

(

)

333222

1ψψλ+ψψλ

−.

RESULTS AND DISCUSSION

The analysis presented above shows that the cone of uncertainty has an elliptical not circular cross section. Particularly, the 1st-order matrix per turbation

method shows that the cone of uncertainty is elliptical when the two smaller eigenvalues of the diffusion tensor are not equal to each other. Onl y when

they are equal will the cone of uncertainty have the shape of a right circular cone. Similar observations can be made based on the other tw o methods

discussed h ere. Figur e 1B shows t he cones of u ncertaint y constructe d from a si mulated hum an brain da ta based on t he error pr opagation method [1]

and the nonl inear least squares m ethod [11]; F igure 1A i s the FA map . The imp lications for DTI tract ography ar e clear. Sinc e the medi um and mi nor

eigenvalu es of most tens or estimat es in the brai n are general ly differ ent [12], the ax ial symm etry of the trac t dispersi on is usuall y broken. In s treamlin e

based or probabilistic methods [13,14], the chance of deviating away from a tract in the direction along which uncertainty is low is less lik ely than from

deviating away from the tract i n the dir ection al ong whic h uncertai nty is hig h. Thus, w e would pr edict an asymmetri c spread of tracts tr ajectori es normal

to the mean trajectory in which paths tending to meander along the direction with lower variance will remain closer to the mean tract whereas tracts

meandering along the direction with higher variance will exhibit more dispersion and lie farther from the mean. Generalizing the approach given in [1 5],

we can view the tractog raphy pro cess as a bi ased rand om walk i n which ther e is anisotr opic diff usion of th e tract traj ectory normal to the mean tract. In

fact, such behavior appears to have been observed in bootstrap simulations of tracts generated using a streamline following method. In Lazar et al.,

white mat ter “axial asymmetr y” (i.e., i nequalit y of 2

λ and 3

λ) caused anisotropi c dispersi on patter ns in the estimate d tract tr ajectories [8] but a

theoretical explanat ion was n ot provided for thi s finding .

Figure 1A Figure 1B

REFERENCES [1] Koay CG et al. Pr oc. ISMRM. 14 ( 2006). [2] Chang LC e t al. Proc. ISMRM . 14 (2006). [3] Bas ser PJ. Proc. ISMR M. 5 (1997). [4]

Basser PJ et al . MRM. 200 0; 44: 41-50. [5] Jones DK . MRM 2003; 49(1):7-12 . [6] Anders on AW. MRM 2001; 46:117 4-1188. [7] Lazar M et al .

NeuroImage 2003; 20:1140-1153. [8] Lazar M et al. MRM. 2005; 54: 860-867. [9] Wu YC et al. MRM. 2004; 52:1146-1155. [10] Hext GR. Biometrika

1963; 50: p 353. [11] Koay CG et al. J Magn Reson. 2006; 182: 115-125. [12] Pierpaoli C et al. Radiology 1996; 201:637-648. [13] Mori S et al.Ann

Neurol 1999; 45:265-26 9. [14] Behre ns TEJ et al. M RM. 2003; 50: 1 077-1088. [15] Boguna M et al . New Jour nal of Physics 2005;7.

CONCLUSION We have presented three independent and complementary

methods to show that the cone of uncertainty is generally a right elliptical cone

rather tha n a right circular c one. W e showed th at the ke y geometr ic object f or

constructi ng the loc al measure of tract dispersion is the c ovariance matrix of the

major eige nvector. W e have also p rovided the theoretic al basis on w hich the

observati on of anis tropic dis persion p atterns i n the esti mated tra ct traject ories

caused by ax ial asymm etric diff usion of wat er diffusi on in white m atter can now

be explained.

Proc. Intl. Soc. Mag. Reson. Med. 15 (2007) 1602

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The Cone of Uncertainty is Elliptical: Implications for DTI Tractography

C. Koay^1 , L-C. Chang^1 , and P. J. Basser 1 (^1) NICHD, NIH, Bethesda, MD, United States

INTRODUCTION

The diffusion tensor is always measured in the presence of background noise. Within the framework of recently proposed methods of error propagation and of 1st^ -order matrix perturbation for DTI [1,2], one can easily characterize the noise induced variability of the estimated tensor itself and of tensor- derived quantities, scalar or vector. Of particular interest in tractography is the uncertainty of the eigenvectors or principal directions. With this end in view, the cone of uncertainty was proposed as a local measure of tract dispersion by Basser [3] and later used by Basser & Pajevic [4] and Jones [5] as a tool for visualizing and quantifying uncertainty along the fiber tract. The cone of uncertainty is not the only measure for quantifying tract dispersion, there is another approach that based on the perturbation method, which was proposed by Anderson [6] and used by Lazar [7,8]. None of these studies provided the key geometric object relating to the local measure of tract dispersion, which is the covariance matrix of the major eigenvector of the diffusion tensor. Here we present three different methods based on the error propagation framework [1], the 1st^ -order matrix perturbation method [2], and the dyadic formalism [3,9], to show that the local variability in the purported fiber direction is, in general, not described by a right circular cone of uncertainty but a right elliptical cone. Consequences of this finding are discussed. METHODS Error Propagation Method Based on the framework presented in [1], propagation of errors amounts to finding an appropriate ìcoordinateî transformation between covariance structures of interest. In DTI, the diffusion tensor can be reparametrized by the eigenvalues and the Euler angles in order to construct the covariance matrix of the major eigenvector of the diffusion tensor. The eigenvectors of this covariance matrix whose eigenvalues are nonzero are then the major and minor axes of the ellipse of the cone of uncertainty. 1 st^ -order Matrix Perturbation Method The pioneering work of Hext [10] provided a method to construct the covariance structure of the major eigenvector based on the 1st^ -order matrix perturbation, see Eq[4.17] of [10], but the importance of this formula has not been pointed out in previous DTI studies. Here, we suggest Hextís approach as an alternative to the other two methods proposed here for constructing the elliptical cone of uncertainty. Dyadic Product Formalism This formalism was proposed by Basser et al.[3] to overcome the problem of bias in computing sample mean of the eigenvector. Here, we show how the cone of uncertainty can be constructed from the average dyadics. Let {^ q^11^^ ,L, q^1 N } be a sample of major eigenvectors and let =^ ∑^ = iN i Ti N q (^) 1 q 1 T^^11 q 1 q 1 be the average dyadic [3]. Further, let the eigenvalue decomposition of the average dyadics be

q 1 q 1 T = ∑ 3 i = 1 λ i ψ i ψ Ti where λ 1 ≥λ 2 ≥λ 3. Then, the covariance matrix of the mean major eigenvector can be shown to be NN −^ 1 (λ 2 ψ 2 ψ T 2 +λ 3 ψ 3 ψ T 3 ).

RESULTS AND DISCUSSION

The analysis presented above shows that the cone of uncertainty has an elliptical not circular cross section. Particularly, the 1st^ -order matrix perturbation method shows that the cone of uncertainty is elliptical when the two smaller eigenvalues of the diffusion tensor are not equal to each other. Only when they are equal will the cone of uncertainty have the shape of a right circular cone. Similar observations can be made based on the other two methods discussed here. Figure 1B shows the cones of uncertainty constructed from a simulated human brain data based on the error propagation method [1] and the nonlinear least squares method [11]; Figure 1A is the FA map. The implications for DTI tractography are clear. Since the medium and minor eigenvalues of most tensor estimates in the brain are generally different [12], the axial symmetry of the tract dispersion is usually broken. In streamline based or probabilistic methods [13,14], the chance of deviating away from a tract in the direction along which uncertainty is low is less likely than from deviating away from the tract in the direction along which uncertainty is high. Thus, we would predict an asymmetric spread of tracts trajectories normal to the mean trajectory in which paths tending to meander along the direction with lower variance will remain closer to the mean tract whereas tracts meandering along the direction with higher variance will exhibit more dispersion and lie farther from the mean. Generalizing the approach given in [15], we can view the tractography process as a biased random walk in which there is anisotropic diffusion of the tract trajectory normal to the mean tract. In fact, such behavior appears to have been observed in bootstrap simulations of tracts generated using a streamline following method. In Lazar et al., white matter ìaxial asymmetryî (i.e., inequality of λ 2 and λ 3 ) caused anisotropic dispersion patterns in the estimated tract trajectories [8] but a theoretical explanation was not provided for this finding.

Figure 1A Figure 1B REFERENCES [1] Koay CG et al. Proc. ISMRM. 14 (2006). [2] Chang LC et al. Proc. ISMRM. 14 (2006). [3] Basser PJ. Proc. ISMRM. 5 (1997). [4] Basser PJ et al. MRM. 2000; 44: 41-50. [5] Jones DK. MRM 2003;49(1):7-12. [6] Anderson AW. MRM 2001; 46:1174-1188. [7] Lazar M et al. NeuroImage 2003; 20:1140-1153. [8] Lazar M et al. MRM. 2005; 54: 860-867. [9] Wu YC et al. MRM. 2004; 52:1146-1155. [10] Hext GR. Biometrika 1963; 50: p 353. [11] Koay CG et al. J Magn Reson. 2006; 182: 115-125. [12] Pierpaoli C et al. Radiology 1996; 201:637-648. [13] Mori S et al.Ann Neurol 1999; 45:265-269. [14] Behrens TEJ et al. MRM. 2003; 50: 1077-1088. [15] Boguna M et al. New Journal of Physics 2005;7.

CONCLUSION We have presented three independent and complementary methods to show that the cone of uncertainty is generally a right elliptical cone rather than a right circular cone. We showed that the key geometric object for constructing the local measure of tract dispersion is the covariance matrix of the major eigenvector. We have also provided the theoretical basis on which the observation of anistropic dispersion patterns in the estimated tract trajectories caused by axial asymmetric diffusion of water diffusion in white matter can now be explained.

The Elliptical Nature of the Cone of Uncertainty in DTI Tractography, Lecture notes of Project Management

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Download The Elliptical Nature of the Cone of Uncertainty in DTI Tractography and more Lecture notes Project Management in PDF only on Docsity!

The Cone of Uncertainty is Elliptical: Implications for DTI Tractography

INTRODUCTION

q 1 q 1 T = ∑ 3 i = 1 λ i ψ i ψ Ti where λ 1 ≥λ 2 ≥λ 3. Then, the covariance matrix of the mean major eigenvector can be shown to be NN −^ 1 (λ 2 ψ 2 ψ T 2 +λ 3 ψ 3 ψ T 3 ).

RESULTS AND DISCUSSION

Proc. Intl. Soc. Mag. Reson. Med. 15 (2007) 1602