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A method is presented that is capable of determining more than one fiber orientation within a single voxel in high angular resolution diffusion imaging (HARDI) data sets. This method is an extension of the Markov chain method recently introduced to diffusion tensor imaging (DTI) analysis, allowing the probability density function of up to 2 intra‐voxel fiber orientations to be inferred. The multiple fiber architecture within a voxel is then assessed by calculating the relative probabilities of a 1 and 2 fiber model. It is demonstrated that for realistic signal to noise ratios, it is possible to accurately characterize the directions of 2 intersecting fibers using a 2 fiber model. The shortcomings of under‐fitting a 2 fiber model, or over‐fitting a 1 fiber model, are explored. This new algorithm enhances the tools available for fiber tracking. Magn Reson Med, 2005.

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© 2005 Wiley‐Liss, Inc. Diffusion tensor imaging (DTI) has become a very important tool for investigating tissue microstructure ( -), and is capable of providing information on the nerve fiber directions within imaged voxels. The principal axis of the diffusion tensor coincides with the nerve fiber direction because the anisotropic nature of nerve fibers preferentially allows diffusion in a direction parallel to the fiber. Despite the success of DTI, the standard second order tensor model copes poorly with partial volume effects ( ) as it is unable to characterize the directions of crossing fibers (, ). This has implications for tractography for 2 reasons. First, crossing fibers will lead to erroneous directions being inferred; this will add to other sources of noise in the data and potentially imply false connectivity. Secondly, voxels may appear isotropic with complete loss of directional information ( ).High angular resolution diffusion imaging (HARDI) is an alternative to DTI ( ).

Methods to analyze such diffusion weighted data sets with many gradient encoding directions, which go beyond the traditional second order tensor approach, have been proposed and implemented in the literature. These schemes include the use of tensors above second order (, ); spherical harmonic ( ), harmonic decomposition ( ), and spherical deconvolution ( ) of the signal; as well as least‐square fitting of a Gaussian mixture model of 2 intersecting fibers (, ) and q‐ball imaging ( ). Determination of uncertainty of the diffusion tensor is a problem that has received much attention in the literature ( -), and this knowledge of the uncertainty in fiber orientation has been incorporated into fiber tracking methods ( ).Behrens et al. ( ) introduced Bayesian inference to the problem of determining fiber orientation. Using a single fiber model, their algorithm could produce a probability density function (PDF) for the fiber orientation within a given voxel ( ). Their algorithm explicitly allows for partial volumes of isotropic material and, perhaps more importantly, allows uncertainty in the fiber orientation to be represented. The algorithm presented here is an extension of this analysis to address the problem of partial volumes in voxels containing 2 distinct fiber orientations, thereby allowing us to directly assess the directions and the separate PDFs of more than 1 fiber within each imaged voxel.

Furthermore, we are able to assess the most appropriate model to be applied, thus avoiding the problems of over‐fitting or under‐fitting the data. This is demonstrated with simulated data of crossing fibers.

(1)where f 1 and f 2 are the fraction of the unattenuated signal coming from the first and second nerve fiber bundle, b i is the the applied b‐value, n i is the direction of the applied diffusion weighted gradient, and S 0 is the voxel signal with no diffusion weighting. D 1 and D 2 are assumed to be second order tensors with 2 degenerate eigenvalues D ⟂ and a non‐degenerate eigenvalue D //, with each fiber bundle assumed to have an axially‐symmetric diffusion profile. D isotropic is the diffusion coefficient of the isotropic partial volume. Diffusional movement of water between the two fiber components is assumed to be in the slow exchange regime, such as might be expected for spatially distinct tissues within a single voxel ( ). (2) P(Data X, M) is easily calculated from the Rician noise distribution of MRI data. The prior probability term P(X M) allows knowledge of the model parameters to be incorporated into the analysis, and for our analysis, was not computationally expensive to calculate.

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The prior distributions used in our analysis are given in Table. The problem in calculating P(X Data, M) comes from evaluating the normalization constant ∫ all X d NX P(Data X, M) P(X M). Markov Chain Monte Carlo is a technique that allows us to draw samples from an unnormalized probability distribution function. This enables us to draw samples of X from the probability distribution P(X Data, M) using the function P(Data X, M) P(X M).

A random walk Metropolis algorithm was used for all the following analysis. See the for more details. a U( X) is 0 if X 0. R(A,B) is 1 inside the range bounded by A and B ( A. Single Voxel Digital PhantomA digital phantom was created by summing the signal from 2 diffusion tensors and an isotropic diffusion component together, according to Eq. Each diffusion tensor had either D // = 1.80 × 10 −3mm 2 s −1 and D ⟂ = 0.15 × 10 −3mm 2 s −1 ( ), or D // = 1.00 × 10 −3mm 2 s −1 and D ⟂ = 0.53 × 10 −3mm 2 s −1 with an isotropic diffusion component of D isotropic = 0.70 × 10 −3mm 2 s −1.

These values represent the extreme cases of high and low fractional anisotropy (FA) with FA = 0.9 and FA = 0.4, respectively ( ). FA is calculated using ( ). (3)where λ 1,2,3 are the eigenvalues of the diffusion tensor.

Most anisotropic voxels would have FAs that fall between these values (, ).In the following analysis, the simulated HARDI data sets were generated using a 92 direction icosahedral sampling scheme. This sampling scheme used directions derived by the Tegmark algorithm ( ), which uniformly distributes points over the surface of a sphere. All SNRs quoted correspond to the unattenuated signal to noise, though no S 0 signal is used in the analysis.

Rician Noise in the Low Signal to Noise Ratio RegimeAs arbitrary noise distributions can be modeled using a Markov Chain Monte Carlo method, it was chosen to model the likelihood function using a Rician distribution, which is the most general noise distribution for the MRI signal. This distribution is valid in low signal to noise ratio (SNR) regimes, whereas the Rician noise floor tends to create a downward bias in the diffusion tensor if a Gaussian noise distribution is assumed ( ). This allows the algorithm to be used at any SNR.The Rician distribution is given in Eq.

(9)is a constant for each fiber.Of course if diffusivity information is required, then more than one b‐value will be required, i.e., determining D ⟂ and D ‖. And were used to fit the data set, where the set of parameters used had been reduced by sampling at only one b‐value.Another partial volume effect that must be taken into account is the effect of an isotropically diffusing partial volume, such as gray matter or cerebrospinal fluid. There is still no need to use multiple b‐values if all we are interested in is mitigating the pollutive effect on the signal of the isotropic partial volume. For a constant b‐value, we have the following equation for the signal. (13)where η was chosen to be 0.3. This prevents the 1 fiber local minimum from destabilizing the Markov chain.

There are many possible configurations that can poorly represent the data if either S 1 or S 2 is zero. If this is the case, the parameters θ, ϕ, and Δλ for that fiber are completely unconstrained, causing the 2 fiber PDF to become unstable such that it no longer provides useful information. Demonstrates how the condition in Eq.

Stabilizes the Markov chain. The value of η used can be reduced if the maximum possible value of Δλ is limited to be within realistic bounds, as re‐convergence of the Markov chain to its actual value of Δλ is difficult if the value of Δλ has grown to a very high value. The value of η was chosen to be 0.3 as this would lead to an SNR with a smallest anisotropic component value of 7, which is still high enough to get useful orientational information out of the data keeping the other parameters stable. These plots show the effect of Markov chain stabilization on the angular PDFs of the fibres which are being fitted. The coordinates are spherical polar angles of the fibre directions in degrees. The two black spots represent the actual fibre directions used to generate the phantom.

Subplot ( a) shows the PDF of an unstabilized Markov chain can fall between the true fibre directions. Subplot ( b) shows that only the stabilized two fibre Markov chain is capable of converging to the true fibre directions. Probabilistic Model SelectionThere are disadvantages in applying a 2 fiber model where the underlying architecture is actually a single fiber. In such cases, a 2 fiber model will give a broader angular PDF than a 1 fiber model.

This is because there are more possible combinations of parameters that give solutions of a significant probability. Once again this causes loss of information concerning fiber orientation.

The Markov chain (given a sufficient number of iterations) fully samples the PDF of the model parameters. This allows us to perform a Monte Carlo integration using likelihood of the data to compare the 1 and 2 fiber models. We would like to be able to calculate the evidence for each model. (17)The value of P(Data M i) calculated by Eq. Converges to the true value of P(Data M i) as the number samples from the Markov chain tends to infinity ( ). For a finite number of samples, the calculation of P(Data M i) could be unstable if a sample with an unusually low value of L(θ) is selected, causing a divergence in the estimate of 1/ L(θ). Raferty ( ) quotes ψ 150 as corresponding to very strong evidence for the model M 2.

Bearing in mind that we are estimating the Bayes Factor from the Markov chain, we should be using a more conservative threshold. For this reason we analyzed the population of Bayes Factors that are generated for different noise configurations of the digital phantom for angular separations between fibers varying over the range 0° to 90°. The purpose of this was to see if the populations of Bayes factors that are generated by the digital phantom are distinctly different enough from the case of parallel fibers to allow model selection based on thresholding ψ. RESULTS AND DISCUSSIONFig. Shows the variation of ln(ψ(Data)) versus the angular separation of fibers used to create the digital phantom. Ψ(Data) was calculated at various angular separation for 10 repetitions of the simulation with SNR = 30, D ‖ = 1.80 × 10 −3mm 2 s −1, and D ⟂ = 0.15 × 10 −3mm 2 s −1. There was no isotropic diffusion component in this simulated voxel, but S iso was still fitted for.

It can be seen, though, that for all the points generated in Fig. There is a large divergence in ln(ψ) as the angular separation increases. The point at which 2 fibers can be resolved is a function of angular separation and SNR.

The Relative Confidence Between the 1 and 2 Fiber ModelFor the proposed algorithm to be useful, we need to be able to determine whether the 1 or 2 fiber model represents the best fit. In order to do this, the possible values of ln(ψ(Data)) were analyzed for each angular separation of nerve fibers. The mean and standard deviation of ln(ψ) was calculated for many different simulated data sets based on the noise in the system.

The mean and standard deviation were then used to calculate the PDFs P(ln(ψ 0)) and P(ln(ψ Δθ)), where ψ Δθ is the value of ψ calculated using a digital phantom with an angular separation between the two fibers of Δθ, and ψ 0 corresponds to no separation between the fibers in the digital phantom. A Gaussian distribution for P(ln(ψ 0)), and P(ln(ψ Δθ)) was assumed. To investigate the utility of the measure ln(ψ Δθ), the statistical parameter P(ln(ψ Δθ) ln(ψ 0)) was investigated. If this parameter is equal to 1, then inferred values of ln(ψ Δθ) will always be greater than ln(ψ 0) and the 2 populations can be perfectly distinguished by assuming a double fiber structure if the measured value of ψ is greater than a threshold value somewhere between the distinct populations of ψ 0 and ψ Δθ. In other words, ln(ψ) based on a single measurement is capable of telling us whether or not to use a 1 or 2 fiber model. If the populations of ln(ψ Δθ) and ln(ψ 0) significantly overlap, then a map of ln(ψ) will not tell us which model to use. P(ln(ψ Δθ) ln(ψ 0)) was calculated numerically using the following equation.

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(18)where U( x 0) = 1 and U( x. A and e, these optimal b‐values are 2000 s mm −2 and 1500 s mm −2 for an FA of 0.9 and 0.4, respectively. Shows that the maximal value of ln(ψ) moves away from 90° for higher b‐values.

This probably happens because at lower b‐values, the angular PDF always points between the 2 underlying fiber directions, but at higher b‐values with large fiber separations the single fiber PDF will point along 1 of the underlying fiber directions as the 2 signal minima become very large; this solution will be better than pointing between the 2 fibers, which would create a lower relative probability of the 2 versus 1 fiber model, and shift the maximum value of ψ to a smaller angle than 90°. Angular Resolution of Fibers and SNRThe ability of the algorithm to determine fiber orientation in the presence of noise was investigated. Example plots of the probability density function of the fiber orientation are given in Fig.

These plots are given in terms of spherical polar angles, where the underlying fiber directions are shown by a pair of crosses that indicate the actual fiber orientations. To demonstrate the effect of noise on the angular PDF, 2 orthogonal fibers were simulated. Shows, as expected, that the angular resolution goes down with SNR. Note that in Fig. The angular separation of fibers is not equal to the difference in ϕ in the above plots since the fibers do not lie in the equatorial plane.