We consider several stochastic algorithms for fiber tracking and compute the connectivity matrix from data obtained by magnetic resonance diffusion tensor imaging of the living human brain

Hahn, Klaus

Prigarin, Sergej

English

Universität München: Elektronischen Publikationen

Prigarin, Hahn:
Stochastic Algorithms for White Matter Fiber Tracking
and the Inference of Brain Connectivity from MR
Diffusion Tensor Data
Sonderforschungsbereich 386, Paper 408 (2005)
Online unter: http://epub.ub.uni-muenchen.de/
Projektpartner
Stochastic Algorithms for White
Matter Fiber Tracking and the
Inference of Brain Connectivity from
MR Diffusion Tensor Data
Sergei M. Prigarin ∗
Institute of Computational Mathematics
and Mathematical Geophysics †
Klaus Hahn ‡
Institute of Biomathematics and Biometry
GSF-National Research Centre for Environment and Health §
December 21, 2004
Abstract
We consider several stochastic algorithms for fiber tracking and
compute the connectivity matrix from data obtained by magnetic
resonance diffusion tensor imaging of the living human brain.
Key words: diffusion tensor imaging (DTI), stochastic fiber
tracking, brain connectivity.
∗e-mail: smp@osmf.sscc.ru
†pr. Lavrentieva 6, Novosibirsk, 630090, Russia
‡e-mail: hahn@gsf.de
§Ingolstädter Landstrasse 1, 85764 Neuherberg/München, Germany
1
1 Statement of the problem
Let V be a bounded set in the 3-dimensional space, V ⊂ R3, and T (x),
x ∈ V , be a given tensor field in V . All tensors T (x) (i.e. 3*3-matrices) are
assumed to be symmetric and positive definite. A problem is to generate a
”connectivity map” in volume V ”coordinated” with the tensor field and to
find tracks that connect different parts of the volume V . A track is considered
as a ”compatible” one with the tensor field if principal eigenvectors of tensors
are tangents or ”nearly tangents” to the track. Below we consider two types
of algorithms:
1. ”Single tracking”: to construct a curve ”compatible” with the tensor
field T starting from a fixed point x0 in volume V .
2. ”Connectivity matrix”: In this case, points x1, . . . , xM in volume V
are given and they are supposed to be extremities of tracks. The aim is to
calculate the connectivity matrix of weights W = (Wij), i, j = 1, . . . , M , that
can be interpreted as ”probabilities” of existence of a track connecting points
xi, xj.
Such problems arise in Diffusion Tensor Magnetic Resonance Imaging
(DT-MRI), where, among others, the course and the connections of the nerve
fibers in the brain are studied. The tensor field indicates a local orientation
of nerve bundles. The goal is to decide which parts of grey matter are con-
nected with each other by white matter nerve fibers and to estimate the
location of the fibers. The described problems require additional mathemat-
ical specifications (for example, it is desirable to have a detailed description
of what is the ”compatibility” of the tensor field and the track). The partial
volume effects fiber crossing and bifurcation are not-well recognized because
of the poor resolution of the tensor field, in addition, noise of different nature
introduces more uncertainty to the problem. The uncertainty makes it rea-
sonable to use stochastic methods in addition to deterministic ones to find
probable tracks. Using Monte Carlo methods to simulate a set of stochastic
tracks we expect to get an appropriate connectivity map. The stochastic
approach in studying the brain connectivity using DT-MRI is a significant
direction of the present-day research [1, 2]. In this paper we consider several
stochastic algorithms (including some new methods) for fiber tracking and
the computation of the connectivity matrix.
2
2 Several Algorithms of Stochastic Fiber
Tracking
Below we assume that the tensor field T (x) is defined on a grid of integer
numbers and to interpolate the field we simply take the value in the nearest
grid point. A fixed point x0 is a starting point for tracks. By e(T ) we denote
the principal eigenvector of tensor T and by {b} we denote the normalized
vector b‖b‖ . Tracks will be described by a sequence of points xn ∈ V .
2.1 Tracking on the basis of the field of the principal
eigenvectors
Following the main direction field by a deterministic differential equation
was one of the first tracking methods in DTI, we extend this approach by
additional random perturbations.
Algorithm E.
v0 = {e(T (x0))},
xn = xn−1 + vn−1∆t +
√
∆tσεn,
vn = {e(T (xn))}, (vn−1, vn) > 0.
Here εn are independent standard normal random vectors and ∆t, σ are
parameters of the algorithm. The step parameter ∆t is assumed to be less
than 1 (remind that 1 is the step of the grid were the tensor field is de-
fined). For this algorithm and the algorithms presented below the parameter
σ defines the intensity of artificial noise that is added to generate stochastic
tracks. If σ = 0, then the algorithms become deterministic and the tracks
are defined in an unique non-random way.
The directions of principal eigenvectors can be unstable under noisy fluc-
tuations of the tensors’ elements (especially, when some of eigenvalues are
close to each other) or they can be simply undefined in case of multiple
eigenvalues. This evident consideration encouraged us to develop sophisti-
cated methods, which take into account more characteristics of the tensor
field than only the directions of principal eigenvectors.
3
2.2 A method of tracking using the tensor field
Algorithm T1:
v0 = {e(T (x0))},
xn = xn−1 + vn−1∆t +
√
∆tσεn,
vn = {T k(xn)vn−1}.
Here ∆t, σ and k are parameters of the algorithm (∆t < 1, k > 0).
In Monte Carlo experiments we set k = 1. This algorithm generates more
”smooth and stable” tracks in comparison with the previous method (see.
Fig.1).
Figure 1: Examples of tracks generated by Algorithm E (narrow line) and
Algorithm T1 (wide line) for ∆t = 0.1 with the same starting point and
without random noise (σ = 0).
2.3 Tracking on the basis of a 2nd order SDE
Algorithm T2:
4
v0 = c0{e(T (x0))}, a0 = 0,
an+1 = {T k(xn)vn} − c1vn,
vn+1 = vn + ∆tan+1 +
√
∆tσεn+1,
xn+1 = xn + vn+1∆t.
Here ∆t, σ, c0, c1 and k are parameters of the algorithm (∆t < 1, k > 0).
In Monte Carlo experiments we set k = 1. This method involves not only
the principal eigenvectors but the whole tensors.
If c0 = 0 and σ 6= 0, then the track ”slowly starts” from x0 in a random
direction, but then it ”turns according to the main diffusion stream”.
The tracks strongly depend on the ”braking” parameter c1 (see Fig.2).
Figure 2: Examples of tracks generated by Algorithm T2 with different values
of ”braking” parameter c1: c1 = 1 (dark wide line), c1 = 3 (light wide line),
c1 = 5 (narrow black line). The values of other parameters: ∆t = 0.1, c0 = 1,
σ = 0. The vector field and the starting point for tracking are the same as
for Figure 1.
5
This algorithm can be interpreted as a numerical method to solve the
Cauchy problem for the following stochastic differential equaion
ξ′(t) = A(ξ)ξ(t) + Σε(t),
where
ξ(t) =
[
x(t)
x′(t)
]
,
A(ξ) =
[
0 I
0 aT k(x)− c1I
]
,
Σ =
[
0 0
0 σI
]
,
ξ(t) is a random 6-dimensional vector process, A, Σ are 6*6-matrices, ε(t)
is a 6-dimensional vector of the Gaussian white noise, and I is the three-
dimensional identity matrix.
One of the reasons to construct this algorithm on the basis of the stochas-
tic differential equation was to check in future if the up-to-date results in the
theory of boundary value problems for stochastic differential equations [3]
can be used to build tracks with fixed starting and ending points.
2.4 A method of stochastic tracking from Literature
Algorithm H [3]:
xn+1 = xn + µΩn,
Ωn = {λdn + Ωn−1}, (Ωn−1, Ωn) ≥ 0,
dn = T
α(xn))rn.
Here rn are independent random vectors uniformly distributed over a unit
sphere and µ, α, λ are parameters of the algorithm (0 < λ). The authors
recommend µ = 0.75, α = 2.
In our opinion, one of the main disadvantages of the method is that there
is no deterministic limit of the algorithm: when the stochastic part of the
algorithm vanishes (λ = 0), a track turns into a straight line.
6
If track points xn are assumed to be coordinates of a solid in space, then
the values of vn for algorithms E, T1, T2 can be interpreted as velocity of
the solid, and the values of an can be interpreted as acceleration of the solid
for algorithms T2.
Bundles of stochastic tracks for different algorithms are shown on Figs.3-
6.
Figure 3: A bundle of 25 random tracks generated by Algorithm E with
parameters ∆t = 0.1, σ = 0.2. The vector field and the starting point for
tracking are the same in Figures 3-6.
3 Computation of the connectivity matrices
Assume now, that the coordinates of M terminal points x1, . . . , xM in volume
V are fixed, the problem is then to calculate the ”connectivity matrix” of
weights Wij, i, j = 1, . . . ,M , which ”correspond to probabilities” of existence
of a fiber connecting points xi, xj (see Section 1).
To calculate the connectivity matrix W = (Wij) we used the following
algorithm. A large number of random tracks is started every terminal point.
If a track passes close enough to another terminal point, then the track is
stopped and it is announced that the terminal points are connected by the
track. The weights Wij are computed by the formula Wij = Nij/N. Here
7
Figure 4: A bundle of 25 random tracks generated by Algorithm T1 with
parameters ∆t = 0.1, σ = 0.2. The vector field and the starting point for
tracking are the same in Figures 3-6.
Figure 5: A bundle of 25 random tracks generated by Algorithm T2 with
parameters c0 = 1, c1 = 2.0, ∆t = 0.1, σ = 0.1. The vector field and the
starting point for tracking are the same in Figures 3-6.
8
Figure 6: A bundle of 25 random tracks generated by Algorithm H with
parameters λ = 1000000.0, α = 2, µ = 0.75. The vector field and the
starting point for tracking are the same in Figures 3-6.
Nij is the number of random tracks that connect points xi and xj, and N is
the total number of tracks connecting terminal points (some of the simulated
random tracks do not connect any pair of terminal points).
The connectivity matrix W was computed for 22 terminal points in the
volume with a tensor field T (x). The results for different algorithms of ran-
dom tracking are presented in Figures 7–13. For every numerical experiment
22 ∗ 10000 random tracks were simulated. The Figures demonstrate the in-
fluence of different noise levels on the connectivity matrix. The different
methods gave reasonable and similar results (may be except Hagmann’s al-
gorithm).
To make the analysis of connectivity more detailed, we estimated the
probabilities Pij to hit terminal point xj for a track starting from point xi.
In addition, to get approximate information about the location of a ”real
fiber” between two voxels x1 and x2 (which are assumed to be connected),
we propose to compute the ”fiber density function” D(x) in volume V by the
following algorithm. First, the values D(x) are equal to zero for every grid
point x in volume V . Then, a large number of random tracks starting from
terminal points x1 and x2 are simulated, and for every track that crosses voxel
x (once or several times) the value D(x) increases by 1. So, D(x) is simply
9
0 5 10 15 20
0
5
10
15
20
Figure 7: Example of connectivity matrix: algorithm E with parameters
∆t = 0.1, σ = 0. The vector field is the same for Figures 7–13.
a number of tracks crossing the voxel x. After normalization, D(x) can be
interpreted as a track density distribution. The ridge of D(x) connecting x1
and x2 defines the connection with highest probability. Methods to model
this maximizing connecting track are in progress.
References
[1] Hagmann, P., Thiran, J.-P., Jonasson, L., Vandergheynst, P., Clarke, S.
Maeder, P., and Meuli, R. DTI mapping of human brain connectivity:
statistical fibre tracking and virtual dissection// NeuroImage - 2003. -
V.19, N.3. - P.545-554
[2] Tench, C.-R., Morgan, P.-S., Blumhardt, L.-D., Constantinescu, C., Im-
proved White Matter Fiber Tracking using Stochastic Labeling// Magn
Res Med - 2002. - P.677-683
[3] Prigarin, S.M., Fedchenko, N.V. Solving of boundary value problems for
linear systems of stochastic differential equations// Siberian journal of
numerical mathematics - 2004. - V.7, N.4. - P.301-318
10
0 5 10 15 20
0
5
10
15
20
Figure 8: Example of connectivity matrix: algorithm E with parameters
∆t = 0.1, σ = 0.25.
0 5 10 15 20
0
5
10
15
20
Figure 9: Example of connectivity matrix: algorithm E with parameters
∆t = 0.1, σ = 0.5.
0 5 10 15 20
0
5
10
15
20
Figure 10: Example of connectivity matrix: algorithm E with parameters
∆t = 0.1, σ = 1.0.
11
0 5 10 15 20
0
5
10
15
20
Figure 11: Example of connectivity matrix: algorithm T1 with parameters
∆t = 0.1, σ = 0.25.
0 5 10 15 20
0
5
10
15
20
Figure 12: Example of connectivity matrix: algorithm T2 with parameters
∆t = 0.1, σ = 0.1, c0 = 1.0, c1 = 2.0.
0 5 10 15 20
0
5
10
15
20
Figure 13: Example of connectivity matrix: algorithm H with parameters
µ = 0.75, α = 2, λ = 1000000. The vector field is the same for Fig. 7–13.
12


Stochastic Algorithms for White Matter Fiber Tracking and the Inference of Brain Connectivity from MR Diffusion Tensor Data

Open Access LMU ( Ludwig-Maximilians-Univ. München)

Prigarin, Hahn:Stochastic Algorithms for White Matter Fiber Trackingand the Inference of Brain Connectivity from MRDiffusion Tensor DataSonderforschungsbereich 386, Paper 408 (2005)Online unter: http://epub.ub.uni-muenchen.de/ProjektpartnerStochastic Algorithms for WhiteMatter Fiber Tracking and theInference of Brain Connectivity fromMR Diffusion Tensor DataSergei M. Prigarin ∗Institute of Computational Mathematicsand Mathematical Geophysics †Klaus Hahn ‡Institute of Biomathematics and BiometryGSF-National Research Centre for Environment and Health §December 21, 2004AbstractWe consider several stochastic algorithms for fiber tracking andcompute the connectivity matrix from data obtained by magneticresonance diffusion tensor imaging of the living human brain.Key words: diffusion tensor imaging (DTI), stochastic fibertracking, brain connectivity.∗e-mail: smp@osmf.sscc.ru†pr. Lavrentieva 6, Novosibirsk, 630090, Russia‡e-mail: hahn@gsf.de§Ingolsta¨dter Landstrasse 1, 85764 Neuherberg/Mu¨nchen, Germany11 Statement of the problemLet V be a bounded set in the 3-dimensional space, V ⊂ R3, and T (x),x ∈ V , be a given tensor field in V . All tensors T (x) (i.e. 3*3-matrices) areassumed to be symmetric and positive definite. A problem is to generate a”connectivity map” in volume V ”coordinated” with the tensor field and tofind tracks that connect different parts of the volume V . A track is consideredas a ”compatible” one with the tensor field if principal eigenvectors of tensorsare tangents or ”nearly tangents” to the track. Below we consider two typesof algorithms:1. ”Single tracking”: to construct a curve ”compatible” with the tensorfield T starting from a fixed point x0 in volume V .2. ”Connectivity matrix”: In this case, points x1, . . . , xM in volume Vare given and they are supposed to be extremities of tracks. The aim is tocalculate the connectivity matrix of weightsW = (Wij), i, j = 1, . . . ,M , thatcan be interpreted as ”probabilities” of existence of a track connecting pointsxi, xj.Such problems arise in Diffusion Tensor Magnetic Resonance Imaging(DT-MRI), where, among others, the course and the connections of the nervefibers in the brain are studied. The tensor field indicates a local orientationof nerve bundles. The goal is to decide which parts of grey matter are con-nected with each other by white matter nerve fibers and to estimate thelocation of the fibers. The described problems require additional mathemat-ical specifications (for example, it is desirable to have a detailed descriptionof what is the ”compatibility” of the tensor field and the track). The partialvolume effects fiber crossing and bifurcation are not-well recognized becauseof the poor resolution of the tensor field, in addition, noise of different natureintroduces more uncertainty to the problem. The uncertainty makes it rea-sonable to use stochastic methods in addition to deterministic ones to findprobable tracks. Using Monte Carlo methods to simulate a set of stochastictracks we expect to get an appropriate connectivity map. The stochasticapproach in studying the brain connectivity using DT-MRI is a significantdirection of the present-day research [1, 2]. In this paper we consider severalstochastic algorithms (including some new methods) for fiber tracking andthe computation of the connectivity matrix.22 Several Algorithms of Stochastic FiberTrackingBelow we assume that the tensor field T (x) is defined on a grid of integernumbers and to interpolate the field we simply take the value in the nearestgrid point. A fixed point x0 is a starting point for tracks. By e(T ) we denotethe principal eigenvector of tensor T and by {b} we denote the normalizedvector b‖b‖ . Tracks will be described by a sequence of points xn ∈ V .2.1 Tracking on the basis of the field of the principaleigenvectorsFollowing the main direction field by a deterministic differential equationwas one of the first tracking methods in DTI, we extend this approach byadditional random perturbations.Algorithm E.v0 = {e(T (x0))},xn = xn−1 + vn−1∆t+√∆tσεn,vn = {e(T (xn))}, (vn−1, vn) > 0.Here εn are independent standard normal random vectors and ∆t, σ areparameters of the algorithm. The step parameter ∆t is assumed to be lessthan 1 (remind that 1 is the step of the grid were the tensor field is de-fined). For this algorithm and the algorithms presented below the parameterσ defines the intensity of artificial noise that is added to generate stochastictracks. If σ = 0, then the algorithms become deterministic and the tracksare defined in an unique non-random way.The directions of principal eigenvectors can be unstable under noisy fluc-tuations of the tensors’ elements (especially, when some of eigenvalues areclose to each other) or they can be simply undefined in case of multipleeigenvalues. This evident consideration encouraged us to develop sophisti-cated methods, which take into account more characteristics of the tensorfield than only the directions of principal eigenvectors.32.2 A method of tracking using the tensor fieldAlgorithm T1:v0 = {e(T (x0))},xn = xn−1 + vn−1∆t+√∆tσεn,vn = {T k(xn)vn−1}.Here ∆t, σ and k are parameters of the algorithm (∆t < 1, k > 0).In Monte Carlo experiments we set k = 1. This algorithm generates more”smooth and stable” tracks in comparison with the previous method (see.Fig.1).Figure 1: Examples of tracks generated by Algorithm E (narrow line) andAlgorithm T1 (wide line) for ∆t = 0.1 with the same starting point andwithout random noise (σ = 0).2.3 Tracking on the basis of a 2nd order SDEAlgorithm T2:4v0 = c0{e(T (x0))}, a0 = 0,an+1 = {T k(xn)vn} − c1vn,vn+1 = vn +∆tan+1 +√∆tσεn+1,xn+1 = xn + vn+1∆t.Here ∆t, σ, c0, c1 and k are parameters of the algorithm (∆t < 1, k > 0).In Monte Carlo experiments we set k = 1. This method involves not onlythe principal eigenvectors but the whole tensors.If c0 = 0 and σ 6= 0, then the track ”slowly starts” from x0 in a randomdirection, but then it ”turns according to the main diffusion stream”.The tracks strongly depend on the ”braking” parameter c1 (see Fig.2).Figure 2: Examples of tracks generated by Algorithm T2 with different valuesof ”braking” parameter c1: c1 = 1 (dark wide line), c1 = 3 (light wide line),c1 = 5 (narrow black line). The values of other parameters: ∆t = 0.1, c0 = 1,σ = 0. The vector field and the starting point for tracking are the same asfor Figure 1.5This algorithm can be interpreted as a numerical method to solve theCauchy problem for the following stochastic differential equaionξ′(t) = A(ξ)ξ(t) + Σε(t),whereξ(t) =[x(t)x′(t)],A(ξ) =[0 I0 aT k(x)− c1I],Σ =[0 00 σI],ξ(t) is a random 6-dimensional vector process, A, Σ are 6*6-matrices, ε(t)is a 6-dimensional vector of the Gaussian white noise, and I is the three-dimensional identity matrix.One of the reasons to construct this algorithm on the basis of the stochas-tic differential equation was to check in future if the up-to-date results in thetheory of boundary value problems for stochastic differential equations [3]can be used to build tracks with fixed starting and ending points.2.4 A method of stochastic tracking from LiteratureAlgorithm H [3]:xn+1 = xn + µΩn,Ωn = {λdn + Ωn−1}, (Ωn−1,Ωn) ≥ 0,dn = Tα(xn))rn.Here rn are independent random vectors uniformly distributed over a unitsphere and µ, α, λ are parameters of the algorithm (0 < λ). The authorsrecommend µ = 0.75, α = 2.In our opinion, one of the main disadvantages of the method is that thereis no deterministic limit of the algorithm: when the stochastic part of thealgorithm vanishes (λ = 0), a track turns into a straight line.6If track points xn are assumed to be coordinates of a solid in space, thenthe values of vn for algorithms E, T1, T2 can be interpreted as velocity ofthe solid, and the values of an can be interpreted as acceleration of the solidfor algorithms T2.Bundles of stochastic tracks for different algorithms are shown on Figs.3-6.Figure 3: A bundle of 25 random tracks generated by Algorithm E withparameters ∆t = 0.1, σ = 0.2. The vector field and the starting point fortracking are the same in Figures 3-6.3 Computation of the connectivity matricesAssume now, that the coordinates ofM terminal points x1, . . . , xM in volumeV are fixed, the problem is then to calculate the ”connectivity matrix” ofweightsWij, i, j = 1, . . . ,M , which ”correspond to probabilities” of existenceof a fiber connecting points xi, xj (see Section 1).To calculate the connectivity matrix W = (Wij) we used the followingalgorithm. A large number of random tracks is started every terminal point.If a track passes close enough to another terminal point, then the track isstopped and it is announced that the terminal points are connected by thetrack. The weights Wij are computed by the formula Wij = Nij/N. Here7Figure 4: A bundle of 25 random tracks generated by Algorithm T1 withparameters ∆t = 0.1, σ = 0.2. The vector field and the starting point fortracking are the same in Figures 3-6.Figure 5: A bundle of 25 random tracks generated by Algorithm T2 withparameters c0 = 1, c1 = 2.0, ∆t = 0.1, σ = 0.1. The vector field and thestarting point for tracking are the same in Figures 3-6.8Figure 6: A bundle of 25 random tracks generated by Algorithm H withparameters λ = 1000000.0, α = 2, µ = 0.75. The vector field and thestarting point for tracking are the same in Figures 3-6.Nij is the number of random tracks that connect points xi and xj, and N isthe total number of tracks connecting terminal points (some of the simulatedrandom tracks do not connect any pair of terminal points).The connectivity matrix W was computed for 22 terminal points in thevolume with a tensor field T (x). The results for different algorithms of ran-dom tracking are presented in Figures 7–13. For every numerical experiment22 ∗ 10000 random tracks were simulated. The Figures demonstrate the in-fluence of different noise levels on the connectivity matrix. The differentmethods gave reasonable and similar results (may be except Hagmann’s al-gorithm).To make the analysis of connectivity more detailed, we estimated theprobabilities Pij to hit terminal point xj for a track starting from point xi.In addition, to get approximate information about the location of a ”realfiber” between two voxels x1 and x2 (which are assumed to be connected),we propose to compute the ”fiber density function” D(x) in volume V by thefollowing algorithm. First, the values D(x) are equal to zero for every gridpoint x in volume V . Then, a large number of random tracks starting fromterminal points x1 and x2 are simulated, and for every track that crosses voxelx (once or several times) the value D(x) increases by 1. So, D(x) is simply90 5 10 15 2005101520Figure 7: Example of connectivity matrix: algorithm E with parameters∆t = 0.1, σ = 0. The vector field is the same for Figures 7–13.a number of tracks crossing the voxel x. After normalization, D(x) can beinterpreted as a track density distribution. The ridge of D(x) connecting x1and x2 defines the connection with highest probability. Methods to modelthis maximizing connecting track are in progress.References[1] Hagmann, P., Thiran, J.-P., Jonasson, L., Vandergheynst, P., Clarke, S.Maeder, P., and Meuli, R. DTI mapping of human brain connectivity:statistical fibre tracking and virtual dissection// NeuroImage - 2003. -V.19, N.3. - P.545-554[2] Tench, C.-R., Morgan, P.-S., Blumhardt, L.-D., Constantinescu, C., Im-proved White Matter Fiber Tracking using Stochastic Labeling// MagnRes Med - 2002. - P.677-683[3] Prigarin, S.M., Fedchenko, N.V. Solving of boundary value problems forlinear systems of stochastic differential equations// Siberian journal ofnumerical mathematics - 2004. - V.7, N.4. - P.301-318100 5 10 15 2005101520Figure 8: Example of connectivity matrix: algorithm E with parameters∆t = 0.1, σ = 0.25.0 5 10 15 2005101520Figure 9: Example of connectivity matrix: algorithm E with parameters∆t = 0.1, σ = 0.5.0 5 10 15 2005101520Figure 10: Example of connectivity matrix: algorithm E with parameters∆t = 0.1, σ = 1.0.110 5 10 15 2005101520Figure 11: Example of connectivity matrix: algorithm T1 with parameters∆t = 0.1, σ = 0.25.0 5 10 15 2005101520Figure 12: Example of connectivity matrix: algorithm T2 with parameters∆t = 0.1, σ = 0.1, c0 = 1.0, c1 = 2.0.0 5 10 15 2005101520Figure 13: Example of connectivity matrix: algorithm H with parametersµ = 0.75, α = 2, λ = 1000000. The vector field is the same for Fig. 7–13.12

DTI mapping of human brain connectivity: statistical ¯bre tracking and virtual dissection// NeuroImage -

Improved White Matter Fiber Tracking using Stochastic Labeling// Magn Res Med -

Solving of boundary value problems for linear systems of stochastic di®erential equations// Siberian journal of numerical mathematics -

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Stochastic Algorithms for White Matter Fiber Tracking and the Inference of Brain Connectivity from MR Diffusion Tensor Data

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