Mathematics in Diffusion MRI

Tom Dela Haije

Supervisors: Luc Florack Andrea Fuster

Overview • Diffusion MRI introduction • What I’ve done so far • What I plan to do still

What I’ve been doing • Main project

• Finsler function as signal model

• Developers framework

• Other projects • Optimal curves on

• Riemannian geometry in diffusion MRI

• Investigating sheet structure

• Supervising student

What I’ve been doing • Main project

• Finsler function as signal model

• Developers framework

• Other projects • Optimal curves on

• Riemannian geometry in diffusion MRI

• Investigating sheet structure

• Supervising student

FINSLER FUNCTIONS

Signal Model

Finsler Functions • Finsler function model

• Link to signal

• Reconstruction from raw data

• Processing of the reconstructed data

• Analysis

• More robust and sound analysis

SHEET STRUCTURE

[V,W] = 0

Science paper Wedeen • High quality data • Questionable reconstruction • Illegible writing

Our work • Understanding the math • Looking at validity of approaches

Methods I

Methods text, I Methods text, I Methods text,

References [1] Wedeen et al., Science 335: 1628-1634, 2012; [2] Catani et al., Science 337: 1605, 2012; [3] Wedeen et al., Science 337: 1605, 2012; [4] Spivak, A comprehensive introduction todifferential geometry: Volume I, Publish or perish, 1999; [5] Misner et al., Gravitation, Macmillan, 1973;

Considerations on the theory of sheet structure of cerebral pathways

Chantal M.W. Tax1, Tom C.J. Dela Haije2, Andrea Fuster2, Max A. Viergever1, Luc M.J. Florack2, and Alexander Leemans1

1Image Sciences Institute (ISI), University Medical Center Utrecht, Utrecht, the Netherlands2Imaging Science & Technology (IST/e), Eindhoven University of Technology, Eindhoven, the Netherlands

Introduction The proposition of cerebral white matter being organized ina three-dimensional grid structure of interwoven sheets [1] has recentlykindled a debate in literature [1-3]. Part of the controversy may be causedby ambiguity in certain definitions and statements given in [1]. In thiswork, we aim for a formalization of the terminology, and providemathematical background that clarifies some of the statements made in theliterature [1-3].

Methods We will recall statements from [1-3].

“The property of parallel 2D sheets depends on a 3D relationship amongcrossing planes at different locations (the Frobenius integrabilitycondition).” [3]

In streamline tractography, one integrates a smooth vector field 𝑉:𝑁 → ℝ3

(with 𝑁 ⊂ ℝ3) derived from ODF maxima, obtaining a streamline (orintegral curve) 𝑡 ↦ 𝛾(𝑡) such that 𝑑𝛾(𝑡) 𝑑𝑡 = 𝑉|𝛾 𝑡⁄ (Fig. 1).

This idea of integrability can be extended to two vector fields 𝑉 and 𝑊.An integral surface 𝑆 ⊂ 𝑁, a two dimensional submanifold called the sheetstructure, is a surface whose tangent plane at 𝑝 is parallel to the planespanned by 𝑉|𝑝 and 𝑊|𝑝 for all 𝑝 ∈ 𝑆, see Fig. 2.

The requirements for such a surface to exist are formalized in theFrobenius theorem, which states that the vector fields 𝑉 and 𝑊 shouldinteract in a “nice” way according to their Lie bracket [4].

“In three dimensions, two smooth families of curves can have 2D surfacesin common only if a specific function – the normal component of the Liebracket of the streamlines, representing their mutual twist – is equal tozero at every point in the region in question.” [1]

Formally, the Lie bracket of vector fields 𝑉 and 𝑊 on 𝑁 ⊂ ℝ3 is a bilinearoperator defining a third vector field 𝑉,𝑊 = 𝐽𝑊 ⋅ 𝑉 − 𝐽𝑉 ⋅ 𝑊, with 𝐽𝑋 theJacobian matrix of a vector field 𝑋.

Evaluated at a point 𝑝, [𝑉,𝑊] is most naturally understood to approximatethe deviation from 𝑝 when trying to move around in an infinitesimal loopalong the integral curves of the fields (Fig. 3).

For the sheet structure to exist, 𝑉,𝑊 𝑝 should lie in the plane spanned by𝑉|𝑝 and 𝑊|𝑝 (or the inner product of 𝑉|𝑝 × 𝑊|𝑝 with 𝑉,𝑊 𝑝, i.e. thenormal component of the Lie bracket, should be zero (Fig. 3). Thisrequirement is called involutivity.

“The low angular resolution of DSI has a negative impact on thetractography reconstructions because it does not allow separation of fibersthat cross at nonorthogonal angles, thus making a grid structure ofinterwoven sheets a very likely configuration.” [2] “Neither smoothnessnor orthogonality creates false sheet structure.” [1]

The Lie bracket criterion shows that local orthogonality of two vectorfields is neither necessary nor sufficient for the involutivity condition ofthe Frobenius theorem to hold.

“By the Frobenius theorem, any three families of curves in 3D mutuallycross in sheets if and only if they represent the gradients of threecorresponding scalar functions.” [1]

The Frobenius theorem analogously states that the system of PDEs𝑉 ⋅ 𝛻𝜙 = 0 and 𝑊 ⋅ 𝛻𝜙 = 0 admits a solution 𝜙 = 𝜙𝑉𝑊 such that 𝛻𝜙𝑉𝑊iswell-defined only if 𝑉 and 𝑊 satisfy involutivity. Since 𝜙𝑉𝑊 does notchange along both 𝑉 and 𝑊, the parallel non-intersecting sheets coincidewith the level sets (or equipotential surfaces) of 𝜙𝑉𝑊, see Fig. 4.

Discussion and Conclusion Future work needs to focus on quantitativevalidation of sheet structure. We would like to pose some questions to thecommunity:• What spatial and angular resolution of the data is required for the

detection of sheet structure, and within which (quantitative) marginscan we still assume sheet structure?

• What influence does the dMRI reconstruction method have on theseresults?

• Are there alternatives to the methods used in [1] to numericallyapproximate the Lie bracket?

• How pervasive is this sheet structure exactly?• Can the underlying scalar functions be calculated, and can the

proposition that these are related to the chemotactic gradients ofembryogenesis be validated?

𝑉,𝑊 = 𝐽𝑊 ⋅ 𝑉 − 𝐽𝑉 ⋅ 𝑊 =

𝜕𝑤𝑥𝜕𝑥

𝜕𝑤𝑥𝜕𝑦

𝜕𝑤𝑥𝜕𝑧

𝜕𝑤𝑦

𝜕𝑥𝜕𝑤𝑦

𝜕𝑦𝜕𝑤𝑦

𝜕𝑧𝜕𝑤𝑧𝜕𝑥

𝜕𝑤𝑧𝜕𝑦

𝜕𝑤𝑧𝜕𝑧

⋅𝑣𝑥𝑣𝑦𝑣𝑧

-

𝜕𝑣𝑥𝜕𝑥

𝜕𝑣𝑥𝜕𝑦

𝜕𝑣𝑥𝜕𝑧

𝜕𝑣𝑦𝜕𝑥

𝜕𝑣𝑦𝜕𝑦

𝜕𝑣𝑦𝜕𝑧

𝜕𝑣𝑧𝜕𝑥

𝜕𝑣𝑧𝜕𝑦

𝜕𝑣𝑧𝜕𝑧

⋅𝑤𝑥𝑤𝑦𝑤𝑧

Fig. 1: An integral curve 𝛾 ofthe vector field 𝑉 (red).Vector fields 𝑉 and 𝑊 arederived from ODF maxima.

Fig. 2: The tangent plane of an integralsurface 𝑆 at a point 𝑝 is parallel to theplane spanned by 𝑉|𝑝 and 𝑊|𝑝(indicated by the dashed surfaces) for all𝑝 ∈ 𝑆.

Image Sciences InstituteContact: chantal@isi.uu.nl, t.c.j.dela.haije@tue.nl

Fig. 4: The parallel non-intersecting sheets coincidewith the level sets of 𝜙. A collection of three vectorfields that satisfy involutivity amongst each otherimplies the existence of a set of three distinct scalarfunctions, deemed 𝜙𝑉𝑊 , 𝜙𝑈𝑉 , and 𝜙𝑈𝑊 . In thespecial case when the vector fields are all orthogonal,then 𝛻𝜙𝑈𝑉 = 𝑊.

Fig. 3: For infinitesimal time ℎ , followsubsequently the integral curve of 𝑉 through𝑝, curve of 𝑊, curve of 𝑉 backwards, andcurve of 𝑊 backwards. The Lie bracketrepresents the gap with 𝑝. If the Lie bracketlies in the dashed plane, involutivity issatisfied.

DEVELOPMENT FRAMEWORK

Class[data & meta-information]

What I’ve got planned • Solve Signal – Finsler link • Finalize development framework

• WIAS / INRIA

• Verify sheet structure • UMCU

• …

THANK YOU