Versatile, robust, and efficient tractography with constrained higher-order tensor fODFs

Abstract

Purpose

Develop a multi-fiber tractography method that produces fast and robust results based on input data from a wide range of diffusion MRI protocols, including high angular resolution diffusion imaging, multi-shell imaging, and clinical diffusion spectrum imaging (DSI)

Methods

In a unified deconvolution framework for different types of diffusion MRI protocols, we represent fiber orientation distribution functions as higher-order tensors, which permits use of a novel positive definiteness constraint (H-psd) that makes estimation from noisy input more robust. The resulting directions are used for deterministic fiber tracking with branching.

Results

We quantify accuracy on simulated data, as well as condition numbers and computation times on clinical data. We qualitatively investigate the benefits when processing suboptimal data, and show direct comparisons to several state-of-the-art techniques.

Conclusion

The proposed method works faster than state-of-the-art approaches, achieves higher angular resolution on simulated data with known ground truth, and plausible results on clinical data. In addition to working with the same data as previous methods for multi-tissue deconvolution, it also supports DSI data.

Appendix: Mathematical derivation of H-psd constraint

This appendix presents the formal definition of our H-psd constraint (Definition 4), based on a matrix representation H of higher-order tensor fODFs (Definition 3). According to Corollary 1, H is positive semidefinite if and only if the corresponding fODF can be decomposed into a nonnegative weighted sum of rank-1 terms, which correspond to single fiber compartments in our framework. This provides the theoretical justification of our H-psd constraint, whose practical effectiveness is demonstrated in the main part of the paper.

We represent fODFs as forms or symmetric tensors p of even degree $d=4$ in $n=3$ dimensions. In tensor notation:

$$\begin{aligned} p(\mathbf x ) = p(\underbrace{\mathbf{x , \dots , \mathbf x }}_{d}) = \sum _{i \in \{1,\dots ,n\}^d} p_{i_1,\dots ,i_d} \, x_{i_1} \dots x_{i_d}, \quad \mathbf x \in \mathbb {R}^n. \end{aligned}$$

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We will call the set of these forms $F_{n,d}$. Symmetry allows us to use a different, nonredundant indexing scheme

$$\begin{aligned} p(\mathbf x ) = \sum _{i \in I(n,d)} {d \atopwithdelims ()i} \, p_i \, x^i \end{aligned}$$

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with multi-indices $i \in I(n,d) = \{i \in {\mathbb {N}_0}^n | \sum _k i_k = d\}$, multinomial coefficients ${d \atopwithdelims ()i} = \frac{d!}{\prod i_k!}$ and monomial terms $x^i = \prod _{k=1}^n (x_k)^{i_k}$.

Our constraint stems from the relations of three subsets of $F_{n,d}$:

$$\begin{aligned} P_{n,d}&= \left\{ p \in F_{n,d} : p(\mathbf x ) \ge 0, \forall \, \mathbf x \in \mathbb {R}^n\right\} \nonumber \\&\qquad \text {``positive semidefinite''} \end{aligned}$$

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$$\begin{aligned} \varSigma _{n,d}&= \left\{ p \in F_{n,d} : p(\mathbf x ) = \sum _k h_k(\mathbf x )^2\right\} \nonumber \\&\qquad \text {``sums of squares''} \end{aligned}$$

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$$\begin{aligned} Q_{n,d}&= \left\{ p \in F_{n,d} : p(\mathbf x ) = \sum _k \langle \mathbf a _k, \mathbf x \rangle ^d\right\} \nonumber \\&\qquad \text {``sums of d-th powers''} \end{aligned}$$

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Here, the $h_k(\mathbf x )$ denote forms of degree d / 2, $\mathbf {a}_k$ denote the individual vectors that define the rank-1 terms of a nonnegative decomposition.

As shown in [21], these three subsets obey

$$\begin{aligned} Q_{n,d} \subset \varSigma _{n,d} \subset P_{n,d}. \end{aligned}$$

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These sets cannot be vector spaces, since if $p \ne 0$ is positive, $-p$ is not. So the concept of vector spaces has to be weakened:

Definition 1

A convex cone is a subset C of $F_{n,d}$ that obeys:

• $p,q \in C \quad \Rightarrow \quad p+q \in C$

• $p \in C, \lambda \ge 0 \quad \Rightarrow \quad \lambda p \in C$

$P_{n,d}$, $\varSigma _{n,d}$ and $Q_{n,d}$ are closed convex cones.

The definition of our constraint and its properties depend on the choice of a scalar product for forms.

Definition 2

A scalar product on $F_{n,d}$ can be defined as

$$\begin{aligned}{}[p,q] = \sum _i {d \atopwithdelims ()i} \, p_i \, q_i. \end{aligned}$$

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For $Q_{n,d}$, this scalar product has a particularly simple form:

Lemma 1

For $p = \sum _k \langle \mathbf a _k, \cdot \rangle ^d \in Q_{n,d}$ and $q \in F_{n,d}$ we have

$$\begin{aligned}{}[q,p] = \sum _k q(\mathbf a _k). \end{aligned}$$

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Proof

By the multinomial theorem:

$$\begin{aligned} \langle \mathbf a , \mathbf x \rangle ^d = \left( \sum _{i=1}^n a_i \, x_i \right) ^d = \sum _{i_1 + \dots + i_n = d} {d \atopwithdelims ()i} a^i \, x^i \end{aligned}$$

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And so

$$\begin{aligned}{}[q,p]&= [q, \sum _k \langle \mathbf a _k, \cdot \rangle ^d] = \sum _k [q, \langle \mathbf a _k, \cdot \rangle ^d] \\&= \sum _k \sum _i {d \atopwithdelims ()i} q_i \cdot (a_k)^i = \sum _k q(\mathbf a _k). \end{aligned}$$

$\square$

Also note that $[\cdot ,\cdot ]$ corresponds to the usual scalar product for tensors and [p, p] is the square of the Frobenius norm $\Vert p\Vert _F$.

In order to derive a matrix representation, we want to reduce a form of even degree $d=2s$ to $d'=2$. For this, we need

$$\begin{aligned} L(\mathbf x ,t) = \sum _i x^i \, t_i \end{aligned}$$

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with a vector of variables t indexed by $i \in I(n,s)$. For fixed t, this is a $F_{n,s}$ form in $\mathbf x$. For fixed $\mathbf x$, this is a linear form in t.

Definition 3

For $p \in F_{n,2s}$, the Hankel form is the quadratic form

$$\begin{aligned} H_p(t) = [p, L^2(\cdot ,t)] = \sum _{i,j} p_{i+j} \, t_i \, t_{j} \, \in F_{|I(n,s)|,2}. \end{aligned}$$

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We can put this into matrix form as $H_p(t) = t^T H_p \, t$. For $p \in F_{3,4}$, the matrix is

$$\begin{aligned} H_p = \begin{pmatrix} p_{xxxx} &{} p_{xxxy} &{} p_{xxxz} &{} p_{xxyy} &{} p_{xxyz} &{} p_{xxzz} \\ p_{xxxy} &{} p_{xxyy} &{} p_{xxyz} &{} p_{xyyy} &{} p_{xyyz} &{} p_{xyzz} \\ p_{xxxz} &{} p_{xxyz} &{} p_{xxzz} &{} p_{xyyz} &{} p_{xyzz} &{} p_{xzzz} \\ p_{xxyy} &{} p_{xyyy} &{} p_{xyyz} &{} p_{yyyy} &{} p_{yyyz} &{} p_{yyzz} \\ p_{xxyz} &{} p_{xyyz} &{} p_{xyzz} &{} p_{yyyz} &{} p_{yyzz} &{} p_{yzzz} \\ p_{xxzz} &{} p_{xyzz} &{} p_{xzzz} &{} p_{yyzz} &{} p_{yzzz} &{} p_{zzzz} \end{pmatrix}. \end{aligned}$$

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Definition 4

A form p with positive semidefinite $H_p$ will be called H-psd.

$$\begin{aligned} H_{n,d} = \{p \in F_{n,d} : H_p(t) \ge 0 \; \forall \, t\} \end{aligned}$$

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The method we propose enforces the H-psd constraint on fODFs during deconvolution.

In the rest of this section, we will discuss some properties of the set $H_{n,d}$ that are relevant to our method. The main tool will be duality:

Definition 5

The dual cone of a convex cone C is the set

$$\begin{aligned} C^\star = \{p \in F_{n,d} : [p,q] \ge 0, \; \forall \, q \in C \} \,. \end{aligned}$$

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If C is a closed convex cone, then (see Reznick [21])

$$\begin{aligned} C^{\star \star } = C. \end{aligned}$$

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Theorem 1

$P_{n,d}$ and $Q_{n,d}$ are dual to each other:

$$\begin{aligned} {Q_{n,d}}^\star = P_{n,d}, \qquad {P_{n,d}}^\star = Q_{n,d} \end{aligned}$$

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Proof

$$\begin{aligned} p \in {Q_{n,d}}^\star \;&\Longleftrightarrow \; 0 \le [q,p] = [\sum _k \langle \mathbf a _k, \cdot \rangle ^d, p] \nonumber \\&\qquad = \sum _k p(\mathbf a _k), \quad \forall \, q \in Q_{n,d} \\&\Longleftrightarrow \; 0 \le p(\mathbf a ), \quad \forall \, \mathbf a \in \mathbb {R}^n \\&\Longleftrightarrow \; p \in P_{n,d} \end{aligned}$$

The second equation is a consequence of ${Q_{n,d}}^{\star \star } = Q_{n,d}$.

$\square$

In the special case of $(n,d)=(3,4)$, the H-psd constraint is equivalent to decomposability into rank-1 terms. This can be shown with the following two theorems:

Theorem 2

(Hilbert)

$$\begin{aligned} P_{n,d} = \varSigma _{n,d} \end{aligned}$$

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if and only if $n=2$ or $d=2$ or $(n,d) = (3,4)$.

Theorem 3

$$\begin{aligned} {\varSigma _{n,d}}^\star = H_{n,d} \end{aligned}$$

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Proof

Observe that $t \mapsto L(x,t)$ is a bijection between vectors $t\in \mathbb {R}^{|I(n,s)|}$ and forms in $F_{n,s}$. So:

$$\begin{aligned} p \in {\varSigma _{n,d}}^\star \;&\Longleftrightarrow \; 0 \le [p, q] = [p, \sum _k h_k^2] \quad \forall \, q \in \varSigma _{n,d} \\&\Longleftrightarrow \; 0 \le [p, h^2] \quad \forall \, h \in F_{n,s} \\&\Longleftrightarrow \; 0 \le [p, L(\cdot , t)^2] = H_p(t) \quad \forall \, t \in \mathbb {R}^{|I(n,s)|} \end{aligned}$$

$\square$

A direct consequence for $(n,d)=(3,4)$ is:

Corollary 1

$p \in F_{3,4}$ is a sum of fourth powers iff it is H-psd, since

$$\begin{aligned} Q_{3,4} = {P_{3,4}}^\star = {\varSigma _{3,4}}^\star = H_{3,4}. \end{aligned}$$

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For higher degrees, the relation is weaker:

$$\begin{aligned} Q_{3,n} \subsetneq H_{3,n} \subset P_{3,n} \end{aligned}$$

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Another property of the H matrix is that it can be used to estimate the number of fibers in an fODF.

Definition 6

The rank of $p \in Q_{n,d}$ is the smallest integer ${\text {rank}}(p)=r$ for which $\mathbf a _1,\dots ,\mathbf a _r \in \mathbb {R}^n$ can be found with

$$\begin{aligned} p = \sum _{k=1}^{r} \langle \mathbf a _k, \cdot \rangle ^d. \end{aligned}$$

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For the cases in Hilbert’s Theorem 2, the ranks of p and $H_p$ are equal as shown in theorem 4.6 in [21]. In general, ${\text {rank}}(p) \ge {\text {rank}}(H_p)$.