Joint Morphometry of Fiber Tracts and Gray Matter Structures Using Double Diffeomorphisms

Abstract

This work proposes an atlas construction method to jointly analyse the relative position and shape of fiber tracts and gray matter structures. It is based on a double diffeomorphism which is a composition of two diffeomorphisms. The first diffeomorphism acts only on the white matter keeping fixed the gray matter of the atlas. The resulting white matter, together with the gray matter, are then deformed by the second diffeomorphism. The two diffeomorphisms are related and jointly optimised. In this way, the first diffeomorphisms explain the variability in structural connectivity within the population, namely both changes in the connected areas of the gray matter and in the geometry of the pathway of the tracts. The second diffeomorphisms put into correspondence the homologous anatomical structures across subjects. Fiber bundles are approximated with weighted prototypes using the metric of weighted currents. The atlas, the covariance matrix of deformation parameters and the noise variance of each structure are automatically estimated using a Bayesian approach. This method is applied to patients with Tourette syndrome and controls showing a variability in the structural connectivity of the left cortico-putamen circuit.

Appendix

We compute here the gradient of the criterion for atlas construction. A variation in the white system $\delta \varvec{L}^{W}_{i0}$ produces a variation in the path of control points and momenta $\delta \varvec{L}^{W}_i(t)$ and consequently $\delta \varvec{T}^{W}_i(t)$. In parallel, $\delta \varvec{L}^{All}_{i0}$ produces a variation in $\delta \varvec{L}^{All}_i(t)$ which , together with $\delta \varvec{T}^{W}_{i1}$, induces a variation in the global template $\delta \varvec{T}^{All}_i(t)$ and then in the criterion $\delta E$:

$$\begin{aligned} \delta E = \sum _{i=1}^{N} \nabla _{\varvec{T}^{All}_{i1}}(D_i[\varvec{T}^{All}_{i1}])^T \delta \varvec{T}^{All}_{i1} + \nabla _{\varvec{L}^{All}_{i0}}(R[\varvec{L}^{All}_{i0}])^T \delta \varvec{L}^{All}_{i0} + \nabla _{\varvec{L}^{W}_{i0}}(R[\varvec{L}^{W}_{i0}])^T \delta \varvec{L}^{W}_{i0} \\ \begin{aligned} \delta \dot{\varvec{L}}^{All}_i(t)&= (d_{\varvec{L}^{All}_{it}}F^{All}_i(t))^T \delta \varvec{L}^{All}_{it}&\delta \varvec{L}^{All}_i(0)&= \delta \varvec{L}^{All}_{i0}\\ \delta \dot{\varvec{L}}^{W}_i(t)&= (d_{\varvec{L}^{W}_{it}}F^{W}_i(t))^T \delta \varvec{L}^{W}_{it}&\delta \varvec{L}^{W}_i(0)&= \delta \varvec{L}^{W}_{i0}\\ \delta \dot{\varvec{T}}^{W}_i(t)&= (\partial _{\varvec{T}^{W}_{it}} G^{W}_i(t))^T \delta \varvec{T}^{W}_{it} + (\partial _{\varvec{L}^{W}_{it}}G^{W}_{i}(t) )^T \delta \varvec{L}^{W}_{it}&\delta \varvec{T}^{W}(0)&= \delta \varvec{T}^W_0\\ \delta \dot{\varvec{T}}^{All}_i(t)&= (\partial _{\varvec{T}^{All}_{it}} G^{All}_{i}(t))^T \delta \varvec{T}^{All}_{it} + (\partial _{\varvec{L}^{All}_{it}}G^{All}_{i}(t)] )^T \delta \varvec{L}^{All}_{it}&\delta \varvec{T}^{All}_i(0)&= \delta \varvec{T}^W_{i1} \cup \delta \varvec{T}^G_0 \end{aligned} \end{aligned}$$

As in [5] we denote: $R_{st}=\exp (\int _s^t d_{\varvec{L}_{u}}F(u) du)$ and $V_{st}=\exp (\int _s^t \partial _{\varvec{T}_{u}} G(u) du )$ which are valid for both frameworks W and All and where we have omitted the index i for clarity purpose. The two first ODEs are linear whereas the last two are linear with source term: $\delta \varvec{L}(t) = R_{0t}\delta \varvec{L}_0$, $\delta \varvec{T}(t)=\int _0^t V_{ut} \partial _{\varvec{L}_u}G(u) \delta \varvec{L}(u) du + V_{0t} \delta \varvec{T}_0$. Calling $\varvec{Y}^W$ and $\varvec{Y}^G$ the white and gray matter objects of $\varvec{T}^{All}$:

$$\begin{aligned} \delta \varvec{Y}^W(t)=\left( \int _0^t V_{ut}^{All} \partial _{\varvec{L}^{All}}G^{All}(u)R_{0u}^{All}du \right) \delta \varvec{L}^{All}_0 + V_{0t}^{All}\delta \varvec{T}^W_0 +\\ V_{0t}^{All}\int _0^1 \partial _{\varvec{T}^W}G^W(s) \delta \varvec{T}^W(s) ds + V_{0t}^{All}\int _0^1 \partial _{\varvec{L}^W} G^W(s) \delta \varvec{L}^W(s) ds \end{aligned}$$

Using the Fubini’s theorem, the $3^{rd}$ term is equal to: $\left( V_{0t}^{All} \int _0^1 V_{u1}^W \partial _{\varvec{L}^W} G^W(u)\right.$ $\left. R_{0u}^W du \right)$ $\delta \varvec{L}^W_0$ – $\left( V_{0t}^{All} \int _0^1 \partial _{\varvec{L}^W}G^W(u) R_{0u}^W du \right) \delta \varvec{L}^W_0$ + $(V_{0t}^{All}V_{01}^W) \delta \varvec{T}^W_0$ – $V_{0t}^{All} \delta \varvec{T}^W_0$. The $4^{th}$ term becomes: $\left( V_{0t}^{All} \int _0^1 \partial _{\varvec{L}^W}G^W(s) R_{0s}^W ds \right) \delta \varvec{L}^W_0$. Plugging them into $\delta E$:

$$\begin{aligned} \begin{aligned}&\nabla _{\varvec{L}^{All}_0}E=\left( \int _0^1 (R_{0u}^{All})^T (\partial _{\varvec{L}^{All}} G^{All}(u))^T (V_{u1}^{All})^T du \right) \nabla _{\varvec{T}^{All}_1}D + \nabla _{\varvec{L}_0^{All}} R^{All} \\&\nabla _{\varvec{L}^{W}_0}E=\left( \int _0^1 (R_{0u}^{W})^T (\partial _{\varvec{L}^{W}} G^{W}(u))^T (V_{u1}^{W})^T du \right) (V_{01}^{All})^T \nabla _{\varvec{Y}^{W}_1}D + \nabla _{\varvec{L}_0^{W}} R^{W} \\&\nabla _{\varvec{T}^W_0}E=(V_{01}^W)^T (V_{01}^{All})^T \nabla _{\varvec{Y}^{W}_1}D \\&\nabla _{\varvec{T}^G_0}E=(V_{01}^{All})^T \nabla _{\varvec{Y}^{G}_1}D \end{aligned} \end{aligned}$$

Calling $\theta ^{All,G}(t)= (V_{t1}^{All})^T \nabla _{\varvec{Y}^{G}_1}D$, $\theta ^{All,W}_t= (V_{t1}^{All})^T \nabla _{\varvec{Y}^{W}_1}D$, $\theta ^{All}(t)=$ $\{\theta ^{All,G}(t)$, $\theta ^{All,W}(t) \}$, $\theta ^{W}(t)=(V_{t1}^{W})^T \theta ^{All,W}_0$, $\xi ^{All}(t)=\int _t^1 (R_{tu}^{All})^T(\partial _{\varvec{L}^{All}} G^{All}(u))^T \theta ^{All}(u) du$ and $\xi ^{W}(t)$ = $\int _t^1 (R_{tu}^{W})^T$ $(\partial _{\varvec{L}^{W}} G^{W}(u))^T \theta ^{W}(u) du$ we obtain the results in Eq. 6.