Bifurcated Topological Optimization for IVIM

Abstract

In this work, we shed light on the issue of estimating Intravoxel Incoherent Motion (IVIM) for diffusion and perfusion estimation by characterizing the objective function using simplicial homology tools. We provide a robust solution via topological optimization of this model so that the estimates are more reliable and accurate. Estimating the tissue microstructure from diffusion MRI is in itself an ill-posed and a non-linear inverse problem. Using variable projection functional (VarPro) to fit the standard bi-exponential IVIM model we perform the optimization using simplicial homology based global optimization to better understand the topology of objective function surface. We theoretically show how the proposed methodology can recover the model parameters more accurately and consistently by casting it in a reduced subspace given by VarPro. Additionally we demonstrate that the IVIM model parameters cannot be accurately reconstructed using conventional numerical optimization methods due to the presence of infinite solutions in subspaces. The proposed method helps uncover multiple global minima by analyzing the local geometry of the model enabling the generation of reliable estimates of model parameters.

Keywords: diffusion MRI; diffusion microstructure; global optimization; intravoxel incoherent motion; separable non-linear least squares; simplicial homology; variable projection.

Figure 1

(A) 0-simplex (point), 1-simplex (edge), 2-simplex (triangle), and a 3-simplex (tetrahedron). (B) 0-chain of vertices, a 1-chain of edges and a 2-chain of simplices. (C) Directed 2-simplex in the directed simplicial complex (left), star domain defined by st(v_i) (center), and it's boundary defined as $\partial (\mathrm{\text{st}}(v_i))=\overline{v_2{v}_3}+\overline{v_3{v}_5}-\overline{v_5{v}_4}-\overline{v_4{v}_2}$ (right).

Figure 2

The above figure demonstrates the levels of optimization involved in TopoPro. (A: Level-1) Depicts the steps involved in solving the reduced functional and estimating the parameters. The reduced functional of only the non-linear parameters has been visualized with the simplicial homology corresponding to its objective function. (B: Level-2) Shows the valley function formed in the process of optimizing the full-functional in the second level. It delineates a problem of bi-modality where the points corresponding to different minima have been indicated in red color.

Figure 3

Shows a sketch of the forward mapping φ(x) from the true model coordinates in the model space (A) to the signal measurements in the data space (B) and its inverse denoted by φ⁻¹(x). (B) Notice that some signal measurements in the data space correspond more to perfusion and some correspond more to diffusion. This is a mixing phenomenon. It is important to note that the deviation from the ideal signal is due to different levels of noise (ε). F(x) represents the true model coordinates in the model-space and G(x) is the approximation of F(x) due to noisy measurements. (C) Depicts the randomly oriented capillary geometry for the perfusion process which is modeled as a “pseudo-diffusion” process (D^*). fe^−bD* is the compartment used to represent this random process. (D) Denotes the diffusion process of the water molecules represented by an isotropic compartment fe^−bD.

Figure 4

Showing the use of simplicial homology groups to visualize the two biophysically feasible global minima via a surface plot and contour plot. The chain-complex associated with both the minima is highlighted in each case with arrows showing correspondences. Both sub-domains contain a valley of infinite physically plausible solutions for the IVIM problem, one high perfusion and one low perfusion sub-domain. The minima of the depicted objective function surface and its corresponding contour plot is homologous to 5 connected 2-tori.

Figure 5

Shows results with the simulated Shepp-Logan phantom for different tumors. In (A,B), we show a qualitative comparison of the simulated data at different SNRs by comparing TopoPro against segmented, Bayesian, SHGO and MIX fitting. The ground truth for each of the model parameters f and D^* are depicted at the top of their respective estimates with different methods. (C) Compares the Root Mean Squared Error of the fits for both D^* (C1) and f (C2) and its average for all model parameters (C3) for each SNR. Notice that TopoPro outperforms the other fitting methods by 10X in the estimation of D^*.

Figure 6

Comparisons of TopoPro on two different real human brain datasets vs. 4 different methods: (A) DWI dataset with dimensions: 256 × 256 × 54 × 21 - slice 33 and (B) DWI dataset with dimensions: 140 × 140 × 84 × 49 - slice 42. Notice how TopoPro provides improved estimates of the different parameters of the IVIM model in both data. The results are compared against methods: Segmented, Bayesian, SHGO, and MIX.

Figure 7

(A) Here, we demonstrate the test-retest reliability of the TopoPro algorithm. On the same data, we split the data in half by choosing alternating b-values. We show using an inverse penalty function that we can improve the stability across all parameters. We plot the first half of the data-split along Y-axis and the other along X-axis. Notice how the reliability (Pearson Correlation Coefficient) improves across all parameters calculated for each parameter, i.e., Perfusion Fraction (f), Diffusion Coefficient (D), and Perfusion Coefficient (D^*). (B) We use the simplicial homology optimizer to show the infinite solution space of D^* at a particular value of D when f = 0. This has been depicted with the surface plot of simplicial complex (B1) and its corresponding contour plot (B2). Notice that all values in the solution subspace for D^* are equally good in this case, making it hard for the optimizer to find a solution.