urOMT (unbalanced regularized optimal mass transport)

Introduction

This project works on an unbalanced version of regularized optimal mass transport (urOMT) problem by adding a new relative source variable and its indicator function into the formulation. Specaifically, we deal with a problem as follows:

Given the initial mass distribution function $\rho_0(x)\geqslant0$ and the final one $\rho_1(x)\geqslant0$ defined on a bounded region $\Omega\subseteq\mathbb{R}^3$, one solves

$$\underset{\rho,v,r}{\text{min}}\quad \int_0^T\int_{\Omega}\big(\left\lVert v(t,x)\right\rVert^2\rho(t,x) + \alpha \chi(t,x)r(t,x)^2\rho(t,x)\big)dx dt $$ subject to

$$\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho v) = \sigma\Delta\rho + \chi\rho r, $$

$$\rho(0,x) = \rho_0(x), \quad\rho(T,x) = \rho_1(x)$$

where a temporal dimension $t\in[0,T]$ is added to the transport process. In the above expression, $\rho(t,x)$ is the dynamic density function; $v(t,x)$ is the velocity field defining the flow from $\rho_0$ to $\rho_1$; $r(t,x)$ is the relative source variable; $\chi(t,x)$ is the given indicator function of $r(t,x)$ which takes values either 0 or 1 to constrain $r$ to a certain spatial and temporal location; constant $\sigma>0$ is the diffusion coefficient; $\alpha>0$ is the weighting parameter of the source term in the cost functional.

This work can be considered as the extension of the regularized optimal mass transport (rOMT) problem with code available at https://github.com/xinan-nancy-chen/rOMT and https://github.com/xinan-nancy-chen/rOMT_spdup and with papers available at

-- Visualizing fluid flows via regularized optimal mass transport with applications to neuroscience,

-- Cerebral amyloid angiopathy is associated with glymphatic transport reduction and time-delayed solute drainage along the neck arteries.

The numerical method of current urOMT problem is described in

-- Unbalanced Regularized Optimal Mass Transport with Applications to Fluid Flows in the Brain

and please cite this paper if you would like to use this code.

Contact Xinan Chen at chenx7@mskcc.org for questions.

Pipeline of This Project Applied in Glymphatics in Brain

Sample cases for demonstration

In this section, we use two datasets to briefly show how our method can quantify and visualize the dynamic fluid flows.

(A) Gaussian Spheres

We created five successive Gaussian-based spheres of size $50\times50\times50$ to test our urOMT algorihtm on. One can go to script create_gaussian.m to check how we created the synthetic data if interested. The five input images are plotted in 3D rendering as follows.

To run urOMT algorithm and its Eulerian and Lagrangian post-processing, one can run script driver_gauss.m with default parameters, which takes about 3 hours and 20 minutes on a cluster with 3 CPUs. The visualized results will pop up automatically.

Here we plot the returned Eulerian results, the time-averaged Eulerian speed map (left) and the time-averaged Eulerian relative-source map (right) as follows.

We also plot the returned Lagrangian results, the pathlines (top-left), the velocity flux vectors (top-right), the speed-lines (bottom-left) and the Péclet-lines (bottom-right) as follows.

(A) Rat Brain MRI

We also test out method on real Dynamic Contrast Enhanced MRI (DCE-MRI) data which is from a 3-month-old healthy rat brain. The follows is the sequence of the 15 input images shown in 3D rendering from right-lateral view (colorbar = 'jet', limits = [0,300]).

One can use script driver_RatBrain.m with default parameters to run urOMT algorithm and its post-processings, which takes about 9 hours on a cluster with 3 CPUs. The visualized results will pop up automatically.

The Eulerian results are as follows, the time-averaged Eulerian speed maps (top row) and time-averaged Eulerian speed maps (bottom row).

The Lagrangian results are as follows, the pathlines (top-left), the velocity flux vectors (top-right), the speed-lines (bottom-left) and the Péclet-lines (bottom-right).

One can alternatively use softeware VisIt 3.3.1 (open source) to visualize the Lagrangian results.