Project 12: 1D-3D Blood-Network/Tissue Oxygen Control - NMOPT

Goal¶

Develop a coupled nonmatching 1D-3D optimal control problem for oxygen transport from a vascular network into tissue. The control is a local change of vessel radii, constrained by physiological bounds, and the target is a measured tissue oxygen concentration such as a BOLD MRI proxy.

This is the most advanced project in the list.

Mathematical Formulation¶

Let $\Omega\subset\mathbb R^3$ be the tissue domain and let $\Lambda$ be a 1D vascular network embedded in $\Omega$ . The tissue oxygen concentration is $c:\Omega\to\mathbb R$ , and the vessel concentration is $c_\Lambda:\Lambda\to\mathbb R$ .

The control is a radius field

r=r_0+u \qquad\text{on }\Lambda,

(1)

with box constraints

r_a(s)\le r(s)\le r_b(s).

(2)

A simplified static model is:

-D_t\Delta c+\kappa(c-c_\Lambda\circ\Pi_\Lambda)\delta_\Lambda=0 \qquad\text{in }\Omega,

(3)

coupled to a 1D transport/reaction equation on the network,

-\frac{d}{ds} \left( D_\Lambda(r)\frac{dc_\Lambda}{ds} \right) + q(r)\frac{dc_\Lambda}{ds} + \kappa_\Lambda(r)(c_\Lambda-c|_\Lambda) = g_\Lambda \qquad\text{on }\Lambda.

(4)

Here:

$\delta_\Lambda$ denotes exchange supported on the embedded network;
$\Pi_\Lambda$ maps tissue points to the closest network point in the exchange model;
$q(r)$ is the flow rate obtained from a static Bernoulli or Poiseuille-type network model;
$D_\Lambda(r)$ and $\kappa_\Lambda(r)$ may depend on the radius.

A simple algebraic network-flow closure is

q_e(r) = C_e r_e^4 (P_i-P_j)

(5)

on each vessel segment $e=(i,j)$ , together with Kirchhoff conservation at interior network nodes:

\sum_{e\ni i} \sigma_{ie}q_e(r)=0.

(6)

The optimization problem is

\min_{(c,c_\Lambda,r)} \frac12\|c-c_d\|_{L^2(\Omega)}^2 + \frac{\alpha}{2}\|r-r_{\mathrm{ref}}\|_{L^2(\Lambda)}^2 + \frac{\gamma}{2}\left\|\frac{dr}{ds}\right\|_{L^2(\Lambda)}^2

(7)

subject to the coupled 1D-3D equations and the bounds on $r$ .

The reduced gradient contains three contributions:

g_r = \alpha(r-r_{\mathrm{ref}}) - \gamma r'' + \partial_r\mathcal C(c,c_\Lambda,p,p_\Lambda,r),

(8)

where $\mathcal C$ denotes the coupled 1D-3D residual tested with tissue and network adjoints $(p,p_\Lambda)$ .

The constrained first-order condition is

(g_r,\widetilde r-r)_{L^2(\Lambda)}\ge 0 \qquad \forall \widetilde r\in [r_a,r_b].

(9)

Implementation Hints¶

Start with fixed flow $q$ and fixed radius-dependent exchange.
Then add the radius-to-flow network model.
Use nonmatching coupling ideas from Project 09.
Use projected gradient first; PDAS is a natural extension.
Keep a very small synthetic network for the first tests.

Relevant Examples¶

Course code: codes/dealii/execs/inverse_poisson_kkt.cc for parameter identification and box constraints.
Course code: parabolic and projected-gradient classes for optimization structure.
deal.II: step-60 for nonmatching grid coupling.
deal.II: mixed-dimensional coupling examples and embedded-manifold techniques where available.

Deliverables¶

A minimal 1D network solver.
A 3D diffusion solver with embedded source/exchange terms.
A reduced-gradient or projected-gradient optimization loop for vessel radii.
Synthetic reconstruction tests from manufactured oxygen data.
A discussion of identifiability: which radius perturbations can actually be seen in tissue oxygen measurements?