Fig 1.
Workflow of algorithm for whole brain flow simulation.
A FreeSurfer segmentation of the T1-weighted acquisition is used to define a brain mask from the union of white and grey matter. The ToF and QSM images were used to segment arteries and veins, respectively. Within the vascular network the flow is implemented according to the Hagen-Poiseuille equation. Flow in the brain is represented by Darcy flow, and coupled with Hagen-Poiseuille flow in the vascular tree using locally distributing source terminals. The resulting linear system is solved for the pressure. Flux is directly proportional to the pressure drop, and is used to compute tracer transport as a function of time. Finally, the workflow provides an in-silico model of whole brain perfusion and indicator dilution.
Fig 2.
Geometry of a frog tongue with segmentation labels from dark to bright: Background (black), arterial network (red), venous network (blue), and tongue tissue (white).
The sharp corners of the tongue tissue occur when the tongue is pinned down to the surface prior to imaging. The arterial input and venous outlet are in the bottom of the picture.
Fig 3.
Automatic detection of vascular networks is divided into two consecutive workflows.
(I) In the first part, (A) a connected, binary mask of the vascular network is generated from the input image (i.e. the ToF or the QSM in the case of a real application), here represented as a synthetic image demonstrating a small network of vessels. (B) Segmentation by adaptive thresholding creates a first approximation to the vascular network. However, due to local dropout in the signal, the segmented map also contains a satellite structure disconnected from the root structure. Computing the distance function around the root structure with the image itself as a speed function generates a favorable map which can be used for backtracing from the satellite structure to the root structure. This procedure generates a most probable path connecting these two structures (green path). (C) End points of the resulting, connected vascular network are either root points or leafs. (II) In the second part, from the connected network in (I), we identify leafs, root points, the skeleton, as well as network nodes. (D) Computing a distance function around the binary segmentation generates a map for a second backpropagation. A consecutive backpropagation from leaf 1 and 2 towards the root ensures a connected skeleton of the network. In addition, the procedure provides the nodes as the points of intersection of two paths of backpropagation, here indicated by the red arrow intersecting the green path.
Fig 4.
Illustration of an arterial network with nodes Ni, i = {0, … 3} and connecting edges.
Each edge has an associated length Ljk, radius Rjk and medial axis (dashed lines). In the current example, N0, N2, N3 are terminal nodes, while N0 is also a root terminal mediating incoming fluid into the vascular network. Node N1 is an interior node. Brain tissue is shown as a filled ellipsoid, and arrowheads indicate direction of flow. For the sake of illustration, only the arterial network is shown, but a similar, fluid collecting venous network is also present in the model.
Fig 5.
Vascular network of the frog tongue.
Node centers are indicated with black dots. The medial axis of the network structure is shown as the set of lines connecting the nodes (red lines: arterial network, blue lines: venous network). The grey area within the tongue tissue indicates the support function ηϵ(x − xk) (15) summed up for all terminals. The red rectangle in the lower field is the small FOV used for demonstrating scale invariance.
Fig 6.
Pressure maps pa and pv of arterial (left) and venous compartment (right) of the frog tongue, respectively.
Note the different greyscale range between the plots, applied for visualization purposes.
Fig 7.
Regional variability of the perfusion P [ml/min/100ml] within the frog tongue.
Fig 8.
Average fluid tracer concentration [mmol L−1] as a function of time for the arterial input function (AIF), and for the arterial and venous compartment of the frog tongue.
The average was calculated over the frog tongue.
Fig 9.
Voxelwise volumetric tracer concentration C [mmol L−1] (22) as a function of time for the frog tongue.
Every five second is shown from left to right and top to bottom, T = {0, 5, …, 60} s.
Fig 10.
Average brain perfusion computed within the same FOV but under various multiplicative resolution scales.
All other simulation parameters were identical across the resolution scales. The average perfusion is converging at higher resolution scales.
Table 1.
Scalar simulation parameters for the frog tongue within various domains.
Matrix size was limited by the native input data. Arterial and venous boundary pressure values were applied as Dirichlet boundary conditions to the vascular root terminals NR. The perfusion proportionality factor α(x) (7) was assigned a constant value everywhere. par. = parameter. Field parameter α is only valid for the brain .
Table 2.
Simulation parameters for the human brain geometry within the various sub-domains.
Arterial and venous boundary pressures were applied as Dirichlet boundary conditions to the vascular root terminals NR. Permeability and porosity are field parameters, but were assigned constant values within each tissue and compartment. par. = parameter.
Fig 11.
3D volume rendering of the T1-weighted data including the arterial (red) and venous (blue) vessels.
Fig 12.
Map of the perfusion proportionality factor α(x) used as input to the simulations.
Higher perfusion was assigned to grey matter than to white matter in order to resemble regional distribution of perfusion within a real human brain.
Fig 13.
One slice of the calculated pressure maps pa and pv of the arterial (left) and venous (right) compartment, respectively.
Fig 14.
Obtained voxelwise perfusion P [ml/min/100ml] (7) for one axial slice.
Fig 15.
Average tracer concentration time curves within the arterial input function, as well as the arterial and venous compartments of the human brain dataset.
Table 3.
Obtained average perfusion , arterial
and venous
pressure, time to peak (
), and mean transit time (
) for various brain regions in the whole brain simulation.
Fig 16.
Relative sensitivity coefficients for the perfusion proportionality parameter α, porosities ϕa, ϕv, vascular permeabilities ka, kv, the pressure drop parameter γa, γv, and fluid viscosity μ investigated for the compartmental pressures pa, pv, the perfusion P, time to peak (TTP), and mean transit time (MTT) for the frog tongue.
Brighter values indicate higher sensitivity of the input parameter to the output parameter. Left: Relative sensitivity coefficient averaged over the voxels for the 3D human brain example. Right: Voxelwise, relative sensitivity coefficients from the frog tongue showing the relation between α and P.