4.1 Modelling framework

In order to investigate the effects of micro-emboli on the cerebral microvasculature, a modelling framework must first be developed. There are two main ways in which the cerebral microvasculature can be modelled–either as a network of individual vessels [33–35], or as an averaged porous medium [30, 31, 36, 37]. We model the microvasculature as a porous medium in order to allow large regions of the brain to be modelled computationally feasibly.

The microvasculature will be parameterised following on from previous research developed by our group [32, 36, 38]. Firstly, geometrically representative models are developed which match morphological properties of the human cerebrovascular network (taken from ex-vivo casts of human brains) [39–42]. These statistically generated models match geometric parameters such as length distributions, vessel radius distributions, vessel densities, bifurcation angles and connectivity. The microvasculature is then split into 2 different spatial scales–the capillary bed, and the penetrating arterioles. The capillary bed is generated statistically through a modified spanning tree method, where the aforementioned geometric parameters are matched over an ensemble average of the networks [43]. The penetrating vessels are also generated statistically, matching geometric parameters that fit ex-vivo data of the human cerebral microvasculature [38].

In order to scale up the microvasculature to larger regions, the capillary bed blood flow is parameterised with a permeability, as are the penetrating arterioles. The flow from the penetrating arterioles to the capillary bed is parameterised through coupling coefficients–which represent the conductance of the pre-capillary arterioles. The parameterisation method is explained in Section 4.7. A schematic of the overarching model, along with example networks of the different microvascular scales and their coupling, can be seen in Fig 1.

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Fig 1.

A schematic representation of the overarching model simulating micro-emboli in the brain post-thrombectomy. Two separate models are developed in this paper: a) a micro-emboli model of the capillary bed (based on geometrically representative networks) and, b) a micro-emboli model of the penetrating arterioles. The grey region in b) represents the homogenized healthy capillary bed to which the terminal vessels of the trees couple. The red planes denote the dividing planes between the 6 layers of the voxel over which the blood flow characteristics will be calculated. c) Is a porous multi-compartment representation of the brain which will depend on the parameters derived from the models in a) and b). The full brain is not simulated in this paper.

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4.2 In-vitro emboli quantification

Previous in-vitro studies generated micro-emboli distributions from thrombectomy procedures. In brief, a silicone cerebrovascular model built from clinical imaging data was used in a flow loop with a peristaltic pump to simulate blood flow through the cerebral vasculature [24, 44]. A thrombus was injected into the cerebrovascular replica to form a middle cerebral artery (MCA) occlusion. The thrombus was either defined as a soft or hard clot, and validated for bulk mechanical properties based on clinical specimens [45]. Four different clot removal techniques (thrombectomy) were then deployed to remove the clot from the MCA. Specifics on the 4 thrombectomy techniques–abbreviated as ADAPT, Solumbra, BGC, and GC at cervical ICA–can be found in Chueh et al. [20]. In brief, the Solumbra, BGC, and GC at cervical ICA techniques all retrieve the clot through Solitaire FR devices. These are delivered with a balloon guided catheter (BGC), or with a distal access catheter at the proximal aspect of the clot in the MCA (Solumbra), or with a guide catheter with the tip at the cervical internal carotid artery (GC at cervical ICA). ADAPT was a direct aspiration as a first pass technique.

The number and size distribution of the clot fragments that break off the thrombus during thrombectomy are quantified using a Coulter counter (Beckman Multisizer 4, Brea, California). Clot fragments were binned according to size, ranging from 8.03 μm to 1200 μm. The number of fragments ranged from a few hundred to 20,000 in the smallest bin, but decreased rapidly with increasing bin size. Each removal technique is repeated 8 times for each clot type, resulting in 64 sets of clot fragment data. When sampling from this data for our in-silico models the 8 experimental micro-emboli density distributions (for a given removal technique and clot type) are averaged such that for any given technique and clot type there is only one average set of micro-emboli results relating number density and particle size. The smallest clot measurable, due to the aperture used in the Coulter counter, was 8.034 μm in diameter, with smaller clots binned under this size.

4.3 Simulating blood flow in the penetrating arterioles

Prior work has shown how blood flow can be simulated in a coupled model of the penetrating vessels and a porous representation of the capillary bed [36]. This model is adapted here such that it can be used to determine the permeability of the penetrating arterioles in health and with micro-emboli occluding the vessels.

An example voxel, where the penetrating arterioles are coupled to the capillary bed, is shown in Fig 1B. Flow through the penetrating vessels is assumed to be steady-state, non-pulsatile (Womersley number << 1) and fully developed, hence approximated by Poiseuille flow: (1) where q is the blood flow through the vessel, R is the radius of the vessel, ΔP is the pressure drop across that vessel, L is the length of the vessel, and μ_vivo is the apparent viscosity of blood in the vessel correcting for the Fåhræus-Lindqvist effect with a constant discharge haematocrit of 0.45 [46] (see S1 Text).

As flow is conserved at each bifurcation node i, the net flow at each node in the tree is zero except for the one inlet node and the multiple terminal nodes that end in the capillary bed. Therefore, the flow through each internal node for n nodes can be written as: (2) where G_ij is the flow conductance between nodes i and j and is defined as as in Eq (1). The conductance is zero unless there is a vessel between nodes i and j. P_i and P_j are the pressures at nodes i and j respectively. The inlet flow and terminal flows are handled by the boundary conditions at the inlet to the arteriole and at the coupling points to the capillary bed. This set of n equations can be written in matrix form and solved as in Su et al. [43].

The capillary bed can be modelled as a porous medium using a volume-averaged form of Darcy’s law [32, 47]: (3) where u_c is the volume averaged capillary velocity, K_c is the permeability (as the permeability is isotropic this can be treated as a scalar), and ∇p_c is the pressure gradient across the capillary bed. The permeability K_c encapsulates the micro-scale geometry of the capillary bed in one averaged parameter, hence allowing for large-scale regions to be modelled computationally efficiently. This equation can be combined with mass conservation to give: (4) where S is the volumetric source/sink term [m³ s^-1 m^-3] that comes from the terminal nodes of the penetrating vessels. The healthy capillary permeability value has previously been found be 4.28x10^-4 mm³ s kg^-1 [32]. Eq (4) is the Poisson equation that can be solved for capillary pressure over an appropriate 3D domain.

4.4 Coupling the penetrating vessels and the porous capillary bed

The coupling between the penetrating vessel terminal nodes and the porous capillary bed has a mathematical singularity at the point of coupling between 1D flow and 3D tissue. Previous models have approximated the Dirac delta singularity with a function that is spatially variant yet maintains mass conservation [30]; have approximated the pressure drop in the vicinity of the coupling point to deal with the strong pressure gradients that appear there [48]; or have simply taken the discretised element nearest the terminal node as the outlet region [36, 49]. Here, we use a spatially variant function to approximate the volumetric source terms [30]. A shape function is introduced that replaces the Dirac delta singularity with an exponential in the vicinity of the source. The shape function, η, is defined as (5) where x is the distance measured from the source/sink node and ϵ is the radius of the sphere over which the flow will be distributed from the arteriole source to the capillary bed (here set as 15 μm). C is a constant that acts to normalise η to ensure mass conservation across the capillary bed elements coupled to the terminal tree node i.e. C is set such that ∫η(x)dx = 1 for a given terminal node. Therefore, the volumetric source term for each finite element within ϵ radius of the terminal node will be (6) where S_k is the volumetric source term for one element within ϵ radius of the terminal node and x_k is the coordinate of the terminal node. Note that S_i here is the same volumetric source/sink term as in Eq (4) with the subscript i denoting one of the terminal node sources/sinks that couple to the porous capillary bed. In effect, a summation over the terminal nodes i is used to account for the potential that multiple terminal nodes may feed an individual element. As the shape function η conserves mass, ∫η(x)dx = 1.

4.5 Solving for blood flow under healthy conditions in the penetrating arterioles

The voxel size chosen to simulate the flow is 2 x 2 x 2.5 mm (on the order of an MRI voxel), with 8 arterioles/mm² penetrating into the voxel [50]. This voxel was situated in a capillary bed of 2.25 x 2.25 x 2.5 mm to avoid terminal nodes being at the face of the voxel (Fig 1B). As the parameterisation of micro-emboli occluding the penetrating arterioles was primarily of interest here, the venules were not included in the simulation. As well as this, the permeability of the capillary bed was chosen such that the maximum possible flow could be obtained through the arterioles (in this case K_c was set to 4.2x10^-5 mm³ s kg^-1). The arterioles have a mean diameter of 20 μm ± 5 μm, with a mean length of 1.25 mm ± 0.31 mm. An inlet pressure of 12.3 kPa is imposed at the tree inlets [51]. Dirichlet boundary conditions were imposed at the top and bottom of the voxel with P_top = 12.3 kPa at z = 0 and P_bottom = 8 kPa at z = 2.5 mm, with periodic boundary conditions on the 4 other faces.

The finite element method is used to simulate the capillary bed partial differential equation using FEniCS, an open source finite element solver [52]. Eq (4) is solved with the parameters and boundary conditions using 870,000 linear Lagrangian elements to obtain the pressure in the capillary bed. The equations above were solved iteratively. The pseudo-algorithm below details the solution steps:

1. Set Dirichlet boundary conditions on the top and bottom of the voxel and periodic boundary conditions on the 4 sides;
2. For all tree outlets coupled to the capillary bed, determine finite elements in porous capillary mesh coupled to the outlet node, and determine their η weightings using Eq (5)–this determines how much flow from the pre-capillary arteriole is distributed to each coupled element in the capillary mesh;
3. Set desired inlet pressures for penetrating vessels and set initial terminal outlet pressures = 0
4. Solve for the pressure in the trees using Eqs (1) and (2). Hence solve for the outlet flows from the trees into the capillary bed. These outlet flows act as the volumetric source terms in the capillary mesh and are distributed to the coupled finite elements using weightings from Eqs (5) and (6);
5. Set the volumetric source terms in Eq (4) and solve for the capillary bed pressure using the finite element method;
6. Update penetrating vessel terminal node pressure values using a weighted average of the capillary bed finite element pressures coupled to the given terminal node;
7. Go to step 4 and repeat, unless a steady state is reached where pressures in terminal nodes and capillary bed have converged to a tolerance of 1x10^-3 Pa;

4.6 Simulating micro-emboli post-thrombectomy

In order to simulate the effect of micro-emboli on the microvasculature, an algorithm is presented here that samples from the in-vitro experimental thrombectomy data and occludes vessels sequentially. The algorithm flowchart is schematically represented in Fig 2.

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Fig 2. Flowchart representation of the micro-emboli occlusion algorithm.

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A clot technique and clot consistency are chosen from one of our 8 clot distributions as described in Section 4.2. The algorithm initially runs the healthy flow solver above to establish baseline healthy blood flow. A clot size is then sampled pseudo-randomly using Python’s inbuilt ‘random’ module from the experimental distributions derived from the in-vitro models [20].

A penetrating arteriole is randomly chosen on the pial surface through which the clot is sent. The clot is assumed to be perfectly spherical, impermeable, and does not interact with the vessel walls (which are also assumed rigid). A check is then made to determine whether the clot is larger than the diameter of the vessel it is entering. If the clot is smaller than the vessel, the clot is assumed to travel until the next bifurcation. At the bifurcation, the clot enters one of the two daughter branches based on a probability that is linearly weighted by the blood flow in the daughter branches (based on Eq (1), the volumetric flow rate and hence clot distribution depends on the vessel length, radius, pressure drop and viscosity). For example, if one daughter vessel has 3 times as much flow as the other daughter vessel, the clot has a 75% chance of going down the vessel with more flow.

The clot continues to travel through vessels if its diameter is smaller than the diameter of the vessel it is entering. If the clot enters a vessel with a smaller diameter, it is assumed that the clot completely occludes that vessel. When that occurs, it is assumed no blood can enter the occluded vessels or any vessels downstream of the occluded vessel. The algorithm recalculates a baseline blood flow and resamples a clot to loop through the steps again.

If the clot reaches a terminal vessel, and it is still smaller than the vessel, it is assumed that the clot generates no occlusions in the penetrating vessels and enters the capillary bed. As we are trying to quantify the damage done by micro-emboli to the penetrating vessels, we do not create capillary bed occlusions in this simulation. The effect of micro-emboli on the capillary bed is thus considered separately in Section 4.9. As such, no vessels in the penetrating tree are occluded, and a new clot is sampled to loop through the algorithm again.

The algorithm described, shown in Fig 2, is repeated until only one penetrating vessel remains or until 50,000 clots have been sampled, whichever comes first. The large number of maximum clots is chosen such that we can ensure that all the vessels are eventually occluded, allowing us to track the damage done by micro-emboli sequentially.

4.7 Parameterising the penetrating arterioles

There are 2 specific parameters of interest when attempting to model the penetrating vessels over a large scale–the permeability and the coupling coefficients. The permeability of the penetrating vessels represents the ability of the trees to drive blood flow deeper into the grey cortex due to their 1-dimensional nature [53]. The coupling coefficients represent the pre-capillary arterioles’ ability to deliver blood to the capillary bed. These parameters are necessary to simulate large regions of the human brain blood flow efficiently [31]. However, these parameters also effectively average out the micro-scale fluctuations in the vasculature, replacing a complex network with one parameter. This can be problematic with the penetrating vessels when considering, for example, the terminal nodes of the trees. It is highly unlikely they are distributed evenly through the grey matter with depth. It is also highly unlikely that the permeability of the trees is constant with depth through the voxel. We mitigate this problem by conceptually splitting the voxel into 6 layers, similar to other approaches [35, 54], and calculating the permeabilities and coupling coefficients for each of the 6 layers through the voxel, increasing the resolution of future large-scale simulations.

4.8 Investigating thrombectomy technique, clot composition, and individual microvasculatures

Based on the pipeline previously presented, we now seek to investigate whether the different micro-emboli distributions from thrombectomy have an impact on the permeability (and hence perfusion) of the penetrating arterioles following the introduction of micro-emboli. The micro-emboli algorithm is therefore used to simulate an emboli shower for 8 different scenarios– 4 techniques used on ‘hard’ or ‘soft’ clots.

As well as this, we investigate the impact that differing microvasculatures have on the resulting drops in permeability and coupling coefficients. Each of the 100 statistically accurate voxels can be thought of as an ‘individual’ microvasculature. We randomly choose 3 of these voxels and, for each voxel, we simulate 100 different sampled clot distributions entering the penetrating vessels. These simulations are compared to another simulation where we take our 100 voxels and run the same sampled clot distribution through each voxel–this is done for 3 different sampled clot distributions. The question that we seek to answer is whether it is the microvasculature or the clot distributions that primarily determine microvascular robustness to drops in perfusion. Differences between distributions are quantified using the correlation matrix distance (CMD) on the covariance matrices of each distribution [56] (see S2 Text).

4.9 Micro-occlusions in the capillary bed

Finally, the effect of micro-emboli on capillary beds is considered. Previously, we demonstrated how geometrically representative models of capillary beds can be homogenized into a porous medium [32]. This method is combined here with a vessel occlusion algorithm to simulate the effect of micro-emboli entering the capillary bed. The decrease in permeability as a function of the vessel fraction lost (number of vessels blocked over initial healthy number of vessels), surface area fraction lost (vessel surface area lost over initial healthy vessel surface area), and volume fraction lost (vessel volume lost over initial healthy vessel volume) are all quantified. This allows for the scaling up of partially occluded capillary networks to large regions of tissue.

Details of the blood flow simulation and homogenization of the capillary bed can be found in El-Bouri & Payne [32]. In brief, Poiseuille flow is assumed through the capillary vessels, with the capillary network being periodic. A pressure gradient is imposed in one direction and the permeability–the ratio of the volume-averaged flow and the pressure gradient–is calculated. This is repeated for the 3 principal directions of the capillary network. The permeability is found to be isotropic for healthy capillary networks, hence can be treated as a scalar [32].

As the smallest measurable clot was 8.034 μm, and the average diameter of a capillary is 6.23 ± 1.3 μm [39], the micro-emboli algorithm above could not be used on the networks as they would always occlude the inlet vessel. As such, a random vessel was chosen in the network and this vessel was fully occluded, with the permeability being recalculated after every occlusion. Three cube sizes were used to simulate the micro-occlusions with lengths of 375, 500, and 625 μm. Normalised permeability changes are calculated, and the effect of the micro-emboli on the capillary bed is quantified such that it can be used in full-brain simulations. Constrained least squares optimization is used to fit a line of best fit with the equation (12) where P_f is the permeability fraction, x_i is the change in either vessel fraction, volume fraction, or surface area fraction of the networks, and B is the gradient. P_f will be 1 when there are no occlusions and drops with increasing vessel blockages. All parameters calculated for the capillary bed are based on 500 statistically accurate capillary networks at each cube size (1500 total).

5.1 Healthy blood flow in the penetrating arterioles

Blood flow was simulated in 100 geometrically representative voxels of the cerebral microvasculature. The permeability and coupling coefficients were quantified for the penetrating arterioles in the ‘no-occlusion’ scenario. The permeability results are shown in Fig 3.

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Fig 3.

a) A scatter plot of the permeability of the 100 voxels at each of the 6 depth layers. The cyan triangle indicates the median permeability of the capillary bed. The mean arteriolar permeability is indicated with a red star at each layer. The arteriolar permeabilities (circles) are shaded differently to aid visualisation. b) A quadratic line of best fit over the median permeabilities at each layer–error bars are interquartile ranges.

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The permeability increases with depth until the middle layers before dropping off towards the bottom layers, as not all penetrating vessels penetrate the full depth of the grey matter (Fig 3A), and most vessels tend to branch from the penetrating tree in the middle layers. The mean value of the permeability across the layers is approximately 50 times the capillary permeability. At each layer the average permeability ranges from around 30x at the top layer, to 75x in layer 3, to 9x at the bottom layer. A quadratic curve fit is used to derive a function that models arteriolar permeability with depth, such that it can be used in future porous model implementations (Fig 3B). The permeability distribution at each layer can also be characterised as log-normal (Shapiro-Wilks test, α = 0.05) up to and including layer 4 (S1 Table).

The coupling coefficients demonstrate a similar drop from the top layers to the bottom layers, driven by the fact that there are fewer terminal nodes at the bottom layers [36]. The variability in the coupling coefficients from voxel-to-voxel is larger than the permeability variability (S1 Fig). The median values of the coupling coefficients are 2.8x10^-5, 4.4x10^-5, 4.3x10^-5, 2.7x10^-5, 1.1x10^-5, and 4.3x10^-6 Pa^-1 s^-1 (from layers 1–6). A quadratic curve-fit is used to derive a function that models the coupling coefficients with depth for future porous model implementations (S1 Fig). The coupling coefficient distribution can be characterised as log-normal (Shapiro-Wilks test, α = 0.05) up to and including layer 4 (S2 Table).

The above results parameterise the healthy penetrating vessels so they can be scaled up to large regions of the brain. More importantly, however, these healthy results provide a baseline from which the effect of micro-occlusions can be quantified.

5.2 The arterioles are robust to micro-emboli at the voxel population level

As there are 4 thrombectomy techniques used with 2 clot consistencies to give 8 different micro-emboli distributions to choose from, a baseline must be chosen to conduct the following simulations. Here we choose the ADAPT technique with a hard clot to present the following results. A comparison with other techniques and clot consistencies can be found in Sections 5.4, 5.5.

The variable of interest here is the fractional drop in permeability. This is defined as the permeability divided by the initial healthy permeability. We compare this independent variable against 3 dependent variables–the fraction of vessels occluded, the fraction of vessel surface area occluded, and the fraction of vessel volume occluded. The surface area and volume drops have been calculated as these geometric parameters control oxygen transport in the microvasculature and can in future be used for validation. Fig 4 shows these results for 2 different layers–the top-most grey matter layer, and the middle layer (from 1.25–1.67 mm depth). An example of the permeability decrease against volume fraction decrease in all 6 layers can be found in S2 Fig.

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Fig 4.

a,b) The fractional drop in permeability against the fraction of vessels blocked where a) is in the top layer of the voxel and b) is the middle layer. c,d) The fractional drop in permeability against the fraction of vessel surface area blocked where c) is in the top layer of the voxel and d) is the middle layer. e,f) The fractional drop in permeability against the fraction of vessel volume blocked where e) is in the top layer of the voxel and f) is the middle layer. The middle layer is at a depth of 1.25–1.67mm. A line of best fit is plotted in black, with the fit equation in the top right corner of each graph, with the bottom left of the graph giving the coefficients of the line of best fit and the R² goodness-of-fit metric.

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The permeability is generally robust with vessel fraction blocked, up to around 50% of the vessels before the permeability starts to drop more rapidly. This can be explained by the fact that many small vessels are occluded first–due to 98% of all emboli having a diameter of < 20 μm for this configuration of clot consistency and thrombectomy technique–which has a relatively small impact on the permeability. On the other hand, the permeability drops almost linearly with volume fraction blocked with a 1:1 ratio in the top layer.

Interestingly, in the middle layer for all 3 dependent variables, the permeability’s robustness to occlusions appears to improve over the first 40–50% of vessel/volume occlusions. As the vessels continue to be occluded, however, the drop off in permeability is quicker than in the topmost layers. The variability increases with depth until by the time the bottom two layers are reached, there is almost a dichotomous choice between a permeability of 1 and 0 (S2 Fig). This is due to the sparse number of vessels in the lower layers, leading to them being vulnerable to very few micro-emboli.

The permeability, and its drops with successive micro-emboli, can thus be characterised as a 2-dimensional surface. The equation for the healthy permeability distribution with depth can be multiplied, at each depth layer, by the respective permeability drop equation for that depth layer. An example of a 2-dimensional surface generated with volume fraction as the dependent variable, ADAPT as the thrombectomy technique, and a hard clot can be found in Fig 5 (calculated using the healthy permeability equation from Fig 3B). These 2-dimensional functions can be calculated for all the different possible combinations of thrombectomy, clot, and dependent variable. This provides a simple look-up table to determine the change in permeability for a given technique, clot, depth, and fraction occluded.

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Fig 5. A 2-dimensional surface plot for the ADAPT thrombectomy technique removing a hard clot.

The permeability is plotted against voxel depth and volume fraction of the vessels occluded.

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The coupling coefficients also display non-linear behaviour with micro-emboli occlusions, but this behaviour is far more variable than the permeability results. This is due to blood being rerouted as occlusions occur, leading to coupling coefficients increasing as well as decreasing from the baseline. As well as this, the coupling coefficients are also dependent on the pressure in the capillary beds and terminal arterioles (Eq (10)). An example of the change in coupling coefficients with depth and volume occlusions can be found in S3 Fig.

5.3 The microvascular geometry dictates robustness to micro-emboli smaller than the penetrating vessel inlet diameter

To investigate the role of the microvasculature on robustness to micro-emboli, 3 voxels were chosen from our 100 voxels. Each voxel had 100 micro-emboli simulations with each simulation having a different clot distribution sampled. Fig 6 shows the changes of permeability for these 3 voxels for the middle-layer. As can be seen, it appears that the microvascular topology dictates the response to micro-emboli, with clear banded regions with non-linear jumps for some voxels, whilst other voxels have an almost linear drop with increasing occlusions. These jumps in the permeability appear due to there being ‘pinch points’ in the networks where one occlusion causes a lot of vessels downstream to be lost.

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Fig 6.

A comparison of the permeability drop against volume fraction occluded for 3 different voxel geometries simulated with 100 different clot distributions (for the ADAPT technique, hard clot) a) Voxel 1, b) Voxel 4, c) Voxel 31.

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In comparison, we simulated 100 different voxels but with the same sampled clot distribution entering each voxel (we fix the random seed sampling the clot distribution to be the same for each voxel). The aim of this was to investigate the effect of the sampled clot distribution variability on perfusion characteristics (similar to how we investigated the effect of microvascular variability in Fig 6). There was little visual difference found for the 3 sampled clot distributions (S4 Fig). Similar behaviour is also seen for the coupling coefficients, with linear drops in coupling coefficients seen for some voxels (S5 Fig).

Quantifying the statistical differences between these distributions was complicated by the bivariate nature of the distributions. As such, to provide a simple comparison, we use the correlation matrix difference (CMD) to compare the covariances of each distribution against the baseline distribution (Fig 4F). The CMD has a value of 0 when the covariance matrices are identical and a value of 1 when they are orthogonal. When comparing the 3 constant microvascular simulations the range of the CMD was 1x10^-4 to 2.85x10^-3. For the constant clot simulations, the range was 7x10^-6 to 3x10^-4 indicating that variability and robustness to micro-emboli is dominated by the microvascular geometry.

5.4 Hard clot micro-emboli lead to larger drops in permeability

The above analysis has all been conducted using ‘hard’ clot fragmentation data. We now simulate ‘soft’ clot fragmentation and the effect of these micro-emboli on flow characteristics. We are interested to observe whether clot consistency impacts downstream flow post-thrombectomy.

Visually, there is no discernible difference for the relationship between permeability and the dependent variable on the population level, but certain voxels display changes in their permeability relationship (S6 Fig). Comparing individual microvascular geometries in both hard and soft clot simulations, the CMD score ranges between 3x10^-4 and 7x10^-3, indicating a similar variability to that found between voxel comparisons. The results are similar for the coupling coefficients.

Regardless of whether there was a visual difference or not, all voxels with soft clot simulations required on average twice as many occluding clots to develop the same drops in perfusion as for the hard clots. As well as this, around 4.5% of the clots sampled from the hard clot distributions went into occluding penetrating vessels (the rest passed through the arterioles into the capillary bed). For the soft clots, this was only 2.5% (Table 1). Clearly, for a given number of micro-emboli the smaller clot fragments of the soft clot lead to smaller drops in permeability.

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Table 1. Summary of the differences between thrombectomy technique and clot consistency when considering total micro-emboli sampled and those micro-emboli that occlude vessels.

The values are averaged over the 100 voxel simulations. The percentage of sampled clots occluding vessels is the average of the 100 individual percentages.

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5.5 Thrombectomy technique has a large impact on clot fragmentation and downstream flow characteristics

The last variable of interest here is the thrombectomy technique. In the in-vitro experiments, 4 techniques were used; these are abbreviated as ADAPT, BGC, GC at cervical ICA, and Solumbra [20]. The emboli distributions from these 4 techniques used on hard and soft clots were used to simulate occlusions in the penetrating vessels. The results are shown in Fig 7 where the scatter plots have been binned into deciles with mean and standard deviation for ease of viewing.

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Fig 7.

Comparison of the permeability drops against volume fraction blocked in the middle layer for 4 different thrombectomy techniques: a) removing hard clots and b) removing soft clots.

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Despite there being little visual difference between the 4 techniques, when analysing the numbers of clots required to reach similar drops in permeability, large differences emerge (summarised in Table 1). We measured: the average number of micro-emboli sampled in total to fully occlude the voxel (column 3 of Table 1); the average number of these micro-emboli that occlude a penetrating vessel (column 4 of Table 1); and hence the percentage of sampled clots that occluded the penetrating vessels (column 5 of Table 1). Simulating the Solumbra technique on a hard clot resulted in the fewest clots required to fully occlude the voxel; on average, over the 100 voxels, 18.6 clots (and hence 26% of sampled clots) sent through the penetrating vessels were required to fully occlude the voxel. The BGC technique had the next fewest clots required with 26 clots and 14% of the sampled clots. The GC at cervical ICA technique, on the other hand required 29 clots and 2.9% of the sampled clots.

In contrast, when using the soft clots, the numbers of clots required to achieve similar permeability drops drastically increased–except for GC at cervical ICA which had similar results compared to when using the hard clots. Both the Solumbra and ADAPT techniques saw a doubling of the clots required.

Clearly, there is a trade-off between the number of emboli that fragment off a clot, the size of these emboli, and the impact they have on downstream flow. The BGC and Solumbra techniques were previously reported to minimise the fragmentation of hard clots [20], but this also means that these clots are larger and hence more prone to occlude larger vessels (as indicated by our large percentage of clots that occlude vessels in the above results). On the other hand, the ADAPT and Solumbra techniques used on soft clots lead to twice as many clots being required to occlude the penetrating vessels to achieve similar drops in permeability.

5.6 Occlusions in the capillary bed lead to large drops in permeability

Finally, we simulated micro-emboli occluding capillary bed networks. These were simulated for 500 statistically accurate networks at 3 different cube length sizes (375 μm, 500 μm, and 625 μm). As the results were not statistically different over the 3 sizes, only the 375 μm results are presented in Fig 8. The permeability maintained its isotropy in the presence of micro-emboli hence it is presented here as a scalar.

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Fig 8. The drops in permeability over 500 statistically accurate capillary networks with a cube length of 375 μm.

Lines of best fit are in black, with the gradient B in the top right corner of each graph. a) The fractional drop in permeability with fraction of vessels occluded (B = −3.2), b) the fractional drop in permeability with vessel volume occluded (B = −4.1).

https://doi.org/10.1371/journal.pcbi.1008515.g008

The permeability drop in the capillary bed was large in comparison to the drop in the penetrating vessels: -8% / % surface area lost, -3.2% / % vessel blocked, and -4.1% / % volume blocked. This is in line with a previous study where a -2.5% drop in CBF/ % vessels blocked was found in simulations of occlusions in mouse and human vascular networks, as well as mouse synthetic networks [13]. It should be noted that the synthetic networks used in [13] had uniform diameter and length, unlike the networks in this paper which have length and diameter distributions that match physiological distributions.

Assuming a micro-emboli shower is spread evenly over the MCA territory (approx. 100 mL of tissue), 1.5% of this volume is capillary vessels, and a vessel density of 8000 vessels/mm³, around 120,000 micro-emboli would have to enter the capillary bed to cause a 4% drop in permeability (note that this is an overestimate as micro-emboli are not likely to spread evenly over the brain tissue). Depending on thrombectomy technique, the number of micro-emboli fragmenting off the clot can vary from a few thousand to over 500,000 [20]. It therefore appears that if micro-emboli enter the capillary bed there is significant potential for damage and large drops in perfusion.