Network-driven anomalous transport is a fundamental component of brain microvascular dysfunction

The brain’s microvascular network is essential for distributing blood-borne oxygen and nutrients and clearing metabolic waste. Ageing and disease disrupt this system through abnormal vessel architectures, reduced regulation, and stalling, leading to lower flow, hypoxia, and compromised clearance of neurotoxic species such as amyloid-β. A central open problem is how the non-local network architecture of arterioles, venules, and the capillary bed shapes heterogeneous blood flows and travel times, and how these heterogeneities affect oxygen delivery and waste clearance. Prior empirical models (e.g., indicator dilution and Capillary Transit Time Heterogeneity models) use assumed travel-time distributions calibrated to short measurement windows (<5 s), limiting predictive power for long times that may control the emergence of hypoxic or amyloid-β–laden vessels. This study asks: what physical network mechanisms generate the observed broad distributions of flow and travel times, do they produce anomalous transport with heavy tails, and how do these dynamics impact the early development of critical regions under hypoperfusion? The authors aim to derive physics-based scaling laws and effective transport models directly tied to network structure, and to quantify consequences for oxygen supply and amyloid-β clearance.

Empirical approaches to microvascular transport include classical indicator dilution frameworks and the Capillary Transit Time Heterogeneity (CTH) model, which often assume gamma-distributed travel times. These models capture that increased transit-time heterogeneity can reduce oxygen extraction efficiency and have been used to interpret imaging data, including findings of altered heterogeneity in active cortical layers and in Alzheimer’s disease. However, they lack mechanistic links to microvascular topology and are calibrated over short times due to blood recirculation, potentially underestimating late-time probabilities. Theoretical analyses in random networks show anomalous transport with power-law travel-time tails well described by Continuous-Time Random Walk (CTRW) theory. Although brain microvasculature is not a random network, its architecture—tree-like arterioles/venules feeding a dense capillary mesh—may provide the ingredients for anomalous dynamics. Related work in porous and network flows (dipole flows, spatial Markov models, CTRW for Lagrangian velocities) suggests that network-driven velocity fluctuations can yield heavy-tailed transit-time statistics, but this has not been mechanistically established in realistic brain microvascular geometries.

Data and networks: The primary dataset is an anatomically realistic mouse vS1 cortical microvascular network (~15,000 vessel segments in ~1 mm³), with vessel diameters corrected to in vivo distributions and vessels classified as arterioles, capillaries, or venules. A second mouse dataset and several biomimetic, space-filling networks (including random networks) are used for comparison and generality checks.

Blood flow simulation: A nonlinear network flow model incorporates microcirculatory blood rheology and hematocrit partition at bifurcations, with physiologically based boundary conditions to compute pressures, flows (Q), and hematocrit in each vessel. Validation against in vivo metrics is provided.

Lagrangian transport: 5×10^7 passive particles are injected at arteriolar inlets with probability proportional to inlet flow, advected at local average vessel velocity, and split at junctions in proportion to downstream fluxes. For each trajectory, the number of visited vessels (trajectory length L), the sequence of local advection times (t), and total travel time (T) are recorded. Diffusive cutoff: for each vessel, the local transit time is truncated by diffusion, t = min(t_advection, l^2/D), to capture species-dependent diffusion (oxygen D≈2×10^-9 m^2/s; amyloid-β D≈6×10^-11 m^2/s).

Statistical characterizations: Flow-rate distributions P(Q) and vessel transit-time distributions p_t(t) are quantified. Vessel transit times follow t = lπd^2/(4Q); with weak variability in l and d relative to Q, transit-time variability is dominated by Q. In the network, trajectory-length distributions P_L(L) and mean travel time versus length T(L) are measured.

Mean-field model: Using the observed dipole-driven trajectory-length distribution (power law L^-2 between L0 and Lc with exponential cutoff beyond Lc) and the empirical T(L) relation (linear regime for L0≤L≤L1, then saturation/power-law trend), a mean-field mapping yields a travel-time PDF with a transition from power-law to stretched-exponential behavior, capturing early-to-intermediate times.

CTRW framework: To capture late-time dynamics driven by capillary-scale velocity fluctuations, a CTRW model is developed where (i) the number of steps (vessels) per trajectory is broadly distributed per P_L(L); (ii) the mean local transit time at step n depends on trajectory length L via the measured T(L) structure; (iii) fluctuations around the mean follow a heavy-tailed noise determined by the flow distribution. Analytical expressions are derived in Laplace space for the travel-time PDF p_T(T), predicting a late-time power-law tail p_T(T) ~ T^-α.

Reactive transport coupling: First-order kinetics are assumed for oxygen consumption (dCO/dt = -kO CO) and for amyloid-β clearance towards a tissue equilibrium (dCA/dt = -kA (CA - CA^e)). Integrating these kinetics over the travel-time distribution yields outlet-to-inlet concentration ratios. Parameters are estimated to match established physiological constraints: oxygen extraction fraction ~30% gives τO ≈ 1.5 s; typical intravascular amyloid-β arterio-venous increase ~20% gives τA ≈ 97 s. Critical times τc for defining hypoxic or high-amyloid conditions are then evaluated.

Hypoperfusion scenarios: Average flow ⟨Q⟩ is scaled to mimic hypoperfusion (due to reduced perfusion pressure or increased microvascular resistance, including capillary occlusions). The travel-time PDF under changed ⟨Q⟩ is obtained by time rescaling relative to baseline. Fractions of trajectories or vessels exceeding critical travel times are computed for oxygen (τc ≈ 3.7 s, corresponding to CO drop to 1/12 of inlet) and for amyloid-β (τc = 8, 16, 40 s representing increasing compromise of clearance). Reference CTH predictions (gamma-distributed travel times calibrated to short-time data) are compared.

Oxygen field mapping: Using particle histories and first-order decay, 3D maps of intravascular oxygen concentration are computed, and averaged along trajectories of equal length to evaluate depth-dependent trends and minima relative to hypoxic thresholds (~10 mm Hg).

• Flow and transit-time distributions: Vessel flow rates span ~7 decades around ⟨Q⟩0 ≈ 4×10^-2 nL/s, with a uniform low-flow regime and a Q^-2 power-law tail above a characteristic Qc ≈ 10^-2 nL/s; about 60% of vessels have Q<Qc and 40% have Q>Qc. This is well approximated by a Cauchy distribution, consistent with dipole flow superposition near arterioles and venules.
• Vessel transit times: Transit-time PDF is Cauchy with characteristic tc ≈ 0.1 s; due to network size (~15,000 vessels), the power-law regime extends up to ~10^4 tc, yielding non-negligible probabilities of very long transits (e.g., ~1% of vessels with t>5 s = 50 tc). Diffusion imposes species-dependent cutoffs, strongly limiting oxygen but not amyloid-β.
• Trajectory statistics: Trajectory lengths L range from <10 to ~80 vessels, with P(L) ~ L^-2 between L0 ≈ 12 and Lc ≈ 50, transitioning to an exponential tail with scale L* ≈ 5 beyond Lc. The observed distributions and spatial patterns reflect dipole-driven flows from arterioles to venules and random-like capillary fluctuations.
• Travel time vs length: Average travel time grows roughly linearly with L in a shallow network regime, T(L) ≈ T1(L−L0)+T0 with T0 ≈ 0.2 s and T1 ≈ 1.8 s, then transitions to a slow power-law-like increase characteristic of 3D dipole flow in the deep capillary bed (T(L) ~ L^1/4 behavior). Overall travel times span 2–3 orders of magnitude after diffusion cutoffs (oxygen ~2, amyloid-β ~3).
• Mean-field vs CTRW: The mean-field model (based on P(L) and T(L)) captures early-to-intermediate travel-time statistics and oxygen dynamics; a full CTRW model, incorporating step-time fluctuations, reproduces the late-time heavy tail p_T(T) ~ T^-α (stable anomalous transport). A reference CTH (gamma) model underestimates late-time probabilities.
• Oxygen fields and hypoxic risk: Simulated intravascular oxygen decreases with depth, reaching minima near 1/12 of inlet (~10 mm Hg) in deep capillaries for longest trajectories, consistent with experimental thresholds for hypoxia. Under baseline perfusion, the fraction of travel times exceeding the hypoxic critical time τc ≈ 3.7 s is ~1.3%. With hypoperfusion, this fraction rises nonlinearly (e.g., ~4× increase for a 2× flow reduction). Mean-field predictions match simulations and exceed CTH estimates by up to two orders of magnitude at baseline.
• Amyloid-β clearance sensitivity: Due to low diffusivity, amyloid-β is highly influenced by late-time dynamics; CTRW predictions match non-exponential increases in the fraction of critical travel times for τc = 8, 16, 40 s across hypoperfusion levels, while CTH underestimates these probabilities by orders of magnitude.
• Mechanistic origin: Heavy tails arise from the combination of dipole-like flow topology (broad P(L) and high-flow neighborhoods of arterioles/venules) and random capillary-scale velocity fluctuations (uniform low-flow regime), establishing anomalous transport as a network-driven property.

The study shows that microvascular network architecture enforces two key ingredients of anomalous transport: a broad distribution of trajectory lengths dictated by dipole-like arteriolar–venular organization, and a broad distribution of capillary transit times driven by random-like velocity fluctuations. These combine to produce heavy-tailed travel-time distributions at the network scale. For high-diffusivity species (oxygen), diffusion suppresses late-time fluctuations, making mean-field descriptions adequate over clinically accessible times, yet still revealing a measurable baseline fraction of hypoxic pathways and a nonlinear sensitivity of hypoxic risk to hypoperfusion. For low-diffusivity species (amyloid-β), the late-time power-law tail dominates, greatly increasing the probability of inefficient clearance events relative to gamma-based models. These mechanistic insights address the research question by linking transport statistics directly to network architecture, moving beyond empirical parameterizations. The results are significant for interpreting perfusion and functional imaging data, for understanding early microvascular contributions to Alzheimer’s disease, and for identifying deterministic spatial hotspots (long-trajectory regions) susceptible to hypoxia and amyloid accumulation. The framework suggests that moderate hypoperfusion and sparse capillary occlusions can trigger disproportionate increases in critical regions, potentially initiating pathological feedback loops in neurovascular dysfunction.

This work establishes physics-based laws for transport in cortical microvascular networks and provides analytical models (mean-field and CTRW) that predict full travel-time distributions from network structure and flow statistics. It demonstrates that network-driven anomalous transport (with heavy-tailed travel times) is a fundamental feature of brain microcirculation, leading to earlier and more prevalent hypoxic and high–amyloid-β regions under hypoperfusion than predicted by empirical models. Contributions include: (1) identifying dipole-flow and capillary-scale fluctuations as the mechanistic sources of anomalous transport; (2) deriving predictive transport models that match simulations without fitting; (3) quantifying nonlinear increases in critical travel-time fractions with hypoperfusion; and (4) mapping 3D oxygen distributions consistent with in vivo observations. Future research should extend simulations to larger domains (up to whole brain), incorporate multiphasic blood and more realistic reactive kinetics (e.g., hemoglobin cooperativity), and couple intravascular and tissue transport-metabolism. Experimental validation of late-time travel-time tails in brain microcirculation and integration into clinical imaging analyses could improve diagnosis and staging of vascular contributions to neurodegeneration.

• Empirical travel-time measurements in vivo are limited to <5 s due to recirculation, constraining calibration and external validation of long-time predictions. Direct experimental confirmation of late-time power-law tails in brain microcirculation is lacking.
• Reactive kinetics are simplified as first-order processes; oxygen-binding cooperativity and complex amyloid-β isoform dynamics are not explicitly modeled.
• Blood is treated with average vessel velocities (no transverse Poiseuille profiles), and single-phase representations; micro-scale intra-vessel velocity profiles are neglected.
• Parameter values (e.g., critical times, diffusivities) and network-specific features may vary across species, regions, and pathological states; while scaling laws appear robust, prefactors may change.
• Boundary conditions and network reconstructions, though anatomically realistic, represent limited cortical volumes; finite-size effects and subcortical architectures may alter quantitative outcomes.
• Hypoperfusion is modeled by global scaling or capillary occlusions without fully modeling compensatory mechanisms (autoregulation) or dynamic remodeling.
• Amyloid-β metabolism and clearance pathways are simplified; defining critical thresholds (τc) involves uncertainty.