2.1. Modeling of 0D-1D cardio–cerebrovascular networks with CAM

In the multiscale framework, large arteries in the extracted cerebral vascular network are modeled as axisymmetric and deformable tubes, through which blood flow is governed by the 1D Navier-Stokes equations. When passing through a vascular junction, flows across it are coupled via characteristic (Riemann‑invariant) relations, whereas the inflow and outflow segments within the network are respectively connected to the 0D heart model and the peripheral circulations represented by 0D lumped-parameter models.

The modeled cerebral arterial network (as depicted in Fig 1a), along with the anatomical and geometric parameters used to define its structure and function, was extracted from vessel segmentations of computed tomography (CT) images [14]. The reconstructed cerebral network contains the CoW, which is a ring of interconnected arteries at the base of the brain and shown in Fig 1b. It connects the anterior and posterior circulations and the left and right hemispheres, providing immediate diversion of blood flow in the acute occlusion cases. Fig 1c summarize the coupling of the whole‑body and heart-cerebral circulations.

We introduced the CAM to analyze the blood flow direction changes and redistribution under different pathological scenarios, providing patient-specific adaptability. The following introduces the detailed mathematical formulation of this multiscale model.

2.1.1. 0D vascular resistance network for cerebral arteries.

Cerebral autoregulation ensures stable cerebral blood flow despite fluctuations in systemic blood pressure. The mathematical model employed in this study is based on a network representation of the arterial system, in which each vessel segment is modeled as a resistance, and the flow rate Q across the segment is governed by the pressure drop ΔP according to Poiseuille’s law [15]:

(1)

Here, μ is the blood viscosity. Let N_node denote the total number of nodes in the vascular network, and A(i) is the set of nodes directly connected to node i. For any pair (i, j) with j ∈ A(i), l_ij, r_ij, and R_ij are respectively the length, radius, and vascular resistance of the segment connecting the two nodes i and j.

Next, we aim to achieve overall pressure equilibrium using iterative computations. The cerebral arterial network (Fig 1a) is represented by a nodal resistance network with multiple inlet sources and terminal outlets. With the flow conductance G_ij = 1/R_ij defined for directly connected node pair (i, j) with j ∈ A(i), the pressure distribution across the vascular network is determined by mass conservation at any node k:

where the net nodal flow Q_k is obtained by summing the flows Q_kj through all vessel segments connected to node k as defined in Eq. (1). By separating these connecting nodes into interior and boundary nodes, we obtain

with

where A_int(k) and A_b(k) are subsets of A(k) corresponding to interior and boundary nodes, respectively. It is worth noting that due to conservation of mass, the flow term Q_k should be zero for all nodes other than the inlet and outlet nodes within the network. In this way, the pressure distribution across the entire cerebral vascular network is determined by the mass-balance equation system:

(2)

where the conductance matrix B is defined as

and the conductance matrix G_b associated with boundary nodes is defined as

Here, the pressure and flow rate vectors (P_b, Q_b) correspond to the boundary nodes, including S inlets and E outlets as shown in Fig 1c. The inlet boundary conditions are determined by the upstream resistance network that distributes the cardiac output modeled by the 0D heart model, while the outlet pressure at terminal nodes are regulated by the CAMs, which are introduced in the following subsections.

2.1.2. Coupling the 0D vascular network with the 0D heart model.

The systemic inflow conditions of the cerebral network are provided by a lumped 0D heart model, which drives the entire closed-loop system as illustrated in Fig 1c. Several 0D models for simulating this circulation have been reported in detail by Sun et al. [16] and Liang et al. [17]. In this study, the time-varying elastance model and valve dynamics are employed for providing physiologically realistic pulsatile pressure and flow waveforms as inflow into the cerebral vascular network while maintaining low computational cost. The model simulates Q_av, which represents the flow rate from the left ventricle to the aorta, as follows:

(3)

with

Here, Q_tv and Q_vc denote the flow rates through the tricuspid valve and the vena cava, respectively. R_av represents the viscous resistance of the aortic valve, and B_av is the Bernoulli resistance of the valves, which is associated with convective acceleration and dynamic pressure losses. L_av and R_av respectively signify the coefficients for inertial terms and viscous losses. Cardiac valve function D_av is defined as an ideal diode model, which opens and closes instantaneously depending on the changing of the pressure gradient sign between the left ventricle (P_lv) and aorta (P_ao). The blood pressure (P_cm) in the four cardiac chambers including the right atrium (ra), right ventricle (rv), left atrium (la), and left ventricle (lv), is obtained by [17,18]

(4)

where E_cm,A and E_cm,B respectively represent the active and passive elastances of the four cardiac chambers. In addition, V_cm stands for the current chamber volume. V_cm,0 denotes the dead volume, assumed here as 0. S_v expresses the viscoelasticity coefficient of cardiac wall and e(t) signifies the normalized time-varying elastance. The detailed formulations and parameter values are shown in Supporting Information (S2 Text and S1 Table).

2.1.3. Coupling the 0D vascular network with a CAM.

While the 0D heart model provides the inflow boundary conditions for the 0D vascular resistance network, incorporating autoregulatory response at the outlets is important to physiologically simulate cerebral perfusion stabilized. Therefore, we wholly adopt the iterative CAM algorithm proposed by [15] as the outflow boundary condition for each outlet segment of our constructed 0D vascular network. In their framework, the cerebral vascular bed at each terminal node is modeled as a passive resistance, and provided an iterative scheme to relate the distal pressure P_outlet at iteration step k to the arteriolar compliance C_a and flow rate Q_outlet at iteration step k-1 by:

(5)

with

Here, P_v and P_ic respectively denote the venous and intracranial pressure. R_v and 𝑅_sa are the venous and arteriolar resistances, respectively. 𝑉_sa is the arteriolar volume. G_q is the gain of the flow feedback mechanism. C_a,0 is the baseline arteriolar compliance, Δ𝐶⁺_𝑎 and Δ𝐶⁻_𝑎 define the upper and lower limits of compliance variation. The parameter values used to define the autoregulating part for simulation are given in Supporting Information (S2 Text and S1 Table). The iterative procedure continues until the convergence criterion is satisfied:

(6)

Then, the mass-balance system in Eq. 2 is closed by boundary conditions at the inlets through 0D heart model (Eqs.(3)–(4)) and the outlets through CAM (Eqs. (5)–(6)). After solving it for the nodal pressure {P_i}, and therefore the flow rate in each vascular segment is given by

(7)

with Q_ij positive from i to j, uniquely determining the flow direction throughout the network.

2.1.4. Coupling the 0D vascular network with 1D flow model.

In this subsection, we extend the 0D cerebral vascular network incorporated with CAM proposed by [15] into a multiscale (0D–1D) framework. Specifically, based on the nodal pressures simulated by the 0D models (Eqs.(2)–(6)) and the corresponding flow distribution and direction using Eq.(7), we prescribe them as boundary conditions for each 1D vessel segment and perform 1D hemodynamic simulations to capture spatial and temporal characteristics of blood flow. The 1D flow simulation Q(x,t) is obtainable by averaging the incompressible Navier-Stokes equations over the cross-sectional area A(x,t) under the assumptions that the blood flow is axisymmetric and that it has no swirl [17]. The well-established equations of conservation of mass and momentum is represented as [18, 19]:

(8)

where t represents the time and where x denotes the axial coordinate along the vessel. Variables A(x, t), Q(x, t), and P(x, t) respectively denote the cross-sectional area of the vessel, the volume flux, and the average pressure over a cross-section. Here, ρ represents the blood density, µ represents the blood viscosity, E_s is the Young’s modulus, and h is the wall thickness. P_e stands for the external pressure, A₀(x) = A(x, t₀) signifies a reference cross-sectional area at a given time point t₀, and P₀ denotes the reference pressure for A = A₀.

Based on the simulated A(x,t) by the 1D solver, the resistance R_ij of each segment considered in the 0D vascular network can be updated at time step k by

(9)

where γ is a given relaxation factor to avoid numerical stiffness. The 0D linear system is re-assembled and solved to update node pressures and edge flows, reflecting the elastic response of the vessel to flow rates.

The 0D and 1D models are coupled through a sequential iterative procedure (see Fig 2 and S1 Text). In each outer cycle, the 0D system including the heart model, the cerebral resistance network, and the CAM, is first solved over one cardiac cycle τ to determine the nodal pressures P_i(t) and flow rates Q_ij(t) at each segment. These quantities are then used as boundary conditions for the 1D blood flow solver, which computes the spatially resolved flow Q(x, t) and cross-sectional area A(x, t) within each vessel segment. The resulting 1D solutions are subsequently used to update the effective resistances of the corresponding vessels via Eq. (9). This bidirectional coupling for a cardiac cycle τ is repeated until reaching a quasi-steady state, in which the results of two consecutive cycles (n-1, n) become nearly identical, as defined by:

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Fig 2. Flow chart of the proposed multiscale model incorporated with CAM and the DFS-based path-flow inversion via an L1-regularized convex optimization.

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(10)

Here, N are the set of vessels and where num(N) represents the number of elements of set N.

2.2. Coupling the multiscale model with Stenosis models

While the multiscale model provides a comprehensive framework for vascular hemodynamics and can be applied to various network structures, including those describing anatomical variations that are topologically equivalent to complete occlusions, it does not explicitly account for the localized flow resistance arising from pathological narrowing (stenosis). To address this limitation, we further incorporate an experiment-based stenosis model proposed by Young and Tsai [20] and reformulated equivalently by Liang et al. [19], which characterizes the additional pressure drop across a stenotic segment with highly variable cross-sectional areas. Specifically, the stenosis model is integrated into the 0D vascular network (Eq.(2)) by expressing the effective resistance for vessel segments affected by stenosis as [19,21]:

(11)

Here, the subscripts 0 and S respectively correspond to the reference (healthy) segment and the stenotic segment. D₀ denotes the healthy vessel diameter, and L_s is the length of the stenosis. The remaining coefficients were obtained from an extensive series of in-vitro steady-flow experiments conducted by Seeley and Young [22], as:

The first term in Eq.(11) accounts for viscous losses, while the second term captures inertial effects associated with geometric contraction. Moreover, the degree (severity) of stenosis R_s is quantified by the percentage reduction in the vessel diameter relative to the healthy segment:

(12)

The coupling method between the two models is carried out as follows: the flow rate Q and cross-sectional A obtained from the 0D-1D multiscale model are used as inputs to the stenosis model. The effective resistance R^eff_ij, updated with the stenosis contribution, is then substituted back into the resistance coefficient matrix in Eq.(2) for subsequent computations. In the 1D solver, the stenosis is represented as a localized reduction of the reference cross-sectional area within the affected vessel segment, capturing the local hemodynamic effects of the segment narrowing. This approach enables the multiscale framework to account for the pressure losses across stenoses, thereby improving the physiological fidelity of the simulations.

2.3. DFS-based feasible path set and sparse inversions of path-flow

In this subsection, we introduce a post-processing analysis methodology applied to the converged hemodynamic solution, which does not alter the simulation results, but instead provides an interpretable decomposition of the simulated blood flow at the segments. It is designed to provide further insight into the computed flow distribution and to quantitatively estimate how blood flow is distributed across multiple candidate vascular paths connecting predefined source and sink nodes. This approach may provide useful information for assessing flow redistribution under different pathological conditions, including potential applications in bypass surgical planning. To this end, we formulate a convex optimization-based strategy that combines flow conservation constraints with sparsity-prior regularization. This framework has been widely employed to address various networked flow problems, e.g., estimating traffic matrices from link loads in communication networks [23], detecting blockages in water distribution networks [24], and optimizing the traffic-power flow for the least-cost social optimum states [25]. Since blood transport within the cerebral vascular network also follows conservation principles and tends to minimize the total energy cost as characterized by Murray’s law [26-28], the proposed approach is expected to effectively infer the source-sink flow matrix of the cerebral circulation governed by similar physical constraints.

Let x ∈ 𝑅^m denote the unknown vector of blood flow assigned to each of the m candidate paths to terminal nodes which are first selected by identifying all possible paths using a depth-first-search (DFS) method, and then further refined by discarding those inconsistent with the quasi-steady blood flow directions predicted by the multiscale model. Based on the flow rate values Q computed by the multiscale model throughout the vascular network, we then impose physiological flow conservations at each vessel segment (Q_line), the inflow sources nodes (Q_start) (e.g., LICA), and the outflow sink nodes (Q_end) (i.e., the terminal arteries) to construct the constraint system as follows:

(13)

where variable x_p represents the actual volumetric blood flow through the p-th feasible path, along with L, S, and E respectively denoting the numbers of segments, sources, and sinks. PathsThro(l), PathsFrom(i), and PathsTo(j) respectively represent the sets of the feasible paths passing through the segment l, starting from node i, and terminating at node j.

We concatenate the three constraint matrices into a unified system Wx = b by:

(14)

with

where λ ₁ and λ₂ are weighting factors used to balance the influence of source and sink node constraints relative to segment-based constraints, thereby allowing flexibility in incorporating certain experimental data. For example, if reliable measurements of inflow rates at source nodes (e.g., ICA or VA from phase-contrast MRI) are available, a larger value of λ₁ can be used to enforce closer agreement with these observations.

However, the equation system that describe the conservation of flow across vascular segments is typically underdetermined due to the limited availability of internal measurements. In vascular trees, particularly within the cerebral circulation, blood flow distribution is normally assumed to follow principles consistent with Murray’s law, which is derived by optimizing the total power expenditure required to pump blood through a single segment [26,29]. Motivated by this principle and the physiological preference for energetically efficient trunk-branch routes, we regularize the path-flow inversion by combining a sparsity prior with a pumping-cost penalty and solve it by the following manner:

(15)

where δ>0 controls sparsity, i.e., minimizing the number of active paths, and σ >0 weights the total costs with the path-resistance R_p computed as the series sum of the resistance of all segments along path p. Thus, the inversion of path-flow is formulated as an L₁-regularized convex optimization problem with non-negativity constraints, which is solved in this study using the CVX modeling framework in MATLAB with its default interior-point solver (SDPT3 solver).

To further evaluate robustness of this optimization algorithm, we introduced multiplicative noise into the constraint vector b: b′ = b⋅(1 + ηξ), where ξ∼Uniform[−1,1] and η∈{0.05,0.10} represents a 5–10% perturbation of the flow results from the multiscale model. The perturbed L₁ problem (Eq. (15)) was solved 500 times, each with an independently perturbed input b′, to evaluate the stability of the inferred path flows. The box plot and stability scores are reported in Supporting Information (S3 Text), which consistently show that the dominant flow-carrying paths exhibit narrow confidence bands and high stability scores (close to 1), indicating that the sparse-prior solutions are robust against perturbations. With the solution x, we further define source‑to‑sink attributions by

(16)

providing a quantitative descriptor of how inflow sources and vessel segments within cerebral structures distribute their contributions to distal vascular territories.

2.4. Computational workflow and parameters for models

The cerebral arterial network was adopted from works reported by [14], while the remaining extracerebral vasculature, including connecting segments such as aortic arch that link the heart to the cerebral circulation, was derived from the studies by Toro et al. [30]. The overall simulation process based on this cerebral vascular network is summarized in Fig 2.

Due to the extensive number of vessel segments, detailed geometrical parameters are not listed in this paper. Parameters for the heart model are from papers by Sun et al. [16] and by Liang et al. [17], and illustrated in Supporting Information (Table A in S2 Text). Additionally, the values of parameters related to the CAM refer to the studies conducted by Payne [31] and by [15], and are provided in Supporting Information (Table B in S2 Text). A summary of the network size and model parameters is provided in Supporting Information (S1 Table: Nomenclature table).

Given these parameters, a physiologically realistic simulation of cardiac–cerebral circulation can be implemented. The 1D domains are divided into grids with a length of ∆x = 0.1 cm, ensuring at least ten computational grids per vessel segment with the time step ∆t = 5 × 10⁻⁴ s. The duration of a cardiac cycle τ is set to 1.0 s for iterative computations. Blood viscosity and density were respectively specified as µ = 0.0045 Pa·s and ρ = 1050 kg/m³. The initial conditions were set as Q(x, 0) = 0 and A(x, 0)=A₀.

3.1. Baseline cerebral flow distribution

Fig 3 provides a visualization of blood flow propagation over one cardiac cycle within the cerebral arterial network. The temporal evolution of flow highlights how pulsatile blood is distributed from the inflow arteries into the downstream territories. At early time points (t = 0.1 s), blood enters the cerebral vasculature through the VAs and ICAs. As time progresses, the flow gradually propagates toward distal branches, accompanied by a flow reduction in upstream vessels. This process reflects the spatiotemporal distribution of cerebral perfusion.g

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Fig 3. Snapshots at representative time points illustrate the propagation of pulsatile flow over one cardiac cycle (one second) from inflow arteries to distal branches within the cerebral arterial network.

The line color at each spatial–temporal position (x, t) represents the normalized values of blood flow Q(x,t)/ max_t |Q(x,t)|.

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For clarity, only the baseline case is depicted here, and videos illustrating the time-dependent flow propagation under all three conditions are available in the GitHub repository (see Data Availability Statement for details) to provide intuitive visualizations.

Subsequently, Fig 4 illustrates the baseline blood flow distribution within the entire cerebral vascular network and the complete CoW under normal physiological condition (i.e., no stenosis, with CAM considered).

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Fig 4. (a) Spatial distribution of blood flow in the entire cerebral vascular network with the complete CoW.

The outflow sink nodes are grouped by their upstream segment originating from one of the six major arteries (RACA, LACA, RMCA, LMCA, RPCA, LPCA) and marked with different colors. Black arrows indicate the flow direction in the segments in the CoW. (b) Comparisons between simulated relative fractions of segments in CoW and clinical statistics reported by Zarrinkoob et al. [32]. (c) Normalized relative contributions of the four primary sources to each terminal nodes assigned to the six regions.

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In Fig 4a, terminal nodes are grouped according to their originating major cerebral arteries (RACA, LACA, RMCA, LMCA, RPCA, LPCA) and highlighted with distinct colors. A distinctive topological feature of this cerebral network is that the flows from LACA and RACA merge immediately after passing through their A2 segments, resulting in downstream terminal nodes being equally affected by both sides. Accordingly, outflow sink nodes originating from LACA and RACA are marked with the same color. For the complete CoW, it’s worth noting that the communicating arteries (LPCoA, ACoA, RPCoA) carry small blood flow, especially ACoA shows negligible activation. This observation is consistent with experimental [9] and in-vivo [8] studies. Zhu et al. [9] reported that the pressure difference across both ends of PCoAs and ACoA in a complete CoW are almost zero, indicating that the anterior and posterior as well as the left and right circulations are relatively independent under normal physiological conditions.

In Fig 4b, simulated mean flow rates were compared with the clinical statistics reported by Zarrinkoob et al. [32] for the complete CoW. All simulation results (red filled circles) fall well within the clinical ranges defined by the mean±standard deviation (s.d.), confirming that the multiscale model can reproduce the physiological flow distribution across source vessels and downstream outlet vessels of the cerebral network in a statistically consistent manner.

In Fig 4c, the normalized contribution from the four primary inflow sources (RICA, RVA, LVA, LICA) to each terminal node are quantified using Eq.(16). The results reveal heterogeneity in supply patterns. Specifically, the ACA territories (RACA and LACA) receive comparable contributions from both ICAs and VAs with minimal left–right asymmetry due to the flow merging from RACAA2 and LACAA2. The MCA territories (RMCA and LMCA) are predominantly perfused by their ipsilateral (i.e., same-side) ICAs, with only minor contributions from the VAs. In contrast, the PCA territories (RPCA and LPCA) are almost entirely supplied by the VA inflows, exhibiting a symmetric distribution between the two sides. These observations suggest that the presented model not only reproduces the overall flow distribution but also provides detailed insights into the blood supply within the brain, reflecting how inflow sources and communicating branches of the CoW jointly shape the perfusion of distal vascular territories.

3.2. Anatomical variations with and without CAM

To investigate the hemodynamic consequences of the whole cerebral network with incomplete CoW and to assess the effects of CAM, the multiscale framework was employed with and without incorporating CAM for the CoW structures lacking PCAP1 (Fig 5) and ACAA1 (Fig 6).

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Fig 5. (a) Cerebral blood flow (CBF) changes from the baseline values of the entire cerebral vascular network with a fetal-type PCAP1 in CoW structure, where the PCAP1 segment is disconnected from BA and the PCAP2 segment originates from the ipsi-ICA through ipsi-PCoA [33].

The outflow (terminal) nodes are grouped by their upstream major arteries (RACA, LACA, RMCA, LMCA, RPCA, LPCA), and marked with distinct colors. The symbol ⟳ indicates that the flow direction in the segment is opposite to that under baseline conditions. (b) Comparisons between simulated relative fractions of the CoW segments and the clinical statistics (mean ± s.d.) reported by Zarrinkoob et al. [32]. Red filled circles and green triangles respectively represent simulations with and without considering the CAM. (c) Normalized relative contributions of the four primary inflow sources to each terminal nodes assigned to the six major regions. Ipsi: ipsilateral; cont: contralateral.

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Fig 6. (a) Cerebral blood flow (CBF) changes from baseline condition for the cerebral vascular network with a missing or hypoplastic A1 segment of ACA in CoW structure, where hypoplasia is defined as a vessel diameter smaller than 0.8 mm [34].

Outflow (terminal) nodes are grouped according to their upstream major arteries (RACA, LACA, RMCA, LMCA, RPCA, LPCA), and displayed with distinct colors. The symbol ⟳ indicates that the flow direction in the segment is opposite to that under baseline conditions. (b) Comparisons between simulated relative fractions and clinical statistics (mean±s.d.) by Zarrinkoob et al. [32]. Red filled circles and green triangles respectively represent simulations with and without considering the CAM. (c) Normalized relative contributions of the four primary inflow sources to each terminal nodes assigned to the six major regions. Ipsi: ipsilateral; cont: contralateral.

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Fig 5a illustrates the distribution of cerebral blood flow (CBF) changes from the baseline values presented in Fig 4. In this fetal-type PCAP1 configuration, ipsi-ICA exhibits the largest flow increase, while cont-ICA shows a minor rise, along with flow reductions in both LVA and RVA. Throughout this study, “ipsi-” and “cont-” respectively denote the ipsilateral and contralateral sides relative to the missing segment in the CoW. Importantly, all communicating arteries (PCoAs and ACoA) become activated, carrying markedly increased flows. In particular, both ipis-PCoA and ACoA even display flow reversals (denoted by the symbol ⟳), where their flow direction becomes opposite to that under baseline conditions. These collateral adjustments effectively redistribute blood flow, thereby ensuring that most downstream terminal vessels maintain nearly unchanged perfusion despite the structural variation.

In Fig 5b, the simulated results obtained with and without considering CAM together with potential flow reversals, are compared against clinical measurements [32]. It is evident that the CAM-enabled simulations (red filled circles) align well within the clinical ranges (mean±s.d.), confirming the model’s physiological consistency. In contrast, simulations without autoregulation (green triangles) exhibit substantial deviations in several vascular segments. The discrepancies are most pronounced in ipsi-PCAP2, which is connected to the missing PCAP1 segment, showing extreme ischemia, whereas cont-PCA presents severe hyperperfusion. Such unrealistic deviations arise from the absence of global pressure-balancing mechanisms within the entire cerebral vascular network, underscoring the essential roles of CAM and potential flow reversals in maintaining stable cerebral perfusion by physiologically consistent redistributions.

Moreover, as shown in Fig 5a, all four inflow sources undergo notable changes, illustrating that inflow sources and communicating branches of the CoW jointly determine the perfusion of distal vascular territories. Fig 5c further quantifies these adaptive effects: ACA territories exhibit only minor variations, while MCA and PCA territories undergo substantial reorganization. Specifically, the ipsi-MCA and PCA are almost exclusively supplied by the ipsi-ICA, whereas the cont-MCA and PCA proportions remain nearly unchanged. This highlights the compensatory function of the CoW: the full activation of PCoA and its associated flow reversal modulate perfusion toward MCA and PCA regions, whereas the activation of ACoA and its redirected flow ensure perfusion symmetry between the hemispheres.

Fig 6 provides the simulation results for the case with a missing or hypoplastic A1 segment of ACA. In contrast to the fetal-type PCAP1 case shown in Fig 5, where ipsi-ICA exhibited a pronounced flow increase, the present configuration shows moderate decreases in flow rates across ipsi-ICA and both VAs, while cont-ICA displays a substantial compensatory increase. From the enlarged CoW view, ipsi-ICA only supplies the MCA and PCA territories. Consequently, the total inflow and the flow through the communicating arteries on the ipsilateral side are adaptively decreased to prevent hyperperfusion in these regions. As illustrated in Fig 6a, the ipsi-MCA and PCA, together with their downstream vessels, maintain stable perfusion, indicating that autoregulatory adjustments within the inflow pathways effectively stabilize cerebral blood flow in the ipsilateral hemisphere. This behavior is also supported by Fig 6b. When CAM is neglected, the simulated flow rate of ipsi-MCA (green triangles) greatly exceeds the clinical range, whereas the CAM-enabled results (red filled circles) align well with clinical observations, confirming that cerebral autoregulation is essential for maintaining physiologically consistent perfusion under different CoW structures.

In addition, Fig 6a reveals that the flow rate in the pathways supplying the ACA territory from cont-ICA becomes noticeably increased, which is quantitatively confirmed in Fig 6c indicating that the ACA region becomes almost entirely supplied by the cont-ICA with minor contributions from both VAs. This redistribution fully activates ACoA, which exhibits a great flow increase and flow reversal (denoted by ⟳), thereby ensuring balanced perfusion between the two hemispheres. Moreover, PCoAs play a relatively limited role in this configuration, and ipsi-PCoA even exhibits a slight decrease in flow from BA toward ipsi-MCA, likely reflects an autoregulatory adjustment acting to prevent excessive perfusion in the ipsilateral hemisphere.

In Fig 6b, when CAM is neglected, the simulated cont-ACA (green triangles) exhibits severe ischemia due to the unregulated low inflow across cont-ICA, whereas incorporating CAM restores physiologically reasonable flow rates for both cont-ICA and cont-ACA (red filled circles), in good agreement with the clinical statistics provided by Zarrinkoob et al. [32]. Furthermore, the computed flow through the hypoplasia ACAA1 segment is close to zero due to its small diameter (defined as less than 0.8 mm), and the corresponding measured value is also zero. Therefore, the bar is not visible, and only the markers (circle and triangle) near zero are displayed to preserve axis alignment with the bar plots in Figs 4b and 5b.

In addition to the separate analyses of each anatomical configuration presented before, Fig 7 further compares the hemodynamic differences within the CoW under the two variation scenarios relative to the baseline condition. As shown in Fig 7d, the simulated CBF variations (red for increase and blue for decrease) agree well with the clinical data from Zarrinkoob et al. [32], with all simulated data points (circles) falling within the physiological range. Although minor sign discrepancies are observed in a few vessels (e.g., at cont-MCA), their values remain consistent with the measured uncertainty. confirming that the model accurately captures the redistribution of cerebral blood flow.

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Fig 7. Comparison of the CoW hemodynamic responses under two anatomical variation scenarios relative to the baseline condition.

(a, c) Schematic visualization of CBF redistribution for the CoW configurations with fetal-type PCAP1 and hypoplastic ACAA1, where the symbol ⟳ indicates that the flow direction in the segment is opposite to that under baseline conditions and the color scale follows Figs 5-6. (b) Predicted baseline flow rates (mL/s) in the communicating arteries (PCoAs, ACoA) and their relative CBF changes (%) from the baseline for the two incomplete CoW structures. Bars labeled with “**” denote the same vessels marked by ⟳ in (a, c), indicating flow direction reversal. (d) Comparison of simulated CBF changes (circles) with clinical means±s.d. from Zarrinkoob et al. [32]. Red and blue respectively indicate increase and decrease.

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Given this good agreement in predicting the flow rates of the non-communicating segments in the CoW, the systemically computed results for the communicating arteries that lack direct measurements can therefore be regarded as reliable. Fig 7b presents the baseline flow rate (mL/s) together with the relative CBF variations (%) for the two incomplete CoW structures, where the labels with “**” in the bar plots (e.g., ipsi-PCoA**) correspond to vessels exhibiting a reversal in flow direction and being marked by the symbol ⟳ in the CoW schematics (Fig 7a and 7c). The results reveal that ACoA is the most sensitive collateral pathway, exhibiting significant flow amplification in both configurations, while PCoAs display structure-dependent responses, being strongly activated under the fetal-type PCAP1 configuration but showing reduced participation when ACAA1 segment is missing. These findings are consistent with the clinical observations provided by Zhu et al. [9], which demonstrated that ACoA is the most sensitive index to ipsi-ICA morphological changes and serves as the primary anterior compensatory channel, whereas PCoAs are important for the anterior-posterior collateral coupling.

Overall, these analyses demonstrate that the presented multiscale model incorporating CAM effectively captures the intrinsic capability of the CoW to maintain stable cerebral perfusion through adaptive collateral regulation under different anatomical variation conditions. To further bridge the gap between the physiological baseline and the extreme anatomical variations that are topologically equivalent to complete occlusions, the following section extends this analysis to investigate the continues transition of hemodynamic redistribution under progressive stenosis.

4.1. Flow redistribution under stenosis progression

In this subsection, Fig 8 presents the simulated redistribution of CBF changes within the CoW as the degree of stenosis R_s (defined by Eq. (12)) in the RPCAP1 segment progressively increases. The simulation results were obtained by the multiscale model coupled with the localized stenosis formulation through Eq.(11). By gradually increasing R_s, the model allows quantitative validation of the hemodynamic redistribution process, verifying that the predicted collateral pathways, activation sequence, and flow patterns converge to those observed in anatomical variations. It ensures the physical continuity between normal, pathological narrowing (stenosis), and structurally incomplete CoW states within a unified multiscale framework.

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Fig 8. Flow redistribution under stenosis conditions.

(a–d) Simulated CBF changes from baseline for the fetal-type PCAP1 case and 20%, 50%, and 80% stenosis conditions applied to the ipsilateral feeding artery. Blue and red indicate flow decrease and increase, respectively. The dashed lines surrounded by dotted box denote the segment with stenosis. The table lists baseline flow rates (mL/s) of the major communicating arteries and their corresponding values under different stenosis levels. The symbol ⟳ indicates that the flow direction in the segment is opposite to that under baseline condition.

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As the stenosis increases from 20% to 80%, the overall flow in the ipsilateral circulation gradually decreases, while the contralateral side compensates with enhanced inflows. The hemodynamic asymmetry between hemispheres is progressively balanced through the communicating arteries, reflecting their increasing activations. Table in Fig 8 quantitatively presents the progressive transition from the baseline to the fetal-type PCAP1 configuration as stenosis severity increases, illustrating the continuous adaptation of collateral flows.

As discussed previously for Fig 4, the collateral mechanism of the communicating vessels in the CoW are almost inactive under the normal conditions [9], showing small flow and near-zero pressure differences across their ends. Compared with this state, as shown in Fig 8b, a mild stenosis of 20% produces slight redistribution of flow, mainly affected by the inflow adjustments (ICAs and BA), implying adequate upstream compensation regulation. Both the ACoA and PCoAs display only minor changes in their cross-flow, reflecting the unperturbed hemodynamic balance of the CoW with mild stenosis. Nevertheless, as listed in the accompanying table, although the flow rate through ACoA remains nearly unchanged value, its direction reverses, indicating an early activation. This finding also agrees with the experimental conclusion that the ACoA is the most sensitive index of unilateral morphological alterations and immediately carries flow from the healthy side to the affected side in response to stenosis [9].

When the stenosis degree increases to 50% (Fig 8c), the ACoA begins to exhibit moderate pressure-driven reverse flow, whereas partial activation of the ipsi-PCoA appears, showing the onset of cross-hemispheric redistribution. At 80% stenosis (Fig 8d), both ACoA and PCoAs are fully activated. Flow reversal (signed by ⟳ in Fig 8) emerges in the ACoA and ipsi-PCoA, ensuring sufficient collateral perfusion to the affected hemisphere. In contrast, cont-PCoA shows an increase in flow magnitude while maintaining its original direction, reflecting a secondary activation state that supports, rather than redirects, collateral flow. These findings related to PCoAs are consistent with experiments by Zhu et al. [9] and with clinical transcranial Doppler (TCD) measurements by Hoksbergen et al. [8], both of which demonstrate that when unilateral stenosis occurred, ipsi-PCoA becomes an essential collateral pathway supplying the anterior circulation and is fully activated under moderate to severe stenosis (exceeds approximately 40%), whereas cont-PCoA contributes limited support even under complete occlusion.

These consistencies confirm that the presented model accurately reproduces the physiological collateral availability of the CoW, with early ACoA activation followed by ipsi-PCoA compensation, and thus provides a mechanistic explanation bridging experimental observations and in-vivo hemodynamics.

4.2. Source-to-sink flow contribution mapping and perfusion sensitivity under CoW variations

Figs 4c and 5c illustrated the normalized relative contributions of the four primary inflow arteries (RICA, LICA, RVA, LVA) to each terminal node based on the path–flow inversion framework described in Section 2.3. Here, Fig 9 further visualizes the absolute source-to-sink flow contributions (mL/s) at baseline and the normalized relative changes from the baseline (%) under the two occlusion conditions, thereby providing a comprehensive depiction of how each terminal node responds to structural variations within the CoW.

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Fig 9. Source-to-sink flow contribution mapping across different CoW configurations.

(a) Baseline: flow contribution (mL/s) of inflow arteries (RICA, LICA, RVA, LVA) to each terminal node grouped into six regions (RACA, LACA, RMCA, LMAC, RPCA, LPCA). (b)-(c): relative flow changes from baseline under (b) fetal-type PCAP1 and (c) Hypoplastic ACAA1 configurations. Arrows in each entry represent the contribution changes (mL/s and %) from baseline to different anatomical variations. Entries labeled “occluded” indicate that the corresponding pathway becomes inactive under the altered CoW configuration, whereas “recruited” denotes a newly activated pathway that is absent under baseline condition, reflecting the deactivation or recruitment of specific collateral supply pathways connecting inflow arteries to downstream territories.

https://doi.org/10.1371/journal.pcbi.1013853.g009

Under the baseline conditions (Fig 9a), the regional flow distribution exhibits a clear lateral dominance, with the ACAs and MCAs primarily perfused by the ipsilateral ICAs. The VAs mainly supply the posterior circulation (PCAs) through the BA. These findings are consistent with the well-recognized distinction between the anterior and posterior circulations of the brain [35,36], which are primarily served by the carotid system (ICAs and their branches) and vertebrobasilar system (VAs and BA), respectively. In addition, ACA and PCA regions receive nearly symmetric inflows from both the right and left circulations, highlighting the well-balanced bilateral perfusion with negligible cross-hemispheric compensation under normal physiological conditions.

For the two CoW configurations shown in Figs 9b-c, the redistribution of source contributions reveals how collateral pathways are recruited to maintain cerebral perfusion. When the CoW configurations is altered from the complete CoW (baseline), the ICAs exhibit opposite trends of contribution to the ACA territory. The flow from the ipsi-ICA consistently decreases, whereas that from the cont-ICA increases, reflecting a compensatory redistribution mediated by the ACoA. In contrast, contributions from VAs remain symmetric between the left and right hemispheres, suggesting that the bilateral balance is primarily achieved through the regulation of ICAs and the activation of ACoA rather than by the posterior circulation.

Particularly, in the fetal-type PCAP1 configuration (Fig 9b), the ICAs and VAs show almost opposite and complementary patterns of redistribution across the MCAs and PCAs, implying a strong coupling between the anterior and posterior circulations. On the ipsilateral side, the ICA increases its flow supply to both the ipsilateral MCA and PCA regions, while the contribution of VAs decreases, indicating recruitment of collateral pathways through the ipsilateral PCoA. Conversely, on the contralateral side, the VAs contribution increases as the flow from the contralateral ICA decreases in the MCA territory or remains nearly unchanged in the PCA territory, also reflecting a dynamic posterior-to-anterior compensatory adjustment mediated by the communicating arteries.

In contrast, such anterior-posterior coupling is absent in the hypoplastic ACAA1 configuration (Fig 9c). In this case, the ACoA becomes fully activated and serves as the dominant compensatory pathway, while the PCoAs play a relatively limited role. As a result, perfusion to the ipsi-MCA territories is primarily maintained by the ipsi-ICA, whereas the PCA territories remain predominantly supplied by the vertebrobasilar system with relatively symmetric contributions from both VAs. Consequently, the redistribution of inflow contributions occurs mainly within the anterior circulation with limited interaction between the anterior and posterior pathways.

Importantly, the appearance or disappearance of specific source-to-sink contributions in Fig 9 reflects the recruitment or suppression of collateral supply pathways revealed by the path–flow inversion framework. This decomposition therefore provides an interpretable representation of how structural variations in the CoW translate into functional redistribution of cerebral perfusion across territories. The heterogeneous sensitivities of terminal territories to different inflow sources highlight the adaptive perfusion capacity of the network to anatomical variations, and may thus provide a quantitative basis for future diagnostic assessment and therapeutic planning in cerebrovascular diseases.

4.3. Role and clinical relevance of cerebral autoregulation

The incorporation of CAM is a critical component of the present framework and is essential for achieving physiologically plausible cerebral perfusion under both anatomical variations and pathological stenosis. As demonstrated in this study, simulations that neglected autoregulatory adjustment resulted in marked deviations from clinically observed flow distributions, including unrealistic hypo- or hyperperfusion in specific vascular territories. In contrast, the CAM-enabled simulations consistently reproduced flow patterns that aligned with experimental measurements and in vivo clinical data, underscoring the necessity of autoregulatory control for realistic cerebrovascular modeling.

From a clinical standpoint, cerebral autoregulation represents the brain’s intrinsic capacity to stabilize regional blood flow in the face of fluctuations in perfusion pressure and upstream resistance. The present results suggest that the effectiveness of collateral pathways through the CoW cannot be fully interpreted without accounting for concurrent autoregulatory modulation of peripheral vascular resistance. This finding is consistent with clinical observations that patients with similar degrees of arterial stenosis may exhibit markedly different perfusion states and ischemic tolerance, depending on the integrity of their autoregulatory reserve.

It should be noted that the CAM parameters adopted in this study represent a generalized autoregulatory response derived from previously published models. In clinical populations, however, autoregulatory capacity is known to be heterogeneous and may be impaired in conditions such as advanced age, diabetes mellitus, chronic hypertension, or prolonged cerebral ischemia. While the present formulation captures the fundamental role of autoregulation in shaping collateral flow redistribution, future extensions incorporating patient-specific or condition-dependent autoregulatory characteristics may further enhance the clinical interpretability of the model and improve its applicability to individualized cerebrovascular risk assessment and treatment planning.

4.4. Clinical implications and decision-making relevance

From a clinical perspective, the present framework provides a functional interpretation of collateral circulation that extends beyond anatomical assessment alone. By capturing the sequential recruitment of collateral pathways—such as early flow reversal in the anterior communicating artery followed by engagement of the ipsilateral posterior communicating artery under increasing stenosis—the model identifies hemodynamic transition states that may reflect progressive stress on cerebral perfusion rather than luminal narrowing severity per se.

Importantly, the results indicate that effective cerebral perfusion emerges from the dynamic interplay between proximal collateral pathways, autoregulatory modulation of peripheral resistance, and redistribution of inflow contributions across vascular territories. This finding is consistent with clinical observations that patients with comparable degrees of arterial stenosis may exhibit markedly different ischemic tolerance and outcomes. The source-to-sink flow analysis further offers a quantitative means to interpret which vascular territories are functionally vulnerable and which collateral routes are already maximally recruited under pathological conditions. For example, when evaluating potential bypass strategies for reduced perfusion observed in a specified downstream territory, the path-flow inversion can identify which pathways play a critical role in maintaining perfusion and how surgical modification of the network may redistribute source contributions across territories, thereby helping to restore a physiologically balanced perfusion pattern consistent with the baseline flow distribution. Although explicit clinical thresholds are not proposed in the present study, the framework provides a quantitative platform for future patient-specific investigations aimed at functional stratification of cerebrovascular disease. Such applications may support risk assessment and treatment planning for revascularization strategies by characterizing residual collateral capacity and identifying territories at risk under further hemodynamic compromise.