General discussion

Our study addresses a critical gap in understanding how fine-scale vascular architecture shapes high-resolution fMRI signals. We introduce 3D VAMOS, a biologically informed computational framework that generates synthetic 3D models of the human cortical vasculature based on morphological and topological features derived from histological data. By integrating realistic hemodynamics and biophysical processes, 3D VAMOS simulates BOLD signal formation, providing a versatile tool for mesoscale neurovascular modelling and refining ultra-high-field imaging protocols.

The 3D VAMOS algorithm generates fully controlled, well-interconnected vascular networks. It constructs the microvasculature using an aperiodic Voronoi tessellation and kernel functions and builds the macrovasculature solely with kernel functions, both guided by statistical properties previously quantified in the literature. The framework can produce vascular networks of specific human cortical regions, as demonstrated here for the primary visual and motor cortices. Beyond human applications, it also generates vascular architectures for other species, as shown for the mouse barrel cortex. This versatility enables direct comparisons of species- and region-specific vascular organization and its impact on neurovascular coupling and BOLD fMRI signals.

Using these models, we demonstrate that cortical depth-dependent BOLD profiles are strongly shaped by the underlying vascular architecture. Human vascular networks display consistent features that yield comparable depth-dependent BOLD profiles across cortical regions, whereas mouse models diverge markedly from human profiles. These differences emphasize both the importance of accounting for interspecies variability in vascular topology when interpreting cortical depth-dependent fMRI signals and the need for species-specific vascular modelling in neurovascular research.

We also show that the cortical depth of neuronal activity proxies determines the shape of depth-dependent BOLD profiles. Activity near the pial surface produced stronger detectable signals than activity in deeper cortical laminae, reflecting the disproportionate influence of superficial vasculature. Moreover, GE BOLD simulations were more sensitive to neurovascular effects but less specific to the microvasculature than SE BOLD simulations. These findings highlight the importance of considering both vascular depth-dependent contributions and sequence-specific sensitivities when interpreting cortical depth-dependent fMRI signals.

Comparison of 3D VAMOS mouse model and realistic mouse model

The comparison between the 3D VAMOS mouse model and the realistic vascular model derived from two-photon microscopy data [45] showed that the simulated vascular topology aligns closely with biological data. 3D VAMOS reproduced key features of the realistic vascular architecture, including major vessel trajectories, radius distributions, depth-resolved volume fractions of vascular compartments, and dense microvascular networks. However, some differences emerged. The synthetic macrovasculature in 3D VAMOS appeared more regular and homogeneous than its biological counterpart, particularly at the pial surface. Importantly, vessel labelling in the realistic model directly influences the calculated compartment volume fractions, since vasculature was manually classified using radius thresholds [31]. In addition, the imposed Gaussian distribution in the synthetic microvascular networks produced less skewness than observed in realistic data. While 3D VAMOS exhibited reduced biological heterogeneity, it showed lower inter-model variability than the realistic models, highlighting its reproducibility. These results demonstrate that 3D VAMOS reliably captures key features of cortical vascular architecture and its depth-dependent organization. Although it sacrifices some biological heterogeneity, most notably in pial surface microvasculature, the model provides a consistent, controllable framework for simulating vascular structures, enabling reproducible analyses across multiple scales.

Compared to modelling frameworks developed by Boas and colleagues [75–77], which primarily target oxygen transport and optical signal formation in experimentally imaged microvascular networks, 3D VAMOS offers a distinct advantage: it generates synthetic, anatomically realistic vascular architectures of rodent cortex. Unlike models restricted to small imaged patches, 3D VAMOS scales to larger cortical volumes, integrates hemodynamics and MRI biophysics, and directly simulates mesoscopic neuroimaging signals such as BOLD. This flexibility enables systematic hypothesis testing under diverse anatomical and physiological conditions. Moreover, it is compatible with oxygen transport models, bridging fMRI and optical imaging approaches.

A key benefit of synthetic 3D vascular models lies in overcoming the limitations of microscopy techniques such as two-photon, scanning electron, and light-sheet microscopy [69]. These methods are constrained by finite illumination depth, typically capturing only a few hundred micrometers of cortex. Resulting vascular reconstructions often suffer from deformations, noise, and artifacts, which complicate downstream analyses. Skeletonization—the core step in computational vascular modelling—is highly sensitive to minor boundary perturbations, and even extensive post-processing cannot always distinguish true vascular features from artifacts. Hence, manual vessel classification is often employed using radius thresholds, an approach that introduces some limitations. For example, the apparent lack of pial arteries in the mouse model, can be attributed to the vascular classification methodology employed in the reference dataset [31]. Second, certain venous structures may be misclassified in deeper laminae. These methodological constraints directly affect the estimated compartmental volume fractions. Against this backdrop, the use of such realistic datasets as a benchmark is particularly informative, as it demonstrates the ability of the 3D VAMOS framework to address these gaps by generating vascular networks that are statistically consistent with known histological properties. Thus, the 3D VAMOS framework addresses these issues by rapidly generating fully connected, biologically plausible vascular networks with user-defined parameters at low computational cost. Depending on the required features and volume size, it can generate a complete cortical network within seconds. This capability enables statistical analyses of vascular architectures without relying on a single dataset.

Future improvements in microscopy post-processing will further refine synthetic models like 3D VAMOS. Integrating these advances with machine learning–based segmentation could create a powerful synergy: better experimental data informing more accurate vascular models, which in turn enhance computational tools for analyzing real biological networks.

On the synthetic human 3D VAMOS model

The main motivation for developing a synthetic 3D vascular model is the limited availability of human cortical vasculature samples [54–56]. Current approaches for 3D visualization of ex-vivo human vascular networks rely on immunohistochemistry labelling combined with microscopy, X-ray microtomography, or optical imaging techniques [78]. However, these methods remain technically challenging, especially for large tissue samples, and tissue degradation and deformation further complicate the acquisition of large volumetric datasets (>1 mm³ isotropic) [51,56,59,79]. The 3D VAMOS framework addresses this limitation by computationally generating human-specific vascular networks based on angioarchitectural characteristics reported in histological and imaging studies [51,59,79].

Traditional vascular models used in BOLD simulations, such as RADOC or microsphere models, fail to structurally capture vascular networks when modelling signal formation at high spatial resolutions [41,42] (see Fig 5). At mesoscopic scales, cortical angioarchitecture follows distinct patterns, such as the microvascular mesh-like network. While RADOC models assume uniform or mixed cylinder sizes within a volume, they cannot represent critical topological features such as penetrating arteries and ascending veins [80]. Particularly, to serve as a benchmark, we chose uniform distributions for the RADOC model for the different laminae as used in traditional biophysical modelling approaches, which historically rely on simplified, uniform vascular geometries. By maintaining this standard configuration for RADOC, we can more clearly isolate and quantify the specific impact that the biologically informed vessel distributions (used in 3D VAMOS) have on BOLD fMRI signal generation. Our results demonstrate that the transition from these simplified uniform assumptions to the more realistic, heterogeneous distributions provided by 3D VAMOS is essential for capturing nuanced laminar BOLD profiles. Moreover, hemodynamic simulations are constrained by the absence of interconnections between cylinders.

The 3D VAMOS model reproduces macro- and microvascular volume fractions of the human cortex [59], as reflected in cortical depth profiles (Figs 1 and 3). This feature is essential for investigating cortical depth specificity in fMRI signals [17,81]. A further advantage is its flexibility: 3D VAMOS can generate vascular networks for different brain regions without being constrained by fixed macrovascular topologies or morphologies. It also enables systematic variation of artery-to-vein ratios to probe how macrovascular organization affects hemodynamics and BOLD signal formation (see Fig 5). Regarding the microvascular architecture, the inclusion of small capillary radii, extending down to approximately 1 µm, represents a modelling choice that warrants careful consideration in light of known limitations in histological measurements. It is well recognized that tissue processing can induce shrinkage, potentially leading to underestimation of in-vivo vessel dimensions. Nevertheless, such measurements provide a consistent and widely used statistical basis for characterizing microvascular architecture, and therefore offer a pragmatic foundation for model parameterization. Importantly, the physiological plausibility of these small vessel sizes is supported by the mechanical properties of red blood cells, which exhibit substantial deformability and are capable of traversing microvessels with diameters smaller than their resting size. This enables effective perfusion even at the smallest scales of the capillary network. These considerations support the inclusion of small capillary radii in the model, as they reflect both the available empirical evidence and the functional characteristics of the microcirculation relevant to BOLD signal formation.

Because the vascular network is fully connected, 3D VAMOS supports local hemodynamic simulations of CBF, CBV, and SO₂ [50]. Continuous pathways from pial arteries through capillaries to pial veins allow realistic modelling of blood flow dynamics across the full cortical depth [75]. This connectivity makes it possible to impose physiologically meaningful boundary conditions at inlets and outlets, such as blood pressure or flow rates, thereby enabling detailed simulations of cerebral circulation [48,82–85]. Beyond global flow constraints, the framework allows compartment-specific manipulations, such as simulating vasodilation or vasoconstriction in targeted vessels to model vascular pathology, neurovascular coupling surrogates [47], or impaired vascular reactivity in disease [86]. Future studies could extend these applications to investigate mean transit time heterogeneity [87], track contrast-agent passage in dynamic susceptibility contrast MRI for stroke research [88], or explore cerebrovascular physiology under controlled gas challenges [28,29].

The computational efficiency of 3D VAMOS further enhances its utility. Generating microvasculature typically requires only seconds (up to a minute), while macrovasculature is generated within a few hundred milliseconds. By contrast, other algorithms for realistic vascular architectures, such as Hartung et al. [85], can take hours depending on vessel properties. This computational efficiency enables high-throughput simulations and enhances statistical power in BOLD studies, as each iteration generates a new vascular morphology while preserving key topological features, analogous to voxel averaging in fMRI data pipelines (see Figs 4–6).

On the simulated BOLD fMRI signals

We simulated cortical depth-dependent BOLD signal profiles using 3D VAMOS models of the human primary visual cortex, human primary motor cortex, and the mouse barrel cortex. The simulated profiles resembled those reported in experimental studies of cortical depth-dependent fMRI across these regions [7,8,10–14,16,18,19,21,22,24,29,89]. Simulation results revealed that vascular geometry alone can drive markedly different depth-dependent BOLD profiles. In particular, mouse models diverged substantially from human ones. Because venous compartments dominate BOLD formation, mouse depth-dependent profiles exhibited greater signal decay and stronger surface-weighted responses than human profiles. These interspecies differences primarily stemmed from distinct artery-to-vein ratios.

We also observed cortical thickness–dependent effects. In the thicker human motor cortex (~4 mm), relative BOLD signal changes were lower than in the thinner visual cortex (~2 mm), even at identical simulated imaging resolutions (see Fig 5). By contrast, the mouse somatosensory cortex (~1 mm) produced distinct profiles compared to human vascular models, again reflecting species-specific artery-to-vein ratios. While normalization can help compare cortical depth profiles across species and regions, such approaches must explicitly account for underlying vascular differences. These results highlight the importance of considering both cortical thickness and vascular organization when interpreting laminar BOLD data.

Across all models, GE relaxation rates (ΔR2*) increased toward the pial surface, while SE relaxation rates (ΔR2) rose more moderately because magnetic field inhomogeneities are refocused. Despite this refocusing, macrovascular contributions at the cortical surface remained strong, particularly at higher oxygenation levels. In deeper laminae, GE and SE BOLD signals were dominated by tissue R2*. These results suggest that deep gray matter architecture exerts limited influence on depth-dependent BOLD decay, whereas superficial cortical depths are highly sensitive to variations in large-vessel topology. This asymmetry underscores the need for filtering and normalization techniques to reduce pial vessel biases in laminar fMRI analyses [13,14,90].

We further simulated localized neuronal activation by selectively modulating SO₂ and CBV within defined cortical depths and for vessel type (see Fig 6 and S1 Appendix). This approach isolated the effects of local vascular changes (CBF, CBV, and SO₂) on the BOLD signal. The simulation results demonstrate that vessel-type–specific hemodynamic responses and the spatial distribution of neuronal activity critically shape the cortical depth-dependent BOLD profiles. In condition (1), where vessels changed their CBV and SO₂ according to their type, the resulting profiles most closely resembled those reported in experimental fMRI data. This suggests that realistic interaction (coordination) between arterial, capillary, and venous compartments is essential to reproduce the characteristic depth-dependent BOLD signatures. The observed differences between GE and SE profiles reflect their known sensitivity to vascular scale: GE, dominated by larger vessels, produced higher amplitudes and more pronounced responses in superficial cortical depths, whereas SE, which emphasizes microvascular contributions, showed stronger and more localized responses at middle cortical depths. Conversely, condition (2), where only capillaries were modulated, resulted in relatively flat profiles for both sequences, suggesting that isolated capillary dynamics are insufficient to drive the laminar BOLD contrast typically observed in vivo. In condition (3), where all vessels changed their CBV and SO₂, the markedly larger amplitudes, particularly at superficial depths, highlight the dominant influence of large-vessel dilation on measurable GE signals. These findings emphasize the importance of modelling vascular heterogeneity and depth-dependent reactivity to interpret laminar fMRI data accurately [15,16,26]. The differential sensitivity of GE and SE sequences highlights the need to consider acquisition-specific biases when linking BOLD contrast to underlying neuronal processes. GE sequences, while more sensitive to signal changes, are influenced by draining veins and thus may overrepresent activity in superficial layers. SE sequences, by contrast, better isolate microvascular effects but with reduced amplitude, making them more suitable for mapping activity at the level of cortical microcircuits [9,17,73,74].

To date, five computational frameworks share the fundamental goal of bridging mesoscopic vascular anatomy and hemodynamic responses to predict empirical fMRI changes, whether at the voxel level or across cortical depth [31,41,42,48,80]. These pipelines incorporate varying MRI sequences (e.g., GE, SE, bSSFP), dimensionalities (2D vs. 3D), and field perturbation mechanics (analytical or Fourier-based magnetic field calculations). A shared objective among the laminar frameworks is accounting for how signal contributions across cortical laminae are modulated by large intracortical veins [31,80]. While 3D VAMOS shares this core objective with established tools like Markuerkiaga et al. [80] and Hartung et al. [48], it diverges significantly in dimensionality, species origin, and pipeline customization. While earlier iterations heavily leveraged idealized or static vascular geometries, 3D VAMOS integrates fully customizable, anatomically realistic microvascular networks with dynamic, depth-dependent adjustments [80]. Furthermore, in contrast to our previous work [31], 3D VAMOS introduces the simulation of synthetic, realistic human vascular models, similar to the work of Hartung et al. [48]. This structural advancement highlights the distinct uniqueness of our framework. A current limitation of the 3D VAMOS model, in comparison to the framework by Hartung et al. [48], is the lack of time-dependent hemodynamics across the vascular network. Nevertheless, 3D VAMOS remains advantageous because it allows users to systematically isolate and manipulate intricate spatial parameters, including local artery-to-vein ratios, vessel densities, vessel-dependent oxygen saturation, and tortuosity, within a highly accessible and flexible simulation pipeline.

Computational demands and simulation runtimes

The simulation runtimes were benchmarked using a standard desktop workstation configuration. The hardware environment utilized a 22-core (Intel Xeon CPU) computer paired with 62 GB of memory RAM. Under these baseline computational conditions, executing a single, complete depth-dependent BOLD simulation experiment for a single brain region required an average runtime of approximately 6 hours. This processing time scales with the structural complexity and spatial dimensions of the target 3D human vascular network, demonstrating that detailed laminar BOLD modelling can be achieved on standard local infrastructure. Nonetheless, the framework is highly scalable; leveraging high-performance computing clusters can drastically reduce overall simulation times and efficiently facilitate the parallel processing required for the statistical averaging of multiple simulation runs.

Methodological considerations

While the vascular architecture in-vivo is inherently finite, tortuous, and characterized by complex branching networks, the use of the infinite cylinder approximation remains a well-justified simplification within the context of BOLD signal modelling. At the spatial scales relevant for diffusion-mediated signal formation, vessel segments can be considered locally straight relative to the diffusion length of spins, such that the magnetic field perturbations are accurately captured by the infinite cylinder solution. Importantly, the BOLD signal reflects mesoscopic field variations that are effectively averaged over stochastic spin trajectories, thereby reducing sensitivity to localized geometric features such as vessel endpoints or bifurcations. Configurations in which spins reside near vessel end-points or along the longitudinal axis of finite segments constitute only a small fraction of the sampled space and contribute minimally to the overall signal, particularly within a Monte Carlo framework, as used in this work, where spatial averaging and random sampling further mitigate potential biases. Moreover, the inherent heterogeneity of the vascular network acts to suppress systematic errors arising from localized deviations from the idealized geometry. The validity of this approximation has been supported by multiple prior studies, including our own previous work [31,49,50,91], which demonstrate consistency between model predictions, theoretical expectations, and experimental observations [10,47,92]. These considerations suggest that, despite its geometric simplifications, the infinite cylinder model captures the dominant physical mechanisms underlying BOLD contrast and remains appropriate for the scale of the present study.

The current implementation of the 3D VAMOS framework adopts a step-wise approximation of SO₂ across arterial, capillary, and venous compartments, rather than modelling a continuous physiological gradient along the vascular tree. While oxygen extraction in-vivo occurs progressively, this compartmentalized representation remains a well-established and widely accepted approach in biophysical BOLD modelling [2,47]. The validity of this approximation is supported by the fact that the BOLD signal is predominantly governed by the paramagnetic effects of deoxygenated hemoglobin, which are most pronounced within the capillary and venous compartments. By assigning discrete yet physiologically plausible SO₂ values to these vascular segments, the model enables isolation of the influence of vascular architecture on laminar BOLD signal formation without introducing the additional complexity of dynamic oxygen transport processes. Moreover, this approach maintains consistency with prior modelling studies and experimental observations derived from compartment-specific pO₂ measurements [63,69]. Although the step-wise approximation may modestly underestimate intra-capillary heterogeneity in oxygenation, it provides a robust and validated baseline for capturing the dominant spatial characteristics of the BOLD response across cortical depth.

A further limitation of the current framework lies in the assumption of impermeable vessel walls, whereby exchange between intra- and extravascular compartments is not explicitly modelled. While this simplification is appropriate for studies focused on BOLD signal formation, where susceptibility effects dominate, it may become restrictive in the context of more comprehensive physiological modelling, particularly when incorporating cerebral blood flow (CBF) or detailed oxygen transport mechanisms. In such settings, transvascular exchange of water and dissolved oxygen can contribute to the observed signal and should be accounted for explicitly. To address this limitation, ongoing work is focused on extending the framework to include numerical modelling of exchange processes between compartments, thereby relaxing the impermeability assumption [93,94]. This development will enable a more physiologically complete representation of tissue–blood interactions and improve the applicability of the model to a broader range of hemodynamic and metabolic studies.

While the current 3D VAMOS framework focuses on the biophysical mechanisms underlying BOLD signal formation, it does not yet incorporate spatial encoding effects, such as T2* dephasing during readout. As a result, the simulated laminar profiles remain idealized and do not capture the spatial blurring (i.e., point-spread function, PSF) and signal decay characteristics present in experimental fMRI data. Recent studies have shown that acquisition parameters, including echo-train length (ETL) and k-space sampling in EPI readouts, can substantially shift and blur laminar signals, particularly in SE-EPI acquisitions [91]. Therefore, incorporating full spatial encoding into the 3D VAMOS framework represents an important direction for future development, enabling more direct comparisons with experimental 7T data and improving the assessment of the effective spatial specificity and fidelity of different imaging approaches.

Regarding angioarchitecture, although pial arteries form anastomoses to support regions of high perfusion or collateral flow [64], the current model includes only primary pial vessel segments from large feeding arteries. Pial veins, by contrast, do not exhibit anastomoses at any cortical depth [51]. Future algorithm enhancements will incorporate these vascular features to better capture cortical perfusion dynamics.

Future studies and computational improvements

A key characteristic of the 3D VAMOS framework is its reliance on a comprehensive set of empirically defined vascular and hemodynamic parameters to simulate depth-dependent percent signal changes. While this multi-parameter architecture allows for highly detailed modelling, a rigorous sensitivity analysis evaluating the isolated and joint impact of variables such as the artery-to-vein ratio, vessel radius, baseline CBV, and Grubb’s constant remains a critical next step to fully map the model’s behavior. Given the high dimensionality of this parameter space, a brute-force sensitivity analysis represents a massive computational undertaking that falls outside the foundational scope of the current validation study. To address this, ongoing work is leveraging a biophysics-informed machine learning approach to systematically map the entire parameter space and learn how complex parameter combinations converge to resemble empirical BOLD responses. Furthermore, the utility of the framework will be extended to explore alternative contrast mechanisms, such as vascular space occupancy (VASO) fMRI signals to characterize the sensitivity and specificity of CBV and SO2 alterations. Exploring these interconnected parameters will be vital for refining the predictive accuracy of future laminar fMRI interpretations.

Current functional neuroimaging methods primarily detect hemodynamic (BOLD) signal changes in response to neuronal activation. Our work addresses a fundamental challenge in high-resolution functional neuroimaging: understanding how fine-scale vascular architecture shapes the BOLD fMRI signal. While directly relevant to interpreting cortical depth-dependent fMRI, our computational framework also opens new avenues for investigating cerebrovascular dysfunction in clinical contexts. For example, cerebral small vessel disease (cSVD) [95]—a leading cause of stroke and dementia—is characterized by microvascular damage that disrupts perfusion and oxygen extraction, yet remains poorly understood at the biophysical level [96]. Using 3D VAMOS, we can explore how such vascular alterations influence the hemodynamic response, providing mechanistic insight into observed pathological changes.

Planned developments of the 3D VAMOS framework aim to expand its scope and enhance its physiological realism. Future iterations will simulate additional cortical regions under varied experimental conditions, systematically comparing outcomes to high-quality experimental data for improved validation. As more high-resolution histological and vascular datasets become available, we will refine anatomical assumptions to enhance model accuracy. A key next step involves scaling the model to larger cortical volumes, incorporating the macro-scale geometry of sulci and gyri to capture their specific effects on microvascular architecture and hemodynamics. Furthermore, we plan to broaden time-dependent hemodynamic changes and fMRI simulation capabilities to include diffusion-sensitive regimes, addressing current assumptions of isotropic diffusion across cortical depth [91].

An important avenue for future development lies in extending the framework to incorporate additional imaging contrasts such as VASO and arterial spin labeling (ASL), which are often associated with improved spatial specificity relative to conventional BOLD imaging. In the current implementation, activation-induced changes in physiological parameters, including cerebral blood flow and volume, are introduced as prescribed inputs rather than dynamically simulated, allowing the model to focus on the relationship between vascular architecture and signal formation. Building on this foundation, future work will explicitly model the primary contrast mechanisms underlying VASO and ASL, namely blood volume–dependent signal changes and the delivery and exchange of magnetically labelled arterial blood, respectively. Importantly, such extensions will also account for secondary contributions, particularly BOLD-related signal contamination, which can significantly influence both modalities and interact with the underlying vascular geometry.

4.1 Generation of a synthetic human 3D VAscular MOdel based on Statistics – 3D VAMOS

A synthetic vascular model is generated using custom MATLAB code (MathWorks, v.2024a). The statistical properties of human cortical vasculature are derived from literature that estimates these properties through histological analysis [51,52,54,55,57]. First, the microvasculature (which in this manuscript includes arterioles, capillaries, and venules) and the macrovasculature (comprising pial arteries and veins, penetrating arteries, and ascending veins) are generated separately. The vascular networks are then integrated by connecting the macrovascular endpoints of both the arterial and the venous compartments to the microvasculature through their nearest proximal vessel in space (i.e., the nearest vessel neighbor), thereby forming a complete and fully connected 3D VAMOS structure. The generation process for each vascular compartment is described in the following sections.

Generation of the microvascular architecture based on Voronoi tessellation and kernel functions.

To account for regional variability in cortical thickness and vascular volume fraction across the human cortical gray matter, we considered that different cortical areas exhibit different cortical thicknesses and cerebrovascular volume fractions [97]. For instance, the human primary visual cortex exhibits a cortical thickness of approximately 2 mm [64,66], while the human primary motor cortex is approximately 4 mm thick [98]. Therefore, the initial parameters of the algorithm included the definition of a customized three-dimensional space with specified x, y, and z dimensions (in millimeters), along with a designated vascular volume fraction. The x and y dimensions represent the in-plane parallel to the cortical surface, and the z dimension corresponds to cortical thickness. Within this volumetric space, we assumed that the cortical vasculature extends from the superficial/pial vessels to the cortical gray-white matter (GM-WM) border.

The 3D VAMOS allows for the definition of any desired volumetric vascular dimensions, ranging from hundreds of micrometers to millimeters. This versatility enables modelling not only human vascular networks but also rodent or other species by adjusting vascular properties accordingly (see Figs 1 and 3).

The microvasculature was generated using Voronoi tessellation, resulting in a mesh-like topological network [54,65]. Voronoi tessellations have been shown to effectively represent the capillary bed [99,100].

The simulated volumetric space was divided into a number S of equidistant slabs in the xy-plane. S was calculated based on the desired vascular volume fraction and vessel radii. An aperiodic Voronoi tessellation was generated by fragmenting each of the slabs into tiles that encompass a given set of seed points [62]. The seed points can be distributed within the slab according to any specified probability distribution to simulate different capillary densities across cortical depth [52]. For example, a Gaussian distribution can be simulated in the xz-plane to create larger capillary densities at middle cortical depths, i.e., a larger density of Voronoi tiles in the middle part of the slab. This would reflect histological observations that vascular density can be slightly higher in the middle layers compared to the superficial and granular layers. [52]. Simpler distributions, such as uniform seed distributions, can also be implemented depending on the modelling needs.

Each slab, then, was tessellated using the linear inequalities formed by perpendicular bisectors between any two connected points in the Delaunay triangulation, employing an adapted version of the polytope-bounded aperiodic Voronoi diagram algorithm [62]. Once all the slabs contained tessellations, vessel joints in the i-th slab were connected with their nearest neighbor vessel joint in the (i + 1)-th slab using the shortest Euclidean path. This results in a fully interconnected network structure consisting of vertices, i.e., microvessel joints, and lines connecting those vertices, i.e., microvessel strands.

To further enhance the realism of the microvascular networks, the vertices (microvessel joints) generated through tessellation were displaced orthogonally to the slab by a small distance, typically on the order of tens of micrometers. This displacement introduced a volumetric distribution to the components of each slab, more closely mimicking the three-dimensional structure of real capillary networks.

To generate a closer resemblance to actual capillary beds, an important characteristic to include is the tortuosity of the microvessels [82–84]. The tortuosity () was defined as the ratio of the vessel length between two vessel joints (vertices), to the Euclidean distance between joints (see Fig 2),

(1)

To generate vessel morphologies consistent with a desired tortuosity, we developed an iterative curve generator that produces anatomically plausible shapes from predefined mathematical functions. Specifically, sinusoidal functions (α-functions; e.g., A sin(ωr + δ), with A, ω, δ randomly sampled) were combined with weighting functions such as Gaussian, gamma, or double-gamma distributions (β-functions; e.g., N(ζ, η), with ζ, η randomly sampled). The product of these α- and β-functions defines a kernel function, which governs the curve generation process and ensures that the resulting morphologies satisfy the imposed tortuosity constraints. Representative examples of microvascular tortuosity generated using this approach are shown in Fig 2.

In addition to tortuosity, each line in the Voronoi network was assigned a value representing the vessel radius. This value was drawn from a Gaussian distribution with a mean and standard deviation selected based on histological data; here for the human primary visual and motor cortex with a mean of 3.235 µm and a standard deviation of 0.850 µm [57,66], and the mouse somatosensory cortex with a mean of 1.750 µm and a standard deviation of 0.300 µm [45]. Other cortical regions or species can be simulated by selecting distribution parameters derived from the corresponding histological data. Moreover, the synthetic capillary network maintains radius continuity across adjacent segments. During the generation of the 3D VAMOS network, we implemented a local “smoothing” constraint ensuring that neighboring vessel segments are assigned similar radii. This approach avoids unphysiological, abrupt changes in vessel diameter at bifurcations or along continuous paths, thereby ensuring that the morphological properties of the capillary bed remain consistent and biologically plausible.

Generation of the macrovascular architecture based on kernel functions.

Building upon the predefined three-dimensional space for the microvascular network, the macrovascular architecture was subsequently constructed. The process begins with generating the pial arteries and veins, whose quantity is determined by the predefined initial number of penetrating arteries and ascending veins, based on literature values for a particular cortical region. For the human brain, the artery-to-vein ratio was set to approximately 3:1 [51], corresponding to 42 penetrating arteries and 14 ascending veins for the primary visual cortex, and 85 penetrating arteries and 25 ascending veins for the primary motor cortex [57,66]. For the mouse barrel cortex 5 penetrating arteries and 15 ascending veins were set [45]. At the top plane of the volumetric space in the z-direction (i.e., the maximal z-cross-section), seed points were randomly placed in the xy-plane, constrained by a predefined minimum “proximity distance” between penetrating and ascending vessel seed points, set to ~120 μm [51]. Each seed point was then designated as either an artery or a vein. This designation followed the principle that veins in humans must be surrounded by arteries: if a seed point was designated as a vein, its nearest neighbors were preferentially labelled as arteries, consistent with the histological artery-to-vein ratios [51,64]. For the mouse models the artery-to-vein ratio was reversed [45]. Next, the labelled pial artery seed points at the maximal z-cross-section were interconnected using kernel functions, with vessel tortuosity fixed at ~1.25[-]. The same process was applied to the labelled pial veins (see Fig 2).

After creating the pial vasculature, the subsequent step generated the main branches of the penetrating arteries and ascending veins. The definition of the cortical penetration depth for these vessels was based on Duvernoy et al. [51]. Penetrating arteries or ascending veins were classified into five cortical depth equidistant bins, as Artery (A1-A5), or Vein (V1-V5) from the pial surface (A1/V1) to the GM-WM boundary (A5/V5 (see Fig 2). Each endpoint of a penetrating vessel was aligned parallel to its corresponding surface seed points, i.e., at the maximal z-cross-section, and subsequently connected by another set of kernel functions.

Furthermore, the number of sub-branches, or daughters of the main vessel segment, can be predetermined as an initial parameter for each penetrating or ascending vessel. These sub-branches were randomly positioned along the main vessel and generated using a set of kernel functions. The vessel radius of both the penetrating or ascending vessels and their sub-branches follows a branching exponent adhering to Murray’s law [67,68]:

(2)

with k ranges between 2 and 3 in both humans and rodents; here we selected k = 2. While Murray’s law typically utilizes an exponent of k = 3 [67], we utilized an exponent of k = 2 for the 3D VAMOS framework. This choice follows the area-preservation law, which dictates that the total cross-sectional area remains constant across a bifurcation. From a physiological perspective, an exponent of k = 2 is characteristic of larger vessels where it is necessary to maintain constant blood flow velocity and minimize pulse wave reflections [68]. From a computational and modelling standpoint, utilizing k = 2 serves as a robust simplification that prevents vessel radii from tapering too aggressively as the network depth increases. This ensures a stable and physiologically plausible CBV across the simulated volume, avoiding the ‘thinning’ artifacts that can occur in complex synthetic trees when k = 3 is applied strictly to every microvascular junction. Radii values were initialized at the pial surface seed points (at the maximal z-cross-section), and gradually decreased across cortical depth, reaching their smallest size at the endpoints of the main branches and their sub-branches (see Fig 2). For the human models, intracortical vessel radii ranged from 13 µm to 23 µm for arteries, and 15.65 µm to 31.65 µm for veins [57,66]. For the mouse models, intracortical vessel radii ranged from 7 µm to 12 µm for arteries, and 10 µm to 15 µm for veins.

Consequently, the 3D VAMOS algorithm currently generates for each parent penetrating or ascending vessel (main branch) a specified number of daughter vessels expanding in a radial pattern (sub-branches), resembling a topological tree-like structure [54]. The number of sub-branches, their length, and tortuosity can be set to different values dependent on the cortical region simulated.

Another feature of 3D VAMOS is the option to connect V5 ascending veins at the GM-WM border. When this parameter is enabled, all V5 veins are interconnected at their endpoints by using a set of kernel functions. The inclusion of connections between V5 veins (large intracortical veins) is motivated by the known presence of deep venous plexuses and collateral connections at the gray-white matter border. Anatomical studies of the human cerebral cortex have identified principal intracortical veins (PICVs) that serve as the primary drainage route for deep cortical layers [63,90,101]. These vessels often exhibit horizontal anastomoses or connections at deeper laminar depths, which facilitate the efficient transport of deoxygenated blood from the deep parenchyma toward the pial surface. By including these connections, 3D VAMOS accurately represents the ‘sink’ effect of the venous system, which is crucial for modelling the spatial specificity of the BOLD signal and the influence of large-vessel ‘blooming’ artifacts across cortical depth.

4.2 Simulation of the BOLD signal using the 3D VAMOS accounting for intravascular and extravascular signal contributions

In this study, BOLD fMRI signals consisted of both extravascular and intravascular signal components influenced by the vascular architecture, baseline CBV (see Fig 1), oxygen saturation (SO₂) in vessels, and hematocrit. The total MRI signal, , was calculated by summing the extravascular signal contribution, , with both arterial, , and venous, , intravascular signal contributions:

(3)

Simulations were performed for gradient-echo (GE) and spin-echo (SE) pulse sequences at seven tesla (7T), assuming an instantaneous readout, i.e., without spatial encoding such as echo-planar-imaging (EPI). Additionally, we assumed that the cortical surface normal of the models was parallel to the main magnetic field [10,47,50,92].

Arterial and venous intravascular signal contributions.

The intravascular signal contribution for each vascular compartment was approximated as follows:

(4)

and

(5)

Where is time, and are baseline arterial and venous blood volume fractions respectively (see Fig 1 for representative bCBV’s), and the relaxation rates and for GE and SE, respectively, are given by:

(6)

where, or , are the intrinsic relaxation rates of blood, and the relaxation rate dependent on SO₂.

We assumed that the arterial and venous intravascular signal contribution to the BOLD signal is non-zero. This assumption is based on the observation that at high magnetic fields strengths, the intravascular arterial and venous signals, particularly in the veins, are not negligible at higher (≥ 80%) SO₂ levels [31,34,102]. On the other hand, we assumed the intravascular signal contribution of the microvascular compartment to be null, given that the and of the capillaries has not been well-characterized, and the intravascular contribution of capillaries is not expected to be as significant as that of larger vessels [31].

Therefore, for GE we calculated (≈ 10.00 ms), and component dependent on SO₂ using the linear (Eq. 7 in [32]) and quadratic (Eq. 8 in [32]) relationships, respectively (see Tables 1 and 2), weighted by the corresponding blood volume fraction (see Fig 1). For SE we calculated (≈ 53.82 ms), and component dependent on SO₂ [32] (see Tables 1 and 3).

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Table 1. Biophysical and pulse sequence parameters used to compute the relaxation rates and BOLD signal changes.

https://doi.org/10.1371/journal.pcbi.1014576.t001

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Table 2. GE: intravascular venous T2_dHb* at 7T.

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Table 3. SE: intravascular venous T2_dHb at 7T.

https://doi.org/10.1371/journal.pcbi.1014576.t003

Simulation of the extravascular signal contribution.

The extravascular signal contribution for both macro- and micro-vessels was computed by modelling the interaction of moving spins within the local inhomogeneous magnetic field induced by the varying SO₂ levels [34,102].

The local frequency shift caused by a vessel segment was modeled as the dipolar induced-response of an infinite cylinder, presuming negligible effects on the cylinder extremities [38,50], given by (in [1/s]):

(7)

where γ is the hydrogen gyromagnetic ratio = 267.5x10⁶ [rad∙(sT)^-1], is the main magnetic field (7 [T]), [-] is the susceptibility difference produced by the in the vessel [39] and the vessel size-dependent hematocrit [104], is the vessel radius in [µm], is the Euclidean distance from the center line of the vessel to a particular spatial position in the simulation volume in [µm], is the angle between the vessel and the spatial position in [rad], and is the angle between the orientation of the vessel and the main magnetic field in [rad].

The dephasing experienced by a population of diffusing spins, , was simulated using a Monte Carlo approach with 30 repetitions per SO₂ state, and a diffusion coefficient of D = 1.0 [µm² ∙ ms^-1], assuming isotropic diffusion across cortical depth [37]. In each repetition, a new 3D VAMOS model was generated (30 repetitions x 9 SO₂ states = 270 models), and 5x10⁷ spins were simulated. Spin dephasing was obtained through:

(8)

where is the phase acquired during the simulation time and is the local frequency shift at spin position at each time-step (= 0.020 ms). For SE sequences, the acquired dephasing after TE/2 was multiplied by -1 (change in polarity) simulating the effect of the 180-degree refocusing radiofrequency pulse. Using equation (9) we can obtain the extravascular MR signal contribution as follows:

(9)

where represents the MR signal component induced by the interaction of diffusing spins within the local magnetic field inhomogeneities (i.e., the signal subject to decay). This signal component depends on the specific pulse sequence and is defined as,

(10)

and = 1/T2₀* and = 1/T2₀ are the intrinsic relaxation rate of cortical tissue for GE and SE, respectively. We used the intrinsic tissue T2₀* ≈ 28 ms for GE and T2₀ ≈ 48 ms for SE according to the nonlinear relationship given by Khajehim et al., [105] for gray matter at 7T (see Table 1).

To confine the ensemble of spins within the simulation space, voxel boundary conditions were set to infinite space. Spins exiting the voxel re-entered the voxel volume on the opposite side, preserving their dephasing history. However, spins reaching the pial surface and GM-WM border were considered invalid iterations and were recomputed. Additionally, spin exchange between extra/intra-vascular compartments was prohibited, establishing an impermeable vascular network [32].

Relaxation rates and BOLD signal changes.

Given that the behavior of in (Eq. 2) presents oscillations due to its multi-exponential nature, we simply approximated the global relaxation rate and by fitting a polynomial of degree one, i.e., a linear fit, on the natural logarithm of , i.e., , for GE and SE, respectively. In this work, two metrics are reported:

(1) the relative relaxation rate in [1/s]:,

(11)

where represents the varying SO₂ states, and is the intrinsic relaxation rate of tissue and (2) the relative BOLD signal change, in [%]:,

(12)

where represents the varying SO₂ states, and () is the tissue signal at TE and is defined as the baseline condition for GE and SE respectively.

4.3 Simulating neurovascular effects with phenomenological approaches

To investigate neurovascular responses within this synthetic vascular environment, we employed a phenomenological modelling strategy based on Grubb’s law [106], which characterizes the empirical relationship between cerebral blood flow (CBF) and cerebral blood volume (CBV). This relationship is given by:

(13)

where α is typically ~0.38 for cortical gray matter [106], but recent studies suggest smaller values that better capture hemodynamic behavior across cortical depth. We adopted α = 0.25, based on the rationale that, according to Poiseuille’s law—which relates blood flow changes to vessel radius—the radius scales with a fourth-power law. This choice is further supported by findings from Ito et al. [107] and Chen & Pike [108], which better reflect cortical depth-dependent vascular dynamics at the mesoscale [107,108].

In our simulations, CBF changes were modeled as a surrogate for neuronal activity. Global increases in CBF were simulated within a range of 0% to 40% relative change and applied uniformly across all 3D VAMOS vessels. Corresponding CBV changes were then computed using the established power-law relationship [109], which uniformly alters the CBV of all vessels, regardless of vessel type. For example, a vessel with a radius of 10 μm undergoing a 20% relative increase in CBF would exhibit a ~ 4.6% relative change in CBV, resulting in a final radius of approximately 10.46 μm.

To mimic localized neuronal activation, we structured the simulations into six conditions, comprising three cortical depth locations and three vessel-specific hemodynamic response types. Simulated neuronal activity was positioned at superficial (0.35 mm), middle (1.0 mm), and deep (1.5 mm) cortical depths. For each location, vessel-specific changes in CBV and SO₂ were modeled under three conditions: (1) capillaries and veins dilated locally, with CBV varying between 0–30%, and arterial and capillary SO₂ changing locally within the same range; (2) only capillaries dilated locally, with both CBV and SO₂ each varying between 0–30% at the respective activity site; and (3) all vessel types dilated locally, with both CBV and SO₂ changing from 0–30% at the corresponding location (see Fig 6 and S1 Appendix). No permutations between CBV and SO₂ changes were included. The hemodynamic point spread function was modelled with a Gaussian spatial profile with a of 0.150 mm [73,110]. For instance, consider a vessel with an initial radius of 10 μm. A global 20% increase in CBF would increase its radius to approximately 10.46 μm. If, in addition, a 10% local increase in CBV is applied (reflecting the neuronal activation intensity), the final vessel radius would reach about 11.50 μm.

Simulations were performed with 3D VAMOS models of the human primary visual cortex, using baseline hemodynamic conditions, including flow rates and SO₂ levels, from the literature [73,74,110]. BOLD signal changes were computed relative to a baseline condition of CBF = 0% across cortical depth, thereby isolating the BOLD effect associated with the simulated neuronal activity (see Fig 6 and S1 Appendix). This phenomenological approach allowed to generate neurovascular responses without explicitly modelling neuronal activity.