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Thoughts and Theory

Virtual Angiogenesis: A new set of tools for creating in silico blood vessels

In the field of cardiovascular research, many circulatory diseases found in global or regional circulations begin with a dysfunctional chemo-mechanical pairing that governs the interaction between peripheral (arteriolar and capillary) blood flow and cellular functions. Mathematical models and computational simulations of the circulation and its coupling with cellular function provide powerful tools to push the boundaries of cardiovascular research.

Sequential vascular growth (Image by Author).

However, the realisation of the domain of definition at a microvascular level is required for the blood flow models and cell coupling. This can be achieved by resorting to imaging techniques, which are capable of delivering high-resolution images of small vessels in animal models. But even with the power of these tools, their applicability to different vascular territories is rather constrained, as is the translation to humans. A solution to circumvent the lack of information about the vascular architecture of peripheral beds was found through the development of methodologies aimed at generating vascular networks in an automatic manner, driven by physiological criteria and in compliance with a set of morphometric constraints and empirical laws. The so-generated networks of vessels can be statistically consistent with the anatomical descriptions in the sense that they can reproduce the main topological features and the functional response observed in vascular networks.

Hierarchical vascular growth: (left) vascularisation obtained with a standard CCO algorithm; (right) vascularisation using DCCO algorithm (Image by Author).

Scale-specific vascular growth: (left) vascularisation obtained with a standard CCO algorithm; (right) vascularisation using DCCO algorithm (Image by Author).

There are two classes of algorithms for the automatic generation of vascular beds: Fractal algorithms and space-filling algorithms. The first class of algorithms can easily follow statistical models of the main morphometric measures (vessel radius, length, aspect ratio and bifurcation angle), regardless of the shape of the vascular territory. The second class of algorithms allows one to generate a network of vessels inside a vascular territory defined in (2D or 3D) space. A particular class of such space-filling algorithms is termed Constrained Constructive Optimisation (CCO) because the sequential generation of the vessel network is driven by the constrained minimisation of a cost function.

Concurrent vascularisation of circular and spherical domains with equiradial vessel inlets (Image by Author).

We test the aDaptive CCO (DCCO) algorithm for the generation of vascular networks in academic scenarios with simple domains and also in realistic scenarios involving multi-stage vascular outgrowth, starting from existing vascular networks taken from detailed human models in anatomically representative territories of the human body.

Inner Retinal Vascularisation

The inner retinal vascularisation, one of the vascular layers of the eye, is homogeneously vascularised with exception of the foveal region which is avascular. Interestingly, the DCCO vascularisation generates four branches near the insertion point of the retinal artery creating a quadrant-based arrangement in the same manner that is observed in vivo. In more distal sites, a uniform vascular pattern is delivered by the algorithm in agreement with anatomical descriptions. This simple example shows the ability of the DCCO to reproduce the in vivo vascular patterns from a small set of constraints and stages.

Inner retinal vasculature generated by a two-staged growth process: (first row) initialisation S0; (second row) generated tree discriminating vessels given as initial conditions (blue), generated at the stage S1 (light blue) and stage S2 (red); (third row) variation of vessel radii along the generated network (Image by Author).

Brain Cortex Vascularisation

In this second case, we vascularise the grey matter tissue in the left frontal gyrus of a prototypical human brain. As initial condition S0, an arterial tree rooted at the left anterior cerebral artery (l.ACA) was used.

The generated vascular tree reproduces several architectural features observed in experimental studies. All main branches perfusing the gyrus are well developed providing significant blood flow to the tissue. Qualitatively similar distributions are observed for the different branches, with some quantitative differences resulting from the size of the corresponding vascular territories that each supply. Also, the cortical vessels presented a homogeneous coverage of the gyrus surface, where its major pial vessels branch into penetrating vessels, which run inwards, nearly orthogonal to the gyrus surface. This results in the first generation of vessels which reach the grey matter. In turn, these penetrating vessels give rise to local arborisations of deep vessels. The final vascular tree homogeneously perfuses grey matter volume as is evidenced by the dispersion of the vessels generated in S3 and S4.

Grey matter vasculature in the superior frontal gyrus generated by a four-staged growth process: (first row) initial tree S0, illustrating the inflow and outflow constraints; (second row) from left to right, vessels corresponding to the pial network (S1 and S2), penetrating arterioles (S3) and deep arterioles (S4); (third row) superior-posterior part (left) and coronal slice at a middle section of the rostral-caudal axis (right), describing the architectural organisation of pial (red), penetrating (green) and deep (blue) vessels across the tissue; and (fourth row) final vascular tree depicting the vessel radius (Image by Author).

Stomach Vascularisation

As a final case, we built a vascularisation for the stomach following the anatomical description presented in the literature.

The resulting vascular tree is in qualitative agreement with the vascular architecture reported in the literature where the subserosal plexus delivers homogeneous perfusion to the serosal layer, while perforators arising from this subserosal network feed the submucosa plexus. Lastly, the blood to the muscularis layer is provided simultaneously by such serosal perforators and by the submucosa plexus. The complex architecture of this organ rendered a more complex distribution of vessel radii at all stages due to the presence of vessels transporting blood to the neighbour at all regions (subserosal plexus vascularise submucosa, muscularis vascularise submucosa, submucosa vascularise muscularis).

Stomach vasculature generated by a five-staged growth process: (first row) initial tree S0, including the inflow and outflow constraints; (second and third rows) from left to right and top to bottom, vessels corresponding to the serosa layer (S1 and S2), muscularis perforators (S3), mucosa layer (S4) and muscularis layer (S5); (bottom row) final vascular tree depicting the vessel radius from anterior and posterior views of the stomach (Image by Author).

We introduced the DCCO as an approach for the generation of vascular networks with a set of functional and architectural traits. This work describes the algorithms that materialise DCCO and, also the rationale for modelling the necessary stages to achieve different and diverse anatomically-realistic representations of the vasculature.

In addition, further extensions were made to integrate the automatic vascularisation process with available data, such as that obtained from medical images or detailed models of the cardiovascular system. These features allow us to use DCCO in patient-specific geometries, enabling new research opportunities to explore the role of small-scale vessels and their physiological characteristics in different clinical scenarios.

This article was written by Dr Maso Talou, Dr Safaei, DProf. Hunter, Prof. Blanco and originally published at https://www.nature.com/articles/s41598-021-85434-9.