Mathematical models of developmental vascular remodelling: A review

Abstract

Over the past 40 years, there has been a strong focus on the development of mathematical models of angiogenesis, while developmental remodelling has received little such attention from the mathematical community. Sprouting angiogenesis can be seen as a very crude way of laying out a primitive vessel network (the raw material), while remodelling (understood as pruning of redundant vessels, diameter control, and the establishment of vessel identity and hierarchy) is the key to turning that primitive network into a functional network. This multiscale problem is of prime importance in the development of a functional vasculature. In addition, defective remodelling (either during developmental remodelling or due to a reactivation of the remodelling programme caused by an injury) is associated with a significant number of diseases. In this review, we discuss existing mathematical models of developmental remodelling and explore the important contributions that these models have made to the field of vascular development. These mathematical models are effectively used to investigate and predict vascular development and are able to reproduce experimentally observable results. Moreover, these models provide a useful means of hypothesis generation and can explain the underlying mechanisms driving the observed structural and functional network development. However, developmental vascular remodelling is still a relatively new area in mathematical biology, and many biological questions remain unanswered. In this review, we present the existing modelling paradigms and define the key challenges for the field.

Fig 1. Schematic of vascular development.

(A) Each stage of development is shown from left to right. On the left, the formation of blood islands join together to form the initial components of the vascular network (vasculogenesis). This initial network is rapidly extended upon during angiogenesis, creating a dense unstructured network (middle). Finally, this dense network organises and matures during remodelling and regression (right). (B) Vascular remodelling can be subdivided into several distinct categories: vascular regression, vascular identity formation, calibre control, and vascular stabilisation. The papers cited in this figure are [–5,10].

Fig 2. Vascular remodelling is a multiscale process.

Vascular remodelling should be considered from a multiscale approach, considering the subcellular (left), multicellular (second across), vessel level (third across), and network left (right). The 3D printed vascular network shown on the far right has been reproduced with permission from [11].

Fig 3. Schematic of vascular pruning.

(A) Initial selection of a vessel with low wall shear stress, ${\overrightarrow{\tau }}_{\mathrm{\Omega }}$. (B) The endothelial cells from the pruning branch polarise and migrate up the wall shear stress gradient and against flow towards the neighbouring vessel, as shown by the black arrows. (C) The endothelial cells of the pruning branch are incorporated into the neighbouring vessel, leaving an empty basement sleeve in place of the pruning vessel. The coloured arrows indicate the direction and the intensity of blood flow, with arterial flow shown in red, and venous flow shown in blue. The endothelial cell nuclei and the Golgi apparatus in within each cell are yellow and red, respectively, showing the endothelial cell polarisation against the flow. Figure adapted with permissions from Santamaría and colleagues [12].

Fig 4. Summary of future directions for each modelling paradigm and the remodelling subtypes addressed by each paradigm.

The simulation shown in the left column of row 2 has been reproduced with permission from Secomb and colleagues [65]. The network shown in the left column of row 4 has been reproduced with permission from Zhou and colleagues [66].

Fig 5. A summary of the currently available models of vascular regression and remodelling.

Models of vascular regression and remodelling can be broadly categorised into 4 paradigm: logarithmic models, hybrid model, multicellular models, and flow-focused models. Related models are connected by a solid black line, while unrelated models that are subsequent on the timeline are connected with a dashed black line. A (nonlinear) timeline is shown on the left. The papers cited in this figure are [,,,,,,,,–83].

Fig 6. A sample logarithmic model of vascular calibre determination in which the vessel diameter is logarithmically dependent on the local haemodynamic factors (B) or the metabolic stimuli (C).

The initial network configuration is shown in (A) in which the vessels initially have a random diameter with no adaption. (B) When the vessel diameter is controlled by the local wall shear stress, the diameter of the anastomosis (vessel denoted by the asterisk) decreases in response to the comparatively lower wall shear stress. (C) The diameter of each vessel is responsive to metabolic stimuli, which acts to stabilise the diameter of each vessel in this model. Figure reproduced from Pries and colleagues [62] with permission. Copyright 2001 the American Physiological Society.

Fig 7. Hybrid models of vascular development.

(A) Alberding and colleagues [76] presented the first 3D hybrid model of vascular development in the cerebral cortex. The initial simulation configuration is shown (i) from which vessels angiogenically sprout (ii) to from a dense network (iii) with variation in vessel diameter. Finally, vessels falling below a given diameter threshold are removed from the network to form an optimised cortical vasculature (iv). In each figure, the partial pressure of oxygen is coloured according to scale at the right in (iv). (B) McDougall and colleagues [75] adapted the Secomb hybrid model to simulate retinal vascular development, presentation development at P2.7 (i), P5.2 (ii), and P7.7 (iii). The green shading of each vessel depicts the vessel radius, with larger vessels shown in lighter green, and smaller vessels shown in darker green (scale on right). Subfigure A reproduced with permission from Alberding and colleagues [76], and subfigure B reproduced with permission from McDougall and colleagues [75].

Fig 8. The Peirce (A), Tabibian (B), and Edgar (C) models are the 3 cell-based models currently available in the literature.

(A) The Peirce and colleagues [52] cellular automaton model examining vascular stabilisation and identity determination; those vessels composed of endothelial cells (red) surrounded by smooth muscle cells (yellow) are becoming arterioles, while the vessels composed of only endothelial cells (yellow) are still capillaries. (B) Tabibian and colleagues [77] published a self-propelled particle model of vascular calibre determination; red circles are components of the capillary, and blue circles are components of the surrounding tissue. (C) Edgar and colleagues [78,79] developed a sphere-based model of vascular calibre determination and regression; each endothelial cell is depicted by an ellipse with the polarity given by the green line. Subfigures A, B, and C are reproduced from Peirce and colleagues [52], Tabibian and colleagues [77], and Edgar and colleagues [79], respectively, with permissions.

Fig 9. The Chen (A), PolNet (B), and Ghaffari (C) approaches are the 3 flow-focused models of vascular remodelling currently available in the literature.

(A) Chen and colleagues [15] coupled long-time series confocal imaging of the developing zebrafish midbrain vasculature (in vivo) with a fluid-structure interaction model to predict which vessels would undergo pruning during vascular development. (B) The PolNet group [7,8,13,18] compared live in vivo images of the retinal vasculature in neonatal mice with a 3D lattice-Boltzmann reconstruction of the blood flow to show that endothelial cell polarity (bi) aligns against the direction of flow, with polarisation strength inversely proportional to the wall shear stress (Bii). (C) Ghaffari [80,81] coupled time-lapse microscopy images of flow dynamics in ex ovo quail embryos with a finite element model of the 2D Navier–Stokes equations to understand how the flow evolves during development. Subfigures A, B, and C are adapted with permission from Chen and colleagues [15], Franco and colleagues [13], and Ghaffari and colleagues [80], respectively.

Fig 10. Key challenges for the development of mathematical models of vascular remodelling.

The 3D printed vascular network shown on the top row has been reproduced with permission from [11].