Dynamics of vascular branching morphogenesis: The effect of blood and tissue flow

Vascularization of embryonic organs or tumors starts from a primitive lattice of capillaries. Upon perfusion, this lattice is remodeled into branched arteries and veins. Adaptation to mechanical forces is implied to play a major role in arterial patterning. However, numerical simulations of vessel adaptation to haemodynamics has so far failed to predict any realistic vascular pattern. We present in this article a theoretical modeling of vascular development in the yolk sac based on three features of vascular morphogenesis: the disconnection of side branches from main branches, the reconnection of dangling sprouts (“dead ends”), and the plastic extension of interstitial tissue, which we have observed in vascular morphogenesis. We show that the effect of Poiseuille flow in the vessels can be modeled by aggregation of random walkers. Solid tissue expansion can be modeled by a Poiseuille (parabolic) deformation, hence by deformation under hits of random walkers. Incorporation of these features, which are of a mechanical nature, leads to realistic modeling of vessels, with important biological consequences. The model also predicts the outcome of simple mechanical actions, such as clamping of vessels or deformation of tissue by the presence of obstacles. This study offers an explanation for flow-driven control of vascular branching morphogenesis.

Figure 1

Progressive contraction of the southern hemidisk (Stokes approximation) transforms the initial blastodisk into a figure 8 (from Ref. 3). Left, the contraction-vector field lines. Right, the progressive deformation of the disk in the field given to the left. The motion of cells towards the hyperbolic point (the center of contraction) creates a tensile stress on the extra-cellular matrix on which cells move.

Figure 2

Scheme of the ingression. The flow of cells in the blastodisk creates a tension which opens a crack. Cells ingress in the crack. Cells which ingress make a U turn (in the vertical plane), move away from the crack, and form the large disk around the figure 8, which is the yolk sac. The crack closes because the stress becomes compressive.

Figure 3

The initial landscape inside the egg, at the moment when our work starts (2.5 days). The very first blood islands are visible along the edge of the yolk sac.

Figure 4

Aspect of the “navel” of the chicken. The aspect of the dorsal aortas in the region of the navel is obtained by transporting in the “flow” of Fig. 1 the mosaic of the embryo [5]. In the center of the embryo, between arm and leg rudiments, one finds a turn at right angles of the mosaic (from Ref. 5). This forms a template of vessels in the region of the navel. This gives us the boundary condition for the arterial outflow: two rudiments of arteries oriented to the right, for one, to the left, for the other (convention of Fig. 1).

Figure 5

(Color) (a) Scheme of the primary circulation stage. Veins are in blue, arteries in red. (b) Scheme of the secondary circulation stage. (c) The true yolk sac, in the final state which we wish to model.

Figure 6

Boundary and pressure conditions at start of the problem from a physicist’s point of view. There is one large circular vein, isopotential at low pressure. There are two arterial rudiments at high pressure (P1). There is a random lattice of capillaries in between the embryo and the large vein. Right, an image of the true capillary lattice.

Figure 7

Examples of small vessel disconnection (arrows), including capillaries, and even one venule. The capillary-free zone appears clearly on the left of the main venule (bottom right) and also around the top-left arteriole.

Figure 8

Morphology of a chick yolk sac vessel, by day 3.5, seen from end to end (approximately $2\mathrm{cm}$), showing that the density of connected capillaries is larger in distal parts than in proximal parts. In the first two segments shown, there are simply no connections to the capillary plexus; in the third, they are hardly visible, if any. Perfused capillaries connected to the vessel are found mostly in distal parts (fourth segment). In proximal parts, one would be able to cut vessels and separate them from the mesoderm; distal parts, vessels are intimately connected to the tissue by capillaries.

Figure 9

Time lapse imaging of capillaries which are disconnected from arteries, then reconnected to veins, and again perfused (reprinted from Ref. 9). This image shows first a vitelline artery [VA in (a)] with several capillary loops which are connected to it; others are already disconnected. From (a) to (f), more capillary loops are disconnected. As a consequence, the dead ends appear as stagnant blood islands, which lie along the artery, but are no longer in contact with the artery lumen. The dead ends behave as sprouts; especially, from (f) to (i), black arrows show a dead end which succeeds to grow over the artery and to reconnect to the other side. As an instantaneous consequence, the blood is flushed from the stagnant islands and flow is resumed in the strands along the artery. Note that the direction of blood flow in these strands is reversed. Now, venules can form in the capillaries which are disconnected from the artery and can be incorporated in the venous bed, in the proximal to distal order, following centrifugally capillary strands which are oriented parallel to the arteries and growing in the direction of arterial tips.

Figure 10

Vessel displacement is readily observed on time-lapse movies [(a), photos 1–4]. The photo sequence shows a developing part of the chick embryo yolk-sac vasculature between days 2 and 3. In photo 1 four branch points are selected and followed over time; photos 2–4 show the fate of this region over a period of time of $8\mathrm{h}$. We visualize the differential growth of the tissue, by overlaying a polygon, whose corners are at four chosen vessel bifurcations, considered as absolute reference points. We follow the rectangle during time (panel right). It is observed that the rectangle is deformed and displaced. The motion is not a mere homothetic dilation, and it is completely self-organized by the branches as they grow.

Figure 11

(a) (Left) We have studied the flow induced elastoplastic deformation in the region where a side branch is connected to a larger main branch using a physical model. In the model we have solved the Navier-Stokes equation in a cavity representing the connection (a), for both arteries and veins. As suggested by in vivo observations, we started with a high flow velocity in the main branch [horizontal, in (a)] and a smaller flow velocity in the side branch. The flow velocity was assigned positive when separating from the main artery to a lateral smaller arteriolar capillary (if the two flows separate at the connection Vy is oriented downwards in the lateral vessel) and negative in the other case (if the two flows converge at the connection, the flow Vy in the lateral venous capillary is oriented upwards). We also did the calculation for a flow velocity equal to zero (stagnant flow—e.g., the side branch is a dead end as is the case for an angiogenic sprout). Once the flow chart is computed, we use the pressure and velocity along the boundary in order to calculate the elastic deformation of the surrounding tissue. In order to find the deformation, we solve Hooke’s law outside the vessel using as boundary condition the pressure and shear exerted by the flow (a). These calculations are performed with the finite-element method in 2D. This gives the incremental elastic deformation induced by the flow [two examples are given in (a), bottom]. When compared to the initial configuration, this gives either a closing (bottom left) or an opening rate (bottom right), a negative (positive) rate for the width of the side branch. (b) (Right) We plot here the “first step” or initial opening-closure rate as a function of the flow velocity in the side branch and as a function of the branching angle of the side branch. The results show the domain of parameters in which arterioles or venules either tend to remain connected or to become disconnected, as a consequence of the elastoplastic behavior of the cells at the bifurcation. The plot is therefore formed of four parameter domains [on the $x$ axis: veins or arteries corresponding to negative or positive flow velocities: on the $y$ axis, closing (negative) or opening (positive) rate]. For each angle, we find that the domain of arteriolar disconnection is much wider than the domain of vein disconnection (see text). In the arterial domain, side branches remain connected to the main branch for sharp angles only.

Figure 12

Using a simple elastoplastic model, we perform iteratively the growth of a small vessel sprout (SP) or tip, which faces a tube (a). The calculation is performed in 2D by the finite-element method. In this figure, a segment of the small disconnected tube is seen facing two bigger tubes. We show here simultaneously (a) one tip facing a tube in a symmetrical situation [(a), bottom part] and one tip facing a tube in an asymmetrical situation [(a), top part]. Stars show the arterial lumens. Pressure is exerted from the inside of the small sprout. This creates a stress field around the tip. We select an ad hoc threshold of shear. The colored line shows the region above this threshold. Many (up to 50) elastoplastic steps of growth are performed iteratively by letting the sprout deform and by considering that only the deformation above a certain threshold is plastic (deformation for good), below the threshold the deformation is elastic. The results (b) show that the disconnected vessel preferentially sprouts out from the tips and the sprouts tend to avoid the large tubes and grow towards the most favorable region, hence swerving around the tube. This effect, however, is dependent on the magnitude of the threshold, for the elastoplasticity of the tissue and, above all, on the components of the deformation which are chosen for the plasticity level (this translates in biological terms in whether cells divide and adapt in response to tensile deformation or to shear, or to a combination of them; in this calculation, only shear is taken into account).

Figure 13

In our stochastic model, random walkers model the effect of interstitial pressure. When these random walkers arrive on the tree, they shift it sideways by one pixel (two examples of hits are shown to the left), except if such a move would disconnect the tree (one example of forbidden hit to the right).

Figure 14

(a) Typical true yolk sac. $\mathrm{SV}=\mathrm{sinus}$ vein. $\mathrm{Stars}=\mathrm{arteries}$. $\mathrm{VA}=\mathrm{vitelline}$ artery. $\mathrm{V}=\mathrm{cranial}$ vein. The embryo is clearly visible in the center. (b) Numerical modeling by Monte Carlo simulation of the morphogenesis of a vascular tree (B1)–(B4). The model is defined by an initial boundary condition and capillary plexus (a matrix of pixels). In this simplified simulation, we put two arterial rudiments at the center and two venous rudiments at the head only (we do not include the caudal vein). Next, we use random walkers (RW) on the lattice to model diffusion fields around the vessels and hence the effect of both shear inside the tube and push outside the tubes (see text). When a growth RW hits an existing vessel, itself part of a tree, the RW is aggregated to it and the vessel extended by one pixel. Such a random aggregation is a stochastic version of capillary enlargement by adaptation to stresses; the growth of the aggregate (seen here as a growing red tree) is formally identical to enlargement of capillaries in the direction of high gradient of pressure (the vasculature grows proportionally to $-\mathrm{grad}P$; see text). When a displacement RW hits the tree, the monomer of the tree is moved sideways by one pixel. The parameters of this simulation are the total fraction of disconnected monomers, at each step, 70% of all segments. (For example, if 1000 monomers are present in the tree, the first 700 are disconnected.) The ration of displacement hits with respect to the growth hits is 4. (This to say that four displacement random walkers are launched between each extension random walker.) The nonretractable monomers are the last 5% which have arrived. In (B1)–(B4), the arterial tree is actually surrounded by a lattice of capillaries which is not represented for reasons of clarity. Since blood flows out of arteries towards veins, the particles driving the growth of veins are launched from the arteries and vice versa. In order to investigate the remodeling of the vasculature in reaction to interstitial tissue shear, we incorporate pressure driven displacements of vessels. We add to the model a second family of RW’s which, coming from vessels, move the vessels. Upon hitting a vessel, a RW simply moves the hit vessel laterally by one step. In some dead ends (tips), the motion amounts to a retraction (see text). The actual motion of vessels is the self-organized result of all these pushes. To model capillary disconnection, vessels are progressively disconnected from the plexus as they grow. In the simulation displayed here, we show only the arteries, to make the comparison of the arterial patterning easier. The oldest arteries tend to progressively form regular boundaries, while distal parts are still more random and bushy. Note the completely spontaneous formation of a crablike pattern, the progressive extension of a stable circular sinus vein, with an angular re-entrance at the head. $\mathrm{SV}=\mathrm{sinus}$ vein; $\mathrm{VA}=\mathrm{vitelline}$ arteries.

Figure 15

(a) shows the calculated prediction of arterial occlusion on patterning by the physical model. The simulation was started with a single artery to the left and a single vein to the right. In this situation the arterial tree grows and tilts towards the right side of the embryo. Similar to the in vivo situation, two main branches pass the head-to-tail axis, one in the head region, one in the tail region. This situation amounts to transforming a quadrupole into a dipole. (b) shows the effect of a mechanical conclusion of the right vitelline artery on global patterning of the arteries and veins in the chick-embryo yolk sac. Note in comparison to the control situation in Fig. 5 that the global patterning has changed for both arteries and veins. Arteries start to grow from the left-hand side towards the right-hand side, crossing the midline, moving towards the venous plexus on the right side. The right vitelline artery has transformed into a vein. Both the posterior (caudal) and anterior (cranial) vitelline veins are absent. Note the remains of the reentrance of the two cranial veins, along the sinus vein, in the region of the anterior (“north”) pole (by 1 o’clock, in this image). Scale bar $1.5\mathrm{mm}$.

Figure 16

(Color) Typical morphology diagram, as a function of parameters in the simulation. In this plate, we vary the following parameters: the proportion, $N_{\text{displace}}$, of displacement-random walkers vs growth-random walkers. The proportion of disconnected vessels $P_{\text{disconnection}}$, and the probability of a vein to grow across an artery, if a random walker reaches it, $P_{\mathrm{cross}}$. In 16.1 (“control”) $P_{\mathrm{disconnection}}=50\%$, $N_{\mathrm{displace}}=30$, $P_{\mathrm{cross}}=1$, in 16.2 $P_{\mathrm{cross}}=0$, in 16.3 $P_{\mathrm{disconnection}}=80\%$, in 16.4 $P_{\mathrm{disconnection}}=0$, in 16.5 $N_{\mathrm{displace}}=50\%$, and in 16.6 $N_{\mathrm{displace}}=10\%$. In 16.1, a reference configuration in which the probability of crossing is one: veins cross arteries in the proximal regions without shunting and grow in direction of the tips of arteries (the sources of random walkers) where they interlace with arteries. In 16.2, the probability of crossing is now zero; as a consequence, veins are forced to turn around arteries to enter into the “fjords” in an interlace fashion. In 16.3, the capillary disconnection is enhanced; as a consequence, the sources of flow for the veins are more distal; this generates a pattern of very thin and elongated veins. In 16.6 the situation is shifted with respect to 25.1, towards less displacement; vessels are more bushy and exhibit direct $A\text{−}V$ shunts in the distal parts, because the deformation exerted by the interstitial tissue is not strong enough to keep the growing veins away from the arteries. In 25.5 we identify a set of ad hoc parameters which generates an interlace of arteries and veins. It is remarkable that the tendency of arteries to grow towards veins (and vice versa) may be equilibrated by the reaction of interstitial tissue which tends to shift them away; as a consequence, vessels approaching arteries tend to grow in parallel, with capillaries in between them (note that the capillaries are not shown, they form the discrete lattice on which the simulation is performed), as observed in the chick yolk sac, the chorioallantoic membrane, or even the retina vasculature. Such an interlaced vasculature is typical [1]. In 16.5, the physical push by interstitial tissue is very strong and capillary breakage is medium (50% of the most distal arterial strands are still connected). Excess of capillary breakage and increased interstitial force appear as factors of vascular rarefaction, the major architectural feature of hypertension. Conversely, reduced capillary breakage or reduced interstitial force increases the number of $A\text{−}V$ shunts and makes more noisy, wavy, vessels. In simulations 2 and 3, the arteries were grown first and the veins were grown afterwards (as is the case for the secondary circulation stage). In all simulations, the total number of vein monomers which have been launched in order to grow a tree of a given size is bigger than for arteries. This comes from constant shifting and hindering of growth by displacement walkers and of discarded walkers when crossing is not possible. This is qualitatively correct in terms of tissue development: increased pressure or the presence of obstacles hinders growth. It is also an observed fact that veins grow more slowly than arteries, as appears in most images above. Remember that the final shape of a vascular bed is a result of progressive proximo-to-distal regularization of vessels by the self-organizing process; bushy venular beds become progressively elongated and thin strands, as commented above (especially Fig. 10; see also other figures).