# 3D Printable Vascular Networks Generated by Accelerated Constrained Constructive Optimization for Tissue Engineering

## Summary (3 min read)

### Introduction

- By incorporating a vascular network into a cell-ladened scaffold, tissues with clinically relevant dimensions can be developed and maintained, which could then be used as a replacement for a diseased tissue.
- The networks bifurcate from large vessels down to fine vessels and anastomose back to large-sized vessels.
- Section III outlines the novel methods implemented, and presents an argument for their efficiency; this is investigated in detail in Section IV, alongside some demonstrative examples.

### II. CONSTRAINED CONSTRUCTIVE OPTIMIZATION

- The required input parameters (those which do not have default values) are listed in Table II.
- The authors follow the established terminology of Schreiner [13] and refer to nodes in a tree with no children as terminals, and the node which starts the root vessel as the source.
- The authors used a constant-viscosity model, as the vessel sizes they are currently able to manufacture (r≥125 µm) are above the radius at which viscosity correction has previously been applied [14].
- The authors use only nodes with at most two children (i.e. bifurcations), but nodes with higher splitting may be approximated by multiple close bifurcations with low separation.

### III. NEW METHOD: ACCELERATED CCO

- The established CCO approach to producing multiple nonintersecting networks adds terminals into the network by considering multiple candidate topologies, optimising each for volume and then selecting the minimum volume network with no intersections [11], [15], [22].
- The intersection test at each stage leads to poor performance: the most efficient algorithm reported in the literature [15] scales as O(N2 logN), where N is the number of terminal nodes, requiring days of computing time for complex cases.
- This is achieved by introducing a new type of node, the transient (see Table III), which acts to create piecewise approximations to curved branches.
- The authors will use the term depth of a node similarly to its standard usage with respect to tree data structures: the number of edges between a node and the root.
- When multiple networks are created to meet at the same terminal points (e.g. arterial/venous pairs), the authors refer to them as being matched.

### A. Creating a bifurcation

- CCO iteratively adds terminal sites into the network, meaning that the overall complexity is N times the complexity within each iteration.
- At each iteration, a branch must be selected from the existing network from which to create a bifurcation, and the location of this bifurcation must be determined.
- The counted selection method will never do worse than testing all existing branches, giving an upper bound on complexity of O(n), where n is the number of terminals currently constructed, since the number of branches is Θ(n).
- For perfusion spaces where the terminals are arranged as a shell around the inlet, better performance is expected, whereas for long, thin volumes the authors expect the worst case performance to be achieved.
- The approximation of the gradient direction in (26) seems acceptable when considering the bulk movement of nodes, increasingly so at larger depths.

### C. Collision resolution

- Firstly, a decision is made as to whether the start and end nodes of each branch are involved in the intersection, by comparing the distance between the nodes and the point of closest approach, d, to the branch radius, also known as 3) The determinate case.
- The perturbation vector, vp, is therefore given by vp = δkawn̂ (36) where wn̂ is the weighted normal in the relevant direction for each branch.
- After each iteration of resolution, the resolver checks the count of each node against a threshold value, zt, and any nodes with counts greater than this are culled from their networks, alongside their matched partners.
- The radii of vessels are kept above the printer minimum feature size, rmin, by adjusting the overall pressure drop if necessary.

### A. Collision resolution

- A demonstration of collision resolution and terminal smoothing working for a simple case is shown in Fig.
- Whilst not perfect, angles have been mostly reduced to below 90°.
- Regions such as this, where the authors have one or more high-flow branches, are exactly where they expected to need to cull terminals.

### B. Computational scaling

- If S = 0, the order of construction becomes important: if the children of the first bifurcation point away from the target site, the bifurcation will be created at the root.
- The distribution for the CSS case shows a very slight decay (Fig. 6), suggesting that the performance floor would be closer to O(N logN) if the authors were to supply a shell-like geometry for which they may use S = 0.

### D. Pseudo-biomimetic 3D printable vascular systems

- To achieve this, the authors adjusted pressures after construction to give correct dimensions at the inlet.
- A human-sized liver (Fig. 12) was approximated as a triangular prism supplied and drained by a quadruply-matched network.
- Finally, one notable departure from the physiological liver is that the biliary tree connects to the blood vessel trees through each of the terminals.
- This, while not biomimetic, is necessary to clear the 3D printed template material from the channel network (when the bulk matrix is micro-porous).
- It is of note that each unit cell in the CCO model is supported with its own terminal, therefore producing a uniform pressure distribution in the perfusion space, which leads to biomimetic vascular networks.

### V. CONCLUSION

- The success of tissue engineered constructs largely depends on the incorporation of perfusable vascular networks which can support the biological functions of the embedded cells.
- M. Schneider et al., “Physiologically Based Construction of Optimized 3-D Arterial Tree Models,” in Medical Image Computing and ComputerAssisted Intervention – MICCAI 2011, Lecture Notes in Computer Science, pp. 404–411, Springer, Berlin, Heidelberg, Sept. 2011. [11].
- J. S. Miller et al., “Rapid casting of patterned vascular networks for perfusable engineered three-dimensional tissues,” Nature Materials, vol. 11, pp. 768–774, Sept. 2012.

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##### References

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...important that the network displays hierarchy [2], [3]....

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### "3D Printable Vascular Networks Gene..." refers background in this paper

...Since the focus is on the higher flow branches, Murray’s exponent was set to γ = 2 to give more realistic radius decay in these vessels [3]....

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...important that the network displays hierarchy [2], [3]....

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