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

TL;DR: An approach of alternating stages of construction and batch optimization of all node positions is introduced and shown to yield consistently more optimal networks, and a biomimetic, liver-like tissue model is demonstrated.
Abstract: One of the greatest challenges in fabricating artificial tissues and organs is the incorporation of vascular networks to support the biological requirements of the embedded cells, encouraging tissue formation and maturation. With the advent of 3D printing technology, significant progress has been made with respect to generating vascularized artificial tissues. Current algorithms to generate arterial/venous trees are computationally expensive and offer limited freedom to optimize the resulting structures. Furthermore, there is no method for algorithmic generation of vascular networks that can recapitulate the complexity of the native vasculature for more than two trees, and export directly to a 3D printing format. Here, we report such a method, using an accelerated constructive constrained optimization approach, by decomposing the process into construction, optimization, and collision resolution stages. The new approach reduces computation time to minutes at problem sizes where previous implementations have reported days. With the optimality criterion of maximizing the volume of useful tissue which could be grown around such a network, an approach of alternating stages of construction and batch optimization of all node positions is introduced and shown to yield consistently more optimal networks. The approach does not place a limit on the number of interpenetrating networks that can be constructed in a given space; indeed we demonstrate a biomimetic, liver-like tissue model. Methods to account for the limitations of 3D printing are provided, notably the minimum feature size and infill at sharp angles, through padding and angle reduction, respectively.

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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GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2019 1
3D Printable Vascular Networks Generated by
Accelerated Constrained Constructive
Optimization for Tissue Engineering
Andrew A. Guy, Alexander W. Justin, Dulce M. Aguilar-Garza and Athina E. Markaki
Abstract One of the greatest challenges in fabricating
artificial tissues and organs is the incorporation of vascular
networks to support the biological requirements of the
embedded cells, encouraging tissue formation and matura-
tion. With the advent of 3D printing technology, significant
progress has been made with respect to generating vas-
cularized artificial tissues. Current algorithms to generate
arterial/venous trees are computationally expensive and
offer limited freedom to optimize the resulting structures.
Furthermore, there is no method for algorithmic generation
of vascular networks that can recapitulate the complexity
of the native vasculature for more than two trees, and
export directly to a 3D printing format. Here, we report such
a method, using an accelerated constructive constrained
optimization approach, by decomposing the process into
construction, optimization and collision resolution stages.
The new approach reduces computation time to minutes
at problem sizes where previous implementations have
reported days. With the optimality criterion of maximizing
the volume of useful tissue which could be grown around
such a network, an approach of alternating stages of con-
struction and batch optimization of all node positions is
introduced and shown to yield consistently more optimal
networks. The approach does not place a limit on the num-
ber of interpenetrating networks that can be constructed in
a given space; indeed we demonstrate a biomimetic, liver-
like tissue model. Methods to account for the limitations
of 3D printing are provided, notably the minimum feature
size and infill at sharp angles, through padding and angle
reduction, respectively.
Index Terms Constrained Constructive Optimization,
vascular networks, tissue engineering, 3D printing
AA Guy is supported by an EPSRC Doctoral Training Partner-
ship Award (EP/N509620/1). AW Justin is supported by EPSRC
(EP/R511675/1 & EP/N509620/1), the Isaac Newton Trust and the
Rosetrees Trust (M787). DM Aguilar-Garza is supported by The Cam-
bridge Trust, CONACyT (Mexico) and the EPSRC Cambridge & Cran-
field Doctoral Training Centre in Ultra Precision (EP/K503241/1). Corre-
sponding author: Athina E. Markaki (email: am253@cam.ac.uk).
AA Guy, AW Justin, DM Aguilar-Garza and AE Markaki are with
the Department of Engineering, University of Cambridge, Trumpington
Street, Cambridge CB2 1PZ, U.K.
AAG conceived of and executed the development and analysis of the
code.
Copyright © 2017 IEEE. Personal use of this material is permit-
ted. However, permission to use this material for any other purposes
must be obtained from the IEEE by sending an email to pubs-
permissions@ieee.org
I. INTRODUCTION
V
ASCULARIZATION is one of the key challenges for
engineering new tissues and organs in the lab [1]. Tissue
constructs without a perfusable vascular network lack a mech-
anism for supplying cells with nutrients and removing CO
2
and cellular waste; for this supply mechanism to be efficient,
it is important that the network displays hierarchy [2], [3]. 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. Furthermore, tissues and organs consist
of multiple interpenetrating tubular systems, including those
specific to the tissue function, such as the biliary system in
the liver. Until such complex systems are constructed in the
lab, fabrication of engineered tissue equivalents for complex
organs (e.g. liver, kidney) will remain a formidable challenge.
By algorithmically generating biomimetic and architec-
turally complex vascular network models, one can rapidly pro-
duce vascular networks capable of uniformly supporting high
densities of metabolically active cells in a 3D environment.
Such models can be used to simulate fluid flow (e.g. wall
shear stresses, velocity field), metabolite diffusion, and the
cellular environment. Furthermore, template structures from
these models can be fabricated via 3D printing techniques
(e.g. inkjet, extrusion-based, stereolithography) and incorpo-
rated into tissue engineered scaffolds to yield perfusable chan-
nels [4]. Such approaches involve a solid sacrificial template
around which biomaterials (frequently loaded with cells) are
cast. These sacrificial structures are removed (e.g. chemically,
thermally, mechanically) to produce a vascular network in the
biomaterial with channels of the same diameter as the 3D
printed feature.
Since manually constructing vascular structures using com-
puter aided design (CAD) packages is unfeasible for the
level of complexity required, procedural generation of vascular
networks is greatly preferable. Other applications for such
models include comparisons of diseased and physiological
tissues [5], surgical planning [6], [7], and the validation of
vessel segmentation algorithms [8]–[10].
A number of algorithms have been developed over the
past 25 years for arterial tree representation, considering the
physiological constraints associated with blood flow and the
hierarchical vasculature in the human body. The most common

2 GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2019
approach employed to produce realistic vascular networks
involves iterative growth [8], [10], [11]. Constrained Con-
structive Optimization (CCO) is a widely accepted method for
the generation of vascular trees [12]–[17]. Other approaches
include the Lindenmayer systems (L-systems) [9], [18], Global
Constructive Optimization [19] and the Random Walk Algo-
rithm [20]. In order to build a physiologically relevant arterial
tree, an optimality criterion must be fulfilled. The principle
of minimum intra-vascular volume is most commonly used to
ensure an optimal build [8], [11]–[14], [16]–[19]. In addition,
several assumptions are frequently applied to simplify these
models. The most common include laminar (steady-state) flow,
flow conservation, equal terminal flow and pressure, Newto-
nian fluid flow, and Murray’s law for optimal bifurcations;
networks that conform to this law maximize flow conductance
per unit volume [2]. Some algorithms simulate additional
phenomena such as the Fåhræus-Lindqvist effect (viscosity
changes with vessel diameter) [15], [20] and vascular growth
driven by oxygen demand [8], [20]. Most algorithms are used
to produce single vascular trees [8]–[10], [12], [13], [17]–[19],
[21], while only a few simulate full vascular networks [11],
[15] or include capillaries [16], [20].
Here, we describe a new CCO approach, capable of produc-
ing multiple and independent interpenetrating networks (e.g.
arterial, venous, epithelial, lymphatic). The networks bifurcate
from large vessels down to fine vessels and anastomose back
to large-sized vessels. The model allows for any volumetric
region to be uniformly supported with vessels and the spacing
between the vessels is also controllable. Thus, multiple inde-
pendent networks can be packed into the space. Further, our
approach is computationally efficient, and engages with vessel
collision detection after the vascular tree has been formed.
Alongside physical flow constraints, such as pressure and
flow rate, our approach incorporates a number of biomimetic
principles, including Murray’s law, which relates the parent
and daughter branch diameters, and minimal network volume
considerations.
Section II gives a brief overview of the fluid mechanics
and physiological laws underlying CCO, and the essential
inputs required to create a network. 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. The mathematical
notation used in this paper and parameters included in the
model are summarised in Table I.
II. CONSTRAINED CONSTRUCTIVE OPTIMIZATION
The required input parameters (those which do not have
default values) are listed in Table II. We 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. We used a constant-viscosity
model, as the vessel sizes we are currently able to manufacture
(r125 µm) are above the radius at which viscosity correction
has previously been applied [14]. Considering steady laminar
flow, the pressure difference over a branch, P , is given as
P =
8µL
πr
4
Q = RQ, (1)
TABLE I
SUMMARY OF NOTATION
Physical and derived values
µ Dynamic viscosity [kPa·s
1
]
γ Murray’s law exponent [-]
x Position vector in R
3
[mm]
Q Volumetric flow rate [mm
3
·s
1
]
P Pressure [kPa]
L Branch length [mm]
r Branch radius [mm]
R Laminar flow resistance of a branch [kPa·s·mm
3
]
R
Reduced resistance [kPa·s·mm]
f Child branch radius fraction of parent [-]
Collision resolution
Calculated values
s
r
Radial slenderness, L/r [-]
χ Branch length fraction of intersection [-]
w Weighting [-]
ˆn Unit normal vector in R
3
[mm]
v
p
Perturbation vector in R
3
[mm]
δ Overlap between branches [mm]
Input parameters and default values
k
r
Radial capture factor 1.2 [-]
k
a
Resolution aggression factor 1.5 [-]
C Relative compliance of a network 1.0 [-]
|v|
min
Minimum perturbation distance 50 µm
z
p
Terminal node cull penalty 4 [-]
z
t
Terminal node cull threshold 20 [-]
β
c
Critical flow ratio for immediate culling 100 [-]
r
+
Radial padding to ensure separation 125 µm
r
min
Minimum feature size (radial) 125 µm
Optimization
Calculated values
˜g(x) Approximation to the direction of the gradi-
ent of the volume of the entire network with
respect to the position, x, of a bifurcation
[mm
3
·mm
1
]
V (x) Finite-difference estimation of the volume
gradient with respect to the position, x, of
a bifurcation, with stride
[mm
3
·mm
1
]
The stride length to take, having determined
the direction: x x · ˜g(x)/|˜g(x)|
[mm]
Input parameters and default values
η Step length fraction 0.1 [-]
τ Termination length fraction 0.2 [-]
β Step fraction reduction ratio 0.5 [-]
z
b
Iterations between step fraction reduction 5 [-]
Construction
S Skip limit: the number of attempts allowed where no progress
is made. The most ‘lazy’ selection is achieved with S = 0,
and traditional CCO selection is with S = .
Computational scaling
N Total number of terminal nodes in the perfusion space.
n Number of terminal nodes currently in tree.
d(i) Terminal depth distribution: for a positive integer i, the
fraction of terminal nodes which have i parent segments to
the root node.
Sets and indexing
B All branches in the tree.
C Children, the immediate downstream elements.
U The chain of upstream elements to the root. Indexed as
{1...M}, with 1 being the first child of the root and M being
the immediate upstream element.
T Bifurcation triad, the elements surrounding a bifurcation:
T = {C, U
M
}.
0 Subscript for the root branch/node.
p Subscript for the parent branch/node in a given context.

GUY et al.: 3D PRINTABLE VASCULAR NETWORKS... 3
TABLE II
THE ESSENTIAL INPUTS TO THE CCO ALGORITHM, AND THEIR UNITS
Global
P
s
Pressure drop from source to terminals kPa
x
s
Source node position mm
Per Terminal
x Position mm
Q Volumetric flow rate mm
3
·s
1
where L and r are the branch length and radius respectively,
µ is the dynamic viscosity (taken to be 4×10
6
kPa·s
1
for blood), Q is the volumetric flow rate and R the laminar
resistance. Physical constraints are provided by flow and
pressure consistency between children (C) and their parent (p):
Q
p
=
X
i∈C
Q
i
, (2)
P
p
= P
i
+ Q
i
R
i
i C. (3)
In calculations, the reduced resistance, R
:
R
= Rr
4
, (4)
is used [12], as radii are only determined when needed.
The relationship between parent and child radii is fixed
by Murray’s law [2] to minimise power dissipation over the
network:
r
γ
p
=
X
i∈C
r
γ
i
, (5)
where γ = 3, the standard result for minimum work, is
the default value. Values in the range 2 γ 3 are
physiologically relevant, with lower exponents being observed
in high-flow vessels. [3].
In this implementation, we 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. At each bifurcation, the child radii are set by their
fraction, f, of the parent radius:
r
i
= f
i
r
p
, (6)
from which follows:
f
γ
1
+ f
γ
2
= 1, (7)
and, by dividing (5) by r
γ
1
:
f
1
=
"
1 +
r
1
r
2
γ
#
1
γ
, (8)
with a similar form for f
2
. By defining the terminal pressures
to be uniform, we can determine the global resistance from the
total flow rate, Q
0
, and pressure drop from source to terminals,
P
s
, allowing the root radius, r
0
, to be calculated with (1)
and (4):
r
0
=
R
0
R
0
1
4
=
Q
0
R
0
P
s
1
4
, (9)
which is propagated down the tree using (6).
TABLE III
NODE TYPES, THEIR RELATIONSHIPS, AND DEPTH MODIFIERS.
Type Parents Children Depth modifier
Source 0 1
def
= 0
Bifurcation 1 2 + 1
Transient 1 1 + 0
Terminal 1 0 + 1
Since terminal pressures are uniform, we can use the
consistency constraint in (3) to calculate the child radius ratio
at a bifurcation,
r
1
r
2
=
R
1
Q
1
R
2
Q
2
1
4
, (10)
with which the radius fractions may be calculated using (8).
The reduced resistance of the parent branch at a bifurcation is
then calculated from its downstream components as
R
p
=
8µL
p
π
+
f
4
1
R
1
+
f
4
2
R
2
1
. (11)
III. NEW METHOD: ACCELERATED CCO
The established CCO approach to producing multiple non-
intersecting 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(N
2
log N), where N
is the number of terminal nodes, requiring days of computing
time for complex cases.
We propose a novel method, breaking the process into
3 separate stages: Construction, Optimization, and Collision
Resolution (see Fig. 1). With the reasoning that the diameter
of any practical vasculature is small compared to the space it
serves, networks are constructed in parallel with no knowledge
of each other, and collisions between them are resolved by
making small excursions around the contact points. This is
achieved by introducing a new type of node, the transient
(see Table III), which acts to create piecewise approximations
to curved branches. Such nodes have previously been consid-
ered in the context of single arterial trees [10] to introduce
tortuosity.
We 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. The difference is that
we do not regard the transient as increasing logical depth, as
we can package segments connected by transients into single
branches, which we view as the fundamental unit of the tree.
The tissue volume around a single terminal will be referred
to as a unit cell, and will be considered to be supplied by
the network if there is a terminal vessel linking this node to
the network (the terminal is constructed), or if a much larger
vessel is occupying the unit cell. The volume supplied by the
entire network is referred to as the perfusion space. When
multiple networks are created to meet at the same terminal
points (e.g. arterial/venous pairs), we refer to them as being
matched. There is no limit to the multiplicity of matching.

4 GENERIC COLORIZED JOURNAL, VOL. XX, NO. XX, XXXX 2019
(b)(i)
(b)(ii) (b)(iii)
(a)
Fig. 1. The Accelerated CCO process: starting from (a), there are
three actions available. (b)(i) Construction: adding new terminals into the
network. The network topology is altered, and the only geometry which
is altered is that of the branches connecting the new bifurcation (the
bifurcation triad, T , shown dashed). (b)(ii) Optimization: The network
topology is fixed, and the geometry of the entire network is altered to
minimize the total network volume. (b)(iii) Collision Resolution: All net-
works are considered, and an attempt is made to resolve any disallowed
intersections geometrically. If this fails, the network topology is altered
by removing terminals to make space for larger vessels, reducing the
network density in regions of unresolvable intersections.
We demonstrate techniques for these stages which scale as
O(N log N) in the best case, no more than O(N
2
) at worst,
with an expected average best case performance for volumes
of O(N
4/3
).
A. Creating a bifurcation
CCO iteratively adds terminal sites into the network, mean-
ing 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 standard method is to evaluate all existing
branches and pick the best, which must give at least O(N
2
)
scaling. A new counted selection process is proposed, which
moves down the tree until it encounters a given number of
consecutive branches which make no further progress to the
target site than their parent (Algorithm 1).
1) Candidate evaluation: The selector is supplied with an
evaluation function: this takes a branch and a terminal, and
returns a structure containing a reference to the branch, a
non-negative cost against which branches are compared, and
a flag indicating suitability. A branch being unsuitable does
not prevent its children from being tested, but does prevent it
from being passed back upstream if a less optimal but suitable
alternative is available. If the evaluator wishes to reject the
terminal (e.g. it is completely contained within a branch) the
cost is set to be negative and the suitability flag is set - this is
then guaranteed to be returned from the search process. The
terminal is rejected if no suitable candidates were found or
the cost is negative. For the investigations in this paper, we
consider only the Euclidean distance from the terminal node to
the branch as the cost function: with prescribed flow through
branches and pressure being fixed over the network, we expect
that longer paths will require larger radii. The suitability test
is whether the triangle formed by the set of nodes around the
bifurcation (the bifurcation triad, T ) is likely to degenerate
into a line. Extensions which we do not investigate here (but
are briefly touched upon in supplementary material) include
augmenting the cost with a flow rate term and providing a
maximum permissible flow asymmetry.
Algorithm 1 Counted selection method
Require: terminal, skipLimit : int[0,], evalFunc
Function Select(current, count) :
best current
for all child c of current.Branch do
candidate evalFunc(c, terminal)
if candidate.Cost < current.Cost then
downstream Select(candidate, 0)
if downstream.Suitable
and downstream.Cost < best.Cost then
best downstream
end if
else if count < skipLimit then
downstream Select(candidate, count + 1)
if downstream.Suitable
and downstream.Cost < best.Cost then
best downstream
end if
end if
end for
return best
2) Performance: 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). When a low number of skips, S, is permitted, the
performance should scale as the mean depth of the branches
currently in the network. Furthermore, we can guarantee that
the depth of the bifurcation created will be less than or equal to
the depth produced by searching all existing branches, which
will help maintain the good balance required to achieve better
query performance. The minimum scaling would be achieved
if the tree had uniform depth balance, which would scale as
O(log n).
We expect that the depth distribution along these extremal
paths will be approximately uniform, with a mean that scales
with the characteristic number of terminals in any dimension,
for the following reasons: (1) the average path to the most
extreme terminals of a perfusion space will pass a number of
internal terminals that scales with the characteristic number of
terminals in any dimension; (2) minimum volume principles
suggest that the path should bifurcate into these terminals as

GUY et al.: 3D PRINTABLE VASCULAR NETWORKS... 5
soon as they are passed (observed in [16], [19], [23]–[25]).
We therefore expect performance of volumes to scale no better
than O(n
1/3
). For perfusion spaces where the terminals are
arranged as a shell around the inlet, better performance is
expected, whereas for long, thin volumes we expect the worst
case performance to be achieved.
Perfect depth balance can be achieved by using a space-
filling fractal [26], but the suitability of fractals for large-scale
organs is debatable [27]–[29]. Notably, manufactured models
which have been designed to achieve uniform depth balance
have only been shown to supply a line or plane of points [30],
[31].
B. Volume optimization
The network volume is approximated as being the sum of
cylindrical volumes over the the set of existing branches, B,
V (B) = π
X
b∈B
r
2
b
L
b
. (12)
Previous implementations have optimized the position of bi-
furcations as they are added into the network, which we will
refer to as ‘incremental’ optimization.
1) Accelerated incremental optimization: Karch et al. [32]
considered that when a bifurcation is moved, only the reduced
resistances and radii fractions in the direct path upstream
to the source must be re-evaluated. We introduce a new
derived property, the effective length, L
, which allows the
approximation to the volume to be re-evaluated in a similar
manner, requiring only calculation along the path to the root,
avoiding the costly evaluation of the sum. By expanding the
sum and replacing radii with the root radius multiplied by their
chained fractions from (6) we get:
V (B) = π
L
0
r
2
0
+ L
1
f
2
1
r
2
0
+ L
2
f
2
2
r
2
0
+ L
1,1
f
2
1,1
f
2
1
r
2
0
+ ...
,
(13)
from which common factors may be recursively extracted:
πr
2
0
L
0
+ f
2
1
L
1
+ f
2
1,1
(L
1,1
+ ...) + ...
+ f
2
2
(L
2
+ ...)
.
(14)
We can now define the effective length relationship at termi-
nals and bifurcations:
L
term
= L
term
, (15)
L
p
= L
p
+ f
2
1
L
1
+ f
2
2
L
2
. (16)
The total volume downstream of a branch is therefore
V (B) = πr
2
0
L
0
. (17)
The update rule can be generalized to any cost function of the
form considered by Schreiner et al. [23]:
C(B, λ, ρ) =
X
b∈B
r
ρ
b
L
λ
b
, (18)
where λ, ρ are arbitrary constants. Using the following substi-
tutions: L
i
L
λ
i
, f
2
i
f
ρ
i
, the general form of the evaluation
becomes:
C(B, λ, ρ) = r
ρ
0
L
0
. (19)
2) Batch optimization: The fundamental limitation of in-
cremental optimization is that the effect of moving a single
bifurcation on the entire network becomes negligible as the
total number of nodes becomes large. Since we will deal with
any intersections at a later point in the algorithm, we may
interrupt the build at various stages to optimize every single
bifurcation at once.
Karch et al. [32] demonstrated that the volume of the whole
network when perturbing a single bifurcation has a single
minimum somewhere within the bifurcation triad, permitting
the use of the simplest method of minimization; gradient
descent of the volume with respect to the position of each
bifurcation:
x
0
= x ηV (x), (20)
where η is some small parameter. Whilst we could use the
accelerated volume calculation method to estimate the gradient
at each node, we would pay a computational cost per iteration
of N times the mean depth of the whole tree, giving an
expected O(N
4/3
) scaling for perfusion spaces.
A constant time approximation to the gradient, giving an
overall scaling per iteration of Θ(N ), is justified here. Starting
by differentiating (17) with respect to a bifurcation position,
V (x) = πr
2
0
L
0
(x) + 2πL
0
r
0
r
0
(x), (21)
we note from (9) that the root radius is a function of only
root reduced resistance, R
0
, given that the flow rates and total
pressure drop are fixed by the logical configuration of the tree.
Since changes in R
propagate upwards from the bifurcation
to the root, we see from (11) that there is a scalar gradient
chain through the upstream nodes, U
m
: m {1 . . . M}, where
1 denotes the first child of the root and M the immediate
upstream node of the bifurcation, and the vector terms are
introduced at the bifurcation triad via their physical lengths
(using C to denote the set of child branches):
R
0
(x) =
dR
0
dR
U
1
M1
Y
i=1
dR
U
i
dR
U
i+1
!
dR
U
M
dR
p
R
p
(x), (22)
R
p
(x) =
8µ
π
L
p
(x) +
X
i∈C
dR
p
dR
i
L
i
(x)
!
. (23)
In order to avoid vanishing gradients due to deep chains, we
wish to approximate the gradient direction and work out the
step size later. Defining β = r
1
/r
2
, the bifurcation relationship
yields:
dR
p
dR
1
=
f
4
1
R
1
+
f
4
2
R
2
2
(1 + β
γ
)
4
γ
R
2
1
β
γ
1 + β
γ
1
(1 + β
γ
)
4γ
γ
β
γ
R
2
R
1
!
.
(24)
The expression for the second child can be found by exchang-
ing the indices. Considering that
z
1 + z
1 0 z 0, (25)
we know that the gradient chain elements are all positive.
R
0
(x) therefore points in the same direction as R
p
(x).

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Abstract: The branching behavior of vascular trees is often characterized using Murray's law. We investigate its validity using synthetic vascular trees generated under global optimization criteria. Our synthetic tree model does not incorporate Murray's law explicitly. Instead, we assume it holds implicitly and investigate the effects of different physical constraints and optimization goals on the branching exponent that is now allowed to vary locally. In particular, we include variable blood viscosity due to the F{\aa}hr{\ae}us--Lindqvist effect and enforce an equal pressure drop between inflow and the micro-circulation. Using our global optimization framework, we generate vascular trees with over one million terminal vessels and compare them against a detailed corrosion cast of the portal venous tree of a human liver. Murray's law is implicitly fulfilled when no additional constraints are enforced, indicating its validity in this setting. Variable blood viscosity or equal pressure drop leads to deviations from this optimum, but with the branching exponent inside the experimentally predicted range between 2.0 and 3.0. The validation against the corrosion cast shows good agreement from the portal vein down to the venules. Not enforcing Murray's law explicitly reduces the computational cost and increases the predictive capabilities of synthetic vascular trees. The ability to study optimal branching exponents across different scales can improve the functional assessment of organs.

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References
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Proceedings ArticleDOI
01 May 1990
TL;DR: The R*-tree is designed which incorporates a combined optimization of area, margin and overlap of each enclosing rectangle in the directory which clearly outperforms the existing R-tree variants.
Abstract: The R-tree, one of the most popular access methods for rectangles, is based on the heuristic optimization of the area of the enclosing rectangle in each inner node. By running numerous experiments in a standardized testbed under highly varying data, queries and operations, we were able to design the R*-tree which incorporates a combined optimization of area, margin and overlap of each enclosing rectangle in the directory. Using our standardized testbed in an exhaustive performance comparison, it turned out that the R*-tree clearly outperforms the existing R-tree variants. Guttman's linear and quadratic R-tree and Greene's variant of the R-tree. This superiority of the R*-tree holds for different types of queries and operations, such as map overlay, for both rectangles and multidimensional points in all experiments. From a practical point of view the R*-tree is very attractive because of the following two reasons 1 it efficiently supports point and spatial data at the same time and 2 its implementation cost is only slightly higher than that of other R-trees.

4,686 citations


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TL;DR: A new bioprinting method is reported for fabricating 3D tissue constructs replete with vasculature, multiple types of cells, and extracellular matrix that open new -avenues for drug screening and fundamental studies of wound healing, angiogenesis, and stem-cell niches.
Abstract: A new bioprinting method is reported for fabricating 3D tissue constructs replete with vasculature, multiple types of cells, and extracellular matrix. These intricate, heterogeneous structures are created by precisely co-printing multiple materials, known as bioinks, in three dimensions. These 3D micro-engineered environments open new -avenues for drug screening and fundamental studies of wound healing, angiogenesis, and stem-cell niches.

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TL;DR: 3D printed rigid filament networks of carbohydrate glass are used as a cytocompatible sacrificial template in engineered tissues containing living cells to generate cylindrical networks which could be lined with endothelial cells and perfused with blood under high-pressure pulsatile flow.
Abstract: In the absence of perfusable vascular networks, three-dimensional (3D) engineered tissues densely populated with cells quickly develop a necrotic core [1]. Yet the lack of a general approach to rapidly construct such networks remains a major challenge for 3D tissue culture [2–4]. Here, we 3D printed rigid filament networks of carbohydrate glass, and used them as a cytocompatible sacrificial template in engineered tissues containing living cells to generate cylindrical networks which could be lined with endothelial cells and perfused with blood under high-pressure pulsatile flow. Because this simple vascular casting approach allows independent control of network geometry, endothelialization, and extravascular tissue, it is compatible with a wide variety of cell types, synthetic and natural extracellular matrices (ECMs), and crosslinking strategies. We also demonstrated that the perfused vascular channels sustained the metabolic function of primary rat hepatocytes in engineered tissue constructs that otherwise exhibited suppressed function in their core.

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TL;DR: At constant flow, f (that is, for any given steady state), and at constant length of arterial section, l, the total energy, E, is a minimum when:
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  • ...important that the network displays hierarchy [2], [3]....

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  • ...The most common include laminar (steady-state) flow, flow conservation, equal terminal flow and pressure, Newtonian fluid flow, and Murray’s law for optimal bifurcations; networks that conform to this law maximize flow conductance per unit volume [2]....

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  • ...The relationship between parent and child radii is fixed by Murray’s law [2] to minimise power dissipation over the network:...

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TL;DR: A large part of the branching vasculature of the mammalian circulatory and respiratory systems obeys Murray's law, and a functional relationship exists between vessel radius and volumetric flow, average linear velocity of flow, velocity profile, vessel-wall shear stress, Reynolds number, and pressure gradient in individual vessels.
Abstract: A large part of the branching vasculature of the mammalian circulatory and respiratory systems obeys Murray's law, which states that the cube of the radius of a parent vessel equals the sum of the cubes of the radii of the daughters. Where this law is obeyed, a functional relationship exists between vessel radius and volumetric flow, average linear velocity of flow, velocity profile, vessel-wall shear stress, Reynolds number, and pressure gradient in individual vessels. In homogeneous, full-flow sets of vessels, a relation is also established between vessel radius and the conductance, resistance, and cross-sectional area of a full-flow set.

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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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Frequently Asked Questions (1)
Q1. What contributions have the authors mentioned in the paper "3d printable vascular networks generated by accelerated constrained constructive optimization for tissue engineering" ?

Here, the authors report such a method, using an accelerated constructive constrained optimization approach, by decomposing the process into construction, optimization and collision resolution stages. The new approach reduces computation time to minutes at problem sizes where previous implementations have reported days. With the optimality criterion of maximizing the volume of useful tissue which could be grown around such a network, an approach of alternating stages of construction and batch optimization of all node positions is introduced and shown to yield consistently more optimal networks. The approach does not place a limit on the number of interpenetrating networks that can be constructed in a given space ; indeed the authors demonstrate a biomimetic, liverlike tissue model. Methods to account for the limitations of 3D printing are provided, notably the minimum feature size and infill at sharp angles, through padding and angle reduction, respectively.