Angiogenesis as a model for the generation of load-bearing networks

Abstract

This research suggests an algorithm to generate structural networks based on discreet elements for given locations of support points and point loads. Previous research attempted to achieve this by using a computational growth simulation of venation systems, which form the structure of leaves. However, such networks always start from a single point and therefore cannot be used to form arches or beams. In order to generate networks that are based on two or three support points, an algorithm has been developed that is inspired instead by angiogenesis, the process by which vascular systems develop. The algorithm is based on a spring system with a variable network graph that connects the support points and is pulled upwards and split sideways into multiple veins by a given set of load points. The algorithm has been used to grow architectural structures. Different networks have been tested using finite element analysis and compared with both venation and column-and-beam structures. The angiogenesis networks as well as the venation network are shown to perform well and may be suitable as architectural structural systems.

Keywords

Architecture angiogenesis structure network growth

Introduction

The aim of this research is to generate structural networks based on given support points and point loads. An algorithm is proposed that develops those networks based on discreet structural members. The networks are grown or refined from an initial basic connection between the support points.

The algorithm is a further development of the author’s previous work of generating structural networks¹ by utilizing a variant of the venation algorithm as developed by the University of Calgary.^2–4 The algorithm proposed here is inspired instead by the process of angiogenesis. It suggests the use of a spring system with a variable network graph so that additional structural members can be inserted into the network during its formation.

Venation

Venation in the leaves of plants forms both their structural and their circulatory system,⁵ and venation development has been shown to relate to physical stress within the lamina.⁶ The University of Calgary has developed an algorithm for its simulation which uses a set of seed points from which the venation network grows toward target points which are distributed in the lamina.^2–4 The author’s previous work showed that the networks generated with this algorithm can perform well as structural systems.¹

However, a difficulty of using leaf and tree structures and other venation systems in load-bearing applications is their topologic behavior of always connecting back any cells or loads to a single support point, the petiole. Structural components that form connections between two support points, such as beams or arches, cannot be generated with the venation algorithm.

The previous work attempted to solve this by growing separate structures together from different support points. Those structures are still each supporting the loads from a single support point, but are in effect leaning against each other. According to the venation algorithm used, the actual connecting members between the separate segments always have the thinnest cross sections of the network.¹

In order to create structural networks that grow toward target points similar to venation systems, but which allow for the development of beams or arches, a network generation based on angiogenesis is proposed as an alternative.

Angiogenesis

Rather than just connecting back to one seed point, every point reached by the network needs to connect back to two seed points, and every member of the network becomes a connection between those two seed points. The growth of the network does not start from the seed points directly, but from an initial connection between the seed points, which then divides and splits up in order to reach the target points of the system and to develop into the network (Figure 1).

Figure 1.

Angiogenesis network connecting two seed points instead of just one in a venation network. The network is developed through a division and refinement of one initial connection between the two seed points.

Angiogenesis, the process by which new blood vessels form from existing ones, differs from venation systems in that it grows from, and generates, unidirectional circulation systems. On the contrary, the circulation through leaf venation systems flows in both directions through the same veins, which contain both xylem cells to transport water toward the leaf cells and phloem cells to move sap away from them. Where venation systems develop from a single seed point, the petiole of the leaf, vascular systems have a start and an end point, the atria and ventricles of the heart.⁷

Therefore, it is possible to use a growth simulation similar to angiogenesis in order to develop structural systems that connect multiple support points. In a vascular system, the vein network needs to reach the proximity of every biologic cell in order to provide it with circulation. This requirement can be replaced by the requirement to transfer loads from defined locations toward the support points. The load points used in the simulation correspond to the biologic cells of the vascular system that need to be reached from the support points. The load points as well as the support points are given and structural networks can be generated based on grown arch systems between the support points.

Angiogenesis occurs in two modes: sprouting angiogenesis, in which a new vein is formed by connecting two already existing veins, and intussusceptive angiogenesis, in which a single existing vein splits up into two parallel veins. The methodology presented in this article is based on intussusceptive angiogenesis. The initial formation of the first veins in an embryo is a separate process called vasculogenesis.⁸

The intent for the simulations is to generate, between the support points, networks that are pulled upwards by the load points to form efficient structural support systems. The resulting networks are expected to create shell-like formations or may have similarities to the rib systems beneath Gothic domes.

It has not been attempted to develop an algorithm that realistically simulates the development of vascular systems as they occur in nature, rather the biological precedent of angiogenesis has been the driving concept for the development of venation systems into growing spring systems.

Related work

Different designers have used venation patterns for the design of products and architecture,^9–11 using digital simulations similar to those developed by the University of Calgary.^2–4

In parametric design, structural performance often drives the parameters of the relational model or algorithm.^12,13 In structural engineering, the voxel-based evolutionary structural optimization (ESO) and bidirectional evolutionary structural optimization (BESO) algorithms have been developed in order to generate efficient structural systems;^14,15 however, those are based on voxels and not on discreet structural members. Networks with discreet elements have been developed through genetic algorithms,^16–19 which then require pre-defined setups or components.

A development of structural systems through growth simulations has been attempted with various algorithms^20–24 or through an assemblage of physical components.^25–27

The study of the growth of organisms on a cellular level is the field of developmental biology.²⁸ In the field of cancer research, the growth of angiogenesis and the factors influencing it are of special interest, as an inhibition of the angiogenesis has been shown to stop the growth of a tumor. Simulations of angiogenesis in this field are used for both research and visualization.^29,30

Variable graph spring system

In the computational model, the vein network is abstracted into a set of points, the cells or nodes of the network, with connections between them that simulate small segments of the veins. Let those cells be called $C_{1}$ , $C_{2}$ , etc. The connections are calculated as springs between cells which act to always pull two neighboring cells together.

Similar to the venation systems tested previously,¹ the simulation is calculated in three dimensions with the z-axis in the vertical direction. As the system is aimed at structurally supporting loads, the support points are placed at the bottom level and the load points are placed at a higher level, with gravity also acting in the z-direction.

During each iteration, the spring system is relaxed by a step and the positions of all cells are updated. The directionality and forces of every cell are re-evaluated. The cells are then tested for division to cause a splitting of the veins: The cell with the largest force acting on it is divided into two cells, and the parent and child cells are reconnected to the network with new spring connections.

Setup

Input

The inputs for the simulation are a set of seed points, the point supports of the final structure, and a set of target points, the load points that are to be reached by the network. Let the two support points be $S_{A}$ and $S_{B}$ . Let the load points be $L_{1}$ , $L_{2}$ , etc.

In a first step, an initial beam line is identified by connecting the two seed points. This line is used to define the flow axis between $S_{A}$ and $S_{B}$ . This step is functionally comparable to the vasculogenesis of a vascular system, which provides a basic initial connection from which the network can proliferate. Angiogenesis, in biology as in the simulations in this article, always requires a pre-existing vein to divide from.

Cells are placed equidistantly along the initial beam line and connected with springs (Figure 2).

Figure 2.

Initial setup with two support points.

Load points

Each load point identifies the cell closest to it and establishes a spring connection to this cell. The spring connections to the load points, the load-point springs, are treated differently from the spring connections between cells. The number of load-point springs is fixed and equal to the amount of load points, while the springs between cells proliferate when the network grows.

Flow directionality

As in a vascular system, a flow direction through the cells and their connections is defined, which is used to guide the division of the veins. This ensures that the cells and connections divide sideways in relation to the flow direction or flow axis and that the new veins follow the direction of the previous vein.

Every cell is placed on a connection between two support points. In the initial distribution, there is always one neighboring cell that is closer to $S_{A}$ and one that is closer to $S_{B}$ . Those neighbors are tagged accordingly for each cell. Let those neighbors be $P_{A}$ and $P_{B}$ . In the diagrams, the connection toward $P_{A}$ is identified by a red arrow and the connection toward $P_{B}$ by a green arrow.

The flow axis is then described as the vector from neighbor $P_{A}$ to neighbor $P_{B}$ . The sideways direction is defined as the vector orthogonal to the flow axis within the horizontal xy plane (Figure 3).

Figure 3.

Flow axis and sideways direction at cell C are in black. The colored arrows indicate the spring connections toward the neighbors $P_{A}$ (red) and $P_{B}$ (green).

In later iterations it is possible that more than one neighbor is tagged as leading in the direction of a certain support point. Let those neighbors be $P_{A 1}$ , $P_{A 2}$ , $P_{A 3}$ , etc. In such cases, the flow axis is described by the vector from the midpoint $P_{A m i d}$ of all neighbors $P_{A n}$ to the midpoint $P_{B m i d}$ of all neighbors $P_{B n}$ (Figure 4).

Figure 4.

Flow axis for a cell C with multiple neighbors ( $P_{B 1}$ – $P_{B 3}$ ) in the B-direction. The colored arrows indicate the spring connections toward the neighbors $P_{A}$ (red) and $P_{B}$ (green).

Every spring connection is now part of a link between the two support points and connects two cells. Of those two cells, the first will have the second tagged as connecting toward one support point, while the second cell will have the first cell tagged as connecting toward the other support point. In the diagrams, every spring connection will have two colored arrows pointing along its length in opposite directions, indicating in which direction each support point is located.

Iterative calculations

Relaxation

The spring system is relaxed by a step. All springs are calculated as having a rest length of 0. The stiffness of the load-point springs is calculated as being significantly lower than the stiffness of the springs between cells.

The movement of the cells is calculated as an acceleration–velocity system: in each iteration, the acting spring forces are added as vectors to the acceleration of a cell, the acceleration is added to the velocity of the cell, and the velocity is added to the position of the cell. All cells are calculated as having the same mass.

The forces acting on a cell are recalculated, as well as the main flow axis and the sideways direction at every cell.

Angiogenesis

Angiogenesis, the splitting of a vein, is triggered by the accumulation of forces acting on a cell, independent of their direction. The cell with the largest force acting on it will split. Neighboring cells are prevented from splitting, sometimes with a delay of several iterations, so that the system can readjust before the next division occurs.

When a cell splits, a new cell is inserted into the network. The dividing parent cell and its child cell are moved apart from each other in the sideways direction.

If there is only one cell $P_{A}$ or one cell $P_{B}$ toward one of the two directions of the support points, this neighbor is added to both parent and child cells. Otherwise, the neighboring cells are distributed between the parent and the child cells along the flow axis, with at least one neighbor in each direction being assigned to each parent and child cell. The spring connections are altered accordingly. The distribution of the springs along the flow axis will pull the parent and child cells apart from each other in the following iterations (Figures 5 and 6).

Figure 5.

Cell with two connections, before and after division.

Figure 6.

Cells with multiple connections in the $P_{B}$ direction, before and after division. The cells in the $P_{B}$ direction are split along the flow axis.

The load points which connect to the dividing cell are likewise distributed between the parent and child cells along the flow axis, so that both cells receive an equal number of load-point connections.

Cells can only divide if they are connected to two or more load points. Otherwise, a cell will lose all its connections to the load points, thereby losing any spring in the vertical direction. This will cause the cell to drop downwards and it will be unable to reconnect to other load points, as there are other cells higher up which are closer to any possible load point (Figure 7).

Figure 7.

Development of the network during the simulation.

Reconnection of load points

A possible variant of the algorithm allows a reconnection of the load points at every iteration. If a cell other than the currently connected cell moves closer to the load point, the load point will change its connection to this cell. In order to guarantee a gradual adjustment, a load point may only reconnect to a cell if a neighbor of the load point is already connected to it. A cell must keep a minimum amount of load points connected to it in order to avoid losing all vertical springs.

Sets with multiple support points

The algorithm which connects two support points can also be applied to sets of multiple support points. In this case, the support points are triangulated in plan by means of a Delaunay triangulation and split up into separate systems which each have two seed points only. Any two connected support points are then calculated separately as an independently acting network (Figures 8 –11).

Figure 8.

A set of five support points is triangulated and subdivided into eight separate networks of two support points each.

Figure 9.

Set of eight connected two-support-point systems: plan.

Figure 10.

Set of eight connected two-support-point systems: side elevation.

Figure 11.

Set of eight connected two-support-point systems: perspective.

Connection to three support points

Lateral loads will cause bending moments at the supports of networks with only two support points. In order to avoid this, the network needs to have a minimum of three support points. It has been attempted to adjust the algorithm accordingly. Let the three support points be $S_{A}$ , $S_{B}$ , and $S_{C}$ .

The initial base connections are set by connecting the three support points to their midpoint and by dividing those lines as described above for two-support-point systems (Figure 12).

Figure 12.

Initial setup with three support points.

Flow directionality

In the algorithm with two support points, every cell is placed on a path which leads from $S_{A}$ to $S_{B}$ , and the neighboring cells along this path are clearly identifiable. When translating the algorithm to use three support points, it is no longer possible to trace a single direction at a given cell. However, it is still possible to identify which neighbors of a cell are positioned closer to a certain support point. It is also possible to identify a main axis along which the forces are acting at a cell.

The neighbors of each cell are tagged if they are positioned in the direction toward $S_{A}$ , $S_{B}$ , or $S_{C}$ . Let those neighbors be $P_{A n}$ , $P_{B n}$ , and $P_{C n}$ . In the initial setup, every cell except the midpoint cell has two neighbors. One of those has a tag into the direction of one support point, the other neighbor has tags into the direction of two support points, so there can be an overlap between the groups $P_{A n}$ , $P_{B n}$ , and $P_{C n}$ of a cell (Figure 12).

In the algorithm with two support points, every connection is a link between those two support points, the two connected cells have tagged each other as connecting toward the opposite support point.

In the system with three support points, every spring connection is now a link between all three support points, and still connects two cells. Of those two cells, one cell will have the second cell tagged as connecting toward one support point, while the second cell will have the first cell tagged as connecting toward both of the other support points.

In the diagrams, every spring connection will therefore have three colored arrows pointing along its length, indicating in which direction each support point is placed. The connection toward $P_{A}$ is identified by a red arrow, the connection toward $P_{B}$ by a green arrow, and the connection toward $P_{C}$ by a blue arrow. One arrow will start at one end of the spring, while two arrows both start at the other end (Figure 12).

In order to identify the main flow axis at a cell, the angles between $P_{A m i d}$ , $P_{B m i d}$ , and $P_{C m i d}$ are calculated. The flow axis at the cell is defined by a vector through the end points of the largest angle. The sideways axis is defined as orthogonal to the flow axis. The remaining point $P_{A m i d}$ , $P_{B m i d}$ , or $P_{C m i d}$ which is not defining the flow axis will be sideways from this axis and it is tagged for the cell as the sideways neighbor (Figure 13).

Figure 13.

Flow axis for cells with neighbors in three directions. The largest angle between the connections defines the flow axis: at cell $C$ , it follows the direction of the vector from $P_{A}$ to $P_{B}$ .

Angiogenesis

During cell division, the parent and child cells are again set apart along the sideways axis, and the load points are divided along the flow axis between the cells as in the algorithm with two support points (Figure 16).

However, the neighboring cells need to be divided so that there are no diagonally crossing connections after the division. Therefore, the algorithm evaluates how the three directions are distributed between the cell’s neighbors and if there are any overlap neighbors which are part of more than one of the groups $P_{A n}$ , $P_{B n}$ , and $P_{C n}$ . One of the following situations will occur:

The group of neighbors in one direction is identical to the group of neighbors in a second direction.

This situation is effectively the same as in the algorithm with two support points. Neighbors are positioned in one of the two directions. The neighbors are divided along the flow axis as in the two-support-point system (Figure 5).

There is no overlap between the groups of neighbors.

The neighbors in the two directions that define the flow axis are split between the parent and child cells along the flow axis.

The sideways neighbors connect only to the parent cell.

A new connection is created between the parent and the child cells (Figure 14).

There is a partial overlap between the groups of neighbors.

The neighbors in the two directions which define the flow axis are split between the parent and the child cells along the flow axis. Neighbors with overlap between those two directions are only assigned to the child cell. Neighbors with overlap between the sideways direction and one of the other directions are only assigned to the parent cell.

The sideways neighbors connect only to the parent cell.

A new connection is created between the parent and the child cells (Figure 15).

Figure 14.

Cell with no overlap between neighbors, before and after division.

Figure 15.

Cell with overlap between neighbors, before and after division.

Figure 16.

Development of the network with three support points during the simulation.

The special behavior for situations with partial overlap was introduced as, otherwise, there is a possibility for cells to repeatedly divide and be pulled in the same direction, resulting in excess cell strings.

Finite element analysis

Analysis setup

The resulting models were tested using finite element analysis (FEA). An identical load case was defined to develop and test the structural networks: the load points to be supported are positioned in a grid within a horizontal right triangle with 10 m width and 15 m length, spaced at 250 mm. The loads are to be supported by three point supports in the corners of the triangle, positioned 5 m below the load points.

In order to simulate a slab instead of separate point loads, structural members were defined within the triangle between the load points. The FEA was tested with three different strengths/sizes of the structural members in the slab.

Three load cases were tested: a load of 1 kN/m² in the x-direction, a load of 1 kN/m² in the y-direction, and a load of 1 kN/m² in the z-direction, all applied evenly at the load points in the triangle.

Steel was used as the material for the analysis. In order to compare the performance of the structural systems, the members were scaled so that each model, excluding the structure of the slab, had an identical mass of 10 t.

Network systems

The following networks were tested as structural systems (Table 1):

A column and beam structure, with vertical columns above the three support points and three horizontal beams connecting the columns;

A reticulate venation system as described by Klemmt¹ (Figure 17);

A two-support-point angiogenesis network, consisting of three systems which each form a connection between two of the three support points (Figure 18);

Variations of three-support-point angiogenesis networks (Figure 19).

Table 1.

Comparison of the structural networks.

	Member length (m)	Minimum diameter (mm)	Maximum diameter (mm)
Column beam	74.84	130.46	130.46
Venation	625.21	20.75	207.52
Angiogenesis two supports	1689.55	9.98	99.78
Angiogenesis three supports: high, one member per load point	918.02	11.80	118.02
Angiogenesis three supports: mid, one member per load point	1311.12	11.71	117.06
Angiogenesis three supports: low, one member per load point	1684.81	12.46	124.58
Angiogenesis three supports: high, all members per load point	9600.13	8.45	84.53
Angiogenesis three supports: mid, all members per load point	13,817.40	7.64	76.44
Angiogenesis three supports: low, all members per load point	15,831.95	7.55	75.49

Figure 17.

Venation system.

Figure 18.

Two-support-point angiogenesis system.

Figure 19.

Three-support-point angiogenesis system.

The variations tested were:

Three different heights of the main connecting arches (the arches are then connected by members along the load-point springs to the load points at the 5-m level); In the algorithm, the stiffness of the load-point springs relative to the springs between cells can be set. By increasing their stiffness, the network is being pulled closer toward the load points.

Every load-point spring of the simulation used as a structural member versus only one load-point spring per load point used as a structural member.

The connections of the networks were used as the structural members. In the venation model, different sizes of the members are defined by the number of load points that are supported by a member.¹ In the angiogenesis models, the springs are stretched further if a larger load is acting on them, so the length of a member was used to define its size. The load-point springs were treated equally as all having the smallest member size.

All networks form a similar arching overall geometry. However, in the mid- and low three-support-point angiogenesis models, the apex is positioned about 400 and 800 mm lower respectively.

Deflection results and evaluation

Between the different structural networks tested, the column-and-beam structure performs least well. Its greatest weakness is the central area of the slab which is only supported from the neighboring slab members and consequently deforms the most, especially with a low strength of the slab members (Tables 2 –4).

Table 2.

Deflection with low slab strength.

	Load direction
	x (mm)	y (mm)	z (mm)
Column beam	238.88	167.88	2,147,483
Venation	91.15	83.76	76.89
Angiogenesis two supports	6751.71	781.30	211.04
Angiogenesis three supports: high, one member per load point	930.80	193.79	578.70
Angiogenesis three supports: mid, one member per load point	1613.02	374.42	875.94
Angiogenesis three supports: low, one member per load point	1995.13	597.89	624.40
Angiogenesis three supports: high, all members per load point	96.77	56.28	27.71
Angiogenesis three supports: mid, all members per load point	65.88	53.79	11.03
Angiogenesis three supports: low, all members per load point	57.67	41.25	14.01

Slab member length: 629.14 m; slab member diameter: 1 mm; slab mass: 4.938 kg.

Table 3.

Deflection with medium slab strength.

	Load direction
	x (mm)	y (mm)	z (mm)
Column beam	168.19	136.28	6008.83
Venation	72.98	66.14	43.74
Angiogenesis two supports	207.11	111.31	58.17
Angiogenesis three supports: high, one member per load point	124.60	71.78	37.83
Angiogenesis three supports: mid, one member per load point	109.70	60.21	20.73
Angiogenesis three supports: low, one member per load point	74.52	48.28	13.73
Angiogenesis three supports: high, all members per load point	50.80	38.63	15.86
Angiogenesis three supports: mid, all members per load point	37.03	27.41	6.17
Angiogenesis three supports: low, all members per load point	33.35	20.79	6.83

Slab member length: 629.14 m; slab member diameter: 10 mm; slab mass: 493.877 kg.

Table 4.

Deflection with high slab strength.

	Load direction
	x (mm)	y (mm)	z (mm)
Column beam	79.68	67.63	49.07
Venation	21.84	11.98	5.20
Angiogenesis two supports	16.68	12.60	5.92
Angiogenesis three supports: high, one member per load point	42.86	29.59	6.59
Angiogenesis three supports: mid, one member per load point	35.30	30.65	3.38
Angiogenesis three supports: low, one member per load point	25.62	16.67	3.94
Angiogenesis three supports: high, all members per load point	33.02	24.46	4.01
Angiogenesis three supports: mid, all members per load point	28.90	19.46	1.01
Angiogenesis three supports: low, all members per load point	22.27	14.77	3.85

Slab member length: 629.14 m; slab member diameter: 100 mm; slab mass: 49,387.65 kg.

A similar behavior can be observed with the two-support-point angiogenesis model, where the central area of the slab is not very well supported. However, a high slab member strength can act against this, in which case this system performs very well especially for the lateral load cases.

All of the three-support-point angiogenesis structures perform well, especially those where all load-point springs have been used as structural members. This greatly increases the cumulative length of the network members and results in a morphology which could be described as a space frame below the slab.

Among the three-support-point angiogenesis structures with a single structural member to any point load, the structural performance increases if the slab strength is higher and the main members are positioned lower.

It was expected that the angiogenesis structures would perform better than the venation structure as the algorithm used to generate the venation systems does not have a logic based on load transfer. However, the venation structure also performs very well, especially considering that the overall cumulative length of the network is relatively small because of fewer structural members, meaning that it could be built with less effort during construction (Figure 20).

Figure 20.

Deflection under loading in the z-direction. Columns show low, medium, and high slab strength. Rows show the nine structural systems which were compared. Magenta indicates the largest deflection within each model.

Conclusion and future work

The proposed algorithm can generate networks which develop from two or three seed points, similar to the way that a venation system develops from a single seed point. The algorithm has been used to generate networks that fulfill structural load-bearing functions. The tests using FEA have shown that the networks perform well as structural systems. However, the comparative structure based on a venation simulation performs similarly well, although the venation growth does not follow a structural logic.

If applied to construction, some of the three-support-point angiogenesis models have a large amount of individual members as indicated by their overall member length (Table 1). This will increase costs and may cause difficulties in their constructability.

The proposed algorithm utilizes the logic of a spring system to suggest the positioning of structural members. A constraint of the networks is that every cell needs to have at least one connection to a load point for it to not drop down and become non-load bearing. While a spring system simulates forces, it does not simulate the transfer of loads well and does not take bending moments into account. It is therefore proposed to develop a growth algorithm which is based on a more realistic calculation of structural forces and behaviors.

Variable graph spring systems, which are altering, adjusting, or extending their network graph during the relaxation simulation, are expected to satisfy a variety of other purposes and should be explored further. The proposed algorithm is based on a sideways division of cells, which resulted in arch and dome formations but did not yield functional results for multi-level arrangements of the load points. Different arrangements of support and load points and different rules of division are expected to result in other network typologies, which may be suitable for different structural or other applications.

It would also be of interest to explore further applications of the algorithm, for example as a tool to generate circulation systems, to propose programmatic relations and densities, or to develop architectural geometries, arrangements, and patterns.

Footnotes

Acknowledgements

This article was written by Christoph Klemmt as part of a doctorate under the supervision of Prof. Dr Klaus Bollinger at the University of Applied Arts Vienna.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article

References

Klemmt

Compression based growth modelling. In: Gerber

Huang

Sanchez

(eds) ACADIA 2014 design agency: proceedings of the 34th annual conference of the association for computer aided design in architecture. Toronto, ON, Canada: Riverside Architectural Press, 2014, pp. 565–572.

Runions

Lane

Prusinkiewicz

Modeling trees with a space colonization algorithm. In: Proceedings of the Eurographics workshop on natural phenomena, 2007, pp. 63–70.

Runions

Fuhrer

Lane

. Modeling and visualization of leaf venation patterns. ACM T Graphic 2005; 24(3): 702–711.

Runions

Modeling biological patterns using the space colonization algorithm. MSc Thesis, University of Calgary, Calgary, AB, Canada, 2008.

Roth-Nebelsick

Uhl

Mosbugger

. Evolution and function of leaf venation architecture: a review. Ann Bot 2001; 87: 553–566.

Laguna

Bohn

Jagla

EA.

The role of elastic stresses on leaf venation morphogenesis. PLoS Comput Biol 2008; 4(4): e1000055.

Birbrair

Zhang

Wang

. Type-2 pericytes participate in normal and tumoral angiogenesis. Am J Physiol Cell Physiol 2014; 307(1): C25–C38.

Risau

Flamme

Vasculogenesis. Annu Rev Cell Dev Bi 1995; 11(1): 73–91.

Tamke

Nicholas

Riiber

The agency of event: event based simulation for architectural design. In: Gerber

Huang

Sanches

10.

Andraos

Generating adaptable structural systems using natural growth algorithms. Master’s Thesis, ENSA Paris Malaquais, Paris, 2015, http://andraos.co.uk/GeneratinganAdaptableStructuralSystem_LR.pdf

11.

Seepersad

CC.

Challenges and opportunities in design for additive manufacturing. 3D Print Addit Manuf 2014; 1(1): 10–13.

12.

Makris

Gerber

Carlson

. Informing design through parametric integrated structural simulation: iterative structural feedback for design decision support of complex trusses. In: eCAADe 2013: computation and performance—proceedings of the 31st international conference on education and research in computer aided architectural design in Europe, Delft, 18–20 September 2013. Delft: Delft University of Technology.

13.

Turrin

von Buelow

Stouffs

Design explorations of performance driven geometry in architectural design using parametric modeling and genetic algorithms. Adv Eng Inform 2011; 25(4): 656–675.

14.

Querin

Steven

Xie

YM.

Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comput 1998; 15(8): 1031–1048.

15.

Huang

Xie

Burry

MC.

Advantages of bi-directional evolutionary structural optimization (BESO) over evolutionary structural optimization (ESO). Adv Struct Eng 2007; 10(6): 727–737.

16.

Byrne

Fenton

Hemberg

. Combining structural analysis and multi-objective criteria for evolutionary architectural design. In: Di Chio

Brabazon

Di Caro

. (eds) Proceedings of the 2011 international conference on applications of evolutionary computation—volume part II. Berlin: Springer-Verlag, 2011, pp. 204–213.

17.

Clune

Connor

Ochsendorf

. An object-oriented architecture for extensible structural design software. Comput Struct 2012; 100–101: 1–17.

18.

Fahlbusch

Hofmann

Heise

. Skylink am Flughafen Frankfurt. Stahlbau 2012; 81(8): 638–642.

19.

Preisinger

Linking structure and parametric geometry. Archit Design 2013; 83(2): 110–113.

20.

Sugihara

Guidelines for the agent based design methodology for structural integration—inanimate material physics simulation and animate generative rules. In: Gerber

Ibanez

(eds) Paradigms in computing: making, machines, and models for design agency in architecture. Los Angeles, CA: eVolo Press, 2015, pp. 94–107.

21.

Richards

Dunn

Amos

. An evo-devo approach to architectural design. In: Proceedings of the 14th annual conference on genetic and evolutionary computation, Philadelphia, PA, 7–11 July 2012, pp. 569–576. New York: ACM.

22.

Richards

Amos

Evolving morphologies with CPPN-NEAT and a dynamic substrate. In: ALIFE 14: the fourteenth conference on the synthesis and simulation of living systems, vol. 14, 2014, pp. 255–262.

23.

Kicinger

Arciszewski

De Jong

Evolutionary computation and structural design: a survey of the state-of-the-art. Comput Struct 2005; 83(23–24): 1943–1978.

24.

Kicinger

Arciszewski

De Jong

. Parameterized versus generative representations in structural design: an empirical comparison. In: Proceedings of the 7th annual conference on genetic and evolutionary computation (GECCO’05) (ed Beyer

), Washington, DC, 25–29 June 2005, New York: ACM.

25.

Dierichs

Menges

Functionally graded aggregate structures: digital additive manufacturing with designed granulates. In: Proceedings of the 32nd annual conference of the Association for Computer Aided Design in Architecture (ACADIA) (ed Cabrinha

Johnson

Steinfeld

), San Francisco, CA, 18–21 October 2012, pp. 295–304. ACADIA.

26.

Dierichs

Menges

Aggregate architectures, observing and designing with changeable material systems in architecture. In: Change, architecture, education, practices—proceedings of the international ACSA conference (ed Costa

Thorne

), Barcelona, 20–22 June 2012, pp. 463–468. ACSA.

27.

Zhao

Liu

Zheng

. Packings of 3D stars: stability and structure. arXiv 2015, preprint arXiv:1511.06026.

28.

Wolpert

Tickle

Jessell

. Principles of development. Oxford: Oxford University Press, 2011.

29.

Shirinifard

Gens

Zaitlen

. 3D multi-cell simulation of tumor growth and angiogenesis. PLoS ONE 2009; 4(10): e7190.

30.

Szczerba

Székely

Macroscopic modeling of vascular systems. In: Dohi

Kikins

(eds) Proceedings of the 5th international conference on medical image computing and computer-assisted intervention part II (MICCAI’02). London: Springer-Verlag, 2002, pp. 284–292.