Wolfram Computation Meets Knowledge

Free-Form Bioprinting with Mathematica and the Wolfram Language

In past blog posts, we’ve talked about the Wolfram Language’s built-in, high-level functionality for 3D printing. Today we’re excited to share an example of how some more general functionality in the language is being used to push the boundaries of this technology. Specifically, we’ll look at how computation enables 3D printing of very intricate sugar structures, which can be used to artificially create physiological channel networks like blood vessels.

Let’s think about how 3D printing takes a virtual design and brings it into the physical world. You start with some digital or analytical representation of a 3D volume. Then you slice it into discrete layers, and approximate the volume within each layer in a way that maps to a physical printing process. For example, some processes use a digital light projector to selectively polymerize material. Because the projector is a 2D array of pixels that are either on or off, each slice is represented by a binary bitmap. For other processes, each layer is drawn by a nozzle or a laser, so each slice is represented by a vector image, typically with a fixed line width.

In each case, the volume is represented as a stack of images, which, again, is usually an approximation of the desired design. Greater fidelity can be achieved by increasing the resolution of the printer—that is, the smallest pixel or thinnest line it can create. However, there is a practical limit, and sometimes a physical limit to the resolution. For example, in digital light projection a pixel cannot be made much smaller than the wavelength of the light used. Therefore, for some kinds of designs, it’s actually easier to achieve higher fidelity by modifying the process itself. Suppose, for example, you want to make a connected network of cylindrical rods with arbitrary orientation (there is a good reason to do this—we’ll get to that). Any process based on layers or pixels will produce some approximation of the cylinders. You might instead devise a process that is better suited to making this shape.

The Fused Deposition Modeling Algorithm

One type of 3D printing, termed fused deposition modeling, deposits material through a cylindrical nozzle. This is usually done layer by layer, but it doesn’t have to be. If the nozzle is translated in 3D, and the material can be made to stiffen very quickly upon exiting, then you have an elegant way of making arbitrarily oriented cylinders. If you can get new cylinders to stick to existing cylinders, then you can make very interesting things indeed. This non-planar deposition process is called direct-write assembly, wireframe printing or free-form 3D printing.

Things that you would make using free-form 3D printing are best represented not as solid volumes, but as structural frames. The data structure is actually a graph, where the nodes of the graph are the joints, and the edges of the graph are the beams in the frame. In the following image, you’ll see the conversion of a model to a graph object. Directed edges indicate the corresponding beam can only be drawn in one direction. An interesting computational question is, given such a frame, how do you print it? More precisely, given a machine that can “draw” 3D beams, what sequence of operations do you command the machine to perform?

3D beams

First, we can distinguish between motions where we are drawing a beam and motions where we are moving the nozzle without drawing a beam. For most designs, it will be necessary to sometimes move the nozzle without drawing a beam. In this discussion, we won’t think too hard about these non-printing motions. They take time, but, at least in this example, the time it takes to print is not nearly as important as whether the print actually succeeds or fails catastrophically.

We can further define the problem as follows. We have a set of beams to be printed, and each beam is defined by two joints, . Give a sequence of beams and a printing direction for each beam (i.e. ) that is consistent with the following constraints:

1) Directionality: for each beam, we need to choose a direction so that the nozzle doesn’t collide with that beam as it’s printed.

2) Collision: we have to make sure that as we print each beam, we don’t hit a previously printed beam with the nozzle.

3) Connection: we have to start each beam from a physical surface, whether that be the printing substrate or an existing joint.

Let’s pause there for a moment. If these are the only three constraints, and there are only three axes of motion, then finding a sequence that is consistent with the constraints is straightforward. To determine whether printing beam B would cause a collision with beam A, we first generate a volume by sweeping the nozzle shape along the path coincident with beam B to form the 3D region . If RegionDisjoint[R, A] is False, then printing beam B would cause a collision with beam A. This means that beam A has to be printed first.

Here’s an example from the RegionDisjoint reference page to help illustrate this. Red walls collide with the cow and green walls do not:





w1=Hyperplane[{1,0,0},0.39]; w2=Hyperplane[{1,0,0},-0.45];





Mimicking the logic from this example, we can make a function that takes a swept nozzle and finds the beams that it collides with. Following is a Wolfram Language command that visualizes nozzle-beam collisions. The red beams must be drawn after the green one to avoid contact with the blue nozzle as it draws the green beam:




For a printer with three axes of motion, it isn’t particularly difficult to compute collision constraints between all the pairs of beams. We can actually represent the constraints as a directed graph, with the nodes representing the beams, or as an adjacency matrix, where a 1 in element (, ) indicates that beam must precede beam . Here’s the collision matrix for the bridge:

Precedence Matrix

A feasible sequence exists, provided this precedence graph is acyclic. At first glance, it may seem that a topological sort will give such a feasible sequence; however, this does not take the connection constraint into consideration, and therefore non-anchored beams might be sequenced. Somewhat surprisingly, TopologicalSort can often yield a sequence with very few connection violations. For example, in the topological sort, only the 12th and 13th beams violate the connection constraint:



ordering=TopologicalSort[AdjacencyGraph[SparseArray[Specified elements: 2832 Dimensions: {135,135}]]]

Instead, to consider all three aforementioned constraints, you can build a sequence in the following greedy manner. At each step, print any beam such that: (a) the beam can be printed starting from either the substrate or an existing joint; and (b) all of the beam’s predecessors have already been printed. There’s actually a clever way to speed this up: go backward. Instead of starting at the beginning, with no beams printed, figure out the last beam you’d print. Remove that last beam, then repeat the process. You don’t have to compute collision constraints for a beam that’s been removed. Keep going until all the beams are gone, then just print in the reverse removal order. This can save a lot of time, because this way you never have to worry about whether printing one beam will make it impossible to print a later beam due to collision. For a three-axis printer this isn’t a big deal, but for a four- or five-axis robot arm it is.

So the assembly problem under collision, connection and directionality constraints isn’t that hard. However, for printing processes where the material is melted and solidifies by cooling, there is an additional constraint. This is shown in the following video:

See what happened? The nozzle is hot, and it melts the existing joint. Some degree of melting is unfortunately necessary to fuse new beams to existing joints. We could add scaffolding or try to find some physical solution, but we can circumvent it in many cases by computation alone. Specifically, we can find a sequence that is not only consistent with collision, connection and directionality constraints, but that also never requires a joint to simultaneously support two cantilevered beams. Obviously some things, like the tree we tried to print previously, are impossible to print under this constraint. However, it turns out that some very intimidating-looking designs are in fact feasible.

We approach the problem by considering the assembly states. A state is just the set of beams that has been assembled, and contains no information about the order in which they were assembled. Our goal is to find a path from the start state to the end state. Because adjacent states differ by the presence of a single beam, each path corresponds to a unique assembly sequence. For small designs, we can actually generate the whole graph. However, for large designs, exhaustively enumerating the states would take forever. For illustrative purposes, here’s a structure where the full assembly state is small enough to enumerate. Note that some states are unreachable or are a dead end:

Full assembly state structure

Note that, whether you start at the beginning and go forward or start at the end and work backward, you can find yourself in a dead end. These dead ends are labeled G and H in the figure. There might be any number of dead ends, and you may have to visit all of them before you find a sequence that works. You might never find a sequence that works! This problem is actually NP complete—that is, you can’t know if there is a feasible sequence without potentially trying all of them. The addition of the cantilever constraint is what makes the problem hard. You can’t say for sure if printing a beam is going to make it impossible to assemble another beam later. What’s more, going backward doesn’t solve that problem: you can’t say for sure if removing a beam is going to make it impossible to remove a beam later due to the cantilever constraint.

The key word there is “potentially.” Usually you can find a sequence without trying everything. The algorithm we developed searches the assembly graph for states that don’t contain cantilevers. If you get to one of these states, it doesn’t mean a full sequence exists. However, it does mean that if a sequence exists, you can find one without backtracking past this particular cantilever-free state. This essentially divides the problem into a series of much smaller NP-complete graph search problems. Except in contrived cases, these can be solved quickly, enabling construction of very intricate models:




Precedence Matrix

So that mostly solves the problem. However, further complicating matters is that these slender beams are about as strong as you might expect. Gravity can deform the construct, but there is actually a much larger force attributable to the flow of material out of the nozzle. This force can produce catastrophic failure, such as the instability shown here:

However, it turns out that intelligent sequencing can solve this problem as well. Using models developed for civil engineering, it is possible to compute at every potential step the probability that you’re going to break your design. The problem then becomes not one of finding the shortest path to the goal, but of finding the safest path to the goal. This step requires inversion of large matrices and is computationally intensive, but with the Wolfram Language’s fast built-in solvers, it becomes feasible to perform this process hundreds of thousands of times in order to find an optimal sequence.

Use Cases

So that’s the how. The next question is, “Why?” Well, the problem is simple enough. Multicellular organisms require a lot of energy. This energy can only be supplied by aerobic respiration, a fancy term for a cascade of chemical reactions. These reactions use oxygen to produce the energy required to power all higher forms of life. Nature has devised an ingenious solution: a complex plumbing system and an indefatigable pump delivering oxygen-rich blood to all of your body’s cells, 24/7. If your heart doesn’t beat at least once every couple seconds, your brain doesn’t receive enough oxygen-rich blood to maintain consciousness.

We don’t really understand super-high-level biological phenomena like consciousness. We can’t, as far as we can tell, engineer a conscious array of cells, or even of transistors. But we understand pretty well the plumbing that supports consciousness. And it may be that if we can make the plumbing and deliver oxygen to a sufficiently thick slab of cells, we will see some emergent phenomena. A conscious brain is a long shot, a functional piece of liver or kidney decidedly less so. Even a small piece of vascularized breast or prostate tissue would be enormously useful for understanding how tumors metastasize.

The problem is, making the plumbing is hard. Cells in a dish do self-organize to an extent, but we don’t understand such systems well enough to tell a bunch of cells to grow into a brain. Plus, as noted, growing a brain sort of requires attaching it to a heart. Perhaps if we understand the rules that govern the generation of biological forms, we can generate them at will. We know that with some simple mathematical rules, one can generate very complex, interesting structures—the stripes on a zebra, the venation of a leaf. But going backward, reverse-engineering the rule from the form, is hard, to say the least. We have mastered the genome and can program single cells, but we are novices at best when it comes to predicting or programming the behavior of cellular ensembles.

An alternative means of generating biological forms like vasculature is a bit cruder—just draw the form you want, then physically place all the cells and the plumbing according to your blueprint. This is bioprinting. Bioprinting is exciting because it reduces the generation of biological forms into a set of engineering problems. How do we make a robot put all these cells in the right place? These days, any sentence that starts with “How do we make a robot…” probably has an answer. In this case, however, the problem is complicated by the fact that, while the robot or printer is working, the cells that have already been assembled are slowly dying. For really big, complex tissues, either you need to supply oxygen to the tissue as you assemble it or you need to assemble it really fast.

One approach of the really fast variety was demonstrated in 2009. Researchers at Cornell used a cotton candy machine to melt-spin a pile of sugar fibers. They cast the sugar fibers in a polymer, dissolved them out with water and made a vascular network in minutes, albeit with little control over the geometry. A few years later, researchers at University of Pennsylvania used a hacked desktop 3D printer to draw molten sugar fibers into a lattice and show that the vascular casting approach was compatible with a variety of cell-laden gels. This was more precise, but not quite free-form. The next step, undertaken in a collaboration between researchers at the University of Illinois at Urbana–Champaign and Wolfram Research, was to overcome the physical and computational barriers to making really complex designs—in other words, to take sugar printing and make it truly free-form.

We’ve described the computational aspects of free-form 3D printing in the first half of this post. The physical side is important too.

Eiffel Tower

First, you need to make a choice of material. Prior work has used glucose or sucrose—things that are known to be compatible with cells. The problem with these materials is twofold: One, they tend to burn. Two, they tend to crystallize while you’re trying to print. If you’ve ever left a jar of honey or maple syrup out for a long time, you can see crystallization in action. Crystals will clog your nozzle, and your print will fail. Instead of conventional sugars, this printer uses isomalt, a low-calorie sugar substitute. Isomalt is less prone to burning or crystallizing than other sugar-like materials, and it turns out that cells are just as OK with isomalt as they are with real sugar.

Next, you need to heat the isomalt and push it out of a tiny nozzle under high pressure. You have to draw pretty slowly—the nozzle moves about half a millimeter per second—but the filament that is formed coincides almost exactly with the path taken by the nozzle. Right now it’s possible to be anywhere from 50 to 500 micrometers, a very nice range for blood vessels.


So the problems of turning a design into a set of printer instructions, and of having a printer that is sufficiently precise to execute them, are more or less solved. This doesn’t mean that 3D-printed organs are just around the corner. There are still problems to be solved in introducing cells in and around these vascular molds. Depending on the ability of the cells to self-organize, dumping them around the mold or flowing them through the finished channels might not be good enough. In order to guide development of the cellular ensemble into a functional tissue, more precise patterning may be required from the outset; direct cell printing would be one way to do this. However, our understanding of self-organizing systems increases every day. For example, last year researchers reproduced the first week of mouse embryonic development in a petri dish. This shows that in the right environment, with the right mix of chemical signals, cells will do a lot of the work for us. Vascular networks deliver oxygen, but they can also deliver things like drugs and hormones, which can be used to poke and prod the development of cells. In this way, bioprinting might enable not just spatial but also temporal control of the cells’ environment. It may be that we use the vascular network itself to guide the development of the tissue deposited around it. Cardiologists shouldn’t expect a 3D-printed heart for their next patients, but scientists might reasonably ask for a 3D-printed sugar scaffold for their next experiments.

So to summarize, isomalt printing offers a route to making interesting physiological structures. Making it work requires a certain amount of mechanical and materials engineering, as one might expect, but also a surprising amount of computational engineering. The Wolfram Language provides a powerful tool for working with geometry and physical models, making it possible to extend free-form bioprinting to arbitrarily large and complex designs.

To learn more about our work, check out our papers: a preprint regarding the algorithm (to appear in IEEE Transactions on Automation Science and Engineering), and another preprint regarding the printer itself (published in Additive Manufacturing).


This work was performed in the Chemical Imaging and Structures Laboratory under the principal investigator Rohit Bhargava at the University of Illinois at Urbana–Champaign.

Matt Gelber was supported by fellowships from the Roy J. Carver Charitable Trust and the Arnold and Mabel Beckman Foundation. We gratefully acknowledge the gift of isomalt and advice on its processing provided by Oliver Luhn of Südzucker AG/BENEO-Palatinit GmbH. The development of the printer was supported by the Beckman Institute for Advanced Science and Technology via its seed grant program.

We also would like to acknowledge Travis Ross of the Beckman Institute Visualization Laboratory for help with macro-photography of the printed constructs. We also thank the contributors of the CAD files on which we based our designs: GrabCAD user M. G. Fouché, 3D Warehouse user Damo and Bibliocas user limazkan (Javier Mdz). Finally, we acknowledge Seth Kenkel for valuable feedback throughout this project.

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  1. I admired what Wolfram’s team has done so far with Wolfram and Math as I am not good with Math.

  2. Really great stuff, guys, my respect! Cheers, Andy