A mathematical three-step rocket helps biologists study tumour blood vessels
Koen Keijzer combined three mathematical models into one unified system capable of making meaningful predictions about how cells form blood vessels. This helps biologists studying the chaotic, leaky blood vessels found in and around tumours.
It may sound unusual: a group of mathematicians collaborating with biologists to explore how cells form blood vessels, particularly around tumours. PhD candidate Koen Keijzer explains: ‘We create computer models based on the rules these cells follow and a set of assumptions.’ A good model aligns as closely as possible with reality. ‘It’s never perfect but it is precisely the model's deviations from reality that help biologists to form better ideas or predictions in their research into tumour-related blood vessels.’ Blood vessel formation plays a key role in tumour growth and metastasis.
A good model aligns as closely as possible with reality.
Building on previous research
In the fourth year of his PhD, Keijzer reached an important milestone: his first scientific publication as lead author. He successfully integrated three mathematical models into one, capable of reliably predicting how cells form blood vessels in specific environments. He credits earlier work for laying the groundwork. ‘The first building block is a 30-year-old computational model that predicts cell behaviour,’ he explains.
From fibrous soup to stiff matrix: Leiden models describe cells in context
The second component is a model developed at Leiden University by Erika Tsingos and Bente Hilde Bakker, who were postdoctoral researchers when Keijzer joined the team. ‘Their model, which I also contributed to, describes how cells behave in their surroundings — a complex mixture of fluids and protein fibres.’ Biologists call this fibrous soup the extracellular matrix, and it comes in many forms. In bone, for instance, it’s extremely rigid.
The third model Keijzer built upon focuses on how cells behave differently in soft versus stiff environments. ‘In a stiffer matrix, cells can anchor themselves using tiny “feet” — cellular structures I included in the model. This causes the cells to elongate and flatten. In softer environments, they remain rounded.’ Tumours stiffen their surroundings, influencing the shape of blood vessels within and around them. ‘A tumour stretches the fibres of the extracellular matrix, much like pulling apart a plate of spaghetti. These fibres then create a kind of highway along which cells can attach and move.’
‘A tumour stretches the fibres, much like pulling apart a plate of spaghetti.’
Combining three models wasn’t easy
While 1 + 1 + 1 normally equals 3, merging the models was far from straightforward. ‘I had to write intermediate models to bridge the gaps,’ says Keijzer. But he succeeded. ‘The model even predicted that cells can adapt their environment. They deform the “spaghetti” fibres, making it easier to crawl along them. This wasn’t explicitly programmed into the model — it emerged naturally from the rules and assumptions.’
Initial challenges with processing speed
Once the model was functional, it had another issue: it was slow. ‘Running the programme required significant computing power, which is limited. A single run took an entire day on a supercomputer, and you always need multiple runs for reliable results.’ Fortunately, a software engineer from the eScience Center stepped in. ‘They streamlined the model so it produced the same results but much faster.’
Testing in transparent zebrafish embryos
The new model helps biologists form better hypotheses for studying tumour blood vessels. ‘Based on previous experiments, we know the model predicts reality quite well. We’re now further testing it in transparent zebrafish embryos.’ Leiden biologists have extensive experience in selectively activating or deactivating genes in zebrafish larvae. ‘They can adjust the stiffness of the extracellular matrix, allowing us to see even more clearly whether the model accurately predicts blood vessel formation.’
One of the scientists who contributed to the research was Bente Hilde Bakker, who passed away in 2024.