Cancer cells are hungry. That is why tumors are well supplied with blood, enabling them to grow aggressively. How well a tumor is supplied with blood indicates whether and how well a therapy will work. You can imagine this as a road network in which trucks transport goods to individual locations.
In the body, blood vessels form the roads along which important “deliveries” are made: oxygen, drugs, and nutrients are transported via the blood. But just as road networks vary from place to place, there are winding alleys and one-way streets in different directions. Some roads are congested. Similarly, the microscopic vascular networks inside tumors vary from person to person. However, there is currently no clinically applicable, cost-effective method for analyzing blood flow in a tumor precisely and non-invasively.
A research project at the Hamburg University of Technology (TUHH) in cooperation with three renowned US institutions – Stanford University, the Mayo Clinic, and the University of California, San Diego – aims to change this. Using 4D ultrasound images (spatial and temporal) and mathematical models, the researchers are developing a method based on the liver that enables rapid quantitative analysis of tumor vessels – directly at the patient's bedside.
From image to diagnosis: ultrasound reimagined
The principle is both innovative and pragmatic. A modern 3D ultrasound device is used to create time-resolved images. These dynamic image data record how a contrast agent flows through the tumor area. This allows conclusions to be drawn about the vascular structure and blood flow. Especially in the liver, an organ with particularly complex blood flow, this opens up new possibilities for cancer diagnostics and treatment monitoring. This involves determining whether a particular therapy is actually effective or whether a different approach should be chosen.
“Typical ultrasound images provide visual information. What we need are numbers: flow rates, distribution parameters, concrete statements about the vascular structure,” says Dr. Sebastian Götschel, senior researcher at the Institute of Mathematics at the Hamburg University of Technology. “Extracting this information from noisy, temporally and spatially limited image data is challenging. But that is precisely our goal.”
Mathematical models for medicine
Goetschel is a mathematician with broad application-oriented expertise. Before joining the TU Hamburg, he worked at the Zuse Institute in Berlin, earned his doctorate at the FU Berlin, and worked at the Lawrence Berkeley National Laboratory in California. He is a member of the coordination team of the TUHH initiative Machine Learning in Engineering.
In the current project, he is working on a so-called inverse problem – one of the trickiest classes of mathematical problems. Known effects, such as the brightness of pixels in ultrasound, are used to infer unknown causes, such as blood flow velocity or vascular resistance. This requires models that are not only mathematically solvable but also physiologically meaningful.
“We first worked with a simple diffusion model,” explains Götschel. “It was mathematically very ingenious, but physiologically unconvincing because blood does not simply spread diffusely.” Instead, the current model considers arterial and venous blood separately. This results in two coupled equations – one for inflow and one for outflow – whose solution provides a meaningful indicator of blood flow.
From mice to humans: a long way to go
The method will first be tested on animal models and then validated in preclinical studies. A patient study is scheduled for completion in 2029. The imaging data will be compared with tissue samples (histology) and combined with other methods such as super-resolution ultrasound (SRUS). The aim is to develop an analysis method that is robust, reliable, and fast enough to be integrated directly into everyday clinical practice—without the need for complex large-scale equipment such as MRI or CT scanners. These are expensive, not universally accessible, and CT scans also involve radiation exposure.
“An ultrasound device fits on a trolley. That's a huge advantage, especially where resources are scarce,” says Götschel. The computing times should also remain within reasonable limits: a powerful workstation could deliver meaningful results within about an hour. To achieve this, calculations are performed in parallel across many processor cores.
It is also important that the results remain explainable: “We need models that we understand in order to know that the results are credible.” So while neural networks could be part of the solution in the future, the focus for now is on classic modeling.
A Hamburg contribution to international cutting-edge research
TU Hamburg is involved in the project with a compact team. In addition to Götschel, a postdoctoral researcher specializing in computational engineering and a doctoral student focusing on mathematical modeling, simulation, and optimization are working on the implementation. Together with partners in the US, they form an interdisciplinary bridge between mathematics, medicine, physics, and engineering. The project is funded for five years with a total of around three million US dollars from the US National Institutes of Health (NIH).
For Götschel, the project is the perfect combination of his interests: “I think it's great when I can make a real difference with my work. Mathematics is often very abstract. Sometimes you spend years researching something, and in the end, only a few experts around the world are interested. Here, it's different, and we can see directly what our work is good for.”
