Mathematical Oncology

Scans of the human brain

Mathematical Oncology is an interdisciplinary collaboration to develop direct and inverse mathematical models for brain cancer growth, making use of AI for applications in healthcare.

The group of Mathematical Oncology is timely formed since mathematical oncology is a rapidly growing field, [1], with the advantages of personalization of medicine through mathematics, modelling, and simulation. Personalization is achieved primarily through the use of patient-specific clinical data to make individualized predictions of response to therapy. The research will present data and simulation standards with the goal of creating reproducible models and improve cancer screening to detect cancer earlier. Moreover, it will be complemented by making full use of the exploratory and explanatory possibilities of Deep Neural Networks, Visualization, and the power of Inverse Problems and Mathematics.

Brain Cancer

Brain cancer is a common cancer worldwide; in 2012 with nearly 256 thousand new cases, brain cancer contributed to about 1.83% of the total number of new cancer cases diagnosed. Standard treatment includes surgery, radiotherapy and chemotherapy but despite all the therapeutic progress, brain tumours are still rarely fully curable. 

Mathematical Contribution to Brain Cancer research

The 2019 Mathematical Oncology Roadmap, [1], identifies three critical mile- stones along the path to mathematically designed cancer treatment: 

  1. obtaining accurate, rigorous, and reproducible predictions of the spatial-temporal progression of cancer; 
  2. avoiding and mitigating therapeutic resistance; and 
  3. merging mechanistic knowledge-based mathematical models with machine learning.

This shows the pivoting role that mathematics can have in cancer research. The group has expertise in a field of mathematics known as inverse problems for partial differential equations. This is ideally suited for cancer research since models come in the form of differential equations and with techniques from inverse problems parameters in those models can be identified.
Researchers in Mathematics at FEM have during a number of years studied mathematical properties of a model related to the growth of brain tumours with the focus of handling the inverse problem of reconstructing the exact location where the tumour started. A fruitful collaboration has now been established between researcher with background in Deep Neural Network, Image processing and Scientific Visualization. This groups is working on dynamical mathematical models to study the growth of brain tumors forward and backwards in time, for recent publications, see [2,3,4,5]. To show some outcomes in terms of tumor growth simulations from this collaboration, see images below:

Depiction of a tumor in the human brain

You can also see a video of the brain tumor simulation by clicking the image below

 A scan of a human brain

A unique feature here, which has been appreciated by the scientific community, is that we can simulate the growth of a tumour both forward as well as backwards in time. Having the ability to step also backwards in time, which is a notoriously difficult inverse ill-posed problem, can open for several new applications. For example, it can help clinicians to make better judgment on therapies and treatment. Furthermore, such results can be used to generate further synthetic data to train automatic systems for tumor detection. The actual simulations itself are unique in that they are produced with a novel state-of-the-art software for three-dimensional brain data visualization developed at the world leading visualization center at Campus Norrköping.
The time-dependent mathematical model we studied is a non-linear partial differential equation given by reaction-diffusion equations containing a function describing the logistic growth of the tumor, the net rate of growth of cells including proliferation and death, and a spatially dependent diffusion tensor.


Research Description

There are several on-going research project within this group, three of them are:

Parameter identification

Recovering the triplet of the unknown parameters, the diffusion tensor, the net rate of growth and the initial cell density from additional measurement. Over the past few years, diffusion tensor imaging (DTI) has been increasingly used to study pathologic changes in brain tumors. Various DTI metrics can be derived from the imaging data to provide information about the orientation and architecture of tissue microstructure at the voxel level. DTI provides a number of parameters about the shape, magnitude, and degree of diffusion anisotropy, which may be used to differentiate different tumor types. 


Modifying our mathematical model to handle different type of therapies. The standard treatment for most cancers involves a combination of surgery and chemotherapy and/or radiation therapy. Combination therapy is one very promising strategy but this raises its own questions – about how to administer the drugs, which can be combined in a seemingly infinite order and sequence. Mathematical models can be used to study these types of questions, offering the cancer biologist and clinical oncologist powerful new tools to add to their arsenal of laboratory and clinical approaches. These models offer a rational and unique method of searching through a large number of possible strategies to identify the most efficient doses to extend patient survival.


We will apply Artificial Intelligence (AI) methods to build and train a system for extracting information from large data sets to identify the triplet of unknown parameters the diffusion tensor, the net rate of growth and the initial cell density in our mathematical model. The main idea is to use machine learning and unsupervised learning with given sample measurements as inputs, resulting in finding a structure that determines risks of developing or indicating the presence of one or several brain tumors. It will also be possible to use clusters to profile/classify the medical measurements (personalized measurements) and then use these clusters to determine degree/magnitude of risk. Ultimately, methods from AI and Digital Pathology together with GAN (Generative Adversarial Networks) will be used to generate realistic synthetic medical images.


A recent grant from LiU Cancer will make it possible for the group members together with clinicians to get together to further work and develop research on mathematical models for cancer treatment.


  1. Russell C Rockne, Andrea Hawkins-Daarud, Kristin R Swanson, James P Sluka, James A Glazier, Paul Macklin, David A Hormuth II, Angela M Jarrett, Ernesto ABF Lima, J Tinsley Oden, et al. The 2019 mathematical oncology roadmap. Physical biology, 16(4):041005, 2019.
  2. Jaroudi, R., Baravdish, G., Åström, F., Johansson, B.T., Source Localization of Reaction-Diffusion Models for Brain Tumors, Lecture Notes in Computer Science 2016, 9796 :414-425.
  3. Jaroudi, R., Baravdish, G., Åström, F., Johansson, B.T., Numerical reconstruction of brain tumours, accepted in Inverse Problems in Science & Engineering, 2018.
  4. Jaroudi, R., Åström, F., Johansson, B.T., and Baravdish, G., Numerical implementation in 3-dimension of reaction-diffusion models for brain tumour growth, International Journal of Computer Mathematics, 2019.
  5. Ssebunjo, W., Baravdish, G., Svensson, O., and Johansson, B.T., Tumor Origin Localization via Non-linear Conjugate Gradient Methods, to be submitted.
  6. Eilertsen, G., Kronander, J., Denes, G., Mantiuk, R. K., & Unger, J. (2017). HDR image reconstruction from a single exposure using deep CNNs. ACM transactions on graphics (TOG), 36(6), 1-15.

  7. Eilertsen, G., Mantiuk, R. K., & Unger, J. (2019). Single-frame Regularization for Temporally Stable CNNs. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 11176-11185).

  8. Eilertsen, G., Jönsson, D., Ropinski, T., Unger, J., & Ynnerman, A. (2020). Classifying the classifier: dissecting the weight space of neural networks. In Proceedings of the European Conference on Artificial Intelligence (ECAI 2020), pages 1119–1126.

  9. Tsirikoglou, A., Eilertsen, G., & Unger, J. (2020). A Survey of Image Synthesis Methods for Visual Machine Learning. In Computer Graphics Forum.

  10. Stacke, K., Eilertsen, G., Unger, J., & Lundström, C. (2019). A Closer Look at Domain Shift for Deep Learning in Histopathology. MICCAI 2019 Computational Pathology Workshop (COMPAY 2019)

  11. Pocevičiūtė, M., Eilertsen, G., & Lundström, C. (2020). Survey of XAI in digital pathology. In Artificial Intelligence and Machine Learning for Digital Pathology (pp. 56-88). Springer, Cham.

  12. Tsirikoglou, A., Stacke, K., Eilertsen, G., Lindvall, M., & Unger, J. (2020). A Study of Deep Learning Colon Cancer Detection in Limited Data Access Scenarios. ICLR Workshop on AI for Overcoming Global Disparities in Cancer Care (AI4CC 2020).

  13. Cirillo, M. D., Abramian, D., & Eklund, A. (2020). Vox2Vox: 3D-GAN for Brain Tumour Segmentation. arXiv preprint arXiv:2003.13653.

  14. Foroozandeh, M., & Eklund, A. (2020). Synthesizing brain tumor images and annotations by combining progressive growing GAN and SPADE. arXiv preprint arXiv:2009.05946.