Multidimensional multifractal analysis: Theory and applications in ultrasound imaging of pancreatic cancer – MUTATION

The global goals of the project resonate an original and innovative research program, structured into three Challenges that each address major scientific deadlocks: i) complex boundary and ii) non-homogeneous multidimensional multifractal analysis, and iii) quantitative ultrasound imaging for pancreatic cancer.
Challenge 1 studies the largely unexplored topic of multifractal boundary regularity analysis and ad- dresses the need for analysis tools for the complex interfaces characterizing objects & their interactions.
Challenge 2 expands on the methodology developed in Ch. 1 and makes it collaborate with texture multifractality in a statistical estimation framework for non-homogeneous multidimensional data.
Challenge 3 addresses the quantitative in vivo analysis of pancreatic tumor development by application of the tools from Ch. 1 & 2 to longitudinal US images. It will systematically build up from phantom over different types of cancer images and empirical models and targets establishing in vivo US imaging for pancreatic cancer early detection and follow-up in practice.

We have developed the statistical model for the scale invariance and the multifractality of multidimensional textures. The main originality lies in the joint modeling of the fluctuations of the pointwise regularity, both in space and for several components or modalities. On the mathematical side, we were able to circumvent several difficulties inherent in the multivariate extension of multifractal analysis, by working with an approximate multifractal model which makes it possible to efficiently decouple the parameters of correlation, self-similarity and joint multifractality. On the statistical estimation side, we have proposed a Bayesian framework which allows the parameters of this model to be deduced. The estimation algorithm is based on a joint model of multi-scale statistics of multiresolution coefficients and has been validated on synthetic examples of multidimensional multifractal data. Together, these contributions lay the foundation for robust measurement of joint properties not accessible to classical spectral analysis, or other types of linear analysis.
At the level of echographic imaging of cancer, we have proposed a digital simulation pipeline which makes it possible to study experimentally the link between the multifractal properties of the reflectivity function modeling the tissues, and those of the different types of ultrasound image. The results we obtain show that there is a systematic correlation between the multifractal properties measured on the reflectivity function and on the raw ultrasound image, and suggest that it is indeed possible to reason in terms of multifractal properties at the tissue level. We have studied real ultrasound images of the thyroid, with very encouraging results, showing that our estimates of multifractal parameters make it possible to distinguish several types of tissue.

We are working to incorporate spatial regularization into the statistical model, in order to make it possible to analyze joint multifractal properties locally. This will be the first step towards truly non-homogeneous models of texture multifractality. We are also working on an computationally more efficient estimation algorithm based on expectation-maximization, which will allow us to use this model systematically for large data sets. Also at the methodological level, we are studying a multifractal formalism which incorporates information on the regularity of boundaries between textures, as well as on the regularity of the texture itself. As far as ultrasound imaging of cancer is concerned, we are preparing a simple but realistic experimental test bench, which consists of ultrasound images of breast cancer induced by orthotopic injection of syngeneic breast cancer cells in the mammary glands in a study longitudinal on the mouse.

1. E. Villain, H. Wendt, A. Basarab, D. Kouame, «On multifractal tissue characteriztion in ultrasound imaging,« IEEE International Symposium on Biomedical Imaging (ISBI), Venice, Italy, April 2019.
2. H. Wendt, M. Hourani, A. Basarab, D. Kouame, «Deconvolution for improved multifractal characterization of tissues in ultrasound imaging,« IEEE International Symposium on Biomedical Imaging (ISBI), Iowa City, USA, April 2020.
3. H. Wendt, M. Hourani, A. Basarab, D. Kouame, «Multifractal Characterization of Tissues in Ultrasound Imaging: a Study on the Influence of Deconvolution,« IEEE International Ultrasonic Symposium (IUS), Las Vegas, Sept. 2020, to appear.

Multifractal analysis enables the theoretically sound and practically robust study of the fluctuations in the regularity of image intensity, quantifying complex transient higher order dependence and heterogeneity in image texture.
The use of this mathematical framework yielded many remarkable successes in applications, for data of very different nature and in diverse disciplines ranging from Physics over Art to Biomedical, to name but a few.
The relevant analysis in applications yet remains fundamentally tied to the assumption that data can be characterized using one single homogeneous texture.
However, in many current real-world applications data are multidimensional and represent several objects - hence are naturally non-homogeneous - that may interact in an intricate manner along their boundaries. Ultrasound imaging for pancreatic cancer follow-up, the leading application of this proposal, constitutes an emblematic example for this case.
These inherent rich heterogeneous properties are central to data characterization but question the founding relations of multifractal analysis, linking regularity fluctuations to power laws that can be robustly assessed. This explains why this powerful image analysis tool remains so far rarely used in such contexts, and essentially limited to computing global fractal dimensions and texture statistics. This lack of adequate tools for assessing the rich and complex reality of pancreatic cancrogenesis critically impairs the use of medical imaging for detection and monitoring.
To address this crucial shortcoming, the central thread of the MUTATION project is to
* construct the theoretical foundations for the multifractal analysis of non-homogeneous, multidimensional data with complex interfaces, and build the corresponding practically robust and accurate statistical estimation framework and
* conduct the accurate space-time quantitative analysis of pancreatic cancer progression, from in vivo ultrasound images, for screening, diagnosis, early detection and follow-up, and to assess the functional role and relevance of tumor heterogeneity.
This global goal resonates the core of a novel research theme to be unfolded and is translated into an original and innovative research program, structured into three Challenges that each address major scientific deadlocks: i) complex boundary and ii) non-homogeneous multidimensional multifractal analysis, and iii) quantitative ultrasound imaging for pancreatic cancer.
To lead to success this ambitious scientific project - coupling functional analysis developments with statistical models to unfold their full potential for high-stake pancreatic cancer follow-up - an excellent young research team has been assembled, building up from previous pairwise collaborations, covering all relevant fields of expertise and incorporating world-renown experts for contributing to theoretical and applied developments at the edge of current knowledge.
This trans-disciplinary and transversal project will output robust innovative methods for the analysis of images and volumes, integrated as scientific toolboxes to be shared with the research community and popularized in research schools, tutorials and special sessions, and professional software libraries for dedicated use in medical imaging applications.
MUTATION's declared goal is to yield significant scientific and societal impact beyond methodology by tackling fundamental issues in pancreatic cancer follow-up, including the functional role of irregularity in tumor development, or the correlation of time-space tumor heterogeneity with treatment efficiency, scientifically nourishing, e.g., in vivo screening and early detection, or innovative targeted treatment therapy development.
The developed tools are also of great interest to other scientific communities and are targeted to lead to important socio-economic impact in a range of other high-tech applications, such as remote sensing, cultural heritage and geophysics.