SIAM News Blog

Fokker-Planck Dynamics Improve Multiscale Imaging of Head and Neck Tumors

By Lina Sorg

Head and neck cancers have the sixth highest incidence among cancers worldwide and result in roughly 550,000 new cases and 380,000 deaths per year. Locoregional recurrence of these tumors is a major driver of mortality, in large part because the treatment resistance is not currently well understood. During a minisymposium presentation at the 2022 SIAM Conference on Mathematics of Data Science, which is currently taking place in a hybrid format in San Diego, Calif., Kyle Lafata of Duke University introduced novel data-driven strategies that model the appearance and behavior of such cancers across space and time. Lafata’s research applies data science, mathematics, and imaging physics to medicine and biology. “Our lab interfaces with physicians and biologists to bring a lot of the techniques that are in development to practical fruition,” he said.

In fact, Lafata’s group is currently working on new computational techniques and mathematical methods to capture tumor appearance and behavior across space and time. Because tumors are complex multiscale systems, such approaches will help scientists better understand tumor biology and may ultimately lead to new treatment strategies. A recent clinical trial at Duke directly influenced this research and served as its catalyst. The study enrolled patients who were undergoing definitive chemoradiation for oropharyngeal head and neck cancer. Participants received 70 J/kg of therapeutic radiation over the course of seven weeks. Lafata and his colleagues obtained fluorodeoxyglucose PET images—which are common in oncological imaging—both before treatment (\(t=0\)) and at the end of two weeks of therapy (\(t=2\) weeks), as well as tissue biopsies from before treatment and after treatment recurrence (\(t>7\) weeks, if applicable). They then digitized the biopsies into whole slide images that allow for computational interrogation of the tissue. Finally, they worked with physician colleagues to collect patient outcomes after seven weeks of treatment and detect evidence of disease progression. “Given this data, our goal is to identify imaging biomarkers of treatment response,” Lafata said.

However, tumor characteristics span different length- and timescales, which makes the problem exceptionally challenging. For example, the PET image resolution is approximately 3 mm/pixel, but the digital biopsies are roughly .25 μm/pixel. Lafata thus utilized high-throughput image phenotyping and Fokker-Planck dynamics to model the way in which tumors change both morphologically and metabolically over time in response to treatment. He began by focusing on the macroscopic length scale of 3 mm. After he and his team acquired the images via computational high-throughput image phenotyping, they segmented the tumor in question and rendered it into a three-dimensional (3D) model. They next extracted, clustered, and linked features from the 3D model to patient outcomes. These features collectively describe unique patterns of the tumor and serve as a so-called visual fingerprint of a patient’s disease.

Figure 1. A feature space in the form of a heat map. Three different biclustering solutions are evident at \(t=2\) weeks.

Lafata then transitioned into a brief discussion of morphological features—which describe a tumor’s size, shape, and orientation—and intensity features, which measure the pixel intensity’s distribution within the tumor. But because intensity features fail to record any spatially encoded information, Lafata turned to texture. Fine texture captures the detailed, high-resolution structure at the imaging system’s resolution limit; this comprises small-scale information like overall image brightness and local heterogeneities. In contrast, coarse texture captures the heterogeneities within the image’s approximate structure; this type of texture accounts for large-scale information and long-distance relationships between pixels within the tumor.

Texture is defined based on joint histograms, and Lafata can extract scalar quantitative values—such as image homogeneity—from the texture histogram just as with a one-dimensional histogram. A tumor with a high homogeneity value has a uniform distribution of pixels, while one with a low homogeneity value is very heterogeneous. Extracting the necessary features yields a feature space in the form of a heat map. At \(t=2\) weeks, three different biclustering solutions are evident in the map and associated with patient outcomes (see Figure 1). “We’re capturing information at two weeks that is linked to responses up to five years later,” Lafata said.

Projecting this information back to the image space reveals crucial details about tumor metabolism in the three clusters. For example, the metabolic heterogeneity and high radiotracer (FDG) uptake in cluster 1 is linked to poor prognosis. On the other hand, cluster 3 was enriched with features that capture coarse homogeneities and low image intensity, therefore implying that metabolic homogeneity and a low FDG uptake are linked to a more favorable prognosis (see Figure 2).

Figure 2. Important details about tumor metabolism for clusters 1 and 3 in the image space.

Yet static imaging does not capture everything about the heterogeneity of the images. “It’s only a snapshot of the disease at time \(t\),” Lafata said. “Treatment response is a dynamic process.” He then moved into the timescale portion of the analysis and addressed the use of Fokker-Planck dynamics to model tumors as dynamical systems. “This is kind of cool because it’s an exercise in data assimilation between imaging science and applied analysis of stochastic differential equations,” Lafata said. The two time-separated images (\(t=0\) and \(t=2\) weeks) serve as the boundary conditions of the Fokker-Planck equation. Here, the image at \(t=0\) is the excited state and the image at \(t=2\) weeks is the equilibrium state. 

The depth of the pixel encoding on the images defines a potential landscape that drives the problem’s dynamics; Lafata modeled these dynamics by time evolving the pixels according to the Fokker-Planck equation. “The whole process generates a four-dimensional spatial-temporal manifold,” Lafata said. “The thought is that this will give us more information about the system. We can extract the same kind of textured features as before, but now we can represent them as a continuous function.”

Figure 3. Projection of the tumor’s potential landscape before and after treatment, based on responsiveness to therapy.
To illustrate this concept, Lafta shared a projection of the tumor’s potential landscape before and after treatment (see Figure 3). A tumor that is responsive to treatment in the first two weeks is linked to a landscape that quickly relaxes, while one that is resistant to treatment implies metabolic heterogeneity — a known factor of therapeutic resistance in these tumors. Observing this process in the imaging space divulges even more interesting aspects about the tumor over time. For example, fluctuations in tumor metabolic response during therapy have major biological applications to situations like tumor hypoxia. And a 3D rendering of the shrinking tumor—wherein the texture changes in a nonlinear way— suggests interesting biological implications in tumor vasculature.

Lafata concluded his talk with several comments about the tumor’s microenvironment, especially its immune system. He is particularly interested in this topic because the majority of immunotherapy patients with head and neck cancer will not have a durable survival benefit; in fact, more than 80 percent of patients with metastatic cancers do not respond to immunotherapy. As such, computational interrogation techniques of the tumor’s microenvironment may lead to new forms of biology that allow researchers to address some of the many unknown questions.

Figures courtesy of Kyle Lafata of Duke University.

Lina Sorg is the managing editor of SIAM News.
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