# Images and Embedding Spaces as Dynamical Systems

## Our research in the applied analysis of stochastic differential equations enables us to model and study the time dynamics of disease. We focus on non-linear dynamical systems theory, non-equilibrium statistical mechanics, and numerical simulation. These mathematical methods describe how high-dimensional feature spaces and sparsely acquired images change over time.

Figure 1. Interpretation of feature spaces as a dynamical systems. (Left) A self-consistent potential function, V(x), is constructed directly from data. When V(x) has several distinct minima, if we propagate the trajectories starting from the data points of a given data set, with a high probability, those trajectories will aggregate into several groups within a short time. Further, if the profile of the potential function intrinsically indicates the density distribution of the data, the data points that end up in the neighborhood of the same minima of V (q) after O(1) time can be considered to be in the same cluster. While the forces from the potential surface push data points towards potential minima, corresponding Brownian fluctuations allow them to jump small potential barriers and escape saddle points into locations of the potential surface otherwise forbidden. (Top Right) Numerical verification of ergodic sampling at critical temperature. Marginal sampling distributions obtained across a projection of Hilbert space are shown relative to the analytic marginal Gibbs distribution left. Data are shown to be more localized at sub-critical temperature right due to non-ergodic dynamical behavior, and do not fully sample the potential landscape. (Bottom Right) A 2D projection of Hilbert space demonstrating ergodic behavior at critical temperature left and self-organization of data (i.e., clustering) at sub-critical temperature right. (DETAILED PUBLICATION)

Figure 2. Mathematical modeling of tumor time dynamics as a dynamical system via non-equilibrium thermodynamics. (Top) PET images before and after treatment are interpreted as no-flux boundary conditions of the Fokker-Planck equation, where the initial condition is an excited state and the final condition is an equilibrium state. To simulate the time dynamics as an estiamte of disease progression, each pixel is propagated according to the Fokker-Planck equation. (Left) This transformation is driven by an underlying potential force uniquely determined by the equilibrium state, resulting in a 4th order spatial-temporal manifold. (Right) Clustering of the time dynamics reveal unique tumor habitats that occur during treatment.

# Computational Image Interrogation

## Our research in computational image interrogation enables us to study the macroscopic appearance & behavior of tumors. We focus on quantitative imaging and high-throughput radiomic phenotyping. Radiomic features describe unique patterns on radiology, which serve as computational fingerprints of disease.

Figure 3. High-throughput computational tumor phenotyping. (Top) High-dimensional imaging signatures (radiomics) are linked to treatment outcome in patients with head and neck cancer. (Bottom) Disecting clustering mechanics: illustrating example comparing PET images and radiomic expression of two different patients demonstrating the importance of metabolic heterogeneity to treatment resistance.

# Computational Tissue Interrogation

## Our research in computational tissue interrogation enables us to study the microscopic appearance & behavior of tumors. We focus on deep learning, graph theory, and high-throughput pathomic phenotyping. Pathomic features describe unique patterns on digital pathology, which serve as computational fingerprints of disease and provide a detailed description of the tumor microenvironment.

Figure 4. Deep learning-based detection of lymphocytes on digital pathology with single-cell resolution validation. The left column shows the H&E image provided as input to the model. The center column shows the model output, where lymphocytes are detected and color-coded as blue relative to other cell types, which are color-coded as yellow. The right column shows corresponding single-cell resolution immunohistochemistry stained with antibodies against CD3/CD20, where brown indicates positive staining for lymphocytes.