Table of Contents
System Architecture
sequenceDiagram
participant User
participant Frontend
participant API
participant Database
User->>Frontend: Submit Query
Frontend->>API: Process Request
API->>Database: Fetch Data
Database-->>API: Return Results
API-->>Frontend: Send Response
Frontend-->>User: Display Results
Mathematical Foundation
The Neural ODE equation: $\frac{d\mathbf{h}(t)}{dt} = f_{\theta}(\mathbf{h}(t), t)$
graph TD
A[Input] --> B[Neural Network]
B --> C[ODE Solver]
C --> D[Output]
The diabetic retinopathy progression model: $$ \begin{align} \frac{d}{dt}\begin{bmatrix} h_{\text{thickness}}(t) \ h_{\text{vascular}}(t) \ h_{\text{inflammation}}(t) \end{bmatrix} &= \begin{bmatrix} f_1(\mathbf{h}(t), \theta_1) \ f_2(\mathbf{h}(t), \theta_2) \ f_3(\mathbf{h}(t), \theta_3) \end{bmatrix} \tag{3} \end{align} $$
where each component models:
- $h_{\text{thickness}}(t)$: retinal thickness progression
- $h_{\text{vascular}}(t)$: vascular density changes
- $h_{\text{inflammation}}(t)$: inflammatory marker evolution
The Neural ODE equation describes continuous dynamics: $$\frac{d\mathbf{h}(t)}{dt} = f_{\theta}(\mathbf{h}(t), t) \tag{1}$$
where:
- $\mathbf{h}(t)$ is the hidden state vector at time $t$
- $f_{\theta}$ is a neural network parameterized by $\theta$
- The tag (1) provides equation numbering
Mathematical Formulas
Inline math: The famous equation is $E = mc^2$.
Display math: $$ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$
Complex equations: $$ \begin{align} \nabla \times \vec{\mathbf{B}} -, \frac1c, \frac{\partial\vec{\mathbf{E}}}{\partial t} & = \frac{4\pi}{c}\vec{\mathbf{j}} \ \nabla \cdot \vec{\mathbf{E}} & = 4 \pi \rho \ \nabla \times \vec{\mathbf{E}}, +, \frac1c, \frac{\partial\vec{\mathbf{B}}}{\partial t} & = \vec{\mathbf{0}} \ \nabla \cdot \vec{\mathbf{B}} & = 0 \end{align} $$
Project Overview
This research project focuses on developing Neural Ordinary Differential Equations (Neural ODEs) to model the continuous-time dynamics of diabetic retinopathy progression. By combining differential equation modeling with deep learning, we aim to predict disease progression trajectories and identify critical intervention points for better patient outcomes.
Research Objectives
- Develop Neural ODE architectures for modeling continuous disease progression dynamics
- Create interpretable models that capture the underlying biological mechanisms of diabetic retinopathy
- Apply Neural ODEs to predict individual patient progression trajectories
- Build computationally efficient methods for real-time clinical decision support
Mathematical Framework
Our approach models disease progression as a continuous dynamical system:
$$\frac{dh(t)}{dt} = f_\theta(h(t), t)$$
where $h(t)$ represents the hidden disease state at time $t$, and $f_\theta$ is a neural network parameterized by $\theta$.
The observed clinical measurements $y(t)$ are related to the hidden state through:
$$y(t) = g_\phi(h(t)) + \epsilon(t)$$
where $g_\phi$ is an observation function and $\epsilon(t)$ represents measurement noise.
For diabetic retinopathy progression, we incorporate multiple biomarkers:
$$\frac{d}{dt}\begin{bmatrix} h_1(t) \ h_2(t) \ h_3(t) \end{bmatrix} = \begin{bmatrix} f_1(h(t), \theta) \ f_2(h(t), \theta) \ f_3(h(t), \theta) \end{bmatrix}$$
where $h_1(t)$, $h_2(t)$, and $h_3(t)$ represent retinal thickness, vascular density, and inflammatory markers respectively.