Discrete Differential Geometry in CAR T Cell Research

Discrete Differential Geometry (DDG) offers powerful mathematical tools to model, simulate, and analyze CAR T cell interactions with tumor environments. This article explores how DDG can be used to study the geometry of tumors, the dynamics of CAR T cell migration, and signal propagation during cancer immunotherapy.

Overview of the Illustration

The simulation covers the following aspects:

• Simulating the tumor surface using a 2D mesh grid.
• Calculating discrete Gaussian curvature to understand tumor surface geometry.
• Simulating CAR T cell signal propagation using diffusion models on the grid.

Python Code for the Illustration

The following Python code demonstrates these concepts step by step:

import numpy as np
import matplotlib.pyplot as plt
from scipy.sparse import diags

def generate_tumor_surface(size, height_variation):
    x, y = np.meshgrid(np.linspace(-1, 1, size), np.linspace(-1, 1, size))
    z = height_variation * (np.sin(3 * np.pi * x) * np.sin(3 * np.pi * y))
    return x, y, z

def calculate_gaussian_curvature(x, y, z):
    dz_dx = np.gradient(z, axis=1)
    dz_dy = np.gradient(z, axis=0)
    d2z_dx2 = np.gradient(dz_dx, axis=1)
    d2z_dy2 = np.gradient(dz_dy, axis=0)
    dz_dxdy = np.gradient(dz_dx, axis=0)
    numerator = d2z_dx2 * d2z_dy2 - dz_dxdy**2
    denominator = (1 + dz_dx**2 + dz_dy**2)**2
    return numerator / denominator

def simulate_signal_propagation(grid_size, source_position, diffusivity, steps):
    n = grid_size**2
    laplacian = diags([-1, -1, 4, -1, -1], [-grid_size, -1, 0, 1, grid_size], shape=(n, n))
    signal = np.zeros((grid_size, grid_size))
    signal[source_position] = 1
    signal_flat = signal.flatten()
    for _ in range(steps):
        signal_flat = signal_flat + diffusivity * laplacian.dot(signal_flat)
    return signal_flat.reshape((grid_size, grid_size))

grid_size = 50
height_variation = 0.5
source_position = (25, 25)
diffusivity = 0.01
steps = 100

x, y, z = generate_tumor_surface(grid_size, height_variation)
gaussian_curvature = calculate_gaussian_curvature(x, y, z)
signal_propagation = simulate_signal_propagation(grid_size, source_position, diffusivity, steps)

fig = plt.figure(figsize=(15, 5))
ax1 = fig.add_subplot(131, projection='3d')
ax1.plot_surface(x, y, z, cmap='viridis')
ax1.set_title("Tumor Surface")
ax2 = fig.add_subplot(132)
c2 = ax2.imshow(gaussian_curvature, cmap='jet', origin='lower')
ax2.set_title("Gaussian Curvature")
fig.colorbar(c2, ax=ax2)
ax3 = fig.add_subplot(133)
c3 = ax3.imshow(signal_propagation, cmap='plasma', origin='lower')
ax3.set_title("Signal Propagation")
fig.colorbar(c3, ax=ax3)
plt.tight_layout()
plt.show()
    

Visual Outputs

The code generates three key visualizations:

1. Tumor Surface: A 3D plot representing the tumor’s geometry.
2. Gaussian Curvature: A heatmap highlighting areas of high and low curvature.
3. Signal Propagation: A heatmap showing the diffusion of CAR T cell signals across the tumor grid.

Tumor Surface Visualization

Gaussian Curvature Heatmap

Signal Propagation Heatmap

Conclusion

By combining Discrete Differential Geometry and computational simulations, researchers can gain deeper insights into CAR T cell behavior and improve therapeutic strategies. The Python illustration provides a foundation for exploring these concepts further in cancer immunotherapy.