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Module 8: Geometry, Disease & Health

Disease is geometric disorder. Tumour vasculature deviates from the optimal WBE network of Module 3–4, showing fractal dimensions 1.8–2.3 instead of 2.9. Ventricular fibrillation is a spiral-wave instability of the heart's reaction-diffusion sheet. Neurodegenerative diseases disrupt the small-world topology of neural networks. Medical imaging increasingly uses fractal and topological biomarkers as diagnostic tools. This final module surveys how the geometric principles of the earlier modules inform modern medicine — and how tissue engineering tries to recreate normal geometry to restore function.

1. Cancer as Geometric Disorder

Normal vasculature is a WBE fractal with dimension \(D \approx 2.9\) (Module 3). Tumour angiogenesis, driven by VEGF overexpression in a hypoxic environment, produces chaotic, Murray-violating networks. Baish & Jain (2000, Cancer Research) compiled box-counting measurements of tumour vasculature across species and tumour types:

Healthy microvasculature: \(D \approx 2.70\text{--}2.90\).
Mammary carcinoma: \(D \approx 2.12\).
Melanoma: \(D \approx 1.85\).
Glioma: \(D \approx 1.98\).

1.1 Consequences of Altered Geometry

Low-\(D\) vasculature fails to space-fill, causing:

Hypoxia: diffusion-limited regions \(>100\,\mu\text{m}\) from capillaries.
Elevated interstitial pressure: poor lymph drainage; impedes drug delivery.
Chaotic flow: pulses and reverses; shear-stress heterogeneity.
Impaired metastasis filter: leaky walls allow circulating tumour cells.

1.2 Vessel Normalisation as Therapy

Jain (2005, Science) proposed anti-angiogenic therapy (anti-VEGF) works not only by pruning vessels but by normalising the remaining network — restoring higher \(D\), improving oxygenation, and paradoxically enhancing chemotherapy delivery. Clinical trials of bevacizumab + chemotherapy confirmed this “vessel normalisation window”.

2. Cardiac Arrhythmia: Geometric Origin of Fibrillation

The heart is an excitable medium — a 2D/3D sheet where electrical action potentials propagate as waves. In normal rhythm, a single wave originating at the sinoatrial node sweeps across the atria and ventricles. In ventricular tachycardia or fibrillation, this normal wave breaks into self-sustaining spiral waves (Winfree 1972; Davidenko et al. 1992).

2.1 FitzHugh-Nagumo Model

A minimal two-variable model of the cardiac action potential:

\[ \partial_t u = D\nabla^2 u + u(1-u)(u-b) - v, \quad \partial_t v = \varepsilon(u - a v) \]

\(u\) = fast voltage variable, \(v\) = slow recovery variable, \(\varepsilon \ll 1\).

2.2 Spiral Core & Winding

A broken wave front develops into a rotating spiral. The wave rotates about a “core” where tissue is transiently unexcitable; rotation period \(T \sim \pi/\omega\) is determined by the local refractory period. Spiral waves:

Re-entrant circuits: wave meets its own tail; feedback loop.
Spiral breakup: at high excitability heterogeneity, spiral fragments → fibrillation.
Topological charge: number of winding defects (phase singularities) ≥ 1.

2.3 Defibrillation

An electric shock transiently excites all cells simultaneously, collapsing all spiral phase singularities and allowing the SA node to resume control. The ~100-200 J delivered is enough to reset the cellular potentials across tens of billions of cardiomyocytes.

3. Neural Network Topology

The brain is a network of ~10¹¹ neurons connected by ~10¹⁴ synapses. Characterising its geometric and topological properties is a central problem of neuroscience.

3.1 Small-World Topology (Watts-Strogatz)

Watts & Strogatz (1998) defined a network with: (a) high local clustering \(C\)(neighbours tend to be neighbours) and (b) short global average path length \(L\)(any two nodes separated by few steps). The rewired regular lattice interpolates between a regular ring (\(p=0\): high \(C\), high \(L\)) and random graph (\(p=1\): low \(C\), low \(L\)).

The small-world regime is at intermediate \(p\sim 0.01\text{--}0.1\)where \(C \gg C_{\text{random}}\) but \(L \approx L_{\text{random}}\). This is what brain connectomes look like: C. elegans, macaque cortex, human DTI-derived connectomes all have small-world topology (Sporns 2010).

3.2 Scale-Free Degree Distribution

Many neural networks also show power-law degree distributions \(P(k) \propto k^{-\gamma}\)with \(\gamma \approx 2\text{--}3\): a few “hub” neurons with thousands of connections, and many sparsely connected neurons. Barabasi-Albert (1999) showed that preferential attachment (“rich get richer”) during growth generates such distributions.

\[ P(k) \propto k^{-\gamma}, \quad \gamma \approx 2\text{--}3 \]

3.3 Alzheimer's & Schizophrenia

Bassett & Bullmore (2009) compiled evidence that disease disrupts small-world topology:

Alzheimer's: increased \(L\) (longer paths), loss of hub integrity.
Schizophrenia: lower \(C\), higher randomness.
Autism: over-connected local clusters, under-connected distant regions.
Epilepsy: hypersynchronisation → drift toward regular regime.

4. Fractal Analysis in Medical Imaging

4.1 Retinal Vessel Fractals

The retinal vascular fractal dimension (measured by box-counting on fundus images) is approximately 1.70 in healthy eyes. Diabetic retinopathy and hypertension systematically lower\(D\) (Mainster 1990; Liew et al. 2008). Fractal dimension is proposed as an early biomarker for cardiovascular disease.

4.2 Mammographic Parenchymal Patterns

Breast parenchymal density on mammograms correlates with cancer risk (Wolfe 1976). Fractal dimension of the parenchymal pattern is an additional independent risk factor (Caldwell et al. 1990): higher \(D\) (more complex texture) is associated with elevated risk.

4.3 Tumour Margin Roughness

Benign tumours tend to have smooth margins (\(D_{\text{margin}} \approx 1.1\)), while malignant ones have jagged fractal margins (\(D \approx 1.3\text{--}1.5\)). This is often quantified in dermoscopy and histopathology.

4.4 Trabecular Bone

Trabecular bone microarchitecture in CT and MRI shows a fractal dimension\(D \approx 1.6\text{--}1.9\) for healthy subjects. Osteoporosis lowers\(D\), reflecting loss of structural complexity and increased fracture risk.

5. Tissue Engineering: Designing Optimal Geometry

A regenerative medicine goal is to restore normal tissue geometry. Scaffolds with biomimetic pore architectures promote cell infiltration, vascularisation, and tissue regeneration.

5.1 Triply Periodic Minimal Surfaces as Scaffolds

TPMS (Module 5) provide nearly optimal stiffness-to-mass ratios with interconnected pore networks. Gyroid-based 3D-printed titanium implants have been shown to:

Match cortical bone's Young's modulus (reducing stress shielding).
Permit neovascularisation through their interconnected channels.
Support osteoblast adhesion on their curved surfaces.

5.2 Vascular Network Printing

Recent advances in bioprinting (Miller et al. 2019) fabricate hierarchical vascular trees following Murray's law to pre-vascularise engineered tissues. Scale is critical: transported oxygen can only diffuse \(\sim 100\,\mu\text{m}\) from a capillary, so organ-scale engineered tissues require integrated fractal perfusion.

5.3 Organoid Morphogenesis

Self-organising organoids (brain, gut, kidney, retina) recapitulate embryonic morphogenesisin vitro. Understanding their geometric rules — Turing patterns, differential adhesion, mechanical feedback — is the frontier of tissue engineering.

6. Tumour vs Normal Vasculature

7. Cardiac Re-entry Spiral Wave

8. Small-World Neural Network

9. Simulation 1 — Tumour Vasculature D

Generate two synthetic networks with different branching rules (WBE-like vs chaotic) and measure their box-counting fractal dimension. The difference distinguishes normal from tumour vasculature (Baish & Jain 2000).

Python

script.py93 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Fractal dimension comparison: normal vs tumor vasculature
# Normal D ~ 2.7-2.9 (space-filling, Murray-obeying)
# Tumor D ~ 1.8-2.2 (chaotic, Murray-violating, Baish & Jain 2000)

rng = np.random.default_rng(42)

def branching_tree(D_target, N_segments=4000, jitter=0.0):
    # Generate a 2D tree whose fractal dimension approximately matches D_target
    # by tuning branching angle distribution and length scaling
    points = []
    # Use recursive with random perturbations
    def rec(x, y, ang, ln, depth):
        if depth == 0 or ln < 0.3 or len(points) > N_segments:
            return
        # scale factor depends on D: steeper for lower D
        scale = 0.7 if D_target > 2.6 else 0.55
        nx = x + ln*np.cos(ang); ny = y + ln*np.sin(ang)
        n_pts = max(2, int(ln))
        for t in np.linspace(0, 1, n_pts):
            points.append((x + t*(nx-x), y + t*(ny-y)))
        # Chaotic branching for low D
        ang_jitter = jitter * rng.normal()
        n_children = 2
        base_angle = 0.35 if D_target > 2.6 else 1.2  # tumor: chaotic angles
        for i in range(n_children):
            rec(nx, ny, ang + (i*2-1)*base_angle + ang_jitter, ln*scale, depth-1)
    rec(500, 100, np.pi/2, 120.0, 10)
    return np.array(points)

pts_normal = branching_tree(2.8, jitter=0.05)
pts_tumor  = branching_tree(2.0, jitter=0.6)

def box_count_dim(pts):
    eps_values = np.logspace(0.3, 2.0, 15)
    N_values = []
    for e in eps_values:
        xi = np.floor(pts[:,0]/e).astype(int)
        yi = np.floor(pts[:,1]/e).astype(int)
        N_values.append(len(set(zip(xi.tolist(), yi.tolist()))))
    N_values = np.array(N_values)
    log_eps = np.log(1.0/eps_values); log_N = np.log(N_values)
    slope, _ = np.polyfit(log_eps, log_N, 1)
    return slope, eps_values, N_values

D_norm, eps_n, N_n = box_count_dim(pts_normal)
D_tum,  eps_t, N_t = box_count_dim(pts_tumor)

fig, axes = plt.subplots(1, 3, figsize=(18, 6))
fig.patch.set_facecolor('#030712')
fig.suptitle("Tumour vs Normal Vasculature: Fractal Dimension",
             fontsize=15, color='#fde68a', fontweight='bold')

ax = axes[0]
ax.set_facecolor('#1a1200')
ax.scatter(pts_normal[:,0], pts_normal[:,1], s=0.5, c='#fbbf24', alpha=0.6)
ax.set_title(f'Normal vasculature (D = {D_norm:.2f})',
             color='#fde68a')
ax.set_aspect('equal'); ax.tick_params(colors='white')
for sp in ax.spines.values(): sp.set_color('#fbbf24')

ax = axes[1]
ax.set_facecolor('#1a1200')
ax.scatter(pts_tumor[:,0], pts_tumor[:,1], s=0.5, c='#f87171', alpha=0.6)
ax.set_title(f'Tumour vasculature (D = {D_tum:.2f})',
             color='#fde68a')
ax.set_aspect('equal'); ax.tick_params(colors='white')
for sp in ax.spines.values(): sp.set_color('#fbbf24')

ax = axes[2]
ax.set_facecolor('#1a1200')
ax.loglog(1.0/eps_n, N_n, 'o-', color='#fbbf24',
          label=f'normal (D={D_norm:.2f})', markersize=7)
ax.loglog(1.0/eps_t, N_t, 's-', color='#f87171',
          label=f'tumor (D={D_tum:.2f})', markersize=7)
ax.set_xlabel('1/epsilon', color='white')
ax.set_ylabel('N(epsilon)', color='white')
ax.set_title('Box-count scaling (Baish & Jain 2000)', color='#fde68a')
ax.legend(facecolor='#1a1200', edgecolor='#fbbf24', labelcolor='white')
ax.grid(True, which='both', alpha=0.2, color='#fbbf24')
ax.tick_params(colors='white')
for sp in ax.spines.values(): sp.set_color('#fbbf24')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#030712')
print(f"Normal D = {D_norm:.3f};  Tumor D = {D_tum:.3f}")
print("Tumor vasculature deviates from WBE predictions (Module 3)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

10. Simulation 2 — Cardiac Spiral Wave

FitzHugh-Nagumo simulation on a 2D grid. A broken wavefront develops into a sustained spiral — the geometric origin of ventricular tachycardia. Parameter tuning produces sustained re-entry or spiral breakup (fibrillation).

Python

script.py57 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# FitzHugh-Nagumo reaction-diffusion: cardiac action potential
# du/dt = D Lap u + f(u) - v
# dv/dt = eps (u - a*v)
# f(u) = u(1-u)(u-b)   (cubic non-linearity)
# Initialise with broken wave front -> spiral wave

def laplacian(Z):
    return (np.roll(Z, 1, axis=0) + np.roll(Z, -1, axis=0)
          + np.roll(Z, 1, axis=1) + np.roll(Z, -1, axis=1)
          - 4*Z)

N = 150
D = 1.0
eps = 0.01
a = 0.5
b = 0.05
dt = 0.1

u = np.zeros((N, N))
v = np.zeros((N, N))

# Initial broken front: rectangle of excited tissue + half-wave of recovery
u[:N//2, N//3:2*N//3] = 1.0
v[:N//4, N//3:2*N//3] = 0.5

snapshots = []
n_total = 4000
snap_times = [0, 500, 1000, 2000, 3000, 3999]
for step in range(n_total):
    Lu = laplacian(u)
    f = u*(1-u)*(u-b)
    u += dt*(D*Lu + f - v)
    v += dt*eps*(u - a*v)
    if step in snap_times:
        snapshots.append(u.copy())

fig, axes = plt.subplots(2, 3, figsize=(15, 10))
fig.patch.set_facecolor('#030712')
fig.suptitle("Cardiac Spiral Wave (FitzHugh-Nagumo model)",
             fontsize=15, color='#fde68a', fontweight='bold')

for ax, snap, t in zip(axes.flatten(), snapshots, snap_times):
    ax.imshow(snap, cmap='inferno', vmin=0, vmax=1)
    ax.set_title(f't = {t*dt:.1f}', color='#fde68a')
    ax.axis('off')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#030712')
print("Spiral waves underlie ventricular tachycardia and fibrillation")
print("(Winfree 1972; Davidenko 1992)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

11. Simulation 3 — Small-World Measure

Rewire a ring lattice with probability \(p\); measure clustering \(C\)and average path length \(L\) as functions of \(p\). The small-world regime (high \(C/L\) ratio) emerges at intermediate \(p\).

Python

script.py92 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
import networkx as nx

# Watts-Strogatz small-world: start with ring lattice, rewire edges with prob p
# Compare clustering C and path length L across p

N = 200
k = 6
p_values = np.logspace(-4, 0, 20)

def normalise(g):
    # Compute C and L for given graph
    C = nx.average_clustering(g)
    if nx.is_connected(g):
        L = nx.average_shortest_path_length(g)
    else:
        giant = max(nx.connected_components(g), key=len)
        L = nx.average_shortest_path_length(g.subgraph(giant))
    return C, L

# Baseline (p=0)
g0 = nx.watts_strogatz_graph(N, k, 0.0, seed=0)
C0, L0 = normalise(g0)

Cs, Ls = [], []
for p in p_values:
    g = nx.watts_strogatz_graph(N, k, p, seed=1)
    C, L = normalise(g)
    Cs.append(C/C0); Ls.append(L/L0)

Cs, Ls = np.array(Cs), np.array(Ls)

# Small-worldness peak at intermediate p
sw = Cs/Ls
best_p = p_values[np.argmax(sw)]

fig = plt.figure(figsize=(18, 7))
fig.patch.set_facecolor('#030712')
fig.suptitle("Watts-Strogatz Small-World Networks in Neuroscience",
             fontsize=15, color='#fde68a', fontweight='bold')

# Panel 1: C & L vs p
ax = fig.add_subplot(1, 3, 1)
ax.set_facecolor('#1a1200')
ax.semilogx(p_values, Cs, 'o-', color='#fbbf24',
            linewidth=2, markersize=7, label='C(p)/C(0)')
ax.semilogx(p_values, Ls, 's-', color='#22d3ee',
            linewidth=2, markersize=7, label='L(p)/L(0)')
ax.axvline(best_p, color='#86efac', linestyle='--',
           label=f'small-world peak p={best_p:.3g}')
ax.set_xlabel('rewiring probability p', color='white')
ax.set_ylabel('normalized', color='white')
ax.set_title('Clustering & path length', color='#fde68a')
ax.legend(facecolor='#1a1200', edgecolor='#fbbf24', labelcolor='white')
ax.grid(True, which='both', alpha=0.2, color='#fbbf24')
ax.tick_params(colors='white')
for sp in ax.spines.values(): sp.set_color('#fbbf24')

# Panel 2: ring lattice p=0
ax = fig.add_subplot(1, 3, 2)
ax.set_facecolor('#1a1200')
g_ring = nx.watts_strogatz_graph(30, 4, 0.0, seed=0)
pos = nx.circular_layout(g_ring)
nx.draw_networkx_edges(g_ring, pos, edge_color='#fbbf24',
                        alpha=0.7, width=1.2, ax=ax)
nx.draw_networkx_nodes(g_ring, pos, node_size=120,
                        node_color='#22d3ee', ax=ax)
ax.set_title('p=0: regular ring (high C, high L)', color='#fde68a')
ax.axis('off')

# Panel 3: small-world
ax = fig.add_subplot(1, 3, 3)
ax.set_facecolor('#1a1200')
g_sw = nx.watts_strogatz_graph(30, 4, 0.15, seed=5)
pos = nx.circular_layout(g_sw)
nx.draw_networkx_edges(g_sw, pos, edge_color='#fbbf24',
                        alpha=0.7, width=1.2, ax=ax)
nx.draw_networkx_nodes(g_sw, pos, node_size=120,
                        node_color='#86efac', ax=ax)
ax.set_title('p=0.15: small-world (high C, low L)', color='#fde68a')
ax.axis('off')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#030712')
print(f"Small-world peak at p = {best_p:.4g}")
print("C. elegans, macaque cortex, human connectome: all show small-world topology")
print("(Watts & Strogatz 1998; Sporns 2010)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Module Summary

Tumor vasculature

D ~ 1.8-2.2 (vs 2.9 normal); Baish & Jain 2000.

Vessel normalisation

Anti-VEGF partially restores D -> improves therapy.

Cardiac spiral waves

FitzHugh-Nagumo; phase singularities; fibrillation.

Defibrillation

~200 J collapses all phase singularities at once.

Small-world networks

Watts-Strogatz; peak at p ~ 0.01-0.1.

Scale-free degree

Barabasi-Albert P(k) ~ k^(-gamma) from preferential attachment.

Disease topology

Alzheimer (high L), schizophrenia (low C), autism (altered modularity).

Medical imaging

Retina, mammogram, trabecula D as biomarkers.

Tissue engineering

TPMS scaffolds + Murray-law printed vasculature.

References

Baish, J. W. & Jain, R. K. (2000). Fractals and cancer. Cancer Research, 60, 3683–3688.
Jain, R. K. (2005). Normalization of tumor vasculature: an emerging concept in antiangiogenic therapy. Science, 307, 58–62.
Winfree, A. T. (1972). Spiral waves of chemical activity. Science, 175, 634–636.
Davidenko, J. M. et al. (1992). Stationary and drifting spiral waves of excitation in isolated cardiac muscle. Nature, 355, 349–351.
FitzHugh, R. (1961). Impulses and physiological states in theoretical models of nerve membrane. Biophys. J., 1, 445–466.
Watts, D. J. & Strogatz, S. H. (1998). Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442.
Barabasi, A.-L. & Albert, R. (1999). Emergence of scaling in random networks. Science, 286, 509–512.
Sporns, O. (2010). Networks of the Brain. MIT Press.
Bassett, D. S. & Bullmore, E. T. (2009). Human brain networks in health and disease. Curr. Opin. Neurol., 22, 340–347.
Mainster, M. A. (1990). The fractal properties of retinal vessels. Eye, 4, 235–241.
Caldwell, C. B. et al. (1990). Characterisation of mammographic parenchymal patterns by fractal dimension. Phys. Med. Biol., 35, 235–247.
Miller, J. S. et al. (2019). Multivascular networks and functional intravascular topologies within biocompatible hydrogels. Science, 364, 458–464.

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