Tissue Age Modeling on Carbon-14碳-14 组织年龄建模

Overview

This project develops and compares three families of mathematical models for inferring cellular age in human arterial tissues from atmospheric Carbon-14 bomb-pulse data. The work traces the methodological evolution from deterministic partial differential equations (PDE), through stochastic ordinary differential equations (SODE), to stochastic partial differential equations (SPDE), with a unified parameter-fitting and error-analysis pipeline applied to three vascular tissues (ACI, CA, AVM).

Course: MATH-SHU 997 Independent Study (Spring 2026)

Institution: NYU Shanghai

Advisor: Prof. Roberto Fernandez

Repository: GitHub - Carbon-14-Bomb-Pulse-Modeling

Report: Final Report (PDF)

Background

Above-ground nuclear weapons tests in the 1950s–60s produced a sharp, well-characterized spike in atmospheric ¹⁴C — the “bomb pulse.” Biological tissues incorporate ¹⁴C in proportion to atmospheric levels at the time of their formation, so measuring residual ¹⁴C in a tissue lets us infer the average “birth year” of its cells. The method has become a standard tool for studying turnover in long-lived or terminally differentiated cell populations (neurons, cardiomyocytes, vascular smooth muscle, etc.).

This study targets three vascular tissues:

• ACI — internal carotid artery
• CA — cerebral artery
• AVM — arteriovenous malformation (a lesion whose formation time is unknown and must be jointly fitted as an offset τ)

Mathematical Evolution

1. Deterministic PDE Model

The cell density n(a, t) over age a and time t is governed by a McKendrick–von Foerster transport equation, which combined with the ¹⁴C decay and bomb-pulse curve yields an analytical convolution formula for tissue ¹⁴C content.

• Solved via the method of characteristics
• Two parameters: γ (death rate) and b (proliferation rate)
• L’Hôpital limit used for the λ = γ − b = 0 case to keep the solution numerically stable

2. Stochastic ODE Model — Agent-Based Simulation

The deterministic rates are replaced by a stochastic birth–death process on individual cells:

• Track N = 2000 individual agents
• At each step dt = 0.1 year, deaths drawn from a Bernoulli, births from a Poisson
• Multiple replicates averaged to estimate the expected ¹⁴C curve

This captures fluctuations invisible to the deterministic model and is closer to the biological reality of small cell populations.

3. Stochastic PDE Model — Cohort-Based Monte Carlo

A cohort-based Monte Carlo scheme couples age structure with stochasticity:

• Cells are tracked as a dictionary of birth-year cohorts
• Initial population N = 10,000 to reduce variance
• Binomial deaths and Poisson births per step
• Yields smoother stochastic solutions while preserving age-structure information

Computation & Fitting Pipeline

Each of the three frameworks runs the same three-stage pipeline:

1. Nelder–Mead optimization — scipy.optimize.minimize on (γ, b), initial guess [0.10, 0.10]
2. 2D grid search — exhaustive sweep over γ ∈ [0.01, 0.30] and b ∈ [0.00, 0.30] with step 0.01 (930 points)
3. Comparison — read both fits, score by RMSE / AIC, export comparison plots

Joint τ-fitting for AVM

Because the AVM lesion has no clear birth-time, I designed a regularized joint fit for τ:

• Sweep τ candidates at 0.5-year steps up to the patient’s lifetime
• Apply a penalty (λ = 0.1) anchoring τ to the sample-mean τ-fraction so it doesn’t collapse to extremes
• Optimize jointly with (γ, b)

Key Results

1. RMSE Across Tissues (Grid vs. Optimization)

• ACI and CA: grid and Nelder–Mead agree almost exactly
• AVM: grid substantially outperforms optimization — the objective surface has multiple local minima that trap Nelder–Mead in a suboptimal basin

2. AVM Start-Time τ Diverges Between Methods

• Grid solution: τ_age ≈ 28.5 years (late onset)
• Optimization solution: τ_age ≈ 10.4 years (early onset)

The two parameter solutions imply qualitatively different biological histories — a striking illustration of how non-convex objective landscapes matter in medical modeling, not just numerics.

3. Methodological Trade-offs

• Optimization (Nelder–Mead): fast and simple, but sensitive to initialization and local minima
• Grid search: exhaustive and interpretable, but cost grows exponentially with parameter count
• Stochastic models (SODE / SPDE): biologically more faithful, at the cost of extra variance and runtime — SPDE trades 5× initial population for noticeably smoother curves

Contributions

1. A unified record of the deterministic-to-stochastic transition — from a first-order transport PDE to a multiplicative-noise particle system, compared on the same data with the same fitting protocol
2. A reproducible, shared pipeline — the three frameworks share data loading, AVM τ-fitting, and output logic; the only substantive difference is the convo_* function, making cross-model comparison clean
3. A regularization strategy for joint τ-fitting in AVM, where penalizing toward the sample mean prevents τ from overfitting in a small sample
4. Quantitative evidence on method choice — in biomedical small-sample regimes, plain grid search can outperform a classical optimizer because of objective non-convexity
5. Extensibility — the code and modeling framework transfer directly to other ¹⁴C-based turnover studies

Technical Stack

• Mathematical methods: McKendrick–von Foerster transport PDE, method of characteristics, stochastic differential equations, Monte Carlo simulation, Nelder–Mead optimization, grid search, AIC / RMSE
• Scientific computing: Python, NumPy, SciPy, Pandas, Matplotlib
• Data: patient ¹⁴C tissue measurements (data.xlsx), reference bomb-pulse curve (Bombpulse_14C_Jan2026-1.xlsx)
• Writing: LaTeX report with figures

Future Directions

• Enable risk-factor stratification (currently all samples collapse to a single “no risk” stratum)
• Implement a proper 2D age–time SPDE solver instead of the cohort-Monte-Carlo approximation
• Cross-validate against public ¹⁴C datasets on other tissues (neurons, cardiomyocytes)