mangala-bulan 0.0.1

Oxygen diffusion solver with Michaelis-Menten kinetics and myoglobin facilitation

mangala-bulan: 1D Idealized Oxygen Diffusion Solver

A high-performance numerical solver for 1D idealized oxygen diffusion in biological tissues with Michaelis-Menten oxygen consumption kinetics and myoglobin-facilitated diffusion.

Key Features

• Six numerical methods: FTCS, DuFort-Frankel, Crank-Nicolson, Laasonen, ADI, and RK-IMEX
• High performance: Numba JIT compilation with automatic parallelization
• Flexible output formats: NetCDF4, CSV, and animated visualizations
• Stability monitoring: Real-time detection and handling of numerical instabilities
• Comprehensive logging: Detailed computational performance metrics

Requirements

System Requirements

• Operating System: Windows, macOS, or Linux
• RAM: 4 GB minimum (8 GB recommended for large simulations)
• Disk Space: 500 MB for installation + space for outputs

Python Dependencies

Package	Minimum Version	Purpose
numpy	≥1.20.0	Numerical arrays and operations
scipy	≥1.7.0	Sparse matrix solvers
matplotlib	≥3.3.0	Visualization and plotting
netCDF4	≥1.5.0	Scientific data storage
pandas	≥1.3.0	Data analysis and CSV export
tqdm	≥4.60.0	Progress bars
numba	≥0.53.0	JIT compilation and parallelization
imageio	≥2.9.0	GIF animation creation

Governing Equations

The solver implements the coupled reaction-diffusion system:

$$\frac{\partial C_{O_2}}{\partial t} = D_{O_2} \frac{\partial^2 C_{O_2}}{\partial x^2} - \frac{V_{max} C_{O_2}}{K_m + C_{O_2}} + D_{Mb} \frac{\partial^2 C_{Mb}}{\partial x^2}$$

where myoglobin concentration follows Hill equilibrium:

$$C_{Mb} = \frac{C_{O_2} \cdot P_{50}}{P_{50} + C_{O_2}}$$

Physical Parameters

Symbol	Description	Default Value	Unit
$C_{O_2}$	Oxygen concentration	-	mg/ml
$C_{Mb}$	Myoglobin-bound oxygen	-	mg/ml
$D_{O_2}$	Oxygen diffusion coefficient	$5.5 \times 10^{-7}$	cm²/s
$D_{Mb}$	Myoglobin diffusion coefficient	$3.0 \times 10^{-8}$	cm²/s
$V_{max}$	Maximum oxygen consumption rate	$2.0 \times 10^{-4}$	mg/(ml·s)
$K_m$	Michaelis constant	1.0	mg/ml
$P_{50}$	Half-saturation pressure	2.0	mg/ml
$L$	Tissue domain length	$1.0 \times 10^{-3}$	cm

Boundary Conditions

• Left boundary (x = 0): $C_{O_2}(0, t) = T_0$ (arterial oxygen level)
• Right boundary (x = L): $C_{O_2}(L, t) = T_s$ (venous oxygen level)
• Initial condition: $C_{O_2}(x, 0) = 0$ for $0 < x < L$

Numerical Methods

The solver implements six finite difference schemes:

1. FTCS (Forward-Time Central-Space)

• Explicit scheme: $C_i^{n+1} = C_i^n + d(C_{i+1}^n - 2C_i^n + C_{i-1}^n) - \Delta t \cdot R_i^n$
• Stability: Requires $d = \frac{D_{O_2} \Delta t}{\Delta x^2} < 0.5$
• Order: $O(\Delta t, \Delta x^2)$

2. DuFort-Frankel

• Explicit three-level scheme: $\frac{C_i^{n+1} - C_i^{n-1}}{2\Delta t} = D_{O_2} \frac{C_{i+1}^n - (C_i^{n+1} + C_i^{n-1}) + C_{i-1}^n}{\Delta x^2}$
• Stability: Unconditionally stable
• Order: $O(\Delta t^2, \Delta x^2, (\Delta t/\Delta x)^2)$

3. Crank-Nicolson

• Semi-implicit scheme: $\frac{C_i^{n+1} - C_i^n}{\Delta t} = \frac{D_{O_2}}{2} \left[\delta_x^2 C_i^{n+1} + \delta_x^2 C_i^n\right]$
• Stability: Unconditionally stable
• Order: $O(\Delta t^2, \Delta x^2)$

4. Laasonen (Fully Implicit)

• Implicit scheme: $C_i^{n+1} = C_i^n + d(C_{i+1}^{n+1} - 2C_i^{n+1} + C_{i-1}^{n+1})$
• Stability: Unconditionally stable
• Order: $O(\Delta t, \Delta x^2)$

5. ADI (Alternating Direction Implicit)

• Tridiagonal matrix solver with sparse LU decomposition
• Stability: Unconditionally stable
• Order: $O(\Delta t^2, \Delta x^2)$

6. RK-IMEX (Runge-Kutta Implicit-Explicit)

• Second-order IMEX scheme treating stiff diffusion implicitly
• Stability: L-stable for stiff terms
• Order: $O(\Delta t^2, \Delta x^2)$

Installation

Prerequisites

• Python 3.8 or higher
• pip package manager

Install from PyPI

The simplest way to install mangala-bulan is via pip:

pip install mangala-bulan

Development Installation

For development or to get the latest unreleased features:

git clone https://github.com/sandyherho/mangala-bulan.git
cd mangala-bulan
pip install -e .

Install with Poetry (for contributors)

git clone https://github.com/username/mangala-bulan.git
cd mangala-bulan
poetry install --with dev

Verify Installation

import mangala_bulan
print(mangala_bulan.__version__)
# Output: 0.0.1

Quick Start

1. Basic Usage

from mangala_bulan import OxygenDiffusionSolver

# Create solver with default parameters
solver = OxygenDiffusionSolver(nx=51, L=1.0e-3)

# Define configuration
config = {
    'method': 'crank',  # Crank-Nicolson method
    'T0': 70.0,        # Initial O2 concentration [mg/ml]
    'Ts': 10.0,        # Boundary O2 concentration [mg/ml]
    'koef': 5.5e-7,    # Diffusion coefficient [cm²/s]
    'dt': 1.0/300,     # Time step [s]
    'ts': 1.0,         # Total simulation time [s]
    'V_max': 2.0e-4,   # Max consumption rate [mg/(ml·s)]
    'K_m': 1.0,        # Michaelis constant [mg/ml]
    'D_mb': 3.0e-8,    # Myoglobin diffusion [cm²/s]
    'P50': 2.0,        # Half-saturation [mg/ml]
    'save_netcdf': True,
    'save_animation': True
}

# Run simulation
result = solver.solve(config)

# Access results
x_positions = result['x']  # Spatial coordinates [mm]
time_points = result['t']   # Time points [s]
oxygen_conc = result['T']   # O2 concentration [mg/ml]

2. Command Line Interface

Run a single scenario:

mangala-bulan configs/crank_scenario1.txt

Configuration Format

Configuration files use simple key-value pairs:

scenario_name = FTCS Method - Scenario 1
method = ftcs
T0 = 70.0          # Initial O2 concentration [mg/ml]
Ts = 10.0          # Boundary O2 concentration [mg/ml]
koef = 5.5e-7      # Diffusion coefficient [cm²/s]
L = 1.0e-3         # Domain length [cm]
ts = 1.0           # Simulation time [s]
dt = 1.0/2000      # Time step [s]
nx = 51            # Grid points
V_max = 2.0e-4     # Max consumption rate [mg/(ml·s)]
K_m = 1.0          # Michaelis constant [mg/ml]
D_mb = 3.0e-8      # Myoglobin diffusion [cm²/s]
P50 = 2.0          # Half-saturation [mg/ml]

Output Formats

NetCDF4 Files

Complete solution data with metadata:

• Dimensions: time × space
• Variables: oxygen_concentration(t,x), x(x), t(t)
• Attributes: All physical and numerical parameters

Animations

• GIF animations showing spatiotemporal evolution
• Customizable colormap, frame rate, and resolution

Performance Features

• Parallel Computing: Automatic CPU core detection with Numba JIT compilation
• Sparse Matrix Solvers: Efficient tridiagonal systems for implicit methods
• Memory Optimization: Streaming I/O for large datasets
• Stability Monitoring: Real-time detection and handling of numerical instabilities

Numerical Stability

Stability Criteria

For explicit schemes (FTCS): $$d = \frac{D_{O_2} \Delta t}{\Delta x^2} < 0.5$$

For the given default parameters:

• Critical time step: $\Delta t_{crit} = \frac{0.5 \Delta x^2}{D_{O_2}} \approx 3.64 \times 10^{-4}$ s
• Recommended: $\Delta t < 3 \times 10^{-4}$ s for FTCS

Project Structure

mangala-bulan/
├── src/mangala_bulan/
│   ├── core/              # Numerical solvers
│   │   └── solver.py       # Main PDE solver implementation
│   ├── io/                # Input/output handling
│   │   ├── config_manager.py
│   │   └── data_handler.py
│   ├── utils/             # Utilities
│   │   └── logger.py      # Simulation logging
│   └── visualization/     # Plotting and animation
│       └── animator.py
├── configs/               # Scenario configurations
│   ├── ftcs_*.txt
│   ├── crank_*.txt
│   └── ...
├── outputs/              # Simulation results
└── logs/                 # Computation logs

Mathematical Background

The system models oxygen transport in muscle tissue where:

1. Fick's Diffusion: $J = -D \nabla C$ governs molecular transport
2. Michaelis-Menten Kinetics: Models enzymatic oxygen consumption with saturation
3. Myoglobin Facilitation: Enhanced transport via reversible oxygen binding
4. Hill Equation: Describes cooperative oxygen binding to myoglobin

The dimensionless Damköhler number characterizes the reaction-diffusion balance: $$Da = \frac{V_{max} L^2}{D_{O_2} K_m}$$

Authors

• Sandy H. S. Herho
• Gandhi Napitupulu

License

This project is licensed under the MIT License - see the LICENSE file for details.

Citation

If you use mangala-bulan in your research, please cite:

Software Citation

@software{mangala_bulan_2024,
  title = {{\texttt{mangala-bulan}: 1D Idealized Oxygen Diffusion Solver}},
  author = {Herho, Sandy H. S. and Napitupulu, Gandhi},
  year = {2025},
  month = {12},
  version = {0.0.1},
  publisher = {PyPI},
  url = {https://github.com/sandyherho/mangala-bulan},
  license = {MIT}
}