Bayesian optimization workflow for ODE models in SBML

Within this thesis a reproducible parameter optimization approach applicable to a range of pharmacokinetic (PK) computational models was developed. The worflow was applied to three PK models of increasing complexity, from simple compartmental models to physiologically-based pharmacokinetic (PBPK) models. The influence of key hyperparameters consisting of number of samples, timepoints, coefficient of variation and priors was studied on the three models.

Systems Biology Markup Language (SBML) provides an intuitive and reproducible way to define ordinary differential equation (ODE) models in systems biology and systems medicine. Here we outline an attempt to build a Bayesian framework to quantify the uncertainty of estimates associated with physiologically based pharmacokinetic (PBPK) models encoded in SBML.

For documentation see Bayesian optimization workflow for ODE models in SBML [Alvarez2026].

How to cite

To cite the software repository

Alvarez, A. & König, M. (2026). Bayesian optimization workflow for ODE models in SBML. Zenodo. https://doi.org/10.5281/zenodo.19695915

To cite the documentation

Alvarez, A. (2026). Parameter Uncertainty in the Optimization of Pharmacokinetic Models: A Reproducible Bayesian Approach

License

• Source Code: MIT
• Documentation: CC BY-SA 4.0
• Models: CC BY-SA 4.0

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Installation

Dependencies

sudo apt install -y build-essential cmake libhdf5-serial-dev

Running

Create a virtual environment with uv(https://docs.astral.sh/uv/getting-started/installation/)

uv sync

Development

For development setup via the following which installs the development dependencies and the pre-commit.

# install core dependencies
uv sync

# install dev dependencies
uv pip install -r pyproject.toml --extra dev
uv tool install tox --with tox-uv

# setup pre-commit
uv pip install pre-commit
pre-commit install
pre-commit run

Testing with tox

Run single tox target

tox r -e py314

Run all tests in parallel

tox r

Example

ODE model

As an example PBPK model (see figure below), a simple PK model is implemented consisting of three compartments, gut, central and peripheral. The substance y can be transferred from the gut to the central compartment via absorption. The substance y can be distributed in the peripheral compartment via R1 or return from the peripheral to the central compartment via R2. Substance 'y' is removed from the central compartment by clearance.

The SBML of the model is available from simple_pk.xml.

The resulting ODEs of the model are

time: [min]
substance: [mmol]
extent: [mmol]
volume: [l]
area: [m^2]
length: [m]

# Parameters `p`
CL = 1.0  # [l/min]
Q = 1.0  # [l/min]
Vcent = 1.0  # [l]
Vgut = 1.0  # [l]
Vperi = 1.0  # [l]
k = 1.0  # [l/min]

# Initial conditions `x0`
y_cent = 0.0  # [mmol/l] Vcent
y_gut = 1.0  # [mmol/l] Vgut
y_peri = 0.0  # [mmol/l] Vperi

# ODE system
# y
ABSORPTION = k * y_gut  # [mmol/min]
CLEARANCE = CL * y_cent  # [mmol/min]
R1 = Q * y_cent  # [mmol/min]
R2 = Q * y_peri  # [mmol/min]

# odes
d y_cent/dt = (ABSORPTION / Vcent - CLEARANCE / Vcent - R1 / Vcent) + R2 / Vcent  # [mmol/l/min]
d y_gut/dt = -ABSORPTION / Vgut  # [mmol/l/min]
d y_peri/dt = R1 / Vperi - R2 / Vperi  # [mmol/l/min]

Funding

Matthias König is supported by the Federal Ministry of Education and Research (BMBF, Germany) within the research network Systems Medicine of the Liver (LiSyM, grant number 031L0054) and by the German Research Foundation (DFG) within the Research Unit Programme FOR 5151 QuaLiPerF (Quantifying Liver Perfusion-Function Relationship in Complex Resection - A Systems Medicine Approach)" by grant number 436883643 and by grant number 465194077 (Priority Programme SPP 2311, Subproject SimLivA).

© 2023-2026 Antonio Alvarez and Matthias König