Bayesian models for ODE models in SBML

This project implements Bayesian models using PyMC on top of ODE-based models encoded in the Systems Biology Markup Language (SBML).

Motivation

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.

Installation

libraries

Install graphviz library

sudo apt-get -y install graphviz graphviz-dev

virtual environment

Create a virtual environment and install the dependencies defined in the requirements.txt

mkvirtualenv parameter-variability --python=python3.10
(parameter-variability) pip install -r requirements.txt

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]

An example output of the model is provided here

Bayesian model

To generate the toy example, the two-compartment model is fed draws from an idealized random distribution for each parameter. These are called `true_thetas'. A forward simulation is then run to generate a run simulation for each theta.

After adding noise to the simulation(s), a Bayesian model fits the data and draws samples from a posterior distribution. The empirical distribution of these samples should contain the `true_thetas'.

Current modelled parameters:

• k: Absorption constant
• CL Clearance constant

To run the example Bayesian model execute the bayes_example.py script

(parameter-variability) python src/parameter_variability/bayes/bayes_example.py

Outputs

Plots of results for the analysis on the Gut compartment

Figure 1: Sampling random parameters from "true" distribution

Figure 2: Toy Data simulated using values from the true distribution

Figure 3: Graph representing the Bayesian Model

Figure 4: Trace Plot of the parameters sampled from the Bayesian model

Figure 5: Proposed simulations sampled from the Bayesian Model

License

• Source Code: LGPLv3
• Documentation: CC BY-SA 4.0

The parameter-variability source is released under both the GPL and LGPL licenses version 2 or later. You may choose which license you choose to use the software under.

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License or the GNU Lesser General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version.

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. See the GNU General Public License for more details.

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).

© 2023-2024 Antonio Alvarez and Matthias König