paper.md

title	tags	authors	affiliations	bibliography
CirculatorySystemModels.jl - A ModelingToolkit Library for 0D-Lumped-Parameter Models of the Cardiovascular Circulation	Julia ModelingToolkit cardio-vascular lumped parameter circulation acausal patient-specific	name orcid equal-contrib affiliation Torsten Schenkel 0000-0001-5560-1872 true 1, 2 name orcid equal-contrib affiliation Harry Saxton 0000-0001-7433-6154 true 2	name index Department of Engineering and Mathematics, Sheffield Hallam University, UK 1 name index Materials and Engineering Research Institute MERI, Sheffield Hallam University, UK 2	paper.bib

Summary

Within the realm of circulatory mechanics, lumped parameter (0D) modelling [@shi2011review] offers the unique ability to examine both cardiac function and global hemodynamics within the context of a single model. Due to the interconnected nature of the cardiovascular system, being able to quantify both is crucial for the effective prediction and diagnosis of cardiovascular diseases. Lumped parameter modelling derives one of its main strengths from the minimal computation time required to solve ODEs and algebraic equations. Furthermore, the relatively simple structure of the model allows most personalized simulations to be automated. Meaning the ability to embed these lumped parameter models into a clinical workflow could one day become trivial [@bozkurt2022patient; @holmes2018clinical].

CirculatorySystemModels.jl is a Julia [@bezanson2017julia] package, built on the acausal modelling framework provided by ModelingToolkit.jl [@ma2021modelingtoolkit], containing all the common elements plus more needed for effective and realistic lumped parameter modelling. Currently CirculatorySystemModels.jl supports common elements such as a capacitor, resistor, inductance and diodes [@westerhof2010snapshots], which act as simple valve functions. We also make extensions to the common elements to include constant compliance chambers, non-linear and Poiseuille resistances [@pfitzner1976poiseuille]. Plus the Double-Hill, and Shi activation functions which are used as the cardiac driving chamber elastance's [@stergiopulos1996determinants; @korakianitis2006numerical]. We also include non-linear valve functions from Shi, and Mynard [@korakianitis2006numerical; @mynard2012simple]. Alongisde individual components we also have created a collection of sub compartments including, full circulatory, systemic, pulmonary and heart models. We then also break down these full systems into collections of elements such as the famous Windkessel models [@westerhof2009arterial] to give the user full control over their modelling.

Users can easily add new elements to CirculatorySystemModels.jl using ModelingToolkit.jl functions.

Statement of need

Lumped parameter modelling has become an essential part of contributing strongly to our understanding of circulatory physiology. Lumped parameter models have been used in many different contexts across cardiovascular medicine such as aortic valve stenosis, neonatal physiology and detection of coronary artery disease [@laubscher2022dynamic; @sepulveda2022openmodelica; @dash2022non]. There already exists some popular packages within both Simulink and Modelica such as the "Cardiovascular library for lumped-parameter modeling" [@rosalia2021object] and "Physiolibary" [@matejak2014physiolibrary] respectively. These languages operate a block orientated "drag and drop" approach with Modelica been the common choice due to it's acausal modelling approach [@schweiger2020modeling]. Other languages, based on XML, exist for lumped parameter modelling such as CellML [@cuellar2003overview] and SBML [@hucka2003systems], while these are great for exchanging models they are often difficult to implement and model analysis is limited. A common theme within all current lumped parameter modelling software is the systems inability to deal with complex event handling and non-linear components. Being based on ModelingToolkit.jl, CirculatorySystemModels.jl overcomes these limitations by leveraging the wider SciML framework.

CirculatorySystemModels.jl provides the Julia community with a quick and effective way to perform lumped parameter modelling, being the first within the field to leverage both multiple dispatch and JIT compilation. As a result of Julia's architecture, as the complexity of the model increases the model analysis time does not become unreasonable.

Other packages exist which allow users to import models from other frameworks, CellMLToolkit.jl [@CellMLToolKit] and OpenModelica.jl [@tinnerholm2022modular]. CirculatorySystemModels.jl goes beyond these by providing a lumped parameter modelling library with seamless integration with the SciML framework [@Dixit2022; @rackauckas2020universal; @rackauckas2017differentialequations] which allows for extensive and efficient model analysis. Since both the modelling library and the framework it is built on are pure Julia, new components can be developed in a transparent and consistent manner.

Using Julia, CirculatorySystemModels.jl models compute significantly faster than models implemented in Matlab or Python, and at the same speed as specialised C-code (\autoref{tbl:benchmarks}). This allows the models to run in real-time, and opens the possibility of global parameter optimisation, global sensitivity analysis.

The modular, acausal approach also allows quick and straightforward changes to the topology of the model, which can be automated using meta-programming techniques.

Example

Validation and benchmarking was performed on a full, 4-chamber, model of the circulation system proposed by [@korakianitis2006numerical] (\autoref{fig-shi-diagram}). This model was previously implemented in CellML [@Shi2018], which makes it an ideal candidate for validation of the new modeling library^[Note that CellML does not allow the callbacks which are required for the extended valve model, so only the simplified model can be compared. The CirculatorySystemModels.jl implementation of [@korakianitis2006numerical] includes the extended model as well.].

Model results from CirculatorySystemModels.jl model (\autoref{fig:shi-results}) are a perfect match for the CellML model. The CellML model was run in three versions: (1) imported into ModelingToolkit.jl using CellMLToolKit.jl, (2) Matlab code exported from CellML, (3) Python/SciPy code exported from CellML. Speedup against Matlab and Python is 2 and 3 orders of magnitude, respectively (\autoref{tbl:benchmarks}).

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CirculatorySystemModels.jl	CellMLToolkit.jl	Matlab (ode45)	Python (scipy.solve)
1x	1.6x	272x	963x
: Simulation time comparison for a single run of [@korakianitis2006numerical]. CirculatorySystemModels.jl model was implemented from scratch. CellML model was imported into ModelingToolkit.jl using CellMLToolkit.jl, Matlab and Python models were created from the CellML code and downloaded from the CellML Model Repository. \label{tbl:benchmarks}

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Acknowledgements

Harry Saxton is supported by a Sheffield Hallam University Graduate Teaching Assistant PhD scholarship.

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