Windkessel model identification

Tools and example data to identify lumped-parameter models of cardiac afterloads, specifically 2-element Windkessel, 3-element Windkessel, and parallel 4-element Windkessel models from measured pressure and flow. The identification is done with Newton's method using forward-mode automatic differentiation in Julia.

The data and methods presented here were used to generate the results presented in the article "Identification of cardiac afterload dynamics from data".

The identification code uses ControlSystems.jl, Optim.jl, LineSearches.jl, ForwardDiff.jl, GenericLinearAlgebra.jl, and Distributions.jl. Data handling, additional analysis, and plotting is done using CSV.jl, DataFrames.jl, Plots.jl, Plotly.jl, StatsPlots.jl, FFTW.jl, and ToeplitzMatrices.jl. Thank you to the developers behind those packages.

Identification tools

oeid.jl contains the functions used to identify Windkessel model parameters (with user-facing functions identify for solving with Newton's Method, identifyglobal for solving with Nelder-Mead). The models are defined by functions G2, G3, and G for 2, 3, and 4 element Windkessel models, respectively. Estimates of the models' initial state condition are solved using the function solvex0, which leverages the periodicity of the input signals. expm.jl is used by oeid.jl for natively computing matrix exponentials.

Data and identification code

Aortic pressure and flow measurements are provided from three sources. In addition to oeid.jl, idanalysis.jl provides multiple initialization as well as analysis and plotting functions.

stergiopulos1999_data/ contains previously published human data, from Figure 4A type A beat of Total Arterial Inertance as the Fourth Element of the Windkessel Model, Stergiopulos et al. (1999), digitized using Webplotdigitizer v4.4. Pressure in mmHg, flow in mL/s.

invivo_data/ contains data measured in vivo in a 70kg pig. Pressure in mmHg, flow in L/min.

exvivo_data/ contains data measured with a 70kg pig heart beating ex vivo against a synthetic afterload. Pressure in mmHg, flow in L/min.

Windkessel model parameter identification for the three data sources is done in id_stergiopulos1999total.jl, id_invivo.jl, and id_exvivo.jl.

Persistance of excitation of the three input signals is analysed in spectrum.jl.

Getting started

1. Download Julia.
2. Install packages:
   1. Open Julia REPL, e.g. using the command julia in terminal,
   2. Enter the Pkg REPL by pressing ]
   3. add packages using command add DataFrames Dates Plots StatsPlots Distributions LinearAlgebra GenericLinearAlgebra ControlSystems Optim LineSearches ForwardDiff Printf CSV FFTW ToeplitzMatrices
3. Open and run one of the id_*.jl files line-by-line in a text editor with Julia extension (e.g. VS Code, or Atom with Juno)

Note that this code was tested with the following package versions:

  [336ed68f] CSV v0.8.3
  [a6e380b2] ControlSystems v0.9.0
  [a93c6f00] DataFrames v0.22.5
  [31c24e10] Distributions v0.24.12
  [7a1cc6ca] FFTW v1.3.2
  [f6369f11] ForwardDiff v0.10.16
  [14197337] GenericLinearAlgebra v0.2.4
  [d3d80556] LineSearches v7.1.1
  [429524aa] Optim v1.2.4
  [91a5bcdd] Plots v1.10.4
  [f3b207a7] StatsPlots v0.14.19
  [c751599d] ToeplitzMatrices v0.6.3
  [37e2e46d] LinearAlgebra