Pores Bridging

Code to perform the model-based data analysis with the Porous-Fisher to explore pore bridging by osteoblastic cells in 3D printed scaffolds. This repository is supplementary material for the preprint "Model-based data analysis of osteoblastic tissue growth in shallow 3D printed scaffolds" available on bioRxiv.

The majority of the code contains the Julia module Pores (Module/Pores.jl) that exports the experimental data and functions used to perform the analysis. The Results folder contains code that performs the analysis displayed in the main document (the Results/Supplementary folder does the same for the supplementary material document). The Figures folder contains scripts to produce and export a .pdf for each figure in the document.

Installation

Ensure Julia is installed (see Required software) and download the repository, in its entirety, to your machine. You should then run Install_Required_Packages.jl from the Module folder.

Use the following commands to add the module to your current search path, and load the module:

  push!(LOAD_PATH,"/path/to/module/folder/")  # Add to load path
  using Pores                                 # Load module

If using Windows, ensure to escape the backslashes in the path: C:\\path\\to\\module\\folder, or use Unix style forward slashes.

Module

The module Pores provides access to the following functions, each thoroughly documented.

SolvePDE() to solve the Porous-Fisher model numerically using the method of lines. Output is a callable function, i.e. u(t) that can be called to give the full, spatial, solution at any time t as a 2D array.
SummaryStatistics() to summarise the numerical, spatial, solution u(t) with four summary statistics (see main and supplementary documents).
Maximise() perform maximisation on a function (i.e., maximum likelihood estimation).
FitModel() to fit a simple model to data (i.e., the quadratic noise function).
Profile() to profile a log-likelihood function.
LogLikelihood() to calculate the log-likelihood for a given set of parameters θ, data Data, and pore size L.
GetΘ() to convert partitioned parameter space (θ,β) into full parameter space Θ. Here, θ might be unknown parameters (i.e., θ = (D,λ,K,u₀)), and β known, fixed, parameters.
ConfidenceInterval() to compute an approximate confidence interval from a profile likelihood.

The module also exports the following variables:

Data a DataFrame containing the full experimental data set (this is also available at Module/Data.csv).
P an array of all pore sizes.
T an array of all observation times.
S a set of all summary statistics.

Results

All results in the main (and supporting material) document can be obtained by running the corresponding script in the Results folder (or Results/Supplementary folder). The output of each script is stored in a JLD2 file in the /Saved directories (which are called when producing figures).

In general, each JLD2 file of results contains two variables. For the MLE_Individual results:

MLE_Individual contains the result.
MLE_Individual_Settings contains information about the parameters used to produce the result, and the time used to produce the result.

The Options.jl file contains global options used to produce all results (i.e., parameter bounds, maximisation tolerances, etc).

Before running scripts, ensure the working directory of the Julia session, pwd() is set to the results folder:

  cd("/path/to/module/folder")

Approximate runtimes for the full computation of the results for each model (using a 3.7GHz Quad-Code i7 desktop running Windows 10), and figures produced, are given below.

Script	Description	Runtime
`MLE.jl`	Compute individual and combined MLEs	~24 hours
`Profiles.jl`	Compute individual and combined profile likelihoods	~9.5 hours
`Profiles.jl`	Compute bivariate profiles	~16 hours

Note that the code uses the .Threads module to run the analysis simultaneously on CPU threads. Use the nthreads() command (run using .Threads first) to verify the number of threads in the JULIA_NUM_THREADS environment variable. For more information on setting the number of threads in Julia visit julialang.org.

Figures

Each figure in the main document (except for purely experimental or graphical figures) can be produced from the corresponding script in the Figures folder (and Figures/Supplementary), which uses the PyPlot package (which calls matplotlib). The Preferences.jl file contains graphical tweaks to obtain the figure style used in the main document.

Required software