Repository for calibration, simulation, and validity assessment of a Hybrid Discrete-Continuum model of cancer growth with applications in 3D cancer cell cultures (paper).

• Code and data
  
• Prerequisites
  
• Introduction
  
• Structure of repository
  
• Instructions
  
  • Converting 3D cellular coordinates to spatial density
    
  • Test simulation on the continuum level
    
  • Saving the simulation output
    
  • Calibration on the continuum level
    
  • Visualization of parameter posterior distributions
    
  • Simulation of the Hybrid model
    
  • Spatial Analysis
    
• Mathematical model
  
• References
  
• Support or Contact
  
• Citing
  
• A note from the author
  

• The associated code can be found in the repository of HyMetaGrowth.

• The data associated with this study can be found on the FigShare repository.

• Details on the data processing and segmentation pipeline can be found here.

• CUDA version ≥ 10
• Π4U package
• NVidia GPU with memory ≥ 12 GB (otherwise the grid size of the simulations should be reduced)

This framework is related to the spatiotemporal modelling of cancer growth in 3 spatial dimensions and the study of relations between morphological patterns, and the biophysical mechanisms. The PDE model is constructed based on observations from 3D cell culture data of Triple-Negative Breast cancer cells cultured in Matrigel extracellular matrix (ECM). For the purpose of incorporating more details in the spatial organization and progression of the cancer cells, the model is hybridized (Discrete - Continuum or HDC) according to the techniques found in Anderson [1], [2] and Franssen et al. [3].

The model is calibrated on the continuum level. The validation is performed on the hybrid (HDC) level by performing spatial statistical analysis between experimental and simulation results.

Converts points with 3D coordinates to spatial density profiles using the adaptive kernel density estimation via diffusion method [4].

Here we calibrate the continuum model by performing Bayesian Inference using the Transitional Markov Chain Monte-Carlo (TMCMC) algorithm found in Π4U package. The Calibration_Continuum directory contains the following subdirectories.

• TMCMC: Contains the code of TMCMC algorithm.

• model: Contains the source code for the simulation of the continuum model. To compile the code, run make. The doall.sh script is called by the TMCMC code. To modify the total number of GPUs used for your study, open this script and change the last argument.

Here we simulate the hybrid model. The Hybrid directory contains the source code for the hybrid model. Further instructions can be found in the following chapters.

This directory contains Matlab and R scripts that perform spatial analysis between simulated and experimental data.

1. Go to the Points2Density directory, and open the Matlab script points2density.m. The script initially scales the space (default is 2500x2500x917 μm) to the approximate cell size (default is dx=15 μm). Then, the scaled space is interpolated to a 480x480x176 grid size that will be introduced to the simulator.

2. Specify the directory and common prefix of the files that the coordinated of the cells. The default directory and files can be found in the res_coord_scaled directory. You can run the script with the default arguments and it will produce the density profiles of the corresponding coordinates. The output can be found in the New_Density_double_precision.

3. Copy the output files into the Calibration_Continuum/IC to proceed with the next step.

1. Code compilation. Go to Calibration_Continuum/model. Then run make.

2. Model parameters. Include the model parameters in the params_test.txt file. Note that the model takes 5 parameters (see the mathematical formulation in the following chapters) + 1 parameter for the sampling of the experimental error (required for calibration). The simulation takes place by default in 486×486×176 spatial grid point that corresponds to 2.5×2.5×0.9 mm^3 space. The simulation time is set to 14. To change these parameters, feel free to modify the variables found in main.cu and simulate.cu.

3. Initial Conditions. The initial conditions (ICs) for the simulation are found in Calibration_Continuum/IC directory. The ICs can be the density profiles at day 0 of the experimental data or your favorite 3D matrix. The ICs should be given in binary format. In the directory of the ICs you can also include the experimental data for the calibration of the model. The ICs and experimental data files should have the same spatial dimensions as your simulation space. The experimental data that accompany this code correspond to days 0, 2, 5, 7, 9, 12, 14, with day 0 the IC.

4. Execution. Once you have set the ICs, the experimental data, and the parameters open the doall_test.sh file. This file executes 3 files; the main_parent file is responsible for loading and unloading the ICs and experimental data to the shared memory, and the main that executes the simulation. The main_parent file takes the following input format

./main_parent <start or stop> <directory of the ICs and experimental data> <common prefix of the ICs/data files>

The accepted filename format for the ICs and experimental data files is < common prefix >_D#.raw (for example AN_D0.raw for data at day 0, AN_D2.raw for data at day 2 etc).

Once the ICs and experimental data are loaded to the shared memory, the ./main file runs the simulation. The input arguments for the ./main file are the following

./main <parameter file such as params_test.txt> <common perefix of the ICs/data files> <number of GPUs (must be 1)>

the <common prefix of the ICs/data files> is the same as the argument in ./main_parent input. In this setup each simulation takes only one GPU, and this argument should be set to 1.

Once you have set the doall_test.sh you can run it by typing ./doall_test.sh in the terminal (make sure that you have execution priviledges before running it).

To enable saving of the simulation results open the Makefile and uncomment the the line SIMU_OPTION += -DSAVE_DATA. This saves simulation output (3D matrices) in vtk format. To alter the time-points where saving takes place go to the simulate.cu file.

1. TMCMC compilation. Go to the TMCMC directory and compile the code by running make. Then, copy the produced sample file from the TMCMC directory one directory above (ie Calibration_Continuum).

2. Optimizations. Go to Calibration_Continuum/model directory. In the Makefile comment the lines SIMU_OPTION += -DSAVE_DATA and SIMU_OPTION += -DNRMSE to make the simulation faster. Compile the code by running make. You can plot the resulted density in Paraview.

3. The doall.sh file will be called by the TMCMC algorithm. This file runs the simulation. Make sure that the ./main command (in the doall.sh) file has the correct arguments (see previous chapter). Also, make sure to set the correct number of GPUs that will be used in the calibration.

4. Go to Calibration_Continuum directory. To run the calibration in one GPU, simply type

   ./sample

   If you are in a cluster and you can use more than 1 GPUs, then you can run the calibration by typing mpirun -np (number of GPUs, e.g. 4) ./sample

Example of a SLURM submission script is also provided. Note that each simulation may take from 1 to 5 minutes in a V100 GPU by NVidia. Thus, the calibration may take a long time, depending on your resources and the parametrization.

1. Go to the Visualization directory

2. Open the plot_samples.R file (thanks to Π4U package) and modify the fname according the file you want to open and plot. Then run script in R. The files produced by the TMCMC that have the pattern curgen*.txt contain the posterior distributions (PDFs) of the candidate model parameters. The resulted plots are saved in the directory of the curgen*.txt files. The plot_data.R script, also, saves the best parameters found by the TMCMC in a best_params_<name of TMCMC file>.txt file.

1. Go to the Hybrid directory.

2. Compile the code by running make.

3. Run the simulator by typing

   ./main <path/to/parameter_file> <common prefix of data (ICs/experiments)>

   The output of the simulation are the files with the coordinates of the cells and the files containing the number of cells in the corresponding time-points (the default is day 0, 1, 2, ... 14).

1. To plot the centroids of both experiments and simulations use the plot_centroids.m Matlab script. The plot_centroids.m script splits the time-points of the coordinates of the simulated cells and saves them in the scaled_coord directory.

2. The calcEnv.R script calculates the K function given the coordinates of the experimental and simulation results. The results can be ploted using the Kenv_plot_all.m script.

3. To estimate the inter-cellular and nearest-neighbour distance distributions use the dist_all_s12.m script.

The Keller-Segel model (originally Keller-Segel-Patlak) is a system of advection-reaction-diffusion equations. Application of this model can be found in Biology, especially cancer, and bacterial growth. The mathematical expression of the model we use is the following:

Continuum	Discrete
ut = DuΔu + su(1-u) - χ∇⋅(u(1-u)∇f) ft = DfΔf + rfu(1-u) ∇u⋅n⃗ = ∇f⋅n⃗ = 0	un+1i,j,k = uni,j,k P0 + uni+1,j,kP1 + uni-1,j,k P2 + uni,j+1,k P3 + uni,j-1,k P4 + uni,j,k+1 P5 + uni,j,k-1 P6

[1] Anderson, A. R. A. (2005). A hybrid mathematical model of solid tumour invasion: the importance of cell adhesion. Mathematical Medicine and Biology: A Journal of the IMA, 22(2), 163–186. https://dx.doi.org/10.1093/imammb/dqi005

[2] Anderson, A. R. A. (2003). A Hybrid Discrete-continuum Technique for Individual-based Migration Models. In W. Alt, M. Chaplain, M. Griebel, & J. Lenz (Eds.), Polymer and Cell Dynamics: Multiscale Modeling and Numerical Simulations (pp. 251–259). Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8043-5_19

[3] Franssen, L. C., Lorenzi, T., Burgess, A. E. F., & Chaplain, M. A. J. (2019). A Mathematical Framework for Modelling the Metastatic Spread of Cancer. Bulletin of Mathematical Biology, 81(6), 1965–2010. https://doi.org/10.1007/s11538-019-00597-x

[4] Botev, Z. I., Grotowski, J. F., Kroese, D. P., & others. (2010). Kernel density estimation via diffusion. The Annals of Statistics, 38(5), 2916–2957.

For any question feel free to contact me at: nikolaos.dimitriou [at] mail.mcgill.ca

If you intend to use this code in your publications, please cite:

N. M. Dimitriou, S. Flores-Torres, J. M. Kinsella and G. D. Mitsis, "Quantifying the Morphology and Mechanisms of Cancer Progression in 3D in-vitro environments: Integrating Experiments and Multiscale Models," in IEEE Transactions on Biomedical Engineering, 2022, doi: 10.1109/TBME.2022.3216231.

If you made it until here it means that you either seek for more help or you managed to run this code. For the latter case, Congratulations!! In case you didn't make it, I want to let you know that this code was not designed with the intention of portability and user friendliness, but mostly to get the job done with the specific data of the aforementioned paper. However, if you are interested in working on this code, I will be very happy to hear from you. :)