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This app demonstrates the evaluation of the effects of mechanical interactions between liver parenchyma (normal liver tissue) and tumor cells on metastatic progression and growth.
- Liver parenchyma agents: are colored by their relative displacement (normalized by their maximum tolerated displacement).
- 0 ≤ relative displacement ≤ 5% max
- 5% < relative displacement ≤ 25% max
- 25% < relative displacement ≤ 50% max
- 50% < relative displacement ≤ 75% max
- 75% < relative displacement ≤ 100% max
- Apoptotic (due to exceeding maximum displacement)
- Tumor cells: are colored by their cell cycle status:
- Ki67- (quiescent)
- Ki67+ premitotic (entered cycle and prepraing for division)
- Ki67+ postmitotic (finishing cycle after division)
- Necrotic (due to insufficient oxygen)
- Cycle arrested in Ki67- state due to exceeding maximum pressure
In this model , We simulate a small tumor growing in liver parenchyma (maximum size is 1 cm × 1 cm). The liver tissue is elastic-plastic: Elastic: on short times, when it’s deformed it elastically tries to return to its rest position and imparts forces on nearby tumor cells; Plastic: on long times, the resting position of these tissue elements relaxes back to the current position.
We found that:
(1) In high-flow / well-perfused liver tissues, growth substrate limitations (oxygen in our case) alone are an insufficient growth control to prevent non-physical overlap of cells. Mechanobiologic feedbacks are necessary. These are modeled as a pressure that can reduce or completely shut down cycle entry.
Try the default one: rE = 0.05 min-1, rP = 0.0005 min-1, and a maximum parenchyma displacement of 0.75 μm to see tumor growing in the liver tissue, and experiencing both mechanobiologic and insufficient oxygen feedback, eventually apoptosis or necrosis.
(2) In some tissue regimes, this can arrest the growth of micrometastases.
Try rE = 0.2 min-1, rP = 0.0005 min-1, and a maximum parenchyma displacement of 3 μm to see this behavior.
(3) In other tissue regimes, tumors will develop invasive fingers into the surrounding tissue along mechanical weaknesses.
Try rE = 0.05 min-1, rP = 0.0005 min-1, and a maximum parenchyma displacement of 3 μm to see this behavior.
We used agent-based models to simulation the cell-cell interactions, where the liver parenchyma are stress-based apoptosis and no birth, while tumor cells are pressure-regulated cycling and stochastic apoptosis.
In details, if parenchymal agent’s magnitude of displacement exceeds the maximum tolerated deformation, the parenchyma would enter apoptosis immediately. For the tumor agent, its cycling rate is controlled by the pressure between them and surrounding liver tissue, when the pressue is greater than the threshold value (it is 1 in our default), the birth rate would be set to 0.
All chemical signals move by chemical diffusion in the simulated environment, using BioFVM  to solve the reaction-diffusion equations. A key property is the diffusion length scale. Diffusion (with a diffusion coefficient D) helps spread a signal over large distances, while decay (and uptake) (with coefficient λ) eliminates the signal to slow its spread. These effects compete to determine the characteristic distance Lthat a chemical signal travels.
In our model, the only substrate is oxygen. We used BioFVM  to simulate diffusive transport. By investigating a poroviscoelastic (PVE) model, we constructed a quasi-steady single-lobule model, and extended this for the far-field oxygen solution. See  for details.
In details, Outside tumors: it's advection-dominated transport. We useed quasi-steady analytic solution for O2 transport; Inside tumors: it's diffusion-dominated transport. We used BioFVM-based numeric solution for O2 transport.
Key features in the agent-based model:
The agent-based model was created using PhysiCell  as follows:
Birth and death:
Cells can proliferate (divide into two daughter cells of half size), with division rates regulated by cell rules. In our model, cells continuously work to grow towards a "target" mature volume.
Note that birth and death are stochastic events for each cell agent: if such a process occurs at rate r, then between now (t) and soon (t + Δt), the probability of the event occurring for that agent is rΔt.
In our model, liver parenchyma have no birth rate, and their death rate is controlled by the displacement as explained by section of Modeling approach. For the tumor cells, they have normal apoptosis rate, while the cycling rate is regulated by the pressue and oxygen function.
Secretion and uptake:
Cells can secrete chemical factors, or they can remove them (i.e., consume or uptake). This can contribute to gradients in chemical factors, and it's a key method of communication between cells. Moreover, cells can "sample" the chemical state of their surrounding environment. Secreting is sending a signal. Uptaking and sampling is receiving a signal.
In our model, tumor cells uptake oxygen, which is particularly important in the center area of big tumor. Because, if the tumor radius is greater than diffusion length scale the tumor cells in the center of tumor would have limited oxygen, as a result, their cycling rate would decrease to zero and enter necrosis state.
Cell agents can stick to one another within a prescribed interaction distance (some multiple of their radius), and they can exert a pushing force on neighbors. PhysiCell  uses potential functions to implement these simple mechanics. Notably, PhysiCell is an off-lattice model, meaning that cells can have variable sizes, and can move freely without grid artifacts.
They aren't required to move some prescribed number of spatial steps. They also can divide without checking for an open neighbor "site". Instead, they can divide and push their neighbors out of the way.
In the model, two types of cell contacted with each other and produce force/pressue, which regulated their birth rate or death rate. See the details in the following Liver parenchyma cells and Tumor cells.
Liver parenchyma cells (Bright orange)
Each parenchymal agent follows regular adhesion-repulsion mechanics (with a position xi). In addition, mechanical interactions are modeled with underlying ECM. Each agent i is attached to the ECM at xi,ECM, and experiences an elastic force. Let di = xi,ECM − xi, and let di = ∥di∥ − Ri (Ri is cell’s equivalent radius) denote the elastic deformation; let rE and rP denote the elastic rate and plastic relaxation rate respectively, both the elastic and plastic reorganization show as following:
Liver parenchyma cells are originally homeostatic. If parenchymal agent’s magnitude of displacement exceeds the maximum tolerated deformation (dmax), the parenchyma would enter apoptosis immediately.
The built-in " Ki67_advanced " cell cycle model is used in our model, with the default models of apoptosis and necrosis. Thus, each cell agent can be transited from the Ki67- phase to the pre-mitotic Ki67+ cycle phase, Apoptotic, or Necrotic. For the transition rate, the regular "oxygen-dependent" cycle entry rate is scaled by the pressure. See  for more details.
Apoptosis is modeled as microenvironment-independent, with a constant apoptotic death rate dA and a stochastic duration with mean TA. Necrosis is modeled as oxygen-dependent: cells enter the necrotic state if σ < σN (necrosis threshold), and then progress as described in PhysiCell . We also added a rule of pressure-regulated proliferation here, so the cycling rate is:
The first part is transition rate with oxygen levels as described in . The normalized cell pressure in conﬂuent tissue (no gaps in the tissue) is 0.5 (2D). We deﬁned as p2 and p1 as 1.0 and 0 respectively here, which means if pressure the tumor cell experiences is greater than 1.0, we set the tumor cell proliferation rate as zero, also labeled as yellow in the simulation.
- Config Basics tab: input parameters common to all models (e.g., domain grid, simulation time, choice/frequency of outputs)
- Microenvironment tab: microenvironment parameters that are model-specific
- User Params tab: user parameters that are model-specific
- Out: Plots tab: output display of cells and substrates
- Animate tab: generate an animation of cells
Edit the domain size and simulation length in the Run button to start the simulation. (After the run starts, the the Run button is replaced with a Cancel button.)tab. Edit diffusion parameters in the tab, and model-specific parameters in the tab. Once you are satisfied with your settings, click the
As the simulation runs, an "Output" widget can be clicked/expanded to reveal the progress (text) of the simulation. Model outputs can be visualized in thetab while the simulation is running. The "# cell frames" will be dynamically updated as the simulation progresses. Note that the simulation images (SVG files) and raw output data (in MultiCellDS format) can be downloaded in this tab once the simulation is done.
Once the simulation is done running, use thetab to create a movie. Please be patient: it may take a few minutes for the script to finish running before the movie is embedded in the tab. You can download the generated movie once it's done.
About the software:
This model  and cloud-hosted demo are part of the education and outreach for the IU Engineered nanoBIO Node. The models are built using PhysiCell: a C++ framework for multicellular systems biology  for the core simulation engine and xml2jupyter  to create the graphical user interface (GUI).
- A. Ghaffarizadeh, S.H. Friedman, and P. Macklin. BioFVM: an efficient, parallelized diffusive transport solver for 3-D biological simulations. Bioinformatics 32(8):1256-8, 2016. DOI: 10.1093/bioinformatics/btv730.
- A. Ghaffarizadeh, R. Heiland, S.H. Friedman, S.M. Mumenthaler, and P. Macklin. PhysiCell: an open source physics-based cell simulator for 3-D multicellular systems. PLoS Comput. Biol. 14(2):e1005991, 2018. DOI: 10.1371/journal.pcbi.1005991.
- R. Heiland, D. Mishler, T. Zhang, E. Bower, and P. Macklin. xml2jupyter: Mapping parameters between XML and Jupyter widgets. Journal of Open Source Software 4(39):1408, 2019. DOI: 10.21105/joss.01408.
- Y. Wang, E. Brodin, K. Nishii, H.B. Frieboes, S. Mumenthaler, J.L. Sparks, and P. Macklin. Impact of tumor-parenchyma biomechanics on liver metastatic progression: a multi-model approach. bioRxiv preprint: 10.1101/2020.05.04.074989. http://doi.org/10.1101/2020.05.04.074989
This software is powered by PhysiCell, a powerful simulation tool that combines multi-substrate diffusive transport and off-lattice cell models. PhysiCell is BSD-licensed, and available at:
- GitHub releases: https://github.com/MathCancer/PhysiCell/releases
- SourceForge downloads: https://sourceforge.net/projects/physicell/
It is a C++, cross-platform code with minimal software dependencies. It has been tested and deployed in Linux, BSD, OSX, Windows, and other environments, using the standard g++ compiler.
The Jupyter-based GUI was auto-generated by xml2jupyter, a technique to create graphical user interfaces for command-line scientific applications.
To learn more about our work, please visit MathCancer.org.
Randy Heiland, Research Associate, Intelligent Systems Engineering, Indiana University.
Paul Macklin, Ph.D. , Associate Professor, Intelligent Systems Engineering, Indiana University
We appreciate both Randy and Paul's support in debug source code and test the GUI
• National Science Foundation (Fox, 1720625)
• Breast Cancer Research Foundation / Jayne Koskinas Ted Giovanis Foundation for Health and Policy (Ewald, Gilkes, Macklin)
• National Cancer Institute (Agus, Atala, Soker, 1R01CA180149)
• National Cancer Institute (Finley, Macklin, Mumenthaler 1U01CA232137)
Cite this work
Researchers should cite this work as follows:
Y. Wang, E. Brodin, K. Nishii, H.B. Frieboes, S. Mumenthaler, J.L. Sparks, and P. Macklin. Impact of tumor-parenchyma biomechanics on liver metastatic progression: a multi-model approach. bioRxiv preprint: 10.1101/2020.05.04.074989. http://doi.org/10.1101/2020.05.04.074989