Mathematical modeling activities approach the fields of medical and toxicological sciences increasingly from different communities, in particular systems biology, biotechnology, biomedical and biomechanical engineering, control engineering, mathematics, computer sciences and physics. The current systems approaches in the biological and medical sciences span from understanding specific mechanisms at a certain level of organization, often classified as basic research, to the holistic approach combining aspects from multiple levels addressing specific problems in complex diseases, often classified as applied research. The ultimate aim is to contribute and eventually improve patient health by including computational models in the patient management, as e.g. diagnosis, treatment, prognosis of diseases.

The reasons why a systems-approach is becoming increasingly popular are many-fold: (1) the aim of quantification in biology-related disciplines, (2) the complexity of the multi-cellular systems that requires economising experiments instead of performing every possible experiment, (3) the increasing opportunity of computational predictions reducing the expensive wet-leb experiments by guiding their designs, (4) the limitation in performing experiments with patients calling for extrapolation strategies from in vitro to human, and to computational models integrating data from numerous sources (in vitro experiments, animal experiments) to extrapolate to the human situation.

In the last decades the focus was mainly on molecular “omics” data. Now the focus is shifting increasingly to the whole cell, tissue and whole organ level. Agent-based models represent each individual cell as “agent” are perfectly suited to integrate the intra-cellular “omics” data and models as they permit a “direct” representation of the functional modules “cells” and of how these fit into the organ micro-architecture. The cells can grow and divide and change their properties according to internal state changes, or die. Two types of single-cell agent-based models are integrated in our software TiSim.

Our cell models

Our novel “Deformable Cell Model” (DCM) meshes the cell surface and solves the force balance equations for each node. These forces have frictional, elastic, and stochastic components. The mesh spans the cell membrane/cortex and also -if needed- internal structures representing the internal cytoskeleton and nucleus of the cell. Parameterization of DCM parameters can be accompished by direct comparison with optical stretcher, pipette or Atomic Force Microscopy experiments. Therefore, DCM permits highly realistic simulations of cell shape and cell mechanical stresses. The model can also be combined with PDE solvers to address diffusion and advection transport. DCM can also be combined in hybrid approaches with the center-based model (CBM) to significantly reduce computation times.

The Deformable Cell model: Concept, Forces, and experiments to calibrate the model. See this paper.
Example: tumor spheroid growth
Tumor growth simulation with DCM, starting from one cell growing up to 600 cells.
Example: monolayer growth
Monolayer growth simulation with DCM, starting from a few cells growing up to 500 cells. Coloring is according to mechanical pressure.

The older model types that we use are so-called “center-based models” (CBMs). In this model type, cells are approximated as largely spherical-or dumb-bell-shaped homogeneous, isotropic elastic sticky objects. Cell movement is computed from a stochastic equation of motion for each cell, whereby cell-cell forces are modelled as forces between cell centers, and cells are parametrized by physical and biological measurable quantities. Cell micro-motility is captured by an active stochastic force. Simulations can contain up to 500 000 - 1 000 000 cells.

The Center based model used to simulate tumor growth under mechanica stress.

Integrated approach

Our cell models can be coupled to both intracellular pathway models (e.g. by using SBML) for every cell and flow/diffusion solvers (e.g. FreeFem). We also conduct research for being able to couple and start the simulations with an tissue architecture that is directly inferred from experimental imgages: “Image2Modeling”. These aspects result in an integrated approach which is mandatory to realistically simulate multicellular response.

Overview of the Tisim software: Our models are physics-based using two cell models (low resolution and high resolution). Tissue architecture can be read in direclty from images to the models. Each cell can multiple contain intracellular modules. Flow and diffusion around the cell can be taken into account.
Simulation of a growing spheroid (from one cell) consuming a chemical compound. The box is which the spheroid is contained has one layer through which the compound can diffuse. The coloring is according to the concentration.


Micro carrier growth (DCM):

DCM Simulation of Microcarrier growth. A few cells are initially distributed on the carrier and proliferate until confluency. The micro carrier is a triangulation of a sphere.


Tumor growth in a liver lobule

A hybrid center-based/deformable cell model (CBM/DCM) simulation of a growing tumor in a liver lobule. The healthy cells and blood vessel network are represented by the CBM, the tumor is represented by the DCM (for better detail, play the video in full-screen):


DCM in lobule blowing up to reach mechanical equilibrium:

Simulation to estimate the most probable configuration and shape of hepatocytes in a liver lobule. Blood vessel network : CBM, hepatocytes : DCM.


Bile canaliculi growth in between two cells

DCM Simulation of the bile canaliculus dynamics in between two cells. The dynamics is here governed by varying osmotic pressure in the initial gap between cells. The yellow particles mark the salt ions generating the osmotic effects (i.e. attracting water). Cases are simulated for increasing osmotic pressure in the cells. If the pressure is too low, the gap may close; if it is too high, the cells may separate.


Scaffold growth (DCM)

DCM simulation of scaffold growth. The scaffold surface is approximated by triangulation.


Center-based Model (CBM) of Liver Lobule

A multi-cellular lobule model including all important cell types during liver regeneration. Upon the damage due to toxin (e.g. CCl4 or APAP), hepatocytes (yellow) surrounding the central vein (big red sphere in the center) become necrotic and start to release signals (e.g. DAMPs) to attract macrophages (green) and stellate cells (small red spheres with arms) to migrate into the lesion to regulate the process of removal of the debris of necrotic cells. Afterwards, the healthy hepatocytes (beige) around the lesion start to grow and migrate to recover the lesion.

Beige: healthy hepatocytes, Yellow: necrotic hepatocytes, Red: stellate cells, Green: macrophages, Cyan: sinusoids. Spheres in movie represent position of cells with approximate shape in a statistical sense; CBM has not explicit notion of cell shape.

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Technical (incl. this web page)

Tim Johann (IfADo):


Dirk Drasdo (Inria, Paris and IfADo):