Multiscale ComputerAided Tissue Engineering

1
Multiscale Computer-Aided Tissue Engineering

• Victor H. Barocas
• Department of Biomedical Engineering

2
Microstructured Material

• Although appearing continuous on the functional
  scale, the gel is really a discrete fiber
  network.
• Our model should account for fiber-fiber
  interaction.
• We would also like to be able to incorporate
  changes in fiber properties directly into our
  model.

3
Averaging Theory - Equations

• Most averaging theories use constant averaging
  volume.
• We employ a material averaging volume.
• As a consequence, mass conservation is more
  straightforward, but stress balance is less so.

4
Multiscale Method

• Macroscopic deformation field is used to
  determine deformation of microdomain boundary.
• Microstructural problem is solved to determine
  local forces.
• Average stress in microdomain is returned to
  macroscopic equations.
• Macroscopic displacements iterate until
  convergence.

5
Heterogeneous Sample

• A model system was constructed with a highly
  aligned central "wound" region and a more
  isotropic surrounding region.
• Data based on Bowes et al. (1999) paper on wound
  architecture, but no variation in collagen
  density.
• Sample stretched uniaxially to 30 strain.

6
Results

• Highly aligned central region deforms less than
  surrounding isotropic region.
• Image at left is a 2-D projection of 3-D result.

7
Now, to the cells

• I'd like to be able to model the cells also, and
  the problem is inherently multiscale.
• Suppose that I want to model the cells as
  migrating, non-interacting stress generators.

8
Current Thinking

• I like the ideas of Gear and Kevrekidis, but they
  are inherently finite-difference in nature,
  making them ill suited for complex shapes.
• I also find direct reintroduction of particles a
  bit unsettling, even though I accept their
  smoothness argument.
• A more attractive option would be to use a scheme
  that is consistent with the averaging-theory
  ideas of the mechanical model.

9
The Theory

• The essential theory is quite straightforward.
• The key element is NOT applying the divergence
  theorem in the micro-problem.
• Then we just have microscale issues.

10
The vision
Buffer regions maintained based on macro
concentration
Inner region used to calculate flux
11
The vision

• Red particle enters inner region and is counted.
• Blue particle leaves outer region and is
  discarded.
• Keep or discard orange particle?
• Is outer region completely rebuilt or only
  tweaked?

12
Simplest Problem I Could Do

• 1-D (actually 2-D micro, but 1-D macro)
• Non-interacting random walkers
• Initial concentration
• c 1cos(?x)
• Concentration evolves over time