Mathematical Modeling of Cellular Behavior

1
Mathematical Modeling of Cellular Behavior
Math 8803 Discrete Mathematical Biology

• Ken Dupont
• Graduate Student
• (Bio) Mechanical Engineering

2
Introduction Tissue Engineering

• Tissue engineering (TE) aims to create, restore,
  and/or enhance function of biological tissues
  through a combination of engineering and
  biochemical techniques
• Bone TE Aim regrow bone that has been lost due
  to causes such as trauma, congenital defect, or
  removal due to excision of tumors
  
• The basic method of TE is to implant a construct
  consisting of scaffold /- cells /- growth
  factors
  

Human mesenchymal stem cells (green) on PLDL
scaffold (black struts), 20X
(K Dupont, GA Tech)
PLDL Scaffold, 4 mm D x 8 mm L
(R Guldberg, GA Tech)
3
Introduction Tissue Engineering - Cells

• Cells can either be seeded onto scaffolds ex vivo
  (outside the body) prior to implantation or can
  be enticed to infiltrate the scaffold in vivo
  (within the body)
• Stem cells can both differentiate into other
  cells and continue to proliferate (divide)
  mesenchymal stem cells are adult stem cells found
  in marrow cavities of long bones that can become
  muscle, cartilage, or bone cells

4
Introduction Tissue Engineering Modeling

• Mathematical/Computational modeling
• of cell dynamics has the potential to be
• a very useful tool in TE
• Advanced knowledge of the behavior of the
• cells on constructs could help to optimize TE
• construct design and limit the number of
• expensive and time-consuming empirical
• experiments

5
Introduction Processes in TE Constructs

• Sengers has listed the many of the events
  happening at the
  
• cellular level in TE constructs
• Proliferation cells divide during mitosis
• Senescence/Death cessation of division and
  later death
  
• Motility cells adhere to and move throughout
  their environment due to a variety of guiding
  signals (taxis)
  
• Differentiation stem cells turn into other cell
  types
  
• Nutrient transport/utilization nutrient
  concentrations higher outside of constructs than
  inside, and cellular demands may vary
  
• Matrix changes - cells produce extracellular
  matrix proteins (i.e. collagen) and degradation
  of matrix may occur as well
  
• Cell-cell interactions Cells can communicate
  with each other (such as during contact
  inhibition)
  
• NOTE All of the processes can vary with space
  and time

6
Processes - Cell Motility
A moving cell note the ovular nucleus
(Dickinson)
7
Modeling Cell Motility Random Walk Background

• Cell motion can be modeled as a random walk
• Recall the Bridges of Konigsberg/random walks on
  graphs from class
  
• Random walk (RW) - stochastic process made up of
  a sequence of discrete steps of certain
  length(s). A random variable can determine the
  step length and/or walk direction
• A more formal description of a random walk is as
  follows Let X(t) define a trajectory that
  begins at position X(0) X0. A random walk is
  modeled by the following expression X(t t)
  X(t) F(t) , where F is the random variable that
  describes the probabilistic rule for taking a
  subsequent step and t is the time interval
  between steps (Wikipedia)

8
Modeling Cell Motility RW Background

• A random walk is an example of a Markov chain,
  which is a collection of random variables Xt
  (where the index t runs through 0,1,..) having
  the property that, given the present, the future
  is conditionally independent of the past
  (Weisstein)

9
Modeling Cell Motility - 1D RW
Endothelial cell taking a 1D random walk (Jones)
Paths taken for eight separate random walks in 1D
originating at the origin and taking 100 steps
(Wikipedia)

10
Modeling Cell Motility RW Lattices

• The paths allowed during a random walk can be
  restricted to the space of a point lattice
  
• A lattice is a set of connected horizontal and
  vertical for 2D line segments, each passing
  between adjacent lattice points which are
  regularly spaced
• A lattice path is therefore a sequence of points
  P0, P1, Pn with n 0, such that each Pi is a
  lattice point and Pi 1 is obtained by offsetting
  one unit east (or west) or one unit north (or
  south) (Weisstein)

Path created during 2D walk on a point lattice
(lattice not shown) (Weisstein)
11
Modeling Cell Motility RW Lattices
Point lattice unit cells are generally in the
shape of squares, such that the point lattices
are sometimes referred to as grids or meshes
Square lattices helps to minimize memory use
and computation times Cells are far from
squares or points, but their position in the mesh
can be represented by the location of the cells
nucleus
Rat mesenchymal stem cells on a 2D cell culture
dish with nuclei stained by Hoechst dye (K
Dupont, GA Tech)
12
Modeling Cell Motility/Proliferation 2D RW
Simulation

• Endothelial cells forming monolayer on blood
  vessel walls 2D surface
  
• A moving cell will usually stop for a period of
  time before continuing on its walk, or it may
  divide, followed by walks of both daughter cells
  
• As the number of cells fills up the surface
  contact inhibition will dominate the process and
  the cells will no longer move or proliferate
  
• Lee tracked individual EC motion experimentally
  in 2D - average cell speed, duration of time
  remaining stationary, and average direction
  changes were determined for use as parameters in
  simulations

Confluent monolayer of ECs on tissue culture well
(Lee)
Cell paths over 36 hours (Lee)
13
Modeling Cell Motility/Proliferation 2D RW
Simulation
Lee used a 2D discrete cellular automaton model
of the proliferation dynamics of populations of
migrating cells Assumed steady state nutrient c
oncentrations and neglected cell loss
These discrete systems provide an alternative a
pproach to continuous models that use ordinary
and partial differential equation to describe the
dynamics of systems evolving in space and time
(Lee) Discrete models can be used to describe
movements of individual cells rather than
looking at entire populations of cells
2D lattice of square computational sites
Each site size of a cell (28 micron sides)
each site has a finite of possible states and 8
nearest neighbors The size of the total grid was
made to simulate the size of one well of a
96-well in vitro cell culture plate with diameter
of seven millimeters
(Jones)
14
Modeling Cell Motility/Proliferation 2D RW
Simulation

• At each time point a lattice site, automaton i,
  is in a certain state xi (xi 0 means no cell
  present)
  
• If a cell is present, xi needs to specify if the
  cell is moving, the direction of locomotion, and
  the time remaining until a change of direction
  
• Time is viewed as discrete steps with uniform
  increments ?t
  
• The state xi of any automaton takes values from
  the set of 4-digit integer numbers klmn
  
• k is the direction that the cell is moving in k
  can take any value from the set 0,1,2,8, with
  0 ? no motion, 1 ? motion east, 2 ? motion
  northeast, etc..
• l is the persistence counter that tells how much
  time is left until the next change of direction
  (tc l ?t)
  
• mn is the cell phase counter, which tells the
  amount of time left until the next cell division
  (tr (10m n) ?t)

15
Modeling Cell Motility/Proliferation 2D RW
Simulation

• Initial cell direction k assigned randomly
• Experimental measurements of the cell
  trajectories were then used to assign initial
  values of l
  
• The value of the counter decreasing by one after
  each iteration, with the cell direction changing
  when the counter reaches zero
  
• The experimental data showed that cells generally
  change directions in a gradual fashion, so
  transition probabilities of a cell making a large
  angle change in direction are small
• mn is assigned to each cell, again using the
  distribution obtained from experimental
  observations of real cell cycles
  
• 64 of cells divided after 12-18 h passed, 32
  after 18-24 h passed, and 4 after 24 -30 h
  passed
  
• mn also decreases by one with each iteration and
  the cell divides when it reaches zero
  
• l and mn are reset after each direction change
  and division, respectively
  

16
Modeling Cell Motility/Proliferation 2D RW
Simulation

• Example
• Assume a 2D square lattice with N x N sites, with
  time step ?t 0.5 hours
  
• Choosing an arbitrary automaton site i gives a
  value of xi 3319 at to
  
• This means that the site contains a cell moving
  north for three more iterations (1.5 h) and that
  the cell will divide after 19 iterations (9.5 h)
  
• At time to ?t, the cell will have moved to site
  i N, located one site north of site i, and the
  value of xi N 3218
  
• The value of xi will then be equal to zero unless
  another cell moves into the site
  

17
Modeling Cell Motility/Proliferation 2D RW
Simulation

• Each simulation run of the model starts by
  randomly distributing cells at varying densities
  throughout the 2D space
  
• An algorithm is then begun to increment cell
  activity at each site with the motion of a cell
  stopping when it no longer has a free site in
  which to move
• If a cell tries to move into an occupied site
  during one iteration, it will stay in its current
  location until the next iteration
  
• If a cell divides during one iteration it will
  not move, and one daughter cell will remain in
  the current site and the other will be randomly
  assigned to one of the neighbor sites
• The rows and columns are scanned randomly for
  incrementation during each iteration to prevent
  artifacts due to scanning sites in one repeated
  order
• CPU time per run lasts between 50-200 seconds on
  an IBMRS/6000 POWERStation 350 computer, with
  time varying based on grid size, initial density
  of cells, and spatial distribution of cells

18
Modeling Cell Motility/Proliferation 2D RW
Simulation

• RESULTS
• Confluence reached faster when (nonmotile) cells
  were seeded at higher densities (left)
  
• Increasing cell speed (S) decreases time to
  confluence
  
• Less of an effect for cells seeded at higher
  density (0.81, right) than those seeded at lower
  density (0.081, middle)
  
• This behavior due to increased contact inhibition
  in the cells seeded at higher density

(Lee)
19
Modeling Cell Motility/Proliferation 2D RW
Simulation

• RESULTS
• Lees model appears to accurately predict 2D
  endothelial cell population dynamics when
  compared to actual experimental endothelial cell
  counts (n3 per time point) after seeding at
  various initial densities

(Lee)
20
Modeling Cell Motility/Proliferation 3D RW
Simulation

• Cheng, from the same research group as Lee,
  investigated application of random walk model of
  cell motility in 3D
  
• Assumes highly porous scaffold
• Allows unrestricted motion
• A cell at one site can move to any of its 6
  adjacent cubic faces
  
• The algorithm for 3D motion is very similar to
  that of 2D motion, again containing a migration
  index, cell division counter, direction
  persistence counter, waiting time, and varying
  transition probabilities to determine the new
  direction that a cell will move in after
  stopping, colliding, or dividing
• One additional feature of the model is that it
  incorporates a waiting time that a cell will
  remain stationary after colliding with another
  cell, which accounts for the tendency of cells to
  form clusters in 3D

21
Modeling Cell Motility/Proliferation 3D RW
Simulation

• Cell seeding in two modes is considered
• Uniform cell seeding throughout the 3D space
• Wound healing seeding, with cells seeded along
  a edges of a cylindrical wound portion of the
  entire 3D grid
  
• The simulation runs until the cell volume
  fraction, ?(t), increases to the point that all
  available sites are occupied by cells

(Cheng)
22
Modeling Cell Motility/Proliferation 3D RW
Simulation
Uniform Seeding
A)
B)
RESULTS
Wound Seeding
(Cheng)
23
Modeling Cell Motility/Proliferation 3D RW
Simulation

• Chengs model also allowed study of the effects
  of chemotaxis on the amount of time to reach
  confluency
  
• Chemotaxis causes cells to
• A) Migrate preferentially in one direction over
  all others (creating a biased/reinforced random
  walk) (top figure)
  
• B) Proliferate anisotropically (bottom figure
  note that only four nearest neighbors are used in
  this figure)

(Jones)
P1 P2 P3 (Perez)
24
Modeling Cell Motility/Proliferation 3D RW
Simulation
CHEMOTAXIS RESULTS

With chemotaxis, the time to confluence
drastically increased, because most of the cells
bunched up near the end of the grid near the
attractant and became contact inhibited
(Cheng)
25
Conclusion

• The list of individual phenomena occurring during
  tissue repair is a long one even without
  considering the specific spatial and temporal
  interactions between them
• Currently, no model can completely describe the
  tissue growth process, because there are still
  too many unknowns regarding the process itself
  
• Application of discrete models of cell behavior
  and treatment of cells as individual stochastic
  objects can be advantageous compared to
  continuous models because the complex behavior of
  cells can be broken down into constituent
  elements
• In the words of Jones by modeling crucial steps
  as discrete processes, it is then possible to
  develop individual areas independently of the
  rest of the model
• Caution must be used in applying models to living
  systems because theoretical understanding is
  required as a check on the great risk of error in
  software and to bridge the enormous gap between
  computational results and insight or
  understanding (Cohen)
• Until more of the basic biology is known, as well
  as the math to represent that biology, models
  will serve as fair predictors for simplified
  cases of cell dynamics and tissue growth

26
References

• Key Publication References
• Cheng G, Youssef BB, Markenscoff P, Zygourakis K.
  Cell population dynamics modulate the rates of
  tissue growth processes. Biophys J. 2006 Feb
  190(3)713-24. Epub 2005 Nov 18.
• Lee Y, Kouvroukoglou S, McIntire LV, Zygourakis
  K. A cellular automaton model for the
  proliferation of migrating contact-inhibited
  cells. Biophys J. 1995 Oct69(4)1284-98.
• MacArthur BD, Please CP, Taylor M, Oreffo RO.
  Mathematical modelling of skeletal repair.
  Biochem Biophys Res Commun. 2004 Jan
  23313(4)825-33.
• Sengers BG, Taylor M, Please CP, Oreffo RO.
  Computational modelling of cell spreading and
  tissue regeneration in porous scaffolds.
  Biomaterials. 2007 Apr28(10)1926-40. Epub 2006
  Dec 18.
• Biological/PubMed only
• Byrne DP, Lacroix D, Planell JA, Kelly DJ,
  Prendergast PJ. Simulation of tissue
  differentiation in a scaffold as a function of
  porosity, Young's modulus and dissolution rate
  application of mechanobiological models in tissue
  engineering. Biomaterials. 2007 Dec
  28(36)5544-54
• Cohen JE. Mathematics Is biologys next
  microscope, only better biology is mathematics
  next physics, only better. PLoS Biology 2004
  Dec 2(12) e439.
• Deasy BM, Jankowski RJ, Payne TR, Cao B, Goff JP,
  Greenberger JS, Huard J. Modeling stem cell
  population growth incorporating terms for
  proliferative heterogeneity. Stem Cells 2003
  21 536-545.
• Jones PF, Sleeman BD. Angiogenesis -
  understanding the mathematical challenge.
  Angiogenesis. 2006 9(3)127-38.
  
• Perez MA, Prendergast PJ. Random-walk models of
  cell dispersal included in mechanobiological
  simulations of tissue differentiation. Journal
  of Biomechanics 2007 40 2244-2253.
• Mathematical/MathSciNet only
• Cavalli F, Gamba A, Naldi G, Semplice M.
  Approximation of 2D and 3D models of chemotactic
  cell movement in vasculogenesis. Math
  Everywhere deterministic and stochastic modeling
  in biomedicine, economics and industry. Springer,
  Berlin, 2007. Pp. 179-191.
• Sherratt JA. Cellular growth control and
  traveling waves of cancer. SIAM J. Appl. Math.
  1993 Dec 53(6) 1713-1730.
  
• Sleeman BD, Wallis IP. Tumour Induced
  Angiogenesis as a Reinforced Random Walk
  Modelling Capillary Network Formation without
  Endothelial Cell Proliferation. Mathematical and
  Computer Modelling. 2002 36 339-358.
• Jointly Referenced/Other