A Cellular Automata Model of S.
Epidermidis-Neutrophils Interactions
on the Surface of a Medically
I l t dD
Implanted
Device
i
A. Prieto-Langarica,
H. Kojouharov
B. Chen-Charpentier
L. Tang
Technical Report 2010-01
http://www.uta.edu/math/preprint/
A CELLULAR AUTOMATA MODEL OF S. EPIDERMIDIS-NEUTROPHILS
INTERACTIONS ON THE SURFACE OF A MEDICALLY IMPLANTED DEVICE
A. Prieto-Langaricaa, H. Kojouharova, B. Chen-Charpentiera, and L. Tangb
a
Department of Mathematics, The University of Texas at Arlington, P.O. Box 19408, Arlington, TX
76019-0408
b
Department of Bioengineering, The University of Texas at Arlington, P.O. Box 19138, Arlington, TX
76019-0138
Abstract
S.epidermidis infections on medically implanted devices are a common problem in modern medicine due to
the abundance of the bacteria. Once inside the body, S. epidermidis gathers in communities called biofilms
and can become extremely hard to eradicate, causing the patient serious complications. We simulate the
complex S. epidermidis-Neutrophils interactions in order to determine the optimum conditions for the
immune system to be able to contain the infection and avoid implant rejection. Our cellular automata model
can also be used as a tool for determining the optimal amount of antibiotics for combating biofilm formation
on medical implants.
Keywords: medical implants, Neutrophils, S. Epidermidis, biofilms, cellular automata.
1. Introduction
Medically implanted devices are becoming increasingly important in medical practice [2]. Due to the
abundance of skin-colonizing bacterium, infectious reactions on such implants constitute common
problems in modern medicine [3]. Staphyloccocus epidermidis is the most common member of the group
of coagulase-negative staphylococci [1], which are bacterial colonizers of the skin and mucous
membranes of humans and other mammals [3]. It has been characterized as the main pathogen involved in
nosocomial bloodstream infections, cardiovascular infections, and infections of the eye, ear, nose and
throat [1]. Being a common host of human skin and one of the most often isolated bacterial pathogens in
hospitals, S. epidermidis becomes almost impossible to prevent from entering the body while inserting a
medically implanted device [1, 3]. Within the body, S. epidermidis can lead to a wide variety of
complications such as inflammation, thrombosis, infections and fibrosis [2]. Such complications have a
direct effect on the stability of the implanted device because they trigger important immune responses
including a rapid accumulation of phagocytic cells [2].
If the immune system is not able to eradicate S. epidermids during the first several hours after it enters
the body then a biofilm formation is likely to commence. Biofilms appear in many different forms,
including layers, clumps ridges, and even more complex micro-colonies that are arranged into stalks or
mushroom-like formations [5,4]. Once protected by the biofilm, the bacteria become extremely hard to
eradicate by the immune system. Most antibiotics are effective only against the fast growing bacteria in
the outer layers of the biofilm, so the slow growers deep inside of the biofilm formation tend to be spared
and to persist in the body [4]. Therefore, it is very important for the immune system to destroy the
bacteria before a biofilm starts forming on the surface of the medical implant.
1
The immune system attacks bacterial infections with phagocytic cells. One of the most important
phagocytic cells, and the primary immune defense against S. epidermidis, are the greatly abundant type of
white blood cell called Neutrophils. In order to be able to attack S. epidermidis growing on medical
implants, Neutrophils cells adhere to the surface of the device and move towards the bacterial formations
[2]. The strengths of the Neutrophils adhesion to the medical implant depends on the type of protein
present on the surface of the implant. Fibrinogen and Albumin are two of the most common protein
coatings used on medically implanted devices. Fibrinogen facilitates a strong attachment between
Neurophils and the implant since it is recognized as a foreign malign substance by the immune system.
However, it also works as a distraction to the Neutrophils in their fight against S. epidermidis, since the
phagocytes place themselves in one spot attacking the Fibrinogen covered implant without moving
around. On the other hand, Albumin is not recognized by the phagocytes as an enemy and hence allows
Neutrophils cells to move freely around the implant.
Experimental studies suggest that two groups of chemokines - macrophage inflammatory protein
(MIP) and monocyte chemoattactant protein (MCP) - appear to be mainly involved in the phagocyteimplant interactions [2]. By releasing those chemokines, the Neutrophil cells present on the surface of the
implant are able to attract other Neutrophils. These chemotactic interactions create waves of incoming
phagocytic cells into the implant aiding in the fight against a bacterial infection. Fibrogen covered
implants are interpreted as a threat to the immune system and many phagocytes are chemically attracted
to them. On the other hand, since an Albumin coated implant is perceived as no longer a menace to the
immune system, fewer phagocytes are recruited to fight the infection under Albumin coating.
In this paper we examine a variety of mixtures of Fibrinogen and Albumin coatings in order to
maximize the effectiveness of the immune system response. Finding the optimum amounts of each of
these two proteins will help the immune system destroy bacteria before it starts forming biofilm
communities. This will reduce the number of rejections of medically implanted devices and drastically
improve the ability of the body’s immune system to combat bacterial infections. The simulation will also
help determine the appropriate amount of antibiotics to use over the implant area so that an S. epidermidis
infection can be successfully controlled.
2. Cellular Automata Models
Cellular automata models are dynamical systems in which space and time are discrete [4]. A cellular
automaton consists of a regular grid, each of which can be in one of a finite number of possible states
updated synchronously in discrete time steps according to local, identical rules [4]. In this paper, we
employ a cellular automata modeling approach to simulate interactions between Neurophils and S.
epidermidis subject to a variety of coatings of Albumin and Fibrinogen mixtures on a medically implanted
device. A set of rules for the movement of the cells and the growth of the bacteria is given for the two
different types of protein coatings. The amounts of Albumin and Fibrinogen in the mixture are allowed to
be varied. As the amount of Albumin increases, so is also the ability of the Neutrophils to move around
the implant. On the other hand, an increase in the amount of Fibrinogen leads to a rise in the level of the
chemotaxis factor, which helps in recruiting more Neutrophils to the implant in the fight against the
bacterial infection.
Our models are divided into two parts. The first part simulates the complex S. epidermidisNeutrophils interactions between the 4th and the 20th hour after the implant is introduced in the human
body. We consider the reproduction of bacteria at the early stage of a bacterial community formation
which triggers the immune response. We also incorporate in our model a series of chemotaxing waves of
Neutrophils cells. The second part of the model simulates the system dynamics after the S. epidermidis
have started forming a biofilm. During this part of the simulation, bacteria experience an increase in the
reproduction rate while the immune system response is virtually ineffective in the presence of a biofilm.
In addition, we consider two different types of motility in our models: (1) an unbiased motility, in which
2
Neurophil cells move at random on the surface of the implant; and (2) a bias motility, in which the cells
move with greater probability towards larger bacterial concentrations.
3. Numerical Implementation
The new cellular automata models are implemented on an SxS grid. A square in the grid is occupied
by a variable density of bacteria while a Neutrophil cell occupies a cxc square. Each square in the grid is
in any one of the following four states:
•
•
•
•
Empty
Covered with S. epidermidis
Covered with a Neurophil cell and S. epidermidis
Covered with a Neurophil cell but without any bacteria present
Each numerical simulation consists of a series of iterative steps. We initialize the model with two SxS
matrices. Every entry in each matrix represents a square in the grid described above. On the first matrix
we randomly select m blocks of cxc numbered squares, each block representing a single Neutrophil cell.
Each cell is has the ability to move in 8 different directions. Direction i is chosen with probability Pi,
i=1,2,3,...,8. The value of Pi depends on the nature of the movement. Cell movement can be either biased
or unbiased depending on the selected model. In the second matrix, b units of S. epidermidis are placed,
with no limit on the number of bacteria that can reside in a single grid square.
Each block of cxc squares in the matrix that represents the Neurophil cells is uniquely numbered.
Every time step we check the area under each cell for S. epidermidis bacteria. Consequently, one of the
following two cases holds:
•
•
There are some bacteria under the area covered by the Neutrophil. In this case, the cell
consumes one unit of bacteria, which is reflected in the model by subtracting 1 from the
corresponding non-zero number in the bacterial matrix. That number corresponds to an entry
that is occupied by the Neutrophil cell on the cell matrix in T units of time. This step repeats
until all of the bacteria present underneath the Neutrophil has been eliminated.
There are no bacteria under the area covered by the Neutrophil. In this case, the cell moves to
an available, free from other Neutrophil cells, neighboring space i, i=1,…,8, (Fig. 1) with a
probability Pi.
Northwest (NW) North (N) Northeast (NE)
7
8
1
West (W)
East (E)
CELL
6
2
Southwest (SW) South (S) Southeast (SE)
5
4
3
Figure 1. Directions for movement of the Neutrophils cells.
The direction i of cell movement is determined randomly according to specific probabilities assigned
to each direction. For unbiased movement of Neutrophil cells, all eight directions have an equal
probability, i.e., Pi = 1/8, i = 1,...,8. In the case of biased movement, the Neutrophils cells move toward a
higher concentration of bacteria with a greater probability Pi. To compute Pi, we consider a 3x3 grid and
place the Neutrophil cell on the center square in the grid (Fig. 1). Then we calculate according to the
3
formula Pi = Ai/B, i=1,...,8, where B is the total amount of bacteria on each of the 8 squares surrounding
the cell and Ai is the amount of S. epidermidis on each of the surrounding positions.
To take into consideration the chemotaxis interactions between the Neurophil cells, we add N
additional cells to the system every G units of time, where G is a function of time and the amount of
bacteria present, and N is a function of time and the amount of cells present. The new cells are placed
randomly on available spaces of the implant ensuring that no two cells overlap on the implant. In
addition, bacteria attached to the implant reproduce every T units of time. However, doubling the amount
of bacteria in the simulation every T units of time will not be accounting for the bacteria that are
eliminated during the time period T. To account for that, we split T into h subintervals and we reproduce
1/h of the total number of bacteria at time T/h, i.e., every τ =T/h units of time 1/h of the bacteria
population is doubled.
4. Numerical Simulations
Both biased and unbiased motility simulations are run an equal amount of times and with the same
parameters in order to examine the effect of Neutrophils ability to identify bacteria around it on the
progression of the bacterial infection. In addition, we vary the amounts of Fibrinogen and Albumin in the
mixture in order to determine the optimal amounts of each protein in the implant’s coating that facilitates
the best immune system response. We use Matlab® to implement the two biased and unbiased motility
models. The time unit used for the simulation is t=20 seconds, which is the same as the approximate time
that it takes for a Neutrophil cell to ingest a single S. epidermidis bacterium. In our numerical simulations
we model the first 48 hours after the implant is introduced to the body. After the initial 16 hours, S.
epidermidis bacteria start forming a biofilm and the immune system is no longer able to effectively fight
the bacterial infection.
Next, we present the specific functions and parameters that are used in the simulations. The time at
which new Neutrophils are incorporated into the simulation, dx, is given by
dx(A)=floor[2880/(15+10*(1-A))],
where A represents the percentage of Albumin in the protein coating mixture. It depends explicitly on the
amount of Albumin. As the amount of Albumin increases, less Neutrophils cells are recruited and
therefore less chemotaxis takes place leading to longer time in between each cells arrival. When levels of
Albumin decrease more Neutrophils are recruited and therefore chemotaxis becomes stronger which
means that the time in between Neutrophils arrival is shorten. In order to represents this relations we use
G(A) = floor[(2n*(1-0.4*A))/3]
for the function G which represents the amount of new Neutrophil cells that are incorporated into the
system as a function of the remaining bacteria, n, and the amount of Albumin in the protein mixture. The
more Albumin in the mixture the fewer Neutrophils cells are recruited into the implant. According to
experimental data, approximately 40% more Neutrophils are found when Fibrinogen is the only protein
used to cover the implant as opposed to when only Albumin coating is used. The amount of initial
Neutrophil cells at the implant area, m, is modeled by the following function:
m(A,b)=floor[12(1-A)+n*b/9072],
where b represents the initial number of bacteria. The function m depends explicitly on the amount of
initial bacteria on the implant area as well as on the amounts of Albumin in the protein mixture with more
cells recruited when less Albumin is present in the mixture. For the time, Ts, that it takes each Neutrophils
cell to move one unit in space (a square in the grid of the model) we use the function
Ts(A)=floor[100*(1-A)/(1.1+A)].
4
Other parameters used in the simulations include c, the size of the cxc square on the grid that a single
Neutrophil cell occupies, and S, the size of the SxS grid used in the cellular automata models. We use c =
12 since the ratio between the radius of a Neutrophil cell and an S. epidermidis bacterium is
approximately 1:12, and S = 120 which represents a grid for approximately 1/100 of the area of a
biomedical implant used in practice.
The initial and final results of one 16-hour biased motility simulation are shown in Fig. 2. The
simulation was run for a protein-coating mixture of 40% Albumin and 60% Fibrinogen and with an initial
total bacteria population of b=9072 bacterial units. As said before, biofilm will start to form after 16 hours
but only if more than 15% of the implant area is covered with bacteria. Since the initial amount of
bacteria in our simulation covers 63% percent of the implant, that means that in relation to the initial
amount of bacteria, biofilm will form after 16 hours if more than 25% of the initial S. epidermidis
bacterial population remains.
Figure 2. Snapshots of the initial (left) and final (right) state of the system in a 16-hour simulation.
Next we performed a series of 48-hour simulations with the random and biased motility cellular
automata models for all values of Albumin between 10% and 90% in 10% increments (Fibrinogen=100%Albumin). As the graphs shows, the ability of Neutrophils to detect and move towards greater bacterial
concentrations plays a fundamental roll in the fight against S. epidermidis infections. The results show a
decrease of at least 5% of the final bacteria when the movement of the phagocytes is directed as opposed
to random, being the bias motility the one more congruent with the experimental results.
In our third set of simulations, we run both of the cellular automata models with 10%, 30%, 40%
60%, 80%, and 90% of Albumin, in order to determine the percentage of bacteria the immune system is
able to eradicate at the end of 16 hours. Since many of the events in both models are controlled by
probabilities, Monte Carlo simulations were used to average the numerical results. For each percentage of
Albumin, we performed 10,000 simulations. In terms of bacteria left at the end of 16 hours, the
simulations yield a wide variety of results: from 98% left on the random movement simulation with 90%
Albumin concentration to 18% on the directed movement simulation with a 10% Albumin concentration
(Fig. 4). Using the bias motility model 3 different mixtures yield results under 25% of the original
bacteria left after 16 hours. We conclude that 10%, 30%, 40% are optimal protein-coating mixtures for
medical implanted devices, with 40% Albumin, 60% Fibrinogen mixture being the least likely protein
coating that leads to an implant rejection by the human body.
5
Figure 3. Percentage of bacteria remaining versus time, in hours, for 9 different protein-coating mixtures
using both random (left) and biased (right) motility cellular automata models.
In order to find other values of Albumin for which the percentage of final bacteria is under the
biofilm-forming cut-off of 25 percent, additional Monte Carlo simulations with 10,000 iterations were
performed for all values of Albumin from 10% to 90% in 5% increments (Fig.5, left). The results show
that the only Albumin percentages that yield optimal results are 10%, 30% and 40%.
Figure 4. A series of 16-hour Monte Carlo simulations for 6 different protein-coating mixtures for both
random (left) and biased (right) motility cellular automata models.
6
Figure 5. Final percentages of bacteria after 16 hours for different values of Albumin in the protein
mixture (left) using Monte Carlo simulations. A histogram plot of the antibiotic units needed for the
immune system to lower after 16 hours the bacteria levels to below the biofilm-forming cut-off (right).
The biased motility cellular automata model was also used to determine the amounts of antibiotic
needed to bring bacteria after 16 hours to levels below the biofilm-forming cut-off of 25%. Monte Carlo
simulations are run for the same 9 different values of Albumin used in the second set of simulations (Fig.
3). A histogram plot (Fig 5, right) shows the results of the simulations, where on the x-axis we have the
different Albumin percentages (i=1,…,9, where % Albumin=10*i), while on the y-axis we can find the
antibiotic units needed to lower the S. epidermidis level under 25% of the initial population after 16 hours
and with it, avoid biofilm formation. One unit of antibiotic in our simulation kills 5% of the bacteria
every approximately 5 hours. For different antibiotics the units can be rescaled to give approximate values
of the needed amounts to get similar results.
5. Discussion and Conclusions
Using a cellular automata model we investigated mathematically and through a series of computer
simulations the interactions between S. epidermidis and Neutrophils on the surface of a medical implant.
We examined a variety of different protein-coating mixtures of Ablumin and Fibrinogen. Our main goal
was to determine the protein-coating mixture of the implant that (1) maximizes the immune system
reaction while minimizing rejection caused by infection; as well as (2) to prevent biofilms from forming
on the surface of the implant. Using our models we were found that a protein-coating mixture that
consists of 40% Albumin and 60% Fibrinogen is the optimal mixture for which biofilms are not formed
on the surface of the implant and the inflammation in minimal. Under this coating, Neutrophils are able to
eliminate most of the S. epidermidis cells during the first 16 hours, therefore preventing the formation of
biofilms on the surface of the implant. Since no biofilms are formed, the Neutrophils are able to
eventually completely eradicate S. epidermidis from the surface of the implant, therefore eliminating the
risk of an infection that could lead to the eventual rejection of the implant.
The model was also used to determine the amounts of antibiotics needed to cover the implant, for 9
different protein-coating mixtures, before the implant is inserted into the human body in order to prevent
S. epidermidis from forming a biofilm. For different types of antibiotics, our simulation results can be
7
easily rescaled to give accurate amounts of that specific antibiotic that is needed to successfully avoid
biofilm formation under any protein-coating mixture.
References
[1] Cuong Vuong and Michael Otto, Staphylococcus epidermidis infections, Microbes and Infection,
Volume 4 (4), Pages 481-489, April 2002.
[2] Jiaxing Xue, Jean Gao, Liping Tang, Mathematical modeling of phagocyte chemotaxis toward and
adherence to biomaterial implants, IEEE International Conference on Bioinformatics and
Biomedicine, Pages 302-307, 2007.
[3] Michael Otto, Staphylococcus epidemidis-the 'accidental' pathogen, Nature Reviews Microbiology,
Volume 7 (8), Pages 555-567, August 2009.
[4] J. P. Eberhard, Y. Efendiev, R. Ewing and A. Cunningham, Coupled cellular models for biofilm
growth and hydrodynamic flow in a pipe Volume: 3 (4), Pages: 499-516, August 2005.
[5] J. W. Costerton, Philip S. Stewart and E. P. Greenberg, Bacterial Biofilms: A common Cause of
Persistent Infections, Science Volume: 284 (5418), Pages: 1318-1322, May 1999.
[6] D. G. Mallet and L. G. de Pillis. A cellular automata model of tumor-immune system interactions,
Journal of Theoretical Biology, Volume: 239 (3), Pages: 334-350, April 2006.
[7] Simpson M,, Landman K, Hughes B, Multi-species simple exclusion processes, Physica A-Statistical
Mechanics and its applications, Volume 388 (4), Pages 399-406, February 2009.
[8] Joe Harrison, Raymond Turner, Lyrium Marques and Howard Ceri, Biofilms, American Scientist,
Volume: 93 (6), Pages: 508-515, November- December 2005.
[9] Wendy Orent, Slime City: Where Germs Talk to Each Other and Execute Precise Attacks, Discover
Magazine, July-August special issue; published online July 17, 2009.
8