ODEs model of foreign body reaction around peripheral nerve implanted electrode.

Abstract

The foreign body reaction that the neural tissue develops around an implanted electrode contributes to insulate the probe and enhances the electrical and mechanical mismatch. It is a complex interaction among cells and soluble mediators and the knowledge of this phenomenon can benefits of formal and analytical methods that characterize the mathematical models. This work offers a lumped component model, described by ordinary differential equations, that taking into account the main geometrical (size, shape, insertion angle) and chemical (coating surface) properties of the implant predict the thickness of the fibrotic capsule in a time frame when the reaction stabilizes. This tool allows to evaluate different hypothetical solutions for accounting the tissue-electrode mismatch.

Full text

Page 1

Abstract—The foreign body reaction that the neural
tissue develops around
contributes to insulate the probe and enhance the
electrical and mechanical mismatch. It is a complex
interaction among cells and soluble mediators and the
knowledge of this phenomenon can benefits of formal
and analytical methods
mathematical models. This work offers a lumped
component model, described by ordinary differential
equations, that taking into account the main geometrical
(size, shape, insertion angle) and chemical (coating
surface) properties of the implant predict the thickness
of the fibrotic capsule in a time frame when the reaction
stabilizes. This tool allows to evaluate different
hypothetical solutions for accounting the tissue-electrode
mismatch.
an implanted electrode
that characterize the
I. INTRODUCTION
lectrodes and electrode arrays, that are mostly used today,
showed an acceptable capability of recording high quality
neural signals and of properly stimulating for sensorial
feedback delivering, but what more deserves the attention of
neural interfacing technology designers and producers is the
long term endurance of performance that is still not
acceptable for human prosthetic applications. Foreign body
reaction (FBR) against implanted electrode is a general
response of the nervous system to implanted material that
does not present main specificities depending on the type of
implant. It affects both the short and even more the long-
term performance of the device, in registration and in
stimulation cues and it is strongly influenced by the
electrode-tissue interface. FBR is widely considered as a
critical issue in long-term maintaining of intraneural
electrodes [1, 2] and consequently improving comprehension
about this response of the body could give new insights in
the development of more tissue-integrated devices.
Inflammatory reactions of the body, and all the sub-
processes that are involved in it, are an optimal example of

Manuscript received April 1, 2010. This work was partially supported by
the Commission of the European Community within the project TIME
(Contract No.: FP7-ICT-224012 – Transverse, Intrafascicular Multichannel
Electrode system for induction of sensation and treatment of phantom limb
pain in amputees).
G. Di Pino, D. Formica, L. Lonini, D. Accoto, A. Benvenuto, P.M.
Rossini and E. Guglielmelli, are with “Campus Bio-Medico di Roma”
University, 00128 Rome, Italy,
g.dipino@unicampus.it; phone: +3906225419610; fax: +3906225419609).
S. Micera are with Scuola Superiore Sant’Anna, 56127 Pisa, Italy (e-
mail: micera@sssup.it).


(corresponding author e-mail:
complex processes, with multiple interactions among several
cells and soluble mediators, that evolve in time. Their study
can benefit from the appliance of formal analytical methods
typical of the mathematical models [3]. Although the
recognized importance of FBR in determining the
performance of electrodes and other implanted devices (i.e.
orthopedic prosthesis and insulin dispenser), and its
inclination to be modeled literature is unfortunately poor of
FBR mathematical model except for an old work by Nichols
and colleagues about FBR against aspecific material [4], that
is not very useful being missed all the used parameters, and
a technical report that faces in particular the FBR to neural
implants that introduce the argument, but remands to a
future, not yet published, scientific article [5]. Here a simple
model, taking into account the main geometrical (size,
shape, insertion angle) and chemical (coating surface)
properties of the implanted electrode has been developed, in
order to evaluate the thickness of the fibrotic capsule in a
time frame when the reaction stabilizes (about 4 weeks),
while keeping low the computational cost.
II. METHODS
A. Blocks schemes and specifications of the model
To realize the model we chose to adopt a lumped
component model, described by ordinary differential
equations (ODEs model), that assumes some theoretical
main simplifications listed in the following:
• Peripheral nervous tissue that receives the implant
is considered homogeneous
• In the set of points, laying on a section plane
perpendicular to the longitudinal axis of the electrode, that
are equally distant from the electrode the evolution of the
FBR is equal
• Blood and tissues adjacent to the insertion site
(following defined as region D and region C) are an
unlimited reservoir of cells compared to the amount
recruited in the FBR
Furthermore to be able to define each single equation we
went through an intermediate step of blocks conceptual
schematization describe in the following.
The model presents an
corresponding to the electrode insertion, that start the
inflammatory process through the mechanical damage
produced at the insertion site. Variation in size, shape and
insertion angle of the electrode modulates the amount of the
damage, while, given a particular damage, the biochemistry
of the surface coating modulates the further biological
process producing the inflammatory reaction. Once this
reaction is started, a balance among the reinforcement and
the resolution of the inflammation is mainly due to the
counter effects of pro and anti-inflammatory cytokines (PIC
initial on/off stimulus,
ODEs model of foreign body reaction around peripheral nerve implanted electrode
G. Di Pino, D. Formica, L. Lonini, D. Accoto, A. Benvenuto, S.Micera Senior Member IEEE, P.M. Rossini and
E. Guglielmelli, Member IEEE.
E

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and AIC). The fibrotic evolution is responsible for the
stabilization of the electrode encapsulation. Secondary
micromovements of the electrode that act after the insertion
push toward an increase and a chronicity of the
inflammation.
The strategy followed in the model needs to define four
spatial regions where the variables are homogeneous. Going
from the electrode surface to the periphery, region A is the
tissue-electrode interface occupied by FBGC, region B the
peri-electrode tissue made by fibrotic ECM, region C is the
ensemble of tissues adjacent to the implant and region D
represent the blood volume inside the peri-implant vessels.
The electrode is in contact only with region A. Region A and
B are concentric and together form the granulation tissue of
the capsule of which the model proposes to give the
thickness as its output. Region B is in contact with both
region D through the blood vessel wall barrier and with
Region C (Fig. 1).

Figure 1: Four ideal spatial regions involved in the model
seen in a section plane perpendicular to the major axis of the
implanted electrode

In region A temporal evolution of inflammation sees the
immediate adsorption of proteins on the electrode surface
that work as adaptor for the proximal monocytes adhesion
followed by the activation in macrophages cell-type and the
fusion to form FBGC. In region B the insertion damage
produce the blood vessels rupture and the consequent
formation of a blood-based matrix (BBM) that during a four
week time period will be remodeled in extracellular matrix
(ECM), mainly through the deposition of collagen fibers by
activated fibroblast. The model is based upon a
schematization that sees the main cells flows among the
different regions moved by the chemokines gradient through
the permeability of blood vessels walls for the extravasation
and of the blood-based/extracellular matrix for passing
across the tissues (Fig. 2). Once arrived in the destination
areas, the fibroblast collagen production in region B and the
monocytes/macrophages evolution in FBGC in region A are
modeled.
A. Equation of the model
According to the block scheme in Fig. 2 the FBR
mathematical model can be defined by the dynamic
equations of the two main cells in the different model areas:
monocytes/macrophages and fibroblasts in region A, B, C
and D.

Figure 2: Main blocks scheme of the model. Blue thick
arrows represents cell flows

1) Monocytes dynamics and formation of region A
The dynamics of monocytes/macrophages in region B can be
defined by the following equation:
˙
M B=˙
M R
M R
M ADH+˙
where the first three terms are due to cellular flows among
different contiguous regions (from region C and D, and to
region A respectively), while the last two are related to the
cellular turn-over, with
and described by a first order dynamics.
The
models the flow of monocytes going from region C
to region B, generated by the chemokines gradient through
the blood based matrix. Mathematically, this can be
described with a mass transfer equation, having the
chemokines gradient (
) as the motive force which
moves the monocytes flow against the resistance of the
blood based matrix ():
C+˙
D??˙
M BORN
B
?˙
M DIED
B

(1)
modeled by a logistic growth

(2)
In eq. (2), the term
is proportional to the monocytes concentration in region C
(
).
Analogously, the flow of monocytes going from region D to
region B can be defined as follows:
is the specific monocytes flow which

(3)
where
vessel wall, and
is the resistance for extravasation through the
is proportional to the monocytes
concentration in region D (
The term
from region B to region A due to the adhesion with the
adsorbed protein upon the electrode coating surface, is
modelled as:
).
, which represents the monocytes migration

(4)

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where and are the coefficients that represent the
capability of induct monocytes adhesion of respectively IgG
and fibrinogen, that are the two main proteins that adsorb
upon a foreign material surface in the body; these
coefficients depend on the material coating the electrode.
To have a complete description of the system modelled so
far, we need to define the time course of some variables used
in the previous equations; in particular
have to be mathematically defined.
As regards the chemokines gradient (
variation has been defined using a Gaussian function fitting
the experimental values for the chemokine MIP-1? taken
from [6].
The resistance for extravasation
vascular permeability (), which mainly depends on the
mechanical insertion damage (D), and can be expressed by
the equation:
, ,
), its time
is the inverse of the

(5)
where
chosen according with the time-course trend of the pro-
extravasation cytokines (PEC) concentration during the
inflammatory response to the foreign body. This is justified
by the fact that PEC are the major factors accountable for the
increment of the vascular permeability. In particular,
reaches its peak quite immediately after the
stimulus (electrode insertion) and then follows an
exponential decay during
inflammatory response. As regards the mechanical insertion
damage (D), it depends from the electrode size and shape,
and from the insertion angle, defined as the angle between
the longitudinal axis of the electrode and that one of the
nerve:
is a function of the simulation time, and was
the progression of the

(6)
where
defined as 4 times the cross sectional area divided by the
perimeter, and
is the insertion angle.
Similar to the resistance for extravasation
of the blood-based matrix
permeability (
). To properly introduce the definition of
this function, it could be useful to recall the main
physiological mechanisms underlying the formation of the
blood-based matrix.
The mechanical damage made by the insertion produces the
rupture of blood vessels and an organization clots-like of the
extracellular space occupied by a blood-based matrix.
During the evolution of FBR the matrix progressively
changes toward a fibrosis. This network works as structural
scaffold for cells migration through the tissues, thus an
optimal dimension of the loops of the net, reflected by an
optimal matrix density (
), is required for achieving an
high matrix specific permeability (
is the hydraulic diameter of the electrode,
, the resistance
is the inverse of its
). A higher or a lower
value of matrix density results in an overall reduction of the
matrix permeability (
).
Thus, matrix specific permeability (
a non-monotonic function of matrix density (
peak value (
) corresponding to the optimal matrix
density (
). To this purpose we defined the function
as the polynomial function of third
degree.
As regards the time course of the matrix density (
during the inflammation process, it reaches its peak among
day 2 and day 8 after the insertion, when it is constituted by
platelets and organized fibrin. After the pick
decreases because of the cleaving action of the MMPs that
attack the fibrin network. It is reasonable to say that wider
and heavier is the damage (
activation of surrounding tissue and faster is the organization
of the clot that reaches its density peak in few days and
viceversa. Given that the
is maximal for an optimal
(
), that is reached both in the ascending and in the
descending part of the
according with this modeling strategy, presents a double-
hump shape as showed in the second row of Fig. 3.
The value of the pick of the overall matrix permeability
(
) is further influence by the damage(
the last row of Fig 3, because the amount of damage
influences the extension of matrix involved in the cellular
migration: ().
Once we obtain the concentration of adhered monocytes
from Eq. (4) the next equation define the concentration of
FBGC in region A (expressed as the number of FBGC per
surface unit).
FBGC = (k
IgG
) can be described by
), with a
)
start to
) and more important is the

curve, the trend of ,
), as showed in
fus+ k
FIBR
fus)? MADH
(7)
where
and are the constant of fusion for
toward generate FBGC relative to IgG and Fibrinogen
respectively taken from [7, 8] and change during the time of
simulation with a sigmoidal shape with plateau [6, 9].
FBGC are the main component that constitute Region A of
the FBR capsule and in this region are tight packed forming
a dense component.
Their spatial organization can be geometrically schematize,
assuming that the shape of a giant cell is similar to a thin
ellipsoid with its major area (axes:
smashed toward the electrode surface, with the aim of
incorporating the foreign body, and the thickness twenty fold
smaller than the minor axis (
and )
FBGC
Thick
?
=1
20?FBGC
min ).Indeed
FBGCs in region A are packed with one of this
directed toward the
schematization, the thickness of region A can be
mathematically obtained from the next equation:
axis
this electrode. According to

(8)

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where is the minor axis of the ellipsoid representing
the FBGC shape and
FBGC around electrode surface.

is the number of layers made by

Figure 3: In the upper row the time evolution of the density
of the matrix (
), in the second row the matrix specific
permeability () and in the lower row the matrix
permeability (
). Different colors correspond to different
grade of the damage (
).
depends on FBGC concentration as defined in the
following equation:

(9)
Where
maximal surface along a section plane perpendicular to the
longitudinal axis of the electrode and passing through the
major axis of the ellipsoid (
literature [10].
According to the geometrical schematization of FBGC given
above, this area corresponds to:
is the mean value of the area of a FBGC
) as is reported in

(10)
And the minor axis is
?FBGC
maj
= 2??FBGC
min

(11)
Thus we have that:

(12)
Substituting the eq. (9) and (12) in (8) the thickness of
region A is:

(13)
2) 8.3.2 Fibroblasts dynamics and formation of region B
Fibroblasts dynamics and collagen secretion in region B has
been modeled by deriving most of the equations from the
work proposed by Dale et colleagues about the collagen
formation in dermal wound healing [11] . Main actors of this
process are fibroblasts, collagen and the family of its
cleaving enzymes called metalloproteinases (MMPs)
including the collagenase, and the pro-fibrotic cytokine
TGF?. Both MMPs and TGF? are produced by fibroblasts
and adhered monocytes activated in macrophages histotype.
Fibroblast concentration in region B (FB) is described by the
following equation:

(14)
FB is influenced by a logistic growth term, which represents
mitotic generation (first term of eq. (14)). Cell growth is
enhanced by TGF?, where a1 and a2 are the parameters
regulating the growth rate and kc is the carrying capacity of
the environment. Cells die at a constant rate a3. Similarly as
seen for monocytes, fibroblasts are recruited both from
contiguous tissues ( from region C) and from the blood
( from region D) thanks to a CXCL family chemokines,
mainly IL-8, gradient (
blood vessels wall (
(
). The equations describing the dynamics of recruited
fibroblasts are analogous to those used for monocytes (see
eqs. (2) and (3)).
Fibroblast proliferation and collagen synthesis are up-
regulated by the cytokine called TGFa, as described by the
following equation:

), through the resistance of
) and of the extracellular matrix

(15)

Concentration of TGF? is governed by an autocrine
mechanism in the fibroblasts, described by the first term of
Eq. 15. Natural decay of TGF? is modeled as a first order
process with time constant A6 [12]. The last term
corresponds to TGF? production by adhered monocytes.
MMPs dynamics is modeled by the equation:

(16)
MMPs bind to collagen breaking down the fibers. They are
secreted by fibroblasts, but the secretion is inhibited by the
presence of TGF?. Again, the natural decay is taken to be of
first order. The last term corresponds to production by
adhered monocytes.
Collagen concentration depends on the concentration of
fibroblasts and TGF?, as described by the first term of the
following equation. Collagen is degraded by MMPs as
described by the last term of the equation.


(17)

Where, a12FB is the basal production of collagen; a13TGF FB
is the amount of collagen production induced by TGF?.

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Assuming that the typical average concentration of collagen
in region B (
) is known, it is possible to estimate the
superficial concentration of collagen as the product of this
average concentration (
) by the thickness of region B
(
has to be equal to the time integral of the superficial
production of collagen, expressed by the product of collagen
production ( ) by the same thickness
Thus, the following equation is able to describe the relation
between collagen production ( ) and thickness of region B
(
):
). At the same time, this superficial concentration
.

(18)
By differentiation both the terms of eq. (18), the following
equation expresses the growth rate of the thickness of region
B in function of collagen production ( ):

(19)
The comprehensive thickness of the capsule around the
electrode () is given by the sum of the thicknesses
of the region A and region B:

(20)
III. RESULTS
For this preliminary simulation of the model has been
considered an electrode with its geometrical properties
resembling the tf-LIFE4s, but implanted perpendicularly to
the nerve. Adhesion and fusion percentage of monocytes and
chemokines gradients have been gathered from data reported
by [6, 9] for polyethylene terephthalate (PET), that presents
the same water contact angle than polyimide (about 70o).
The model has been implemented in MATLAB/Simulink
(The Mathworks, Natick, MA) using ODE45 function as
solver for the differential equations.
The follow graphs present the time course of the number
of FBGCs for square millimeter, number of fibroblasts in
region B for cubic millimeter and the thickness of region A,
B and the total thickness of the capsule. In agreement with
the fact that the reaction became stable after about 4 weeks
since the insertion values tend to stabilize when overcome
day 25 (Fig 4).
IV. DISCUSSION AND CONCLUSIONS
Aim of this model is to gather an idea of the influence of
geometrical and chemical parameters that characterize an
electrode upon the foreign body reaction produced by the
implantation of the electrode in a peripheral nerve. This tool
is useful even before that the electrode is fabricated and
tested, thus allowing electrode developers to spare time and
economic resources and to evaluate several possible
hypothetical solutions for accounting the tissue-electrode
mismatch.
The dynamics of the processes have been modeled starting
from a deep analysis of the literature about FBR, in
particular in the nervous system. The parameters have been
taken from published articles, favoring especially the data
regarding the PNS, or have been settled according with the
most plausible results of simulations. The output of the
simulation with an electrode that resemble the characteristic
of tfLIFE is consistent with the histological description
reported by Lago et colleagues [13]. Nevertheless the model
will profit from a experimental validation in vitro and in
vivo, adopting different materials as foreign body or
different coating surfaces.
Because biocompatibility is a functional characteristic,
which strictly speaking requires an analysis of tissue
response using an electrically functional implant, the future
version of the model should take into account also the
electrical properties of a stimulating/recording working
electrode.

Figure 4: Time course of the number of FBGCs for square
millimeter, number of fibroblasts in region B for cubic
millimeter and the thickness of region A, B and the total
thickness of the capsule.
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