Ebook Computational biophysics of the skin: Part 2

Chapter 8

Cellular Scale Modelling of the Skin
Barrier
Arne Nägel, Michael Heisig, Dirk Feuchter, Martin Scherer,
and Gabriel Wittum
Frankfurt University, Goethe Center for Scientific Computing,
Kettenhofweg 139, 60325 Frankfurt am Main, Germany


Computational modelling and simulation of penetration processes
in the skin barrier on multiple biological scales in space and time
is increasingly being recognized as a powerful tool to develop and
to refine hypotheses, focus experiments, and enable more accurate
predictions. One area of the ongoing research effort is physiologybased transport models. On the one hand, these are based on first
principles and describe processes in the skin mathematically in
terms of conservation equations. On the other hand, these models
employ detailed morphology information and are thus capable of
exploiting relationships between form and function. Particularly, such
models provide an understanding how microscopic physiological
structure and heterogeneity govern penetration. In this chapter,
we describe microscopic geometry models of the skin cells (e.g.
corneocytes) and the lipid bilayers of the stratum corneum (SC).
Computational Biophysics of the Skin
Edited by Bernard Querleux
Copyright © 2014 Pan Stanford Publishing Pte. Ltd.
ISBN  978-981-4463-84-3 (Hardcover),  978-981-4463-85-0 (eBook)
www.panstanford.com


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Cellular Scale Modelling of the Skin Barrier

The particular focus is on geometries based on tetrakaidekahedra
(TKD). These polyhedra with 14 faces have certain desirable features
for the generic construction of cellular membranes. We provide a
detailed geometric description of these membranes, which is
complemented by examples of computations in the simulation
system UG.

8.1  Introduction

The stratum corneum (SC) (see Fig. 8.1a) is the outermost skin layer
of the epidermis of mammals. It consists of corneocytes and a lipid
matrix. The corneocytes are dead, keratinized, fully differentiated
skin cells that arise from the underlying keratinocytes. The
corneocytes are embedded in a matrix of lipid bilayers (Fig. 8.1b).
The SC incorporates the main barrier function of the skin. It protects
the human body against the ingress of pathogens and preserves it
also from death by dehydration. The protection against the ingress
of xenobiotics is achieved by slowing the diffusion of the substances
through the SC. This is the result of the special arrangement and
geometry of the corneocytes and the chemical properties of the
lipid matrix. Micrographs of the SC show that the corneocytes are
arranged in staggered columns with different overlapping (see
Figs. 8.1 and 8.2).
The overlap of the cells depends on the body region and the
differing mechanical stresses. The corneocytes are closely packed,
flexible and provide an excellent protective function. The lipid
matrix consists of free lipid bilayers and lipids, which are covalently

bound to the corneocytes. The anchoring of covalently bound lipids
occurs via transmembrane proteins [9]. During the keratinocyte
differentiation in the stratum granulosum lipids are extruded into
the intercellular space, and transform into lipid bilayers with
lamellar structure [2,10]. The SC is composed of 10–15 layers of
flattened corneocytes according to the body region and has a
thickness of about 0.02 mm [4] (see Figs. 8.1a,b).
For the numerical simulation of drug diffusion in the SC
many different mathematical models exist. They differ in both the
physical description, and also in the geometry model of the
corneocytes and of the extracellular lipid matrix. One popular twodimensional geometry model is the “brick and mortar” geometry in
which corneocytes are represented as bricks and the lipid matrix


Introduction

as mortar (see Fig. 8.2, bottom left) [4]. This model thus takes into
account the overlap of the corneocytes. Wang et al. [11] gives a survey
of existing brick-and-mortar models (cf. Table 1 of [11]). Obviously,
three-dimensional geometry models are also desirable because they
can represent the SC more realistic than a two-dimensional model.
(a)

Stratum corneum

Stratum lucidum
Stratum granulosum

(b)
Stratum spinosum

Stratum basale

Figure 8.1

(a) Layers of the epithelial skin layer: the epidermis. Modified
from [1]. (b) Magnified sketch of two corneocytes A and B
with lipid matrix. The lamellar structure of the lipid matrix
is indicated, as well as the network of keratins within the
corneocytes. Reprinted from [2] with permission from BioMed
Central.

Figure 8.2

Light micrograph and four different geometric models of the
SC: Top left: Light micrograph. Reprinted from [3] with
permission from Elsevier. Bottom left: 2D-Brick-and-mortar
model [4,5]. Top right: 3D-Cuboid model [6]. Bottom center:
3D-Model with hexagonal prisms [7]. Bottom right: 3DTetrakaidekahedron [8].

219


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Cellular Scale Modelling of the Skin Barrier

The issue “Modeling the human skin barrier—towards a
better understanding of dermal absorption”, which was published
recently in the journal Advanced Drug Delivery Reviews, provides an

overview of different mathematical models as well as the state of
the art of computational research tools that are employed for
modelling dermal absorption. We refer for further details, e.g.
existing two- and three-dimensional geometry models for the SC,
to several articles in this issue (e.g. [12–16]).
In this work, we will focus on geometries based on tetrakaidekahedra (TKD) to model the corneocytes in the SC, because the experimentally observed geometry of the corneocytes is very similar
to the space-filling polyhedron tetrakaidekahedron (see Fig. 8.3b),
which has an almost optimal surface to volume ratio and is a solution of the Kelvin problem (cf. Section 8.3). The geometry model was
suggested in [8,17,53], and later on applied successfully in [18,19].
(a)

Figure 8.3

(b)

(a) Micrograph of soap foam with a tetrakaidekahedron like
structure. Reprinted from [20] with permission from Nature
Publishing Group. (b) Single corneocyte. Reprinted from [21]
with permission from Nature Publishing Group.

This work is organized as follows: Section 8.2 provides an extended motivation for the tetrakaidekahedral geometry model
of the SC. Section 8.3 then describes the geometry of the tetrakaidekahedron mathematically. In particular, we introduce the necessary parameters for characterizing tetrakaidekahedra. Finally, we
provide a mathematical model in Section 8.4 and conclude with
computational results in Section 8.5.

8.2  Motivation for a Stratum Corneum
Geometry Model with Tetrakaidekahedra

Using microscopic examination of frozen sections of the SC in the
60s the shape of the corneocytes was assumed being hexagonal

Motivation for a Stratum Corneum Geometry Model with Tetrakaidekahedra

[3,22]. Recent studies confirm this structure [23–27]. In 1975
Menton [20,28] first presented such three-dimensional models of
the corneocytes with tetrakaidekahedra (see Fig. 8.5). He based this
geometrical arrangement on the similarity of micrographs of the
SC (see Fig. 8.2, top left) with the spatial arrangement of soap foam
(see Fig. 8.3a). The soap foam is arranged, as the SC, in overlapping
columns. This geometry is similar to stacked tetrakaidekahedra
(see Fig. 8.2, lower right) [29].
Physically one can explain the formation of the tetrakaidekahedron form of the corneocyte cells as follows: During the
differentiation the keratinocytes in the epidermis are packed denser
because they are displaced upwards. Due to its surface tension and
mutual pressure, the cells obtain a geometric configuration in which
they can be packed with an almost optimal surface to volume ratio
to minimize the otherwise occurring forces and without gaps.
During the differentiation the cuboidal and irregular arranged
keratinocytes with different size change to similarly sized and
flat corneocytes, which have a columnar arrangement [30,20] (cf.
Figs. 8.2 and 8.4a). Also gaps outside the corneocytes are unlikely
due to the pressure. The cells have a staggered arrangement and are
interdigitated. Such an interdigitating arrangement saves surface.
It is tight, elastic and offers an ideal protection.
(a)

(b)

Figure 8.4

(a) LM-micrograph of the stratum corneum. Reprinted from
[3] with permission from Elsevier. (b) Corneocyte model with
tetrakaidekahedra, adapted from [33].

Menton as well as Pleswig and Marples presented various
micrographs of corneocytes, e.g. [20] cf. Fig. 18, [21] cf. Fig. 3D.
Menton also referred to micrographs of elder pith [31] and cork
cells [32]. Menton also made experiments with soap foam, showing
that foam bubbles are also be arranged in columns and that they
overlap. In case of soap foam bubbles the arrangement is similar

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Cellular Scale Modelling of the Skin Barrier

to highly composite tetrakaidekahedra (see Fig. 8.3a). Based on his
reflections on the origin, size and arrangement of corneocytes and his
experiments with soap foam [28], Menton proposed a model for the
SC geometry with interdigitating cells based on tetrakaidekahedra.
Lord Kelvin introduced the term “tetrakaidekahedron” for a
body with 14 faces. The name can be derived from the Greek in which
“tetra” means four and “deka” means 10. As part of his research in
1887 Kelvin experimented with soap foam on the search for an ideal
geometric arrangement. Kelvin looked for an answer how to divide
the space into equal-sized cells with the smallest possible partition
(“Kelvin problem”) [34–36]. The common solution before Kelvin

was a decomposition with rhombic dodekahedra (see Fig. 8.5c).
Kelvin came to the conclusion that his problem could be solved
with a partition into tetrakaidekahedra with six quadrilateral and
eight hexagonal faces (“Kelvin cells”) (see Figs. 8.4b and 8.5a,b).

D


E


F


=

<

;

F


Figure 8.5

G


H

Three space decompositions without gaps with polyhedra
of equal volume: (a) Kelvin tetrakaidekahedra, (b) paper
model of the Kelvin-decomposition, (c) rhombic dodekahedra,
(d) paper model of the Weaire–Phelan decomposition [40,36,
41] –> Base unit of 3 × 2 Goldberg tetrakaidekahedra and 2
pentagonal dodekahedra, (e) paper model of 2 base units of
the Weaire–Phelan decomposition [40,36,41].

Kelvin built on the work of Joseph Plateau [37]. The “Plateau
rules” state, that in a border of the foam always three surfaces of
the bubble meet at junctions at an angle of 120°, which is the angle
in a regular hexagon (“Plateau border”). On the other hand, at a
node four plateau borders meet at an angle of 109.47°. This angle


Motivation for a Stratum Corneum Geometry Model with Tetrakaidekahedra

is the tetrahedral angle, which is of fundamental importance for
the arrangement of all organic compounds in the nature. Thus,
the four binding partners of carbon are tetrahedrally arranged to
keep the binding energy minimal. Arrangements with greater
than four borders per node, or more than three faces at a border
are considered to be unstable and tend to transform themselves
according to the rules. Thus, the decomposition with the rhombic
dodekahedron in Fig. 8.5c is considered as unstable, since eight
borders intersect at a node [38,39]. The decomposition of the space
with tetrakaidekahedra, however, is stable according to the Plateau
rules since only four edges intersect at a node. As already stated,
tetrakaidekahedra can be packed without gaps, i.e. 100% tight

(see Figs. 8.4b and 8.5a,b).
In search of a solution how to partition the space into cells of a
specified volume, such that the total area of the interfaces between
the cells is minimal, Kelvin found a package with tetrakaidekahedra.
The arrangement proposed by Kelvin has a 0.7% less total area than
the complete rhombic decomposition of dodekahedra and was
assumed to be ideal for a century, until in 1994, when D. Weaire and
R. Phelan presented a solution that has a further 0.3% reduction
in total area (see Figs. 8.5d,e and [40,36,41,39]).
Here, however, a hybrid decomposition of equal sized but not
identical cells is used. This decomposition consists of dodekahedra
with pentagonal faces and so-called Goldberg tetrakaidekahedra.
These tetrakaidekahedra have 14 faces: 2 hexagonal and 12 pentagonal (five-sided) faces. In Fig. 8.5e the Goldberg tetrakaidekahedra
have a columnar arrangement in the X direction, in the Y direction,
and in the Z direction. In the void spaces remaining between these
columns pentagonal dodekahedra fit. A basic unit of the Weaire–
Phelan decomposition consists of 3 × 2 (X,Y,Z) Goldberg tetrakaidekahedra and two dodekahedra.
Recently, Inayat et al. [29] showed that the tetrakaidekahedron
geometry represents the foam structure better than the cubic,
Weaire–Phelan or pentagonal dodekahedron geometry. The TKD
model is the most suitable model to describe the geometrical
configuration of the foam structures.
In addition to the arrangement of and decomposition by
polyhedra, it is important to investigate the ratio of surface to volume
in single polyhedra and, furthermore, which convex polyhedra have
a preferably large volume with a minimum surface. Goldberg [42]

223

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Cellular Scale Modelling of the Skin Barrier

provides a lower bound estimate for the relation between volume
and surface areas in convex polyhedra:


A3
36 pn(n –1) ,
 u(n) =
V2
(n – 2)2

(8.1)

where A3/V2 is a dimensionless ratio between surface area A and
volume V, and n is the number of faces of the polyhedron. The larger
n, the smaller the limit u(n). Asymptotically (n  )u(n) approaches
the surface to volume ratio of a sphere, i.e. 36 p = AS3/VS2.
A small value for the ratio A3/V2 is advantageous, for example,
to have a low evaporation surface or to save expensive surface
material. Table 8.1 presents values for A3/V2 and for the Goldberg
estimate u(n) for the tetrakaidekahedron, the sphere, as well as the
five Platonic bodies: icosahedron, dodekahedron, octahedron,
hexahedron (cube), and tetrahedron (Fig. 8.6). The ratio A3/V2 for the
tetrakaidekahedron is smaller than for the octahedron, tetrahedron
and hexahedron. It is very close to the surface to volume ratio of
the regular dodekahedron.
D

U

K
G

L

D

K
G

Figure 8.6

R

W

Sphere, icosahedron, tetrakaidekahedron, dodekahedron,
octahedron, cube, and tetrahedron.

The last column in Table 8.1 compares the surfaces of the
bodies with identical volume. The surface AS of a sphere S with
volume VS is compared to the surface ATKD of a tetrakaidekahedron
which has the same volume as the sphere S, i.e. VTKD = VS. For
the Platonic bodies appropriate comparisons of the surfaces with
VS = VI = VO = VD = VC = VT are made.

R

AT = 3t 2

AC = 6 h2

Tetrahedron

Cube

AO = 2 3o2

Octahedron

1
2o3
3

VT =

1
2t 3
12

VC = h3

VO =

3
VD = (15+7 5)d

AD = 3 (5(5+2 5))d 2

Dodekahedron

1
4

VTKD = 8 2a3

ATKD = (6 +12 3)a2

Tetrakaidekahedron

AI = 5 3i 2

Icosahedron

5
(3+ 5)i 3
12

VS = __
​ 43 ​ pr3

AS = 4pr2

Sphere
VI =

Volume V

Area A

14
12
8

150.123
149.858
187.061

374.123

4

6

20

136.460

216.000

∞

n

36p

A3/V2
AS = AS ∙ 1.000

VO = VC = VT)

A if (VS = VI = VD = VTKD =

339.292 AT = AS ∙ 1.490

212.058 AC = AS ∙ 1.241

175.929 AO = AS ∙ 1.182

149.288 AD = AS ∙ 1.098

142.942 ATKD = AS ∙ 1.099

132.645 AI = AS ∙ 1.064

36p

u(n)

Surface and volume analysis for the five platonic bodies, sphere and tetrakaidekahedron

Body

Table 8.1

Motivation for a Stratum Corneum Geometry Model with Tetrakaidekahedra

225


226

Cellular Scale Modelling of the Skin Barrier

In this comparison, the tetrakaidekahedron performs very well
and does not even need more than 10% surface area than the sphere.
Although spheres have the best surface to volume ratio, spheres
cannot be packed 100% tight. The
____space filling in the densest
packing of spheres by Kepler is p/​√18 ​ ≈ 74% [43,44]. With regular
icosahedra, regular dodekahedra, regular octahedra and regular
tetrahedra [45] a 3D tesselation is also not possible. Unlike to these
polyhedra, an arrangement with tetrakaidekahedra is without
gaps and has the aforementioned surface-minimization property.
Motivated by this desirable feature, we will introduce a geometry
concept based on a uniform decomposition with tetrakaidekahedra
in the next Section.

8.3  Tetrakaidekahedron Model

This section describes the geometric properties of tetrakaidekahedra. Subsection 8.3.1 presents a parameterization as introduced
earlier in [17]. In Subsection 8.3.2, we introduce nested tetrakaidekahedra, which allow constructing the lipid matrix between the
corneocytes.

8.3.1  Parameters of a Tetrakaidekahedron

A single tetrakaidekahedron is depicted in Fig. 8.7. It is specified

uniquely by information about the length a, the height h and the
distance w of two parallel edges pj and qj. The angles, the volume
and the surface can be deduced from these values. The value s is a
measure for the overlap of two adjacent tetrakaidekahedra.
3
(w – 2a)
3



s=



smax =



w ≥ 2a

The overlap s is maximal for a –> 0:
w

3

(8.2)
(8.3)

For a convex tetrakaidekahedron the following inequality is
necessary:

(8.4)


Tetrakaidekahedron Model

For w = 2a the overlap s vanishes and we obtain an octahedron
with opposite two hexahedral faces. If Eq. 8.4 is violated, the TKD
degenerates, i.e. is no longer convex.

Figure 8.7

Angles and lengths in a unit tetrakaidekahedron [8]. 3D view
(left), top view (right). a: edge length of the base hexagon,
h: height of tetrakaidekahedron, w: distance between two
parallel edges pj and qj, d: largest distance between two edges,
s: overlap of two adjacent tetrakaidekahedra.

The width b is the horizontal distance between two edges of pj
and qj.
3
(2w – a) = 2s + a 3
3
The variable w indicates the relative horizontal overlap [46].


b=

(8.5)

s
1
1
(8.6)
=
<
b 2+ 3 a 2
s
The variable d defines the largest distance between two points of
the tetrakaidekahedron.




w=

d=

1 2
1
h +12(a2 – aw + w 2 )2 = a2 + b2 + h2
3
9

1
1
= a + (2w – a)2 + h2
3
9
2

(8.7)

The angle a includes a side hexagon and a vertically adjacent hexagon.


a = 2p – g – b

(8.8)

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Cellular Scale Modelling of the Skin Barrier

The angle b includes a basis hexagon and a side rectangle.


b=

p
h
+ arccos
2
2
h +3(w – 2a)2

g=


p
2h
+ arccos
2
2
2
4h +3(w – 2a)

The angle g includes a basis hexagon and a side rectangle


(8.9)

(8.10)

8.3.2 Parameter for the Lipid Matrix Tetrakaidekahedron

Feuchter [8] modelled the extracellular lipid matrix of corneocytes by a second tetrakaidekahedron, which is nested to the
tetrakaidekahedron of the corneocyte (see Fig. 8.8). Let Ci denote
tetrakaidekahedron of the corneocyte. Moreover, assume that a
lipid layer thickness ql, i.e. distance between two corneocytes Ci is
given. In order to parameterize the outer tetrakaidekahedron Ti, its
values for al, hl, and wl must be related to the values for a, h, and w
of Ci.
Q

&L

&L

K

7L

7L
KO

KKO

K

 K
 O

 K K

K

 O

 K
 O

 K K

K

 O

Q

7LF 7L
F
L  L 

 K
O

 K
O


K
 KO K

K
KO K KO ª
O
K
 KO K