Development of a Two-Dimensional Model for Predicting Transdermal Permeation with the Follicular Pathway: Demonstration with a Caffeine Study
D e v e l o p m e n t o f a Tw o - D i m e n s i o n a l M o d e l f o r Pr e d i c t i n g Tr a n s d e r m a l P e r m e a t i o n w i t h t h e F o l l i c u l a r P a t h w a y : Demonstration with a Caffeine Study
Panayiotis Kattou 0
Guoping Lian 0
Stephen Glavin 0
Ian Sorrell 0
Tao Chen 0
0 Unilever , Colworth Science Park, Sharnbrook, Bedfordshire MK44 1LQ , UK
Purpose The development of a new two-dimensional (2D) model to predict follicular permeation, with integration into a recently reported multi-scale model of transdermal permeation is presented. Methods The follicular pathway is modelled by diffusion in sebum. The mass transfer and partition properties of solutes in lipid, corneocytes, viable dermis, dermis and systemic circulation are calculated as reported previously [Pharm Res 33 (2016) 1602]. The mass transfer and partition properties in sebum are collected from existing literature. None of the model input parameters was fit to the clinical data with which the model prediction is compared. Results The integrated model has been applied to predict the published clinical data of transdermal permeation of caffeine. The relative importance of the follicular pathway is analysed. Good agreement of the model prediction with the clinical data has been obtained. The simulation confirms that for caffeine the follicular route is important; the maximum bioavailable concentration of caffeine in systemic circulation with open hair follicles is predicted to be 20% higher than that when hair follicles are blocked. Conclusions The follicular pathway contributes to not only short time fast penetration, but also the overall systemic bioavailability. With such in silico model, useful information can be obtained for caffeine disposition and localised delivery in lipid, corneocytes, viable dermis, dermis and the hair follicle. Such detailed information is difficult to obtain experimentally.
bioavailability; diffusion; in silico modelling; pharmacokinetic model; transdermal drug delivery
Department of Chemical and Process Engineering
University of Surrey, Guildford GU2 7XH, UK
A major challenge for scientific research regarding skin
penetration of drugs, cosmetics, etc., is the development of robust
non-animal methods to test percutaneous absorption and
bioavailability. Dermal absorption due to exposure to
agrochemicals and environmental pollutants is also becoming a global
concern. Experimental approaches including in vivo, ex vivo,
in vitro and clinical studies on human volunteers are often
expensive and time consuming (
). Additionally, there is a
general trend in safety regulatory guidelines worldwide to
move away from animal testing of cosmetic products and
ingredients (e.g. the European Commission) (3). In parallel to
the advancement in experimental methods, in silico modelling
of dermal absorption and delivery has been demonstrated to
be useful in refining and reducing the experiments needed, to
enable faster design of new products and more reliable safety
assessment, and to improve the understanding of the transport
). In addition, a mechanistic model can help analyse
the relative importance of different penetration pathways; it
can be used to examine the impact of physico-chemical
properties of the chemical, the physiological variability and
application scenarios on penetration (
). As a result, in silico
modelling has become an important tool in the study of
topical and transdermal delivery.
Within this context, quantitative structure-permeability
relationship (QSPR) models emerged mainly focusing on
estimating the permeability coefficient or maximum flux of the
). The compartmental approach, which treats the
skin layers as different units of uniform concentration (
the next step of in silico modelling. Then, the introduction of
more sophisticated diffusion-based models has attracted
substantial attention since its appearance. These models produce
spatially explicit and time dependent predictions of
transdermal permeation following topical exposure. The majority of
early diffusion-based models do not consider the
heterogeneity of the stratum corneum (
). When the
bricks-andmortar structure of the stratum corneum was introduced
and its importance acknowledged, the main challenge was to
obtain the transport properties of chemical compounds in
different domains of the skin. In some cases the transport
properties are obtained by fitting to experimental skin
penetration data (
). In such cases the main problem is that the
model only works well with that particular experimental
dataset. This limits the range of chemicals the model can be
used for prediction.
Subsequently, Wang et al. (
) created a predictive model
of transdermal permeation that derived transport properties
of the solute from fundamental principles. Chen et al. (
presented a multi-scale approach using a similar
bricks-andmortar structure for the stratum corneum. These later
modelling studies adopted a multi-scale approach where
transport properties of skin lipids and cornecocytes are
determined separately, e.g. through molecular modelling
and QSPRs, achieving improved prediction accuracy.
Later, Dancik et al. (16) and Chen et al. (
) further included
viable epidermis and dermis. Some excellent review articles
have been published to summarise recent progress in this
area; see e.g. (
). Recently we extended the model of
Chen et al. (
) to include absorption into the systemic
circulation and subsequent kinetics (
However, as far as the penetration pathways are concerned
there is a noticeable gap with respect to the follicular pathway.
In the past decade many studies (
) confirmed and
highlighted the important contribution of the hair follicles to
transdermal penetration. Although hair follicles occupy only
ca. 0.1% of the skin surface, their diffusion coefficient can be
orders of magnitude higher than that in the stratum corneum,
and thus the overall effect can be significant (30). This
pathway has been considered in some simple compartmental
) which, however, have limited predictive
capability because of the need for parameter fitting of the
model to experimental data.
The aim of this study is to develop a mechanistic model of
the follicular pathway and integrate the follicular pathway
into our latest multi-scale model of transdermal permeation
). The stratum corneum, viable epidermis and dermis as
well as the systemic circulation kinetics are modelled using the
same methodology as before (
). The follicular
pathway was integrated by considering the physiological and
compositional properties of sebum and hair. The model’s
predictive capability is demonstrated through simulating an in vivo
study of topical application of caffeine on skin with or without
hair follicle blocking (
). Predicted caffeine plasma
concentration-time profiles showed good agreement with
the reported clinical data. A sensitivity analysis has been
conducted to demonstrate the rational of the chosen sebum
partition and transport properties of caffeine and the effect of
variability in such properties on the overall transdermal
delivery of caffeine.
MATERIAL AND METHODS
This section describes the method for modelling the follicular
route of dermal absorption, and its integration with the
previously reported multi-scale model of transdermal permeation
and bioavailability (
). The technical details concerning the
partition and transport properties of chemicals in skin and the
blood circulation kinetics can be found in the appendices.
The Hair Follicle Route: Modelling Approach
The main focus of this study is on the follicular pathway.
Although there exist debate with regard to the actual route of
follicular penetration (
), studies have been conducted to
examine at which phase of the hair growth cycle follicular
penetration occurs. Domashenko et al. (33) reported a study
on mouse skin and human scalp xenograft. They concluded
that transfection of liposomes occurred only at the onset of a
new growing stage of the hair cycle (
). In contrast another
study, one that the assumptions of the current approach are
based on, conducted on 8 human volunteers suggested that
penetration through the shunt pathway only occurs when the
hair follicle is active (
), where active hair follicle is
characterised by hair growth and/or sebum production (
It is also known that sebum is a penetrable medium (
and that the diffusion coefficient in sebum (
) is usually
several orders of magnitude faster than that in hair (38). The
above, alongside with the fact that hair it is a very dense
material mainly made of keratin, support the argument that the
sebum is the main transport route of follicular pathway. The
gap between the hair infundibulum and the skin is assumed to
be filled with sebum.
Based on the above assumption, the hair follicle anatomy
(Fig. 1a) is converted into a 2D domain of heterogeneous
material as illustrated in Fig. 1b, upon which a mathematical
model is developed. Figure 1b shows the different
compartments considered in the computer model, where
follicular penetration is simplified to be through the vertical
sebum layer and the hair follicle itself is impermeable. In
addition, the skin near a certain hair follicle is considered to be
symmetric with respect to the hair follicle, and thus only half of
the anatomy around hair (the unshaded area in Fig. 1a) needs
to be modelled. The dimensions (and density with respect to
skin surface area) of the hair follicle in this model are specific to
body sites and can be set by the users, accounting for the
variability from site to site on human body as reported in
the literature (
Figure 1b also shows the other skin compartments around
the hair follicle, which are needed for the integration of the
follicular pathway with these compartments in modelling. The
bricks-and-mortar structure of the stratum corneum can be
seen at the top. Viable epidermis and dermis are modelled as
homogeneous compartments; in the dermis compartment the
systemic circulation is included as previously reported (
Note that the sebaceous gland is not explicitly modelled.
Further details regarding the integrated modelling framework
are presented in the next section.
The Integrated Mathematical Model
Figure 2 shows the complete modelling framework where the
hair follicle, shown on the right hand side of the schematic, is
integrated with the rest of skin compartments. Briefly, on the
top is a homogenous vehicle layer, followed by the
bricks-andmortar structure of the stratum corneum (bricks: corneocytes;
mortar: lipid), where the number of corneocyte layers, N, is
specific to body site. Further down are the homogeneous viable
epidermis and dermis, and the blood capillaries in the dermis
where the solute clearance into systemic circulation is
calculated. With the full thickness skin, hair and blood capillary
considered, this framework intends to model the transdermal
permeation and kinetics in vivo. Simplifications compared to the
real hair follicle arrangement have been made. The bending of
the stratum corneum in the follicular orifices as well as the
funnel–shaped infundibulum (
) were initially considered.
Separate simulations showed that including such detailed
representation of the hair follicle region had an impact in the local
region but negligible effect on the overall transdermal
permeation. Although in vivo studies concerning the role of follicular
pathway are less common than in vitro ones, there is a general
uncertainty regarding the suitability of in vitro tests for
predicting in vivo situations with respect to the contribution of
the follicular pathway to transdermal permeation mainly due
to pre-processing of the samples (
). Therefore, the focus
of this study is to model the in vivo situations.
The governing equations include the diffusion equation
that describes the transport of solute in all compartments as
Table I Model Input Parameters
* Value scaled down to match the reported ratio of surface area of hair follicle to that of skin; see text for details
illustrated in Fig. 2. Inside dermis, mass transfer due to
convection in blood is also included. Mass transfer calculation
between the interfaces of compartments (e.g. between lipid
and corneocyte in the stratum corneum, or between sebum
and dermis) requires the partition coefficients of the solutes, as
detailed in Appendix A and B. At the four boundaries of the
entire modelling domain (solid lines in Fig. 2), zero flux is
assumed. To solve these partial differential equations, in
which the concentration of the solute changes with both
spatial coordinates and time, the 2D domain in Fig. 2 is
discretised into rectangular grids (
). Within each of the
grids, the partial differential equation is converted to an
ordinary differential equation (ODE) (Appendix C) using the
standard method of lines (
). Due to constraint on space,
detailed gridding is not presented, though it suffices to state
that a large number of grids (1880 in total) are used to ensure
that the results are independent of numerical inaccuracy due
to this discretisation. The model was implemented in C++
with ODEs solved by using the CVODE solver, a part of
the SUNDIALS computational package (
The partition and diffusion coefficients in various
compartments, except those for sebum, are obtained from the QSPR
models calculated from the physicochemical properties of the
solute and skin structures; these models have been published
Fig. 5 Comparison of predicted plasma concentration of caffeine with
published clinical data. In vivo data were obtained with blood samples taken from
six human volunteers with open hair follicles (HF) (
in the literature and are collated in Appendix A and B. With
the lack of reliable QSPR models for sebum, the
experimentally measured partition and diffusion coefficients were used
and will be discussed subsequently.
The vehicle of topical application is an important
compartment that needs to be properly modelled, especially for in vivo
finite-dose applications. In some cases (as in the caffeine case
described below), the applied dose results in the concentration
of the chemical in the vehicle exceeding its solubility in the
corresponding solvent. The excess portion over solubility is
modelled as solids. The rate of diffusion of the chemical from
vehicle into skin is generally slower than the dissolution rate of
solids in the vehicle. Therefore the vehicle is assumed to
remain saturated until the excessive amount is fully dissolved (to
make up the depletion of vehicle due to absorption into skin).
The vehicle is then switched to a finite source in the model.
As demonstration, the model was applied to simulate the
reported clinical study (
) of topically applied caffeine to human
chest. In this experiment an ethanol and propylene glycol
(30:70, v/v) solution containing 2.5% caffeine was applied to
the chest of six healthy volunteers in two different set-ups:
before and after blocking of the hair follicles with a
wax-mixture. The application area was 25 cm2 with a dose of 2 mg
cm−2. The solution was left to evaporate, and since ethanol is
known to evaporate within minutes, in the simulation the
Fig. 7 Predicted kinetics of transdermal delivery of caffeine with open and
blocked hair follicles. (a) Caffeine permeated into the skin (b) caffeine
disposited in the skin and (c) caffeine delivered to systemic circulation.
Table III Predicted Systemic Kinetics Following Topical Delivery of Caffeine
with Open and Blocked Hair Follicles (HF). Percentage Change = (Open HF
– Blocked HF)/(Open HF)
vehicle was simplified to consist of only propylene glycol and
caffeine. During the experiments, blood samples were taken
after each application at different times. The measured
concentration of caffeine in the plasma with and without follicular
blocking was reported to show the contribution of the follicular
pathway to the overall transdermal penetration. This plasma
concentration profile will be compared with model predictions.
The input parameters used for simulation are listed in
Table I, where the physicochemical properties listed are used
to calculate the partition and diffusion coefficients as detailed in
Appendix A and B. The dimensions of skin including the hair
follicle are chest specific. Specifically, the stratum corneum
thickness in chest was set to 14 μm (
). The viable epidermis
thickness is set to 100 μm and dermis thickness 1000 μm (
Regarding the follicular pathway, the vertical depth of the hair
is set to be 610.57 μm (
). The vertical depth of the sebum is set
to be 410 μm which is approximately the depth of the sebaceous
glands in the skin (
). The radius of hair in the thorax was
reported to be ca. 40 μm (
) and that of the hair follicle opening
50 μm (
). This suggests that the lateral width of the sebum
layer in Fig. 1b is 10 μm, by assuming that sebum completely fills
the space between hair and the follicular opening. Furthermore,
Otberg et al. (
) showed that the average area of follicular
orifices in thorax is 0.19% of the skin surface. To directly
represent this ratio using the actual diameter of hair follicle with a
computer model would require a large simulation domain and
computational power. Here, the width of the stratum corneum
(and the viable epidermis and dermis beneath) is kept to three
corneocytes width (i.e. 120 μm) to save computation expenses.
Accordingly the width of sebum is scaled down to 0.046 μm to
meet the above ratio. (Detailed calculation is as follows. Given
the lateral width of stratum corneum of 120 μm and the surface
area ratio of 0.19%, the entire follicle opening radius is
0.19 % × 120/(1 − 0.19%) = 0.228 μm. Furthermore, it was
found that the radius of hair in chest is ca. 40 μm (
) and that
of the hair follicle opening 50 μm (
), suggesting the sebum
annulus radius is 1/5 of the entire follicle opening. Therefore in
the scaled geometry the sebum annulus width is determined to
be 0.228/5 = 0.046 μm.) The dimensions of the hair follicle are
summarised in Fig. 3.
The partition and diffusion coefficients of caffeine in
various skin compartments (except in sebum) were obtained from
the established QSPR equations detailed in Appendix A and
B, using the physicochemical properties of caffeine given in
Table I. The partition and diffusion coefficients in sebum were
reported for some chemicals but the QSPR models developed
had substantial uncertainty in the prediction (
). In the
present study, the diffusion coefficient was set to be the same
as the measured data of butyl 4-hydroxybenzonate
(9.67×10−11 m2 s−1, (36)), which has the same molecular
weight as caffeine; according to Mitragotri (
) the diffusion
coefficient in a certain media is primarily determined by the
molecular weight of the chemical. The sebum:water partition
coefficient of caffeine was obtained from a standard
equilibrium experiment conducted at the China Agricultural
University (private communications). The clearance rate of
caffeine in systemic circulation was based on the reported data
for oral delivery 0.078 L h−1 kg−1 (
). The vehicle:water
partition coefficient is estimated to be 0.87 from the solubility
of caffeine in the vehicle over that in water (
). The diffusion
coefficient of caffeine in the vehicle was estimated to be
9.16×10−10 m2 s−1 using the Stokes-Einstein equation (Eq.
B.1). Table II summarises the diffusion and partition
properties of caffeine in different compartments.
RESULTS AND DISCUSSION
The predicted plasma concentration of caffeine is shown in
Fig. 4 (blocked hair follicle) and Fig. 5 (open hair follicle) in
comparison with the published experimental data (
range of the concentrations was obtained from six subjects
reflecting significant inter-subject variability. In both cases,
the model prediction appears to be in good agreement with
the in vivo data.
Subsequently, sensitivity analysis was conducted to explore
the impact of the parameter variability relating to sebum on
model predictions. Sensitivity analysis respect to other skin
compartments has been reported elsewhere (e.g. (
Figure 6 shows the predicted systemic kinetics when the
diffusion coefficient of caffeine in sebum is subjected to 30%
variability. Clearly, decreasing (increasing) the diffusion
coefficient results in slower (faster) penetration of caffeine into the
blood, since the penetration through the follicular pathway
becomes slower (faster). A similar effect was observed when
subjecting the partition coefficient and sebum width to similar
extent of variability (results not reported).
Figure 7 illustrates the predicted kinetics of caffeine
absorbed from the vehicle (Fig. 7a), disposition in skin (Fig.
7b, all skin compartments except blood), and delivery into
systemic circulation (Fig. 7c), in terms of the percentage of
total dose applied for open and blocked hair follicles.
Caffeine is mildly hydrophilic and thus its partitioning into
oily sebum is substantially less than partitioning into either
lipid or corneocytes in stratum corneum (c.f. Table II).
Fig. 8 Subcellular disposition of caffeine in the stratum corneum of the skin with open (a, b, c) and blocked (d, e, f) hair follicles (HF) at 5 min (a, d), 20 min (b, e)
and 1 h (c, f) after application.
However, the diffusion coefficient in sebum is 10+ times higher
than that in lipid (and several orders of magnitude higher than
in corneocytes). As a result, the overall effect of the follicular
pathway is significant. It can be seen from Fig. 7a that when
hair follicles are open, a greater and faster uptake of caffeine by
the skin is observed. Figure 7b shows that, due to the additional
follicular pathway that bypasses the stratum corneum to reach
viable epidermis and dermis, caffeine resides for less time in the
skin with faster and higher delivery to the blood (Fig. 7c), when
compared with the situation where hair follicles are blocked.
In Table III the relative contribution of the follicular
pathway to caffeine delivery to systemic circulation, as predicted by
the model, is quantified in terms of the maximum plasma
concentration (Cmax), the time to reach Cmax (Tmax) and the area
under curve (AUC). AUC represents the overall systemic
bioavailability of dermal exposure for a given time. These values
clearly show that hair follicles contribute significantly to the
overall transdermal permeation especially at the early stage of
application. Specifically, the AUC in systemic circulation one
hour after application is substantially higher when hair follicles
are open (0.86 ng h mL−1) then when hair follicles are blocked
(0.1 ng h mL−1), with a percentage difference of 88%. Even at
10 h post application, the percentage difference of AUC
between blocked and open hair follicles is still very significant, at
21%. The difference in Cmax and Tmax is also significant. It is
worth noting that the predicted difference also agreed to a large
extent with the clinical study (
). These results highlight the
importance of hair follicles for the bioavailability in the skin and
systemic circulation after dermal exposure to caffeine.
Figure 8 presents the detailed 2D disposition of caffeine in
the stratum corneum predicted by the model at different time
steps after the application of caffeine. The two sections of the
figure represent the simulations with the follicular pathway
open (a-c) and blocked (d-e). The concentration profiles clearly
show the contribution of the follicular pathway in the
permeation process. Caffeine concentration in the corneocytes
is visibly higher than in the lipid and this is due to the high level
of binding of caffeine in corneocytes compared to any other
compartment (Table II). At early times (e.g. up to 20 min), the
concentration profiles are noticeably different with the
follicular route being of significant importance in penetration.
Due to the relatively low partition coefficient in sebum, the
disposition of caffeine in sebum is not apparent in Fig. 8a-c. In
Fig. 9, the concentration profile in sebum is rescaled and shown
for different times. As can be seen from this figure, high
concentrations are observed during the early stage after application
whereas as time proceeds, caffeine concentration in sebum
decreases. This, together with the overall penetration profile
illustrated in Figs. 5, 6, and 7, suggests that the impact of follicular
pathway on caffeine delivery is more significant at the early
times after application. This observation from model prediction
agrees with experimental studies in the literature, e.g. (
This paper presents a new in silico model for transdermal
permeation and systemic bioavailability with the integration of
the follicular pathway. The multi-scale model considers the
important molecular and microscopic principles involved in
skin permeation and systemic absorption. To our knowledge,
this is the first model in the open literature that has the
capability to offer quantitative prediction of the three major
pathways (intercellular, intracellular and follicular) of transdermal
permeation. The model confirms the importance of the
follicular pathway. Prediction of the disposition of chemicals in
various skin compartments enhances our understanding of
the local pharmaco−/toxico-kinetics after skin exposure for
assessing efficacy and toxicity. This model could provide
improved in silico screening for pharmaceuticals and industrial
chemicals and be a valuable tool in extrapolating from in vitro
experiments to in vivo exposure conditions - a key component
to reduce the reliance on animal models. It should be noted
that the model is applicable to topical application of small
molecules (<500 Da) on a given body site, provided that the
appropriate physiological parameters, physicochemical
properties of the solute and vehicle properties are specified. Further
validation of the model with experimental studies of more
compounds and/or exposure scenarios is needed. Currently,
a more comprehensive validation of the developed model
against advanced imaging data is being planned.
ACKNOWLEDGMENTS AND DISCLOSURES
This work was supported by the UK Biotechnology and
Biological Sciences Research Council (grant number: BB/
L502042/1), and Unilever Research Colworth, UK. We
would like to thank Dr. Scott Singleton at the Strategic Science
Group, Unilever, for his valuable insight to this project, and
Professor Lujia Han’s group at the China Agricultural
University for sharing unpublished sebum partition data.
bricks and mortars in the SC are fixed according to our
previous study (
A.3. The hydraulic permeability of the medium (k)
APPENDIX A: PHYSICAL PROPERTIES
These properties are required in the QSPR models in
Appendix B to predict partition and diffusion coefficients in
various compartments of skin, and mass transfer into blood.
A.1. Solute radius
The radius of the solute (Å) is estimated using the equation
below based on the molecular weight (MW) of the solute (
rs ¼ p3ffi3ffiffi=ffiffi4ffiffiffiπffiffiffiffiffiffiffiffi0ffiffiffi:ffi9ffiffi0ffiffiffi8ffiffi7ffiffiffiMffiffiffiffiffiWffiffiffiffi
The hydraulic permeability is estimated from the correlation
derived from Jackson and James (
) given as:
A.2. Volume fraction of water in stratum corneum at saturation
Swelling and shrinking of the stratum corneum (SC) is not
considered in this study (
) thus the porosities of both SC
lipids and corneocytes are considered to be constant. Here
the porosity is defined as the volume fraction of the SC pores
at fully dehydrated state. The volume fraction of water in the
SC at saturation (φsc) is related to the volume of lipid (Vl),
keratin (Vk), and saturated water (Vw), content by φsc = Vw/
(Vw + Vl+Vk). The water phase distributed into the SC lipids
and corneocytes (Vw= Vwl +Vwk) with the corresponding
saturated volume fraction of water (or porosity) in SC lipids and
corneocytes as φm= Vwl/ (Vwl +Vl) and φb= Vwk/(Vwk +Vk).
Since Vw + Vl + Vk= Vwl + Vl + Vwk +Vk, it follows that Vw/
φsc = Vwl/φm +Vwk/φb. Using the relationship V = m/ρ
(where m is the mass and ρ is the density), it follows that
porosity of the SC is related to the porosities of SC lipids
and corneocytes by (
f l=ρl þ f k=ρk
f l=ρl f k=ρk
¼ 1−ϕm þ 1−ϕb
where fl (≈12.5%) and fk(≈87.5%) are the dry mass fractions of
SC lipid and keratin. The overall porosity of the SC can be
related to its saturated water content (mass fraction) by the
ϕsc ¼ ρwð1− f scÞð f l=ρl þ f k=ρkÞ þ f sc
In order to estimate the above parameters, the user needs
to enter the thickness of the SC. The dimensions and offset of
k ¼ βr2f ð1−θbÞγ
where β and γ are fitting parameters that will be explained
later on and θb is the actual fraction of water in the corneocyte
phase. rf is the radius ok keratin microfibril (rf = 3.5 nm (
A.4 Volumetric blood flow
The volumetric blood flow (Q) can be estimated according to
physiology. It is known that the average resting cardiac output
is ca. 5.6 L min−1 for a human male and 4.9 L min−1 for a
). The overall blood flow to skin is estimated to be
5% of the cardiac output (
). Therefore for a male and
female respectively the equations used are shown below:
APPENDIX B: DIFFUSION AND PARTITION
B.1 Vehicle properties
Diffusion coefficient in vehicle and any other medium can be
calculated using the Stokes-Einstein equation (
Dw ¼ 6πηrs
where K is the Boltzmann constant, T is the room
temperature (309 K), η is the viscosity of the medium and rs is the
solute radius as calculated in Eq. (A.1).
Partition coefficient in the vehicle is calculated using the
solubility of the chemical in water and the solubility of the
chemical in the vehicle. If the vehicle is water then it equals 1.
B.2. Lipid properties in stratum corneum
Diffusion coefficient in the SC lipid is related to the solute
radius (A.1) by the equation proposed by Mitragotri (
The partition coefficient of lipid to water can be estimated
using the equation below (
where CA, is the solute concentration in element A, V is the
element volume, ts is the time, qAB is the flux of solute into/out
of element A from neighbouring element B.
When it comes to dermis in addition to diffusion, convection
needs to be considered when modelling the transport between
dermis and the capillaries due to blood flow. The full
explanation of this approach can be found in the respective reference
). For a dermis grid I, the mass balance equation is:
VA dt ¼ −∑qAB þ Q b;A
B.3. Corneocyte properties in stratum corneum
Solute diffusion coefficient in the SC corneocytes (Db) is
estimated in the same way as our previous study (
1 þ prsffikffi þ 3rs2k
where α, β, γ and λ are model fitting parameters (α = 9.47,
β = 9.32×10−8, γ = −1,17 and λ = 1,09), S = (1 - θb) rsþr f ,
k is the hydraulic permeability (Eq. (A.3)) and Dw the diffusion
coefficient in water (Eq. (B.2)). The fitting parameters α, β, γ
and λ were chosen after applying the above QSPR model to
fit steady-state skin permeability data in the respective studies
The partition coefficient of corneocyte to water is estimated
using the following equation (
Kbw ¼ ð1−ϕbÞKkw þ θb
where Kkw is the solute binding constant to SC keratin and is
estimated using the following reported eq. (
Kkw ¼ ρ
where ρk = 1.37 g/cm3 and ρw = 1 g/cm3. In order to get this
coefficient the user is required to enter the Octanol water
partition coefficient of the chemical to be tested.
B.4. Viable epidermis and dermis properties
As literature suggests that viable epidermis and dermis
have similar multiphase compositions (
partition and diffusion properties in these two layers are
assumed to be similar. The partition coefficient is
calculated from the following equation proposed by
Kretsos et al. in 2008 (59) and then modified by Ibrahim
et al. in 2012 (
Kv ¼ 0:7
0:68 þ f u
þ 0:025 f nonKow0:7
where the three terms 0.68, 0.32/fu and 0.025 fnon
account for the chemical disposition in albumin accessible
aqueous phase. fnon is the fraction of solute non-ionized
where Qb , A is the volumetric blood flow into dermis grid A, Cb
is the solute concentration in the blood and Kdb is the partition
coefficient from dermis to blood.
APPENDIX C: OVERALL SIMULATION
The mass transfer equation is solved by a finite difference
scheme. In order to do so the skin is divided into a finite
number of smaller elements.
The mass transferred between neighbouring elements A
and B is given by (
qAB ¼ δA
where qAB is the flux of solute from grid A to grid B, Ai is the
interfacial area between the two grids, δA and δB are the
corresponding diffusion length, DA and DB are the diffusion
coefficients of the two elements, KABis the solute partition
coefficient between the two elements (KAB = 1 if the two grids
are the same element and KAB = KAw/KBw if element A is
different from B), CA and CB are the solute concentrations
in the two elements.
According to mass conservation principles, concentration
of each element A satisfies the following equation:
The systemic circulation and clearance is described by the
¼ N * ∑
where Vb is the volume of the whole body blood vessel, and
KCbis the first order clearance that may include transport into
other tissues and metabolism.
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Commons license, and indicate if changes were made.
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