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Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
149
Chapter 6
A Simple and Accurate Method to Assess
Autonomic Nervous System through
Sudomotor Function
Kamel Khalfallah, Jean-Henri Calvet,
Philippe Brunswick, Marie-Laure Névoret,
Hanna Ayoub and Michel Cassir6
6.1. Introduction
Peripheral neuropathies are assessed mostly by large fiber tests like
vibration perception, pinprick and electromyography. Unfortunately,
large myelinated fibers are damaged later in the progression of peripheral
neuropathies than small fibers, except in diseases specific to myelin.
The autonomic nervous system, mostly managed by small fiber nerves,
is very important for many vital functions like heart rate, blood pressure,
digestion, temperature control and grip, but it is not easily
assessed today.
The current small fiber tests evaluate temperature (hot and cold
sensation) but are subjective and operator dependent, which impacts the
reliability of the results. An interesting paper from Peter Dyck shows that
even world renowned neuropathy experts could arrive at different
diagnoses in the same patients when using the existing sensory tests [1].
The gold standard test for small fiber neuropathies is Epidermal Nerve
Fiber Density (ENFD) measured from punch skin biopsies. This invasive
Kamel Khalfallah
Impeto Medical, Paris, France
Advances in Biosensors: Reviews. Volume 3
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and lengthy process is not appropriate for recurrent assessments and not
recommended for diabetic patients with known impaired healing.
Therefore, starting in the 1990s the Mayo Clinic in Rochester,
Minnesota, developed an interesting concept of testing sudomotor
function to evaluate autonomic neuropathies and small fiber
neuropathies simultaneously and objectively.
First, this chapter will present key information about the physiology of
sweat glands and their control via small C fibers releasing Cholinergic
and Adrenergic neurotransmitters. Then a full physical model of sweat
gland behavior in the presence of low direct voltage applied to the skin
will be developed and its physico-electrochemical properties
demonstrated. This will be followed by a description of the Sudoscan
patented technology. Finally, the main results of the principal clinical
development programs performed over the last 10 years using this very
easy to use technology will be summarized.
6.2. Foundations: The Eccrine Sweat Glands
Two types of sweat glands have been identified: apocrine and eccrine.
All mammals have eccrine sweat glands on their paws (palms and soles
in apes and humans). In addition, humans are unique in that they have
millions of bodily eccrine sweat glands distributed throughout their
hair-bearing skin. Although structurally alike and producing a similar
water-based sweat, these two types of eccrine sweat glands form two
distinct subcategories based on their function and characteristics. Sweat
is made on the palms and soles to increase adherence and grip. In
contrast, bodily sweat glands produce sweat to regulate body
temperature. They are both controlled by the hypothalamus and
primarily respond to cholinergic effectors (as for salivation and
digestion) [2, 3]. Indicative of their different phylogenic origin, paws and
body eccrine sweat glands appear at different times during development
(16
th
vs. 22
nd
week in humans, respectively) [4, 2, 5].
Eccrine sweat glands are independent from pilosebaceous units. They are
made of a coiled tubular epithelium associated in its coiled secretory
portion with myoepithelial cells that allow sweat excretion. Eccrine
sweat glands are solitary and evenly distributed throughout the skin,
although their density varies by skin site [6, 7] with the highest densities
on the palms and soles. Their coil is closely associated in the deep dermis
with nerve endings, adipose pockets, and vasculature.
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
151
6.2.1. Physiological Behavior
Sudomotor function is the only component of the Autonomic Nervous
System that is managed only by the sympathetic arm. As shown in
Fig. 6.1, sympathetic nervous system physiology is comprised of
ganglions near the spine connected to very long thin C fibers innervating
the glands. Fibers to the sweat glands on the soles of the feet are the
longest small nerves in the body, stretching along the entire leg. These
nerves are most often the very first to be damaged in case of metabolic
insult, for instance. Consequently, evaluating sudomotor function on the
soles of the feet could be the earliest way to detect the first signs of small
fiber neuropathies.
Fig. 6.1.
Human nervous system.
One important point to note is that these small nerves can regenerate, as
shown by Christopher Gibbons et al. [8], and do so quite faster than other
small nerves of the sensory nervous system, making them the best tool
to assess treatment efficacy.
In addition, Laure Rittié [9] published some interesting articles
demonstrating that sweat glands initiate the healing process of scar
formation on the palms of human hands.
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During normal physiological function, activation of the eccrine sweat
glands starts with a “chemical” stimulus. For instance, in the cholinergic
pathway (the most important), this leads to the following sequence, or
activation cascade, see Fig. 6.2:
(i) The neurotransmitter acetylcholine binds to its corresponding
muscarinic acetylcholine receptor on the membrane cells of the sweat
gland wall;
(ii) This activates the G proteins coupled to the neuroreceptor;
(iii) The G proteins, or their intracellular messengers, then modulate ion
channels, creating an ion flux through the luminal membrane;
(iv) This polarizes the gland to voltages around 10 mV.
Fig. 6.2.
Cholinergic sequence activation.
6.2.2. Agonist and Antagonist Neurotransmitter Activations
The small C nerves of sweat glands involve mainly cholinergic
neurotransmitters, but some adrenergic fibers can be found in low
percentage. The cholinergic fibers are principally responsible for
activation of chloride ion channels via Muscarinic M3 receptors, while
Adrenergic fibers target the Cystic Fibrosis Transmembrane
Conductance Regulator (CFTR) chloride channels, which are well
known and have been thoroughly studied in Cystic Fibrosis (CF), (see
[10] and Fig. 6.3).
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
153
Fig. 6.3.
A diagram of the two components and two steps required
in the Schwartz-Thaysen mode of exocrine gland secretion to explain
the final composition of excreted gland fluid.
Due to defective chloride CFTR channels (the only ones on the duct
which should recycle chloride ions), the concentration of chlorides in
sweat is elevated in individuals with CF. This unique characteristic is
exploited in the diagnosis of CF: the sweat test measures the
concentration of chloride ions that are excreted in sweat. Sweating is
induced on the skin by pilocarpine iontophoresis. At the test site, an
electrode is placed over gauze containing pilocarpine and an electrolyte
solution that will not interfere with sodium and chloride measurements.
A second electrode (without pilocarpine) is placed at another site and a
mild electric current draws the pilocarpine into the skin where it
stimulates the sweat glands chemically.
Philip Low at Mayo Clinic proposed to use the same pharmacological
stimulation to measure and quantify sweat excretion, leading to the
development of the Quantitative Sudomotor Action Reflex Test
(QSART). Using this approach, he proved that sudomotor function
testing had clinical utility to assess autonomic neuropathies in different
diseases like diabetes. However, this pharmacological stimulation is not
easy to quantify and to reproduce. In addition, very small quantities of
sweat (µl) are difficult to measure and require an environment with very
well controlled temperature and hygrometry. QSART measures sweat
function at several locations on the body (forearm, proximal & distal leg,
and foot) but over an area covering few glands, and normative data
depend on gender and age. Finally, chemical iontophoresis will only
stimulate cholinergic nerves.
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6.2.3. Electrical Stimulations
Sympathetic Skin Response (SSR) has been described as a sudomotor
function test but is not defined as a real stimulation of sweat glands. SSR
measures a change in the skin electrical potential in response to an
arousal stimulus or an electric shock applied to the skin. The test thus
assesses a polysynaptic reflex (spinal, bulbar, and suprabulbar); the
source of the skin’s electrical potential is attributed to sweat glands and
the epidermis, but there is no evidence of a direct stimulation being
applied to the sweat glands, and any global response to the stimulus
applied is not taken into account.
Direct stimulation of the glands
A very important body of scientific work has been published by Vitale
et al. [11] on electrical stimulation of recently biopsied eccrine sweat
glands from the palms of adult monkeys. An Electrical Field Stimulation
(EFS) elicited an immediate secretory response which ceased abruptly
upon its termination. This response was inhibited in a dose-dependent
manner by atropine (a cholinergic antagonist). Although most glands
were inhibited by atropine, a minor component insensitive to atropine
was operative in some preparations. Physostigmine (a cholinesterase
inhibitor) potentiated the response to a subthreshold EFS. More
importantly, lidocaine completely and reversibly blocked EFS secretion
but had no effect on methacholine-induced secretion. These experiments
prove that the electrical stimulation excites nerves which deliver
neurotransmitter to activate secretion.
The conclusion of this article was that “These results confirm that eccrine
sweat glands from the palms are activated predominantly via cholinergic
pathways and that EFS may be used to study the control of secretory
function in normal and diseased states”.
The details of the electrical stimulation’s experimental conditions show
that a minimum of 750 mV to 1 V threshold is necessary to activate
complete secretion of these eccrine sweat glands.
6.3. Insight into the Sudoscan Measure: Electrochemical
Hairless Skin Model at Low Voltages
Sudoscan technology combines reverse iontophoresis, i.e., migration of
some ions from the human skin to the electrodes, and steady multiple
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
155
chronovoltammetry, which is the application of rectangular direct
current (DC) pulses of varying voltage amplitudes.
6.3.1. Global Qualitative Considerations
The measurement relies on the particularities of the outer-most layer of
the human skin, called the stratum corneum (SC). It consists of a lipid
corneocyte matrix crossed by skin appendages (sweat glands and their
follicles). See Fig. 6.4a and [12].
Fig. 6.4.
a) Simplified
representation
of
the
human
skin.
b) Electrodes
and
voltages.
According to [12], at low voltages – less than 10 V- the SC is electrically
insulating (essentially capacitive and negligible) and only the
appendageal pathway is conductive. In the current application, only
eccrine glands on the palms and the soles are concerned. The palms and
soles are where these glands are most numerous and present in
abundance (~ 500/𝑐𝑚
), and therefore, where the electrodes are placed.
Their secretory canal extends straight towards the skin where it leads
to a pore.
The aim here is to present a new and very complete electrochemical
model for the hairless skin at low voltages, previously only partially
published [13]. Recall the most complete and well-known model of
Chizmazdhev & al [12], which is purely electric (no chemical aspects are
considered directly) but remains undoubtedly a starting point and a
compulsory passage for more sophisticated and realistic models.
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Concretely, an electrode (the anode) applies a low positive direct current
(DC) voltage on the skin to the gland, and the current leaves via the
cathode (see Fig. 6.1b). To simplify and because obviously it is the
polarization (i. e. the potential difference) that counts, the reference
voltage will be the one reached by the body, namely the interstitium in
which the gland bathes.
Some minor assumptions are needed and useful: for instance, the
electrodes are supposed to behave “perfectly”, with no overpotential and
the sweat inside the gland (~ salt water) is an incompressible fluid. The
interstitium is made up of water and several ions [14-16], mainly (with
their concentrations in mmol/L): 𝐶𝑙
(120), 𝑁𝑎
(145), 𝐻
(pH = 7.4),
𝐻𝐶𝑂
(20) and lactate (15). So that, as mentioned in 2.3, electro-active
species that will migrate and react with electrodes are essentially:
Chlorides 𝐶𝑙
at the anode (entering current to the gland);
The proton 𝐻
at the cathode (leaving current from the gland),
because 𝑁𝑎
cannot react for considered polarization.
Only these two species will be retained and their reactions and exchanges
of electrons with the electrode are assumed kinetically (much) faster than
physiological kinetics, which is largely the case with nickel or stainless
steel electrodes. Under the electrochemical gradients, the species move
(migrate) inside the gland and some exchanges with the interstitium
across the wall of the gland will occur. The conservation laws of physics
can be applied to eccrine glands in order to understand these
physiological kinetics.
6.3.2. General Law of Conservation
It is important to state clearly the origin and hypothesis of the physical
laws considered; see [17] for details on the construction of the laws of
conservation in classical continuum mechanics. Hence, despite the
presence at a microscopic level of species (ions) moving and interacting,
the medium will be considered continuous as in classical continuum
mechanics, i.e., the geometric parameters of the problem and the
infinitesimal volumes, distances needed to proceed some limits are much
higher than the size of ions …
A complete exposition on the general law presented in this subsection
with concepts necessary to recall, is provided in Appendix 1; we will
solely use the main results below.
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through Sudomotor Function
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6.3.3. Geometrical Model of Eccrine Gland, Variables
and Currents
6.3.3.1. Geometry and Variables
The eccrine gland is composed of two parts: the secretory portion, a coil
where physiologically the sweat is (isotonically) filtered from the
interstitium; and the excretory portion where some species can move in
both directions (entry or absorption according to their electrochemical
gradient) across some ionic channels or not. The excretory portion is a
quasi-straight duct that leads to a pore on the skin surface. These two
regions have lengths of the same order, with the coil being slightly larger
than the duct [14].
The geometrical model consists in unrolling the coil and joining it to the
duct to form, as in the Chizmazdhev model [12], a cylindrical tube, but
here of a finite and more realistic length. See Fig. 6.5.
Fig. 6.5.
Geometrical
model
for
the
eccrine
gland.
The principal variables are the concentrations 𝑐
and the velocities 𝑢
of
the electro-active ions present in the sweat: 𝑘∈𝕊 𝐶𝑙
,𝐻
. We
shall add the electric potential inside the gland Φ. It will be expressed
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from the principal variables, see further. They are all functions of 𝑥,𝑡),
where 𝑥 is the abscissa along the axis and 𝑡 is time.
The geometric parameters are:
ℎ: thickness of the stratum corneum (SC),
𝑟
: radius of the duct, 𝑟
: radius of the coil,
Lengths 𝐿
, 𝐿
of excretory and secretory portions.
The electrical parameters are:
𝜎: conductivity of the electrolyte (sweat),
𝛷
: potential applied at the anode,
𝛷
: potential reached by the body after the application of the anodic
tension. As previously specified, to simplify the analysis and the
calculations, the potential 𝛷
0
, will be chosen as reference.
Surfacic conductances of the wall of the two portions, 𝐺
,𝐺
, depend on
the species 𝑘 and are generally a function of the potential difference
between both sides of the gland wall 𝛷𝛷
. They may involve the
gaping probabilities of some ionic channels and/or electroporation
phenomena, see [12].
Surfacic capacitances of the wall of the two portions, 𝐶
,𝐶
, may depend
on the species 𝑘 and are almost constants, according to
Chizmazdhev [12].
6.3.3.2. The Currents
A. Cross-wall current: is a transverse current due to charges that cross
the wall of the gland, and which depends on the ion 𝑘. Its density (per
unit area on the wall of the gland) depends on the way the crossing
occurs, through an ion channel or not.
The ion channel is related here to the 𝐶𝑙
ion that can go through this
epithelial membrane using its own dedicated ion channel [19-21]. This
ion channel approach is more fecund than a simple conductance model
because it also considers the chemical gradient. The density of this
current 𝐽
for ion 𝑘 is given by
𝐽
𝑧
.𝐺
.𝛷𝛷
𝛷
,
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
159
where 𝑧
is the charge of the ion and 𝐺
the conductance per unit area,
𝛷
is the equilibrium potential of the ion according to Nernst’s law
(see [22] and Fig. 6.6)
𝛷
.
.
.𝑙𝑛
,
in which 𝑅 is the perfect gas constant, 𝑇 is the absolute temperature and
𝐹 the Faraday constant (charge of a mole). In physiological polarizations,
this channel behaves according to Boltzmann function, voltage
dependent.
Here, it is important to mention that the muscarinic channels are driven
by cholinergic neurotransmitters, which are supposed to be voltage
dependent. The duct, meanwhile, consists essentially of CFTR channels
that are minimally voltage dependent and especially sensitive to the
chemical (concentration) gradient.
Fig. 6.6.
Channel across the gland wall.
B. Capacitive wall current: is a transverse current due to charges that
accrue on (or leave) the wall of the gland; it may depend on the ion 𝑘 and
its density (per unit area on the wall of the gland) and is expressed as
follows:
𝐽
𝑧
.𝐶
.
,𝑝∈𝑠,𝑒, (6.1)
where 𝐶
is the capacitance per unit area, which is almost constant
~
0.5 µ𝐹/𝑐𝑚
[12].
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This current is unsteady (i.e., it vanishes at steady states).
C. Axial current: is a current along the gland axis due to the motion
(migration) of ions following this axis inside the gland. The surfacic
density of the axial current, as a function of the main variables, for any
species 𝑘, is by definition:
𝐽
𝐹.𝑧
.𝑐
.𝑢
, (6.2)
Then the axial current is given by:
𝐼
𝜋.𝑟
.𝐽
,
and the total density and current are:
𝐽
𝐹.∑𝑧
.𝑐
.𝑢
, 𝐼
𝜋.𝑟
.𝐹.∑𝑧
.𝑐
.𝑢
6.3.3.3. Ohm’s Law and Electrical Field
Ohm’s law stipulates that the current = conductance * potential
difference; thus conductance = conductivity * surface/length. When
applied to the (previous) axial current:
𝐼
𝜎.𝜋𝑟
(6.3)
The electric field 𝔼 is inferred therefrom, which by definition is:
𝔼≡
∑𝑧
.𝑐
.𝑢
(6.4)
6.3.4. Governing Conservation Equations
It is shown in the following that the whole problem can be solved
sequentially: first the mass equation will lead to the potential, then the
momentum balance gives the constant concentration, and finally the
velocity can be deduced from Ohm’s law.
6.3.4.1. Mass Conservation
As represented in Appendix 1 equation (6.15) and Table 6.1, the
conservation of mass can be written:
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
161
𝜕
𝜕𝑡𝜌 𝜕
𝜕𝑥𝜌 𝑢 𝒜
It remains now to specify the source term 𝒜 which represents, per unit
volume, the loss or gain due to the transverse (cross-wall and capacitive)
current. This is described in Appendix 2.
Finally, with the concentration, the mass balance is given by:
.
.
.𝐽
,
(6.5)
where 𝐽
𝐽
𝐽
.
This is recognized as a classical transport equation with an original
source term.
Thus, by multiplying (6.5) with 𝑧
, using (6.3) and (6.4), the (total) axial
current is written:
𝜋.𝑟
.𝐹
∑
.
2𝜋.𝑟.∑𝑧
.𝐽
,
which at steady state (𝜕/𝜕𝑡 = 0) shows that the variation of the axial
current equalizes the sum of the transverse currents.
Now as 𝐽
𝐽
𝐽
, by using expression (6.1) of 𝐽
and
Ohm’s law (6.3) with the capacitive currents gathered, we end up with:
∑
.
∑𝐶
.
∑𝑧
.𝐽
To compare to the purely electric balance equation
7
of Chizmazdhev
[12]: eq. (8) page 846, in which the first term does not exist! See the
extract in Fig. 6.7.
7
which is the unique equation of the model of Chizmazdhev.
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Fig. 6.7.
Extract from Chizmazdhev [12].
6.3.4.2. Momentum Conservation
Appendix 1 equation (6.15) and Table 6.1 demonstrate that the
momentum conservation is written:
𝜌 𝑢
𝜌 𝑢
ℱ (6.6)
It remains now to specify the source term ℱ which represents, per unit
volume, the resultant of the present external forces. In the following, and
since only one species is involved at each electrode side, the species
index 𝑖 will be dropped.
The ions are supposed to be rigid spheres moving in a continuous
incompressible fluid. Recall that when some charged species are in
motion, they create an electric field 𝔼 (already seen) and a
magnetic one 𝔹.
Assuming that thermal agitation and interactions between species and
with the wall are negligible and that Stokes law [24] is applicable, the
species is submitted to the following forces:
Lorentz force:
𝑧.𝑒.𝔼 𝑧.𝑒. 𝑢 ∧ 𝔹,
the second term is rigorously null in the 1D model because it is
orthogonal to the velocity.
Drag force due to sweat opposition:
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through Sudomotor Function
163
𝜉.𝑢𝑣,
where 𝑣 is the speed of sweat (here 𝑣 0, because the application of
the electrode firmly against the skin plugs its gland pores and blocks the
physiological sweat flow) and 𝜉 is the Stokes coefficient [24] given by
𝜉 6𝜋.𝜇.ℋ,
where 𝜇 is the dynamic viscosity of the sweat (water) and ℋ the
hydrodynamic radius of the ion.
Stokes law is only approximative because the shape of the ions is
certainly not spherical. Moreover, at a microscopic scale, the ions bathe
in a medium far from being continuous but rather filled with particles of
similar size to one of the ions studied. However, the hydrodynamic
radius is the radius of a hypothetical hard sphere that diffuses with the
same speed as the ion. And in fact, deduced from the real mobility of the
ion in the electrolyte, defined by:
ℳ≡ 𝑙𝑖𝑚 𝑢/𝔼 𝑧. 𝑒/𝜉,
which was tabulated, see for example Atkins & al [25].
Thus, the resultant force is:
ℛ 𝑧.𝑒.𝔼 𝜉.𝑢
And per unit volume
ℱ ℛ.
≡ℛ.
.
We infer finally the momentum conservation law:
.
.
.
.𝑐.𝔼
.𝑐.𝑢, (6.7)
6.3.4.3. Momentum Steady Equation: Concentration Constancy
A numerical solution of the whole problem, i.e., the previous partial
differential equations augmented with suitable initial and boundary
conditions, although it may be delicate, is possible. Here it is conceivable
to simplify the problem to obtain analytical results in two ways:
Considering steady solutions, which allow capacitive aspects
and time derivatives to vanish;
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Considering, as assumed in Section 6.3.1, only electroactive
ions, i. e., 𝐶𝑙
at the anode and 𝐻
at the cathode.
At steady state, the equation for the momentum, for chloride near the
anode, reduces to:
.
.𝑐.𝔼
𝑐.𝑢
A main result is the theorem of constant concentration:
Assuming that 𝑐 is constant then:
𝑐
∗
𝑐𝑐
∗
.1𝛼,
with
𝛼
...
.....
.....
,
𝑐
∗
.
.
,
𝒞
.
.
Consequence:
𝑰𝒇 𝜶 ≪𝟏 𝒕𝒉𝒆𝒏 𝒄 𝑪𝒕𝒆 𝒄
∗
The proof is developed in Appendix 3.
Some numerical applications are necessary to obtain the order of
magnitude of the quantities 𝑐
∗
,𝒞 𝑎𝑛𝑑 𝛼
, see Fig. 6.8.
Thus, in both cases, it appears clearly that the error is absolutely
negligible and this leads to a constant distribution of the concentrations
along the gland’s axis, remarkably almost equal, for chloride ions, to the
concentration in the interstitium (bath): ~120 𝑚𝑚𝑜𝑙/𝐿. Fig. 6.9 illustrates
this remarkable result and recalls the shape of the distribution of the
concentration of chlorides in the healthy physiological state: isotonic
filtration in the secretory coil and reabsorption in the
excretory duct.
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
165
Fig. 6.8.
Numerical application for
𝑪𝒍
near the anode and
𝑯
near the cathode.
Uniqueness of this solution is ensured since 𝒞≪1 and all the other
terms in equation (6.20) of Appendix 3 are bounded and of the
order of ~1.
Fig. 6.9.
Distribution of the concentration of chlorides inside the eccrine gland,
along the axis of the gkand.
This simple result is noticeable and will simplify the mass equation
because the equilibrium potential will be null.
6.3.4.4. Mass Steady Equation
For the proton, there is no dedicated channel; it is a “passe-partout”.
For chloride ions, the steady state can be expressed as 𝑐𝑥 𝐶𝑡𝑒
𝑐
, which implies that the equilibrium (Nernst) potential, see 6.3.3.2,
is constant and almost null: 𝛷
𝐶𝑡𝑒0.
µ[k g/(m.s )] 0,001 Cl‐ H+ Cl‐ H+
σ[S/cm] 0,01 M [m2/(s.V)] 7,98E‐08 3,6230E‐07 m[kg] 5,8116E‐26 1,6605E‐27
e[C] 1,602176E‐19 H[m] 1,00E‐10 2,3459E‐11 ζ 2,0077E‐12 4,4222E‐13
F[C/mol ] 96485 M[kg/mol] 0,035 0,001 C3,8964E‐17 1,1133E‐ 18
N 6,022415E+23 c*[mmol/L ] 129,88 28,61
r[cm] 0,001
α
abs[mmol/L ] 1,4211E‐14 ‐1,7764E‐15
Gmax[ µS/cm2] 10
α
relat[ %] 1,0942E‐ 14 ‐6,2095E‐15
Ф
max[ V] 1
Genera lDa ta Io nsDa ta Input Ions Dat aO utput
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So that in all cases, the conservation of mass is reduced to:
.
.
.
.𝐺.𝛷 (6.8)
Taking into account Ohm’s law 𝑐.𝑢
.
, at steady state, the
potential can be depicted using an ordinary (non-linear) differential
equation:
.
.𝐺.𝛷, (6.9)
with 𝐺 𝐺𝛷.
Hence, a complete decoupling between the equations is observed and the
whole problem can be solved sequentially: first, the momentum balance
yields the constant concentration, then the mass equation (6.9) will lead
(see next subsection) to the potential, and finally the velocity can be
deduced from Ohm’s law.
6.3.5. Steady State Additional Analytics for the Conservation of
Mass Equation
Considering the steady conservation of mass equation just obtained,
augmented with suitable boundary conditions, and recalling that no
steady contact discontinuity in a medium with moving species, as seen
in Appendix A1.4, then the model reduces to the system of differential
equations, with notations from Fig. 6.5.
Current continuity at the entry of the duct:
0
Balance in the duct:
.
𝐺
Φ
.Φ
Combine with the coil (continuity of current and potential):
𝑟
𝐿
𝑟
𝐿
,Φ
𝐿
Φ
𝐿
Balance in the coil:
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
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167
.
𝐺
Φ
.Φ
Current continuity at the end of the coil:
𝐿
𝐿
.
To progress in the exploration for the last equations, a minor but quite
obvious assumption is necessary:
∀Φ0,𝐺Φ0 and 𝐺 increasing
8
Then the first lemma can be proven:
∀𝑥,Φx Φ0Φ
and Φ decreasing
The proof is demonstrated in Appendix 4.
6.3.5.1. The Voltage inside the Gland is Almost Constant
In fact, because the human skin is electrically weakly conductive, the
model solution can be shown to be quite close to the linearized problem.
In a first step, we begin by the lemma:
Consider the simplified linearized model (with the same upper boundary
condition):
.
𝐺
Φ
.Φ
𝐶𝑡𝑒,
𝐿
𝐿
.
,
Then:
∀𝑥,Φ
𝑥Φ
and Φ
decreasing,
∀𝑥,ΦxΦ
𝑥
Now, the important corollary can be stated:
8
This hypothesis can be relaxed by assuming it is monotone, which is always observed
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We have
∀𝑥,Φ
.1 𝜀ΦxΦ
,
with
𝐺
𝐺
Φ
and 𝐺
𝐺
Φ
,
𝜀
.
.
𝑟
ℎ𝐿
𝑟
.𝐿
𝑟
.𝐿
𝑟
2𝑟
ℎ𝐿
𝐿
2ℎ𝐿
The order of magnitude is given in Fig. 6.9 where the parameters of the
gland are taken from [12, 14]. No doubt, 𝜀8 % ≪1. To illustrate this,
see the following Fig. 6.10(a) for realistic simulations corresponding to
different patient status.
The proof of the lemma and corollary is demonstrated in Appendix 5.
6.3.5.2. The Axial Current is Almost Piecewise Linear
The important corollary continues as follows:
Let 𝐼
Φ
,𝑥 be the axial current and 𝐼
_
Φ
,𝑥 be the simplified
current, then
𝐼
_
Φ
,𝑥
π.Φ
.2𝐺
Φ
.𝑥𝐿
𝑟
𝐺
Φ
.𝑟
2𝐿
𝑟
, i𝑓 𝑥 𝐿
𝐺
Φ
.𝑟
2𝑥2𝐿
2𝐿
𝑟
,otherwise
And if 𝜀1:
∀𝑥,𝐼
_
Φ
.1𝜀,𝑥𝐼
Φ
,𝑥𝐼
_
Φ
,𝑥
Fig. 6.10(b) confirms this result with realistic simulations.
For the proof, see Appendix 6.
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
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Fig. 6.10.
Along the gland axis a) Voltage is almost constant, b) Current is
quite piecewise linear, Normal and Cystic fibrosis curves are confused because
the voltage imposed here (0.5 V) is before detachment (see Fig. 6.12 below).
6.3.6. Skin Conductance: Gland Wall Ion Permeability
One parameter of the measured signal, among others, see Section 6.4, is
the Electrochemical Skin Conductance (ESC). It is deduced from the
(axial) current crossing the electrode and the applied voltage Φ
.
Because, from the previous formula
𝐼
_
Φ
,0 𝜋.2𝐺
Φ
.𝐿
.𝑟
𝐺
Φ
.𝑟
2𝐿
𝑟
.Φ
The main theorem can be proven:
Notations by definition:
𝐸𝑆𝐶Φ
,
, (6.10)
The (total) conductance of the gland wall:
𝐺𝑊𝐶Φ
2π.𝑟
.𝐿
.𝐺
Φ
2π.𝑟
.𝐿
.𝐺
Φ
π.𝑟
.𝐺
Φ
Result:
1𝜀𝐺𝑊𝐶Φ
.1𝜀𝐸𝑆𝐶Φ
𝐺𝑊𝐶Φ
To demonstrate this, the previous result with the axial current can
be used:
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∀𝑥,𝐼
_
Φ
.1𝜀,𝑥𝐼
Φ
,𝑥𝐼
_
Φ
,𝑥,
⇒𝐼
_
Φ
.1𝜀,0𝐼
Φ
,0𝐼
_
Φ
,0,
and
𝐼
_
Φ
,0 𝜋.2𝐺
Φ
.𝐿
.𝑟
𝐺
Φ
.𝑟
2𝐿
𝑟
.Φ
6.3.7. Normalization
To conclude this section, normalization of the conductances is necessary
to refine results. The range of skin conductance is from a few µS to
several hundred µS. The aim of this normalization is to pack very high
and healthy conductances as well as stretch weak and problematic
conductances. For a gross conductance 𝑌 expressed in µS, the
normalization is defined by the increasing dimensionless bijection from
0,∞ to 0,100:
𝑌
.
,
where 𝑋
40 µ𝑆
The theorem is written:
Given:
𝜃
.
.
.
,
where 𝐺𝑊𝐶Φ
in µ𝑆. Then:
∀Φ
,1𝜃.𝐺𝑊𝐶
Φ
.1𝜀𝐸𝑆𝐶
Φ
𝐺𝑊𝐶
Φ
The order of magnitude is given in Fig. 6.11. No doubt, 𝜃0.8 %
𝜀8 % ≪1.
The second inequality is obvious thanks to the increasing bijection. For
the first one, we have:
𝐸𝑆𝐶
Φ
.
.
.
.
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171
𝐺𝑊𝐶
Φ
.1𝜀.1𝜀𝑋
𝐺𝑊𝐶Φ
.1𝜀
𝑋
1𝜀𝐺𝑊𝐶Φ
.1𝜀
1𝜃.𝐺𝑊𝐶
Φ
.1𝜀
Here, in Fig. 6.12, are displayed realistic simulations corresponding to
different patient status and their conductances. An actual measurement
will be presented in Section 6.4.
Fig. 6.11.
Numerical application with very high and realistic electrochemical
skin conductance.
Fig. 6.12.
Curve current – Applied anodic voltage.
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6.3.8. Section Conclusion
The voltage applied on the anode is nearly the same as the one applied
on the nerves on the secretory portion, which confirms that these
sympathetic nerves are electrically stimulated and can deliver, if they are
present, their neurotransmitters to open the chloride ions channels as
seen in Section 6.2.
ESC measurement gives the possibility of strictly determining the
conductance of the walls of the eccrine glands, i.e., the capability of the
gland to secrete ions. Thus, when these glands are activated and
stimulated by electric polarization, sudomotor function is evaluated very
precisely. ESC measurement, in addition to other parameters of the
signal, (see next section), are measurably impacted in certain pathologies
and treatments, as further described in Section 6.5.
Thus, to conclude here, ESC does not depend on concentrations, sweat
conductivity or stratum corneum thickness. As for prospects, the
technology could be used as an in vivo and non-invasive tool for the
reverse problem: deduce immediately the surfacic conductances of the
two portions, the excretory and secretory, and also link the sudomotor
function and electrical characteristics of glands: 𝐶,𝐺 to structure i.e.,
muscarinic and adrenergic innervation quality and diverse ion channels,
pumps, shunts and co-transporters.
6.4. The Sudoscan Technology
Two PhD research projects have been devoted to the electrochemistry of
the technology, notably, the electrochemical behavior of different types
of electrodes with an electrolyte mimicking sweat, see publications [28]
and [31], in collaboration with the Engineering Institute Chimie
ParisTech, France (part of PSL Research University). Fig. 6.13 displays
the ranges of involved currents.
The first study showed that different types of nickel and/or stainless steel
electrodes generate a very high current consumption in NaCL solutions
of different Cl
-
concentrations and pH as long as the voltage applied is
greater than 500 mV for human physiology-mimicking conditions. This
work also showed the importance of “cleaning” or regenerating the
electrode from cumulative oxyde layers of complex compounds by
applying a reverse negative voltage after each test.
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Fig. 6.13.
(Surfacic) current ranges of in vitro experiments
and in vivo measures.
The second study was able to mimic the condition of an in vivo
measurement considering the real kinetics of the electrochemical
reactions due to the conductance of the sweat glands. A central finding
in [27] showed the very poor result of single voltage voltammetry due to
the large variability of the layers on the electrodes depending on each
patient’s pH and 𝐻𝐶𝑂
concentrations.
6.4.1. Sudoscan Medical Device Description
The simplified principle of Sudoscan technology consists in imposing on
human skin decreasing pulses of low direct current voltages and to
collect, once the capacitive and transient effects have stabilized, the
electrochemical response of the skin. It is therefore a question of multiple
steady chronovoltammetry, possibly thanks to the reverse iontophoresis
through the eccrine sweat glands, essentially chlorides at the anode and
protons at the cathode.
No specific patient preparation (fasting or other) or medical personnel
training is required for Sudoscan testing. The measure is rapid
(approximately 2 minutes) and non-invasive.
Measurements are performed on glabrous skin surfaces where the
eccrine sweat glands are the most numerous: on the palms of the hands
and soles of the feet. The apparatus consists of two sets of two electrodes
for the hands and the feet, connected to a computer for recording and
data management purposes, see Fig. 6.14. To conduct the test, the
patients are required to place their hands and feet on the electrodes. They
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must then stand still for the 2-minute duration of the test, in contact only
with the electrodes.
Fig. 6.14.
Photo of the medical device and its electrodes.
Large area stainless steel electrodes are used alternately as anode
or cathode.
The overall measurement scheme, see patent [26], is presented in
Fig. 6.15.
Fig. 6.15.
Schematic circuit of the measure.
At the low voltages used, the stratum corneum is purely capacitive; the
steady current therefore only passes through the sweat glands. The active
electrodes are the anode where the voltage is imposed, and the cathode
connected to ground by a calibration resistance making it possible to
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175
close the circuit and deduce the current. The other 2 electrodes, Hi1 and
Hi2, are connected to the ground by high impedances, so that no current
flows through them and they serve to recover the voltage reached
by the body.
The voltages imposed on the active anode for the fifteen steps, as well as
their durations, are shown in Fig. 6.16.
Fig. 6.16.
Anode voltage steps.
At the end of the recording, the ESC for all 4 extremities are displayed
immediately. The results are expressed in µS (microsiemens). The test
also evaluates the percentage of asymmetry between the left and right
side, for both hands and feet, indicating whether one side is more
impaired than the other. The results can also be displayed in the form of
a geometric figure that allows for rapid interpretation (see example in
Section 6.5.2.5).
6.4.2. The Signal Measured and Its Main Parameters
For each limb (hand or foot) and on each side (left or right), the raw
signal thus collected is a current-voltage curve at the anode and at the
cathode. An actual measurement is presented in Fig. 6.17.
Only the various voltages
Φ
,
Φ
,
Φ
and
Φ
are measured; the
current in the circuit, between anode, cathode and ground, is
expressed as:
𝐼
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Fig. 6.17.
The current-voltage curve measured.
A first important parameter is the low conductance LC. It is the ESC at
that voltage (see figure), defined previously, in equation (6.10) by the
chord. However, here ESC is defined by the dynamic conductance
between steps 13 and 14, because of overpotential and offset (see next
parameter), and because this low part is linear. It is given by the discrete
slope (here at the anode and similar at the cathode):
𝐿𝐶≡∆𝐶13,14
Φ
Φ
Φ
Φ
Below, the main other parameters considered:
The offset, Oa: results from the overpotential (voltage consumption,
diode effect) omnipresent at the anode in electrochemistry, due to the
electrode surface oxidation and corresponding to the bottom of the
oxidation wall. It depends in particular, see [27], on certain
concentrations of the electrolytes in sweat (chlorides 𝐶𝑙
and
carbonates 𝐻𝐶𝑂
) and the material composition of the electrode,. It can
also be affected by certain pathologies or treatments.
The first voltage step is not used directly, but it stabilizes the
overpotential and offset due to its extended duration of eight seconds
instead of the usual one second. The last level is also not used; instead,
it confirms that the measured signal has not reached the oxidative wall.
Once the offset has stabilized, the entire measured signal is just a simple
shift, without distortion, of the purely physiological signal. On the other
hand, as the lower part of the signal is generally – and is assumed and
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found to be – linear, this offset is obtained by extending the slope of the
low conductance towards the abscissa axis, by (see figure):
𝑂𝑎 Φ
Φ
By shifting the signal of the offset, the conductance 𝐶
for any step 𝑠 is
then deduced simply by the chord:
𝐶
The high conductance HC is defined by:
𝐻𝐶 𝐶
The detachment ratio 𝑟:
𝑟
The coefficient of non-linearity 𝜌, quantifying the non-linearity at the
bottom of the curve (~ second derivative):
𝜌≡
∆,
∆,
So if 𝜌1∶ linear lower part,
If 𝜌 1∶ early detachment case,
If 𝜌 1∶ case of early landing.
The hand to foot ratio 𝑅:
𝑅
At the cathode, the formulas are analogous. Note that the offset on the
cathode side is very low because the first step allows the regeneration
[28] of this electrode.
6.4.3. A Much Desired and Important Property of the Measure:
Reproducibility
Besides being rapid, non-invasive and quantitative, the technology is
able to recognize any movement or tremor during the scan by detecting
a break in the monotony of the signal and will alert the technician to the
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failure of the measurement. In addition, multiple studies were completed
showing that the technology has a good reproducibility [29]. Two test
measurements were assessed on the same day in patients with at least
one cardiovascular risk and in patients with diabetes; in a second study
healthy individuals and patients with diabetes completed 2 tests on each
of 3 different devices. Results were compared using a Bland and Altman
plot. (see 6.5.2.1)
This very good reproducibility can be explained by several factors:
a) The large number of sweat glands measured on the palms and soles
(several thousands) which average the effect of any disease;
b) The application of the electrode firmly against the skin which plugs
the pores of the glands and blocks the physiological sweat flow, thus
minimizing a possible dependence on test conditions such as stress,
temperature, physical effort or exercise [30];
c) The steady measure, i.e., after the stabilization of the transitory
capacitive aspects of the SC, the gland and the sweat/electrode
interface;
d) The active measure with the same constant polarizations, about ten
times higher than physiological ones;
e) The non-dependence of the measure on the stratum corneum
thickness, as noted above, which warrants that even calloused feet
can be accurately measured;
f) The ESC being computed by the slope of the curve (resulting
current over applied voltage), as previously explained and therefore
not polluted by any electrochemical overpotential of the
electrode [27];
g) The internal electronic circuit of the technology, which measures
the voltages precisely with an accurate analog-to-digital converter
having a resolution of 10 bits;
h) The high sensitivity of stainless steel electrodes which creates a
strong current generator due to Cl
-
consumption [31].
6.4.4. Section Conclusion
As a conclusion for this section, the measured Sudoscan signal is very
rich: the linear lower part followed by detachment (or landing) make it
possible to assess with precision and in a reproducible way the
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sudomotor function. Multiple steady chronovoltammetries allow a signal
measurement over a wide range of voltages, without electrochemical
pollution, contrarily to a single voltammetry [27].
6.5. Clinical Applications for Various Pathologies
and Treatments
Sweat glands are innervated by small C fibers that can be affected early
in systemic diseases like diabetes or amyloidosis. Impairment of these C
fibers portends the progression to a more diffuse neuropathy. Small
fibers are often the first to degenerate in diabetic peripheral neuropathy
and are responsible for loss of pain, cold, and heat perception, increasing
the risk for foot trauma. In addition, impaired sweat function, resulting
in dry skin and loss of healing, further accelerates the development of
diabetic foot complications.
Usual clinical tests for peripheral neuropathy mainly explore large fibers
and are operator dependent and time consuming; more importantly, large
fiber deficits – and thus their detection with usual clinical tests – occur
late in the evolution of the diabetic neuropathy process. Therefore, a
small fiber neuropathy (SFN) test can be helpful for at least three
reasons: i) Making the diagnosis of SFN can lead to a focused search
early for its etiology; ii) Disease modifying or symptomatic treatments
can be started early; and iii) Awareness of the disease can increase
patient’s compliance, which is critical in the follow-up of a chronic
condition like diabetes.
In the context of this widespread need for a rapid, non-invasive,
objective, quantitative, accurate and easy to use SFN diagnostic tool, the
Sudoscan device, measuring sudomotor function, was developed.
After a brief review of the method as it is explained to clinicians who use
the device in daily practice, the clinical development strategy and
principal research results will be presented. In this section ESC
correspond to Low Conductances (LC) as defined in the
previous section.
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6.5.1. Principle of the Method as It Is Explained to Clinician Users
“Sudoscan measures the capacity of the sweat glands to release chloride
ions in response to an electrical stimulation. The patient is only asked to
place the palms of the hands and soles of the feet, areas with the highest
sweat gland density, in contact with two large electrodes for 3 minutes.
Gradual voltages less than 4 V are applied to the electrodes. There is an
observable electrochemical reaction between the Cl
-
ions present in
sweat and the electrodes. The device records an electrochemical
conductance (ESC) related to the amount of chloride ions extracted from
the sweat glands and detected by the electrodes. The level of conductance
depends on the number of chloride ions that react with the electrodes. In
patients with normal sweat function, many chloride ions react with the
electrodes when direct current is applied to excite the small nerves on the
glands and open chloride ion channels; this results in a high level of
conductance. In patients with sweat dysfunction, less chloride ions are
attracted and the level of conductance is lower (Fig. 6.18).
Fig. 6.18.
Illustration of the mechanism as presented to clinician users.
6.5.2. Clinical Development
As there was no equivalent device available that had already established
the clinical utility of this sudomotor methodology, the Sudoscan
technology had to be evaluated in clinical studies. Since the test is rapid
and non-invasive – in contrast to the majority of clinical and research
tests used in the diagnosis of SFN – the principal clinical studies were
performed as add-on assessments during standard of care follow-up
of patients.
The aim of the clinical development program was to establish the
following properties of the Sudoscan technology to support its diagnostic
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181
clinical utility: i) Normative data and accuracy of the method;
ii) Performance in the assessment of sweat function and small fiber
neuropathies as compared to usual reference methods; iii) Performance
in the detection of peripheral neuropathy in patients with diabetes;
iv) Performance in the diagnosis of peripheral neuropathy in patients
with neurodegenerative diseases; v) Performance in the follow-up of
patients with peripheral neuropathy.
6.5.2.1. Normative Data and Accuracy of the Method
Normative ESC values in adults were defined in a population of over
1350 healthy subjects. Mean ESC for women or men at the hands
(75 [57-87] vs. 76 [56-89] μS, p = 0.35) or feet (83.5 [71-90] vs.
82.5 [70-91] μS, p = 0.12) [32] were not significantly different. Overall,
there was no effect on ESC of body mass index (BMI), or exercise status;
a very small (and clinically insignificant) decrease with age, and a
significant effect of race/ethnicity. Specific ESC thresholds have thus
been defined for patient race/ethnicity and are applied for proper test
interpretation (Fig. 6.19).
Sudoscan tests completed among 100 healthy children confirmed that
normative values were equivalent to those for adults
[33].
Fig. 6.19.
Evolution of Hands ESC and Feet ESC with age on 1350 healthy
volunteers from [32].
The accuracy of the method was determined according to FDA
guidelines (2 measurements performed on each of 3 devices, for a total
of 6 Sudoscan tests per patient), and evidenced a coefficient of variation
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of feet / hand ESC of 4 % in healthy subjects and 7 % in patients with
diabetes [34].
The results for these studies established that the Sudoscan technology
was robust under a variety of clinical circumstances and for a wide range
of populations; additionally, if the technology is used to monitor patients
over time, the good reproducibility ensures that a change in ESC is a
reliable marker of sudomotor function change and should prompt further
investigation.
6.5.2.2. Performance in the Assessment of Sweat Function
and Small Fiber Neuropathies as Compared
to Usual Reference Methods
Established sudomotor function tests have long been recognized as some
of the most sensitive and specific means to diagnose distal small fiber
neuropathies. However, they are also all affected by ambient
temperature, hydration status, medications, age and gender, previous
exposure of the skin to alcohol, repeated testing over the same region,
and application of moisturizing creams – factors to which Sudoscan, as
discussed above, is not necessarily prone [35]. Though restricted to
specialized centers, QSART is probably the most widely available
sudomotor function test and thus most commonly used clinically.
Sudoscan was compared to the QSART in 3 studies that documented a
larger Area Under the Curve (AUC) (0.77 vs 0.57, 0.71 vs 0.53 for
Sudoscan
compared to QSART, respectively) in the diagnosis of small
fiber or diabetic neuropathy [36, 37] (Fig. 6.20, Table extracted from
[37]). Though established as a standard, validated test of sudomotor
function, QSART appears to have no table diagnostic limitations for
small fiber and diabetic neuropathies – in addition to the fact that the
technology has significant technical requirements, limited availability,
and a high susceptibility to environmental factors.
Two studies established the higher performance of Sudoscan as
compared to Neuropad® [38] (poster Ziegler D et al, Neurodiab 2017,
publication ongoing).
In a comparison with 4 reference diagnostic methods for small fiber
neuropathy, Sudoscan was found to be less sensitive than Laser Evoked
Potential (LEP), a highly specialized and time-consuming research
method; however, Sudoscan had marginally better diagnostic
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
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183
performance than Quantitative Sensory Testing (QST) for warm
detection and much better performance than cold detection and
sympathetic skin response [39].
Controls Diabetes DPN P
value
N = 16 N = 20 N = 27
ESC feet (µS) 83.0 (16.3)
A
76.3 (15.9)
AB
64.4 (15.9)
B
0.017
ESC hands (µS) 71.3 (18.0) 70.4 (17.6) 58.2 (18.2) 0.037
Sweat latency,
distal leg (B)
a
87.2 (49.4) 96.4 (49.5) 90.5 (49.7) 0.909
Sweat volume,
distal leg (µl)
a
0.90 (0.60) 0.70 (0.60) 0.60 (0.60) 0.107
Sweat latency,
distal foream (B) 80.3 (41.1) 89.1 (40.2) 84.9 (42.3) 0.807
Sweat volume,
distal foream
(µl)
a
1.00 (1.40) 1.20 (1.40) 1.30 (1.40) 0.4
Fig. 6.20.
Mean values of ESC and QSART parameters in healthy controls,
patients with diabetes but no diabetic peripheral neuropathy (DPN), and patients
with diabetes and DPN (Data are age- and gender adjusted mean (SD).
a
Log-transformed variable. ESC, Electrochemical Skin Conductance. DPN
Diabetic Peripheral Neuropathy. Unequal superscript letters indicate significant
group differences, Table extracted from [37].
The authors concluded that LEP, QST for warm detection and Sudoscan
provide significant diagnostic sensitivity compared to other tests of small
fiber neuropathy evaluated, and that the combination of all three tests
provided improved diagnostic accuracy. However, it must also be noted
that LEP and QST are limited to research institutions and therefore
cannot be considered practical for real world diagnostic evaluation of
small fiber neuropathy. Finally, the clinical strategy proposed by this
reference center, when a patient presents for small fiber neuropathy
assessment is to perform a Sudoscan. If Sudoscan is positive, SFN is
established and taken into account for patient follow-up. If Sudoscan is
normal, another more specialized test is performed to confirm
absence of SFN.
The current gold standard in the diagnosis of distal small fiber
neuropathy is intraepidermal nerve fiber density (ENFD). This test
requires removal of a small skin punch biopsy at the lower leg, and
shipment to a specialized lab for processing and microscopic
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examination prior to results being available. Several studies investigated
the diagnostic performance of Sudoscan and ENFD in healthy controls
and patients referred to specialized centers for small fiber neuropathy
evaluation. Smith et al. demonstrated that feet ESC and ENFD had
similar AUC (0.761 and 0.752, respectively) [40], while Novak et al.
demonstrated high correlation between ESC and ENFD adjusted for
weight (r = 0.73, p = 0.0001; Fig. 6.21) [41]. These findings suggest that
Sudoscan technology is a robust alternative diagnostic test for small fiber
neuropathy, providing a non-invasive, painless assessment with results
immediately available. In addition, repeated testing for follow-up of
treatment or disease progression is markedly simpler and more
acceptable with Sudoscan than skin biopsy.
Fig. 6.21.
Correlation between FESC and skin biopsy (Epidermal Nerve Fiber
Density (ENFD) and Sweat Gland Nerve Fiber Density (SGNFD) from [41].
Finally, a significant correlation between Sudoscan ESC and corneal
nerve fiber length (CNFL) assessed using in vivo laser confocal
microscopy has been evidenced (r
2
= 0.8) [42].
CNFL is considered a
surrogate marker of denervation in diabetic sensorimotor and autonomic
neuropathy; in this study, the authors evaluated patients with
transthyretin familial amyloid polyneuropathy, a genetic disorder in
which small fiber neuropathy is a prominent feature. The authors showed
that ESC and CNFL – two non-invasive, quantitative tests with
immediately available results – were very highly correlated in a small
population (n = 15) with small fiber neuropathy of varying severity. It
must be noted, however, that confocal microscopy requires complex
ophthalmological equipment and analytical software, as well as
anesthetizing the patient’s cornea for examination.
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185
6.5.2.3. Performance in the Detection of Peripheral Neuropathy
in Patients with Diabetes
Among patients with diabetes, peripheral neuropathy is the most
common complication, affecting up to 70 % of patients during their
lifetime; diabetic peripheral neuropathy (DPN) results in significant
morbidities such as a chronic pain, loss of sensation, foot ulcers,
gangrene, and amputations. Distal sensorimotor peripheral neuropathy is
a dying-back disorder, most commonly affecting small nerves on the
extremities first. Early detection could allow identification of and
aggressive intervention in those at greatest risk for further morbidity.
Considering the vast number of patients with diabetes across the world,
its unrelenting chronicity, and the complicity of care of each patient, a
simple, rapid, but quantitative test for DPN is valuable for the initial
triage and ongoing follow-up of this population. Nine studies involving
more than 1000 patients with diabetes showed sensitivities from 73 to
97 % to detect peripheral neuropathy with negative predictive values
from 83 to 94 % when Sudoscan was compared to reference symptom
scores or usual tests for DPN [43]. The high negative predictive value of
Sudoscan is particularly important as it can reassure clinicians to the
current lack of DPN in patients with normal ESC.
Fig. 6.22.
Feet ESC distribution according to grade of diabetic foot risk
in 2400 patients from 4 French Hospitals.
Boulton et al. [44] initially developed the scale of 4 risk categories based
on results of a comprehensive foot examination; it was to be used to
Advances in Biosensors: Reviews. Volume 3
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determine how frequently patients with diabetes should undergo foot
evaluations and by whom (generalist or specialist). A risk of 0 indicates
no loss of protective sensation (LOPS), no peripheral artery disease, and
no foot deformity. In contrast, a risk of 3 is reserved for patients with
prior ulcer or amputation. Unfortunately, rarely is a comprehensive foot
exam performed on every diabetic patient on a yearly basis, as
recommended by the American Diabetes Association. In addition, even
a risk category of 1 represents relatively advanced DPN, when
LOPS – suggesting myelinated nerve fiber damage – has developed. A
study is ongoing on more than 2400 patients with diabetes to compare
ESC values and gradation of diabetic foot risk (Fig. 6.22, poster
presentation in EASD, 2019, publication ongoing). The figure below
shows that even among patients rated as risk category 0, 40 % have
abnormal Sudoscan results and may be at elevated risk for diabetic foot.
Sudoscan may be more sensitive early in the progression of diabetic foot
– before clinical signs become apparent – and allow for prompt
identification of patients requiring intensified foot care or referral
to podiatry.
6.5.2.4. Performance in the Detection of Cardiac Autonomic
Neuropathy in Patients with Diabetes
As stated in the introduction, the autonomic nervous system is the
primary extrinsic control mechanism regulating heart rate, blood
pressure, and myocardial contractility. Cardiac autonomic neuropathy
(CAN) describes a dysfunction of the ANS and its regulation of the
cardiovascular (CV) system. CAN is an important risk factor for CV
morbidity and mortality. In the diabetes population, the prevalence of
CAN varies from 2.5 % to 50 %; it is implicated in a five-fold increased
risk of CV mortality and in fact is the strongest predictor for mortality in
diabetes mellitus [45].
The American Diabetes Association has recommended screening
patients with diabetes for autonomic neuropathy, but only by assessing
for symptoms and signs in those with microvascular complications.
However, early symptoms of CAN tend to be nonspecific and frank
clinical manifestations of hypoglycemia unawareness, resting
tachycardia, orthostatic hypotension, gastroparesis, constipation,
diarrhea, or fecal incontinence occur too late to be reversible. Thus, a
diagnosis of CAN is frequently delayed or may never occur before the
occurrence of a catastrophic event.
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
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187
Cardiovascular autonomic reflex tests (CARTs) remain the gold standard
for the diagnosis of CAN but are time-consuming and are limited to
highly specialized centers. Therefore, universal screening for and
diagnosis of CAN are widely disregarded in clinical practice. This
dichotomy between the ideal and actual reality highlights the need for
diagnostic procedures of sufficient reliability and accuracy that are at the
same time accessible and easy, and that might also allow for the selection
of persons at higher risk for CAN to be further evaluated with CARTs.
Three studies performed on significant groups of patients evidenced the
performances of feet ESC (FESC) in the screening of CAN in patients
with diabetes. Sensitivities of FESC to detect CAN or confirmed CAN
diagnosed according to CARTs were between 80 and 100 % [46-48]
(Fig. 6.23).
Fig. 6.23.
ROC curve of feet electrochemical skin conductance (FESC)
to diagnose CAN, AUC = 0.86, from [46].
D’Amato et al. studied over 100 patients with diabetes with COMPASS
31 (a simple, self-administered questionnaire), FESC and CARTs; CAN
was defined as ≥ 1 abnormal CART and confirmed CAN as ≥ 2 abnormal
Advances in Biosensors: Reviews. Volume 3
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CARTs. The authors noted that either an abnormal COMPASS
31 and/ or abnormal FESC score has a 92 % sensitivity to identify CAN
and 100 % sensitivity for confirmed CAN [48].
Importantly, CAN was noted to be reversible in a population of severely
obese diabetic patients following bariatric surgery: both FESC and
cardiac autonomic function improved at 12 and 24 weeks; interestingly,
improvements in CAN measures were not correlated with changes in
weight, BMI, body fat, or lipid level. These studies demonstrate that
sudomotor function testing can detect asymptomatic CAN in the clinical
setting, select patients for more advanced testing and/or aggressive
treatment, and may allow for easy assessment of treatment efficacy of an
often-deadly complication [49].
6.5.2.5. Performance in the Diagnosis of Peripheral Neuropathy
in Patients with Neurodegenerative Diseases
Sudoscan was evaluated in several diseases involving small fiber
neuropathies.
Hereditary transthyretin amyloidosis is a rare, autosomal dominant,
progressive and fatal disease caused by mutations in the TTR gene.
Amyloid polyneuropathy (TTR-FAP) is a progressive sensory–motor
and autonomic neuropathy, initially affecting small fibers followed by
larger fibers. Early identification is critical to prevent permanent tissue
damage, in particular cardiac autonomic neuropathy which contributes
to premature demise. Unfortunately, most established small fiber and
autonomic tests have low sensitivity and specificity in TTR-FAP or are
impractical for chronic clinical care. Patients with familial amyloidosis
demonstrate a dramatic decrease in ESC, and AUC of feet ESC to detect
dysautonomia was 0.76 [50] (Fig. 6.24). Feet ESC were decreased in 40
% of pauci-symptomatic patients; in addition, correlations with clinical
symptom scores, Neuropathy Impairment Score (NIS) and Karnofsky
performance status, were 0.62 and 0.61, respectively.
Sudoscan was found to be similarly effective in other disorders where
autonomic neuropathy or SFN is prominent: in Parkinson’s disease the
AUC to detect dysautonomia was 0.82 [51]. A decrease in feet ESC was
observed in 87 % of patients with Gougerot-Sjögren and SFN [52].
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
189
Fig. 6.24.
Values of Feet ESC in healthy controls and patients
with transthyretin-related familial amyloid polyneuropathy (TTR-FAP)
according to presence of symptoms and dysautonomia from [50].
6.5.2.6. Performance in the Follow-up of Patients with Peripheral
Neuropathy
As Sudoscan is non-invasive, quantitative and highly reproducible, it can
be used for the follow-up of patients to assess treatment efficacy. A
significant increase in ESC was demonstrated among subjects with
limited cardiorespiratory performance after implementation of lifestyle
changes [30]. In diabetic patients, an improvement in ESC has been
observed after bariatric surgery [49]. In patients receiving chemotherapy
(oxaliplatin or taxanes), a decrease in ESC could be measured in parallel
with clinical deterioration [53]. Longitudinal studies have been
performed in patients receiving treatment for TTR-FAP [54], Vitamin
B12 deficiency [55]
,
or painful small fiber neuropathy (capsaicin patch
[56]), all supporting the ability of Sudoscan to quantify changes in
nerve function.
Overall, more than 120 papers involving Sudoscan have been published
in peer-reviewed journals. These have established the technology’s
robustness, performance, and normative data, and demonstrated its
clinical utility; but most importantly, Sudoscan has been shown to be
safe (no adverse event related to Sudoscan reported in > 1,000,000 tests
completed) and suitable for use in all clinical settings.
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Fig. 6.25.
Changes observed in feet ESC 3 months after capsaicin patch
application from [56].
6.5.3. New Clinical Developments
As advanced previously in Section 6.4, a Sudoscan test incorporates new
parameters not clinically analyzed in the ESC results currently processed
and expressed. It is understood that some of these parameters may
provide more information on the ratio of adrenergic to cholinergic nerves
on the sweat glands, knowing that these nerves can regenerate at different
speeds [9]. Characteristics of these new parameters (mean, standard
deviation, median, interquartiles) have been calculated on a population
of more than 1200 healthy subjects without peripheral neuropathy. In the
near future, characteristics of these parameters in well characterized
patients including populations with Parkinson’s disease, receiving
chemotherapy, and with cystic fibrosis will be compared to healthy
controls. The next step will be to assess the evolution of these parameters
in the progression of the diseases.
Finally, another way of enhancing the detection and work-up of small
fiber and autonomic neuropathies is to complete the Sudoscan test along
with relevant, internationally recognized questionnaires. Patients can
easily answer questions about their symptoms during or just after
recording a Sudoscan test. Questionnaires can be selected according to
the suspected or known condition: a questionnaire for small fiber
neuropathy in patients where Sudoscan is used to explore SFN, or a
questionnaire adapted to diabetic foot risk in patients with diabetes.
A recent study has evidenced the benefits of combining the
COMPASS 31 questionnaire and Sudoscan to assess CAN. In this study
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
191
involving 102 patients with diabetes, COMPASS 31 and ESC
individually had sensitivities of 75 % and 83 %, respectively, for
confirmed CAN, and specificities of 65 % and 67 %, respectively, for
DPN. When combining the tests, the sensitivity for CAN rose to 100 %
for CAN and the specificity up to 89 % for DPN [48].
6.5.4. Conclusion on Clinical Applications
Assessment of autonomic dysfunction is important in the follow-up of
several diseases like Diabetes and Parkinson’s disease. Sweat glands are
innervated by small C fibers, and thus sweat function measurement is
now recognized as a reliable way to assess autonomic function and small
fiber neuropathy. Several methods have been developed but many of
them are not suitable for daily practice since they are invasive, time
consuming, highly specialized or of low reproducibility. Based on the
presence of chloride ions in sweat, Sudoscan was developed using the
electrochemical reaction between chlorides and stainless-steel
electrodes. Clinical development of this simple, rapid and noninvasive
method was performed to define normative values, reproducibility and
performance compared to established neurophysiologic testing methods.
In dozens of clinical studies, Sudoscan has been shown to have
diagnostic utility to assess diabetic foot risk, and quantify impairment of
autonomic and small fiber nerves in neurologic diseases. As the method
is highly reproducible it is well suited to the follow-up of patients to
evaluate treatment efficacy or neurotoxicity from chemotherapy. New
developments are ongoing to utilize additional test parameters to discern
among different disorders, and to add diagnostic value to Sudoscan with
the concurrent use of disease-specific questionnaires which can be filled
out during the test. The simplicity and performance of Sudoscan
technology has the potential to accelerate early detection of autonomic
and small fiber neuropathies in routine clinical practice, and provide
highly specific data to monitor patients much more precisely and
routinely than has been possible up until now. Ultimately, Sudoscan
could lessen the burden of costly and morbid neuropathic complications.
6.6. General Conclusion
This chapter shows the importance of measuring sweat function on the
palms and soles where the highest concentration of eccrine glands
occurs. Sudoscan patented technology benefits from a conjunction of
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very interesting physiological and physical phenomena. The
physiological one is related to the frailty of these long thin nerves
managing sweat glands, resulting in an excellent sensitivity to diagnose
early neuropathic complications.
The main physical phenomenon is the fact that the range of DC voltages
applied on the skin allows:
– An electrochemical reaction between chloride ions and stainless-steel
electrodes;
– A flow of chloride ions passing only through the sweat ducts (as the
stratum corneum cannot transfer ions at these voltages);
– The stimulation of small nerves on the glands which deliver
neurotransmitters to open muscarinic and CFTR ion channels;
– A flow of chloride ions generating a current related to the density of
small nerves on the gland, producing the exact conductance of the total
amount of glands.
These voltages surpass physiological voltages usually perceived by the
nerves and the ion channels, which leads to a very robust measurement
independent of environmental conditions in real life such as temperature,
emotion, and exercise as demonstrated in many clinical studies.
Overall, more than 120 papers involving Sudoscan have been published
throughout the world in peer-reviewed journals with very
consistent results.
New parameters included in the signal processed by Sudoscan as well as
the concomitant use of adapted questionnaires seem very interesting to
enhance the diagnostic and prognostic qualities of the technology.
Acknowledgements
We would like to thank Abdul Kayoum Moutairou, biostatistician at
Impeto Medical, Paris, France for the data mining results.
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Appendix 1. General Law of Conservation
A1.1. Integral (Global) Form
Let us consider a continuous medium where 𝜌 is the density or volumic
mass and 𝑈 is the velocity, in a material volume Ω(𝑡) 𝑖. 𝑒.: A subdomain
of the medium, of arbitrary size, composed of the same particles,
dependent of time 𝑡, that we will study and follow in his motion, of
boundary 𝜕Ω(𝑡) with its outward normal 𝑛. We note the spatial
coordinate (position) 𝑥∈𝛺𝑡.
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
197
Let 𝑎𝑥,𝑡 be a scalar or vector field enough regular, one can prove the
transport theorem:
𝑎𝑥,𝑡
𝑑𝛺𝑡
𝑑𝑖𝑣𝑎⨂𝑈
𝑑𝛺𝑡, (6.11)
⊗ designates the usual vector product,
is the (total or particular) time
derivative, and
is the partial time derivative.
And for a vector field 𝑓 enough regular, the divergence formula:
𝑓.𝑛 𝑑𝛴𝑡
𝑑𝑖𝑣𝑓𝑑𝛺𝑡
(6.12)
A balance equation for the quantity (𝜌 𝑎) can be written under the general
form:
ρ 𝑎
𝑑𝛺𝑡 𝒜 𝑑𝛺𝑡
𝐴.𝑛 𝑑𝛴𝑡
, (6.13)
where 𝒜 is a (volumic) production-disappearance term and 𝐴 is a
(surfacic) exchange flux across the boundary 𝜕Ω(𝑡).
A1.2. Conservative Differential (Local) Form
Using the transport theorem (6.11) and the divergence formula (6.12),
we get immediately:
𝜌 𝑎𝑑𝑖𝑣𝜌 𝑎⨂𝑈 𝒜𝑑𝑖𝑣𝐴 (6.14)
And in one space dimension (1D) of interest here because it will be
our case:
𝜌 𝑎
𝜌 𝑎 𝑢 𝒜
(6.15)
A1.3. Mass and Momentum
The classical conserved physical quantities are: Mass, momentum and
energy. The last one is of less interest here because it introduces thermic
power, heat exchanges and additional variables of whom at least the
temperature… Moreover it will not bring any thing here because the
measure is isotherm.
Advances in Biosensors: Reviews. Volume 3
198
For the two others, Table 6.1 summarizes:
Table 6.1.
Balance terms.
Balance
𝑎
𝒜
𝐴
Mass
𝟏
See after
𝟎
Momentum
𝑼
𝓕
𝟎
9
where ℱ) is the resultant of the external volumic forces.
A1.4. Case of Discontinuity
Consider a discontinuity surface Λ(𝑡) moving with velocity 𝑆. Let us no te
⟦𝑧⟧ 𝑧
𝑧
the jump of a variable 𝑧 across this discontinuity. Then
the divergence formula and the transport theorem can be generalized:
𝑓.𝑛 𝑑𝛴𝑡
𝑑𝑖𝑣𝑓𝑑𝛺𝑡
⟦𝑓⟧.
𝑛 𝑑𝛬𝑡
𝑎𝑥,𝑡 𝑑𝛺𝑡
𝑑𝛺𝑡
𝑎⨂𝑈.𝑛 𝑑𝛴𝑡
⟦𝑎⟧𝑆.𝑛 𝑑𝛬𝑡
And besides the previous conservation laws under conservative
differential form, one can deduce the Rankine-Hugoniot jump relations
that govern the discontinuities:
⟦𝜌 𝑎 ⊗ 𝑈 𝑆𝐴⟧.𝑛 0,
which gives in 1D because 𝐴 0:
⟦𝜌 𝑎 𝑢⟧ 𝑠⟦𝜌 𝑎⟧,
and then precisely for the mass and momentum
⟦𝜌 𝑢⟧ 𝑠⟦𝜌 ⟧ 𝑎𝑛𝑑 ⟦𝜌 𝑢
⟧ 𝑠⟦𝜌 𝑢⟧
9
In the absence of shear forces (or viscosity) and pressure gradient…
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
199
By eliminating 𝑠, we obtain:
⟦𝜌 ⟧ ⟦𝜌 𝑢
⟧ ⟦𝜌 𝑢⟧
,
and after simplification:
𝜌
𝜌
𝑢
𝑢
0 ⇒ 𝑢
𝑢
𝑠 (6.16)
Thus the discontinuity propagates at the continuous speed of the
medium. It is the so called contact discontinuity (the mediums on both
sides of the discontinuity do not mix). It is the only possible
discontinuity. Hence in this model NO CHOC.
This shows also that there is no steady 𝑠 0 contact discontinuity in
moving medium. What we will use later.
A1.5. Problem Type
We refer to [18] for the definitions and notions used here. To simplify
the analysis, we stay with our 1D case. The model under the classical
vector form is:
𝒮𝑤, (6.17)
𝑤 is the vector of conservative variables, 𝑓 is the flux and 𝒮 is a source
10
term. They are given by:
𝑤 𝜌
𝜌 𝑢,𝑓𝑤 𝜌 𝑢
𝜌 𝑢
𝑤
,𝒮 𝒜
(6.18)
The Jacobian matrix of the flux writes:
𝐴𝑤≡
𝑤 01
01
𝑢
2𝑢
Its eigenvalues verify 𝑑𝑒𝑡 𝐴 𝜆 𝐼 0, thus:
10
See the details further. Here let us precise only that 𝒜 is proportional to the
transverse current, 𝑚 is the mass of a particle and 𝐹 is the resultant force submitted by
this ion
Advances in Biosensors: Reviews. Volume 3
200
𝜆.2𝑢𝜆𝑢
𝜆𝑢
0
This leads to a double eigenvalue:
𝜆
𝜆
𝑢,
and the (UNIQUE) associated (right) eigenvector is:
𝑟 1
𝑢
The matrix 𝐴 is not diagonalizable. The system
11
is not really hyperbolic!
To close this subsection, we end with the link with the usual fundamental
principle of dynamics. For this, we notice the general result: when an
eigenvalue is double; the associated characteristic field is “linearly
degenerate”, that is by definition:
.𝑟 0,
which is easily verified and the corresponding discontinuities are
exclusively of “contact discontinuity”, what we have already noted. In
addition, in this case ℛ≡𝜆 𝑢 is a Riemann invariant:
ℛ
.𝑟 0
By multiplying (at left) the initial system by
ℛ
, we obtain the general
equation for the invariant:
ℛ
𝜆
ℛ
ℛ
.𝒮
Telling that the Riemann invariant verifies an (ODE) ordinary
differential equation
12
on the characteristic curve 𝛤 (which here is
nothing else than the trajectory because 𝜆 𝑢):
𝛤:
𝜆,
ℛ
ℛ
.𝒮
11
We meet such systems in gas dynamics with constant pressure called “pressureless”
and used to study the formation of large-scale structures in the universe!
12
This is the basis of the method of characteristics…
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
201
By developing this ODE, we obtain:
𝑢
≡
𝒜
That is:
𝑚 𝛾 𝐹
𝒜, (6.19)
This is an elegant way to retrieve the “Lagrangian” fundamental dynamic
principle (force = mass acceleration) with an unobvious corrective term
due to the flux of ions of the transverse current, from an “Eulerian”
conservation of momentum!
Appendix 2. Mass Equation Source Term
First, recall that for ion 𝑘, the density 𝜌
is related to our variable: its
concentration 𝑐
by the simple relation
𝜌
𝑐
.𝑀
,
where 𝑀
is the “constant” molar mass.
We will calculate 𝒜 on a slice inside the gland with radius 𝑟 and
thickness 𝑑𝑥. Its volume is 𝑉 𝜋.𝑟
.𝑑𝑥 and its lateral area is
𝑆 2𝜋.𝑟.𝑑𝑥; the total current crossing this lateral surface is
𝑆.𝐽
,
where 𝐽
𝐽
𝐽
.
So if we note 𝑒 the elementary charge and 𝑚
the mass of the ion, the
mass transfer is
𝑆.𝐽
.
≡𝑆.𝐽
.
Hence, per unit volume
𝒜 𝑆.𝐽
.
.
,
Advances in Biosensors: Reviews. Volume 3
202
𝒜
.
.𝐽
Appendix 3. A Result Proof
At steady state, the equation for the momentum, for chloride near anode,
reduces to
.
.𝑐.𝔼
𝑐.𝑢,
which, by using the expression of the electric field (6.4)
𝔼
and Ohm’s law (6.3) 𝑐.𝑢
.
gives
.
.𝑐.
.
.
.
,
or else, after developing and simplifying by
:
𝑐 𝑐
∗
𝒞.
.
.
, (6.20)
𝑐
∗
.
.
,
𝒞
.
.
,
To prove, begin with equation (6.20) above and suppose 𝑐 is constant,
then we have 𝑐
𝑥 0 and
𝑐 𝑐
∗
𝒞.
"
⇔𝑐
𝑐
∗
.𝑐2𝒞.𝛷
"
0
Solving this second order algebraic equation, we get the constant 𝑐
∗
and
an absolute error:
𝑐 𝑐
∗
𝜀
,
𝜀
𝒞.
"
∗
∗
𝒞.
"
For the moment, to finish, just assume, see further the mass steady
equation:
Chapter 6. A Simple and Accurate Method to Assess Autonomic Nervous System
through Sudomotor Function
203
0𝛷
"
.
𝛷
.
𝛷
Appendix 4. A Result Proof
We have Φ
0 ⇒ Φ
increasing.
But Φ
𝐿
𝐿
0 ⇒ Φ
0 ⇒ Φ
decreasing.
Current continuity and Φ
0 ⇒ Φ
𝐿
0 ⇒ Φ
0 ⇒
Φ
decreasing.
As Φ
0 0 and current continuity ⇒ Φ
0Φ
.
Appendix 5. A Result Proof
Let us put 𝐷 Φ
Φ then
.
GΦ
.Φ
𝐺Φ.Φ0⇒
increasing.
But 𝑑D
𝑑𝑥𝐿
𝐿
GΦ
.Φ
𝐺Φ.Φ
𝜎0⇒𝐷 decreasing.
However 𝐷0ℎ
00⇒∀𝑥,𝐷𝑥0:Φ
𝑥 Φx.
We can solve exactly the linearized problem, we find:
Φ
𝑥
⎩
⎪
⎨
⎪
⎧
𝐺
𝑥
2ℎ𝑥𝐿
𝑟
𝜎𝑟
ℎ𝑥𝐺
𝑟
2𝐿
𝑟
, if 𝑥 𝐿
𝑟
𝐺
𝐿
2ℎ𝐿
𝜎𝑟
𝑟
𝐺
𝑥𝐿
𝑟
𝑥𝐿
2𝐿
𝑟
ℎ𝐿
𝑟
2𝐿
𝑟
,otherwise
(6.21)
These formula show easily that:
∀𝑥,Φ
𝑥Φ
and Φ
decreasing.
We can deduce ∀𝑥, ΦxΦ
𝐿
𝐿
because Φ
decreasing.
It remains just to calculate Φ
𝐿
𝐿
to get 𝜀… Because here,
simply
Advances in Biosensors: Reviews. Volume 3
204
𝜀
Appendix 6. A Result Proof
For the proof, we use Ohm’s law, see (6.3):
𝐼
_
𝜎.𝜋.𝑟
.
,
and begin by calculation 𝐼
_
from formula (6.21), we obtain the
desired result.
Next, we continue with the Ohm’s law:
𝐼
𝜎.𝜋.𝑟
.
,
and using the balance equation (6.9), we have
2𝜋.𝑟.𝐺Φ.Φ,
⇒𝐼
Φ
,𝑥 𝐼
Φ
,𝐿
𝐿
2𝜋.𝑟.𝐺Φ.Φ.dx
,
or
𝐼
Φ
,𝐿
𝐿
𝜋.𝑟
.𝐺Φ𝐿
𝐿
.Φ𝐿
𝐿
,
and
𝐺Φ
.1𝜀.Φ
.1𝜀𝐺Φ.Φ𝐺Φ
.Φ
