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ResearchCite this article: Yuan J, Dagdeviren C, Shi Y,Ma Y, Feng X, Rogers JA, Huang Y. 2016Computational models for the determinationof depth-dependent mechanical properties ofskin with a soft, flexible measurement device.Proc. R. Soc. A 472: 20160225.http://dx.doi.org/10.1098/rspa.2016.0225
Received: 30 March 2016Accepted: 16 September 2016
Subject Areas:mechanical engineering, mechanics
Keywords:conformal modulus sensors, skin moduli,epidermis and dermis, electromechanicalcoupling
Authors for correspondence:John A. Rogerse-mail: [email protected] Huange-mail: [email protected]
†These authors contributed equally to thestudy.
Computational modelsfor the determination ofdepth-dependent mechanicalproperties of skin with a soft,flexible measurement deviceJianghong Yuan1,2,3,4,5,6,†, Canan Dagdeviren7,8,†,
Yan Shi1,2, Yinji Ma1,2,3,4,5,6, Xue Feng1,2, John A.
Rogers9,10 and Yonggang Huang3,4,5,6
1Center for Mechanics and Materials and 2AML, Department ofEngineering Mechanics, Tsinghua University, Beijing 100084,People’s Republic of China3Department of Civil an Environmental Engineering, 4Departmentof Mechanical Engineering, 5Department of Materials Science andEngineering and 6Skin Disease Research Center, NorthwesternUniversity, Evanston, IL 60208, USA7The David H. Koch Institute for Integrative Cancer Research,Massachusetts Institute of Technology, Cambridge, MA 02139, USA8Harvard Society of Fellows, Harvard University, Cambridge,MA 02138, USA9Department of Materials Science and Engineering, and 10FrederickSeitz Materials Research Laboratory, University of Illinois atUrbana-Champaign, Urbana, IL 61801, USA
YH, 0000-0002-0483-8359
Conformal modulus sensors (CMS) incorporate PZTnanoribbons as mechanical actuators and sensorsto achieve reversible conformal contact with thehuman skin for non-invasive, in vivo measurementsof skin modulus. An analytic model presented inthis paper yields expressions that connect the sensoroutput voltage to the Young moduli of the epidermisand dermis, the thickness of the epidermis, as wellas the material and geometrical parameters of theCMS device itself and its encapsulation layer. Resultsfrom the model agree well with in vitro experimentson bilayer structures of poly(dimethylsiloxane).
2016 The Author(s) Published by the Royal Society. All rights reserved.
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These results provide a means to determine the skin moduli (epidermis and dermis) and thethickness of the epidermis from in vivo measurements of human skin.
1. IntroductionHuman skin is a complex living tissue consisting of several heterogeneous layers, namelythe epidermis (composed of the stratum corneum and the viable epidermis) [1], the dermis(composed of the superficial papillar dermis and the collagen-rich reticular dermis) [2] andthe hypodermis (i.e. the subcutaneous fat) overlying the subcutaneous tissue (i.e. the muscle)[3]. Although the human skin behaves as nonlinear, viscoelastic, anisotropic, inhomogeneousand incompressible material [4], the effective Young’s modulus remains a key parameter thatcharacterizes its overall mechanical properties [5,6]. Determination of the Young modulus ofeach component layer of the human skin is of critical importance for cosmetic and clinicalapplications, of relevance in the efficacy of cosmetic products such as creams [7], drug deliveryusing microneedles or microjets [1], and diagnosis of various skin diseases such as scleroderma,Ehlers–Danlos and skin cancer [6,8].
For measuring the Young modulus of human skin, various experimental methods have beendeveloped, e.g. indentation [1,3], suction [2,4,7], torsion [5,9], traction [10,11], elastography [6]and surface wave propagation [12]. By performing in vivo suction experiments with differentaperture diameters, Hendriks et al. [2] found a large difference in stiffness between the reticulardermis layer and the upper layer consisting of the epidermis and the papillar dermis. The resultsverified, to a certain extend, the hypothesis that experiments with different length scales cancapture the mechanical behaviour of different layers of human skin. By using a spherical tip witha large diameter compared with the sample thickness in in vitro microindentation experiments,Geerligs et al. [1] observed no significant differences in stiffness between the stratum corneum andthe viable epidermis of human skin. From in vivo indentation experiments using a conical steelindenter with a height of 10 mm, Pailler-Mattei et al. [3] concluded that it is necessary to take intoaccount the effect of the subcutaneous fat and muscle to estimate the Young modulus of humanskin correctly. Hendriks et al. [4] showed, however, that contribution from the subcutaneous fatlayer to the mechanical response of the human skin is negligible in suction experiments, using anaperture size of 6 mm. By measuring the surface waves induced by short impulses via the phase-sensitive optical coherence tomography, Li et al. [12] successfully evaluated the Young moduliof the dermis and the subcutaneous fat of human skin. These experiments did not, however,yield the Young moduli of the thin epidermis, because the excitation frequency and the samplingfrequency they used in in vivo experiments were not sufficiently high.
In recent years, stretchable and flexible electronics have emerged in forms that allow accuratemeasurements of electrophysiological signals and thermal and mechanical properties of thehuman body [13–23]. Dagdeviren et al. [24] reported a microconformal modulus sensor (CMS)system that can achieve soft and reversible conformal contact with the underlying complextopography and texture of the human skin, and surfaces of other organs of the body, to provideaccurate and reproducible non-invasive measurements of the modulus. In this paper, we developan analytical model to extend Dagdeviren et al.’s experimental design [24] to simultaneousdetermination of the Young moduli of the epidermis and dermis layers as well as the epidermisthickness. The paper is organized as follows. The geometrical and material properties of theproposed model are described in detail in §2. The analytic model, which accounts for theinteraction between the CMS system and the multi-layered skin, is discussed in detail in §3. Theanalytic model is then compared with a validation experiment. Section 4 presents a strategy toextract the skin moduli (epidermis and dermis) and the epidermis thickness from data obtainedfrom the CMS system. The paper concludes with some summary remarks in §5. We emphasizehere that this is mostly an analytical paper, rather than an experimental paper, with figures 3–6being the main analytical results of our theoretical model.
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sensor
sensoractuatorxOHencap
Hepidermis
actuatorsurrounding media
encapsulation
encapsulation
epidermis
dermis
epidermis
zz
z z z
–a +a –b +bxs
xsxa
l
2a 2b
Eencap
Eepidermis
Edermis
xa
ta (xa)–ta (xa) –ts (xs)ts (xs)
l
semi-infinite solid
Figure 1. Bilayered structure of the human skin and conformal modulus sensor (CMS) system mounted on the surface withencapsulation, where (xa, z) and (xs, z) are the local coordinates with the origin at the centres of the actuator and sensor,respectively; (x, z) is the global coordinates. The actuator, the sensor and the surroundingmedia interact through the resultant,distributed shear forces τ a and τ s at their interfaces.
2. Model descriptionAs shown in figure 1, by neglecting the extremely soft subcutaneous fat [4], the human skin canbe approximately modelled as a bilayered system: an elastic epidermis layer with the thicknessHepidermis and Young modulus Eepidermis and an elastic dermis layer with the Young modulusEdermis. The dermis layer is much thicker (e.g. approx. 1 mm [12,25]) than the epidermis (e.g.approx. 0.1 mm [12,25]) and is therefore simplified as a semi-infinite solid. Their Poisson’s ratiosare 0.5 because of the incompressibility.
A CMS system with an encapsulation layer (thickness Hencap, Young’s modulus Eencap, andPoisson’s ratio 0.5) is mounted on the surface of the human skin (figure 1). The piezoelectricactuator (length 2a) and sensor (length 2b) in the CMS system are laminates composed of sevenparallel layers with a total thickness of about 5 µm [24]. The fourth (middle) layer among themis made of PZT-4 with a thickness of hPZT (≈0.5 µm [24]), and the poling direction of PZT-4 isalong its thickness. The geometrical and material properties of each constituent layer are listed inappendix A. The spacing between the centres of a pair of sensors and actuators is l.
The model is two-dimensional with plane-strain deformation. The inertia effect is negligibleowing to the low working frequency (below 1000 Hz) of the CMS system [24] such that theharmonic vibrations in the experiments can be treated as a quasi-static problem.
The piezoelectric actuator is subjected to an input voltage Uinput. Under the applied electricfield Ez along the poling direction of the PZT-4 layer, the actuator expands in the thicknessdirection, and shrinks in the length direction owing to the inverse piezoelectric effect. Because thelength (approx. 102 µm [24]) of the actuator (or sensor) is much larger than its thickness (≈5 µm[24]), the length-shrinking mode dominates the deformation. The actuator (or sensor) can thenbe modelled as an electro-elastic line with zero thickness [26] to impose a resultant shear forcedistribution onto the surrounding media along the interface between the encapsulation layer
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and the epidermis layer at the location of the actuator (figure 1). The shear force deforms thesurrounding media and then lengthens the piezoelectric sensor in its length direction, whichresults in an output voltage from the sensor owing to the direct piezoelectric effect under theopen-loop electrical boundary condition.
The analytical solution in §3 gives the relation between the output voltage of the sensor andthe input voltage of the actuator in terms of the Young moduli of the epidermis and dermis layersand the thickness of the epidermis layer, which forms the basis for measuring the skin modulus.
3. Analytical solutionIt should be pointed out that the bending mode also exists during deformations of the laminatedactuator/sensor, but it makes little contribution to the axial strain of the PZT-4 layer, because theneutral axis of bending of the actuator/sensor is inside the PZT-4 layer. The bending effect of theactuator/sensor on the output voltage of the sensor is therefore neglected.
(a) Analysis of the actuator and the sensorAs shown in figure 1, let (xa, z) and (xs, z) be the local coordinates with the origin at the centres ofthe actuator and sensor, respectively, with the z-axis normal to the interface and x-axis directionfrom the actuator to the sensor (figure 1). The resultant, distributed shear force on the actuatorand the sensor, is denoted by τ a(xa) and τ s(xs), respectively. Force equilibrium of the actuator andthe sensor requires
∫ a
−aτa(ξa) dξa = 0 and
∫ b
−bτs(ξs) dξs = 0, (3.1)
which results from the traction-free condition at both ends of the actuator or the sensor,because (i) the thickness of the actuator/sensor is extremely small (≈5 µm [24]) as comparedwith its length (approx. 102 µm [24]) and (ii) the Young moduli (approx. 102 kPa [24]) of thesurrounding encapsulation and epidermis are many orders of magnitude smaller than that ofthe actuator/sensor (approx. 102 GPa [24]).
The equilibrium equations are
∂
∂xa
7∑k=1
σ(k)a (xa)h(k) + τa(xa) = 0 and
∂
∂xs
7∑k=1
σ(k)s (xs)h(k) + τs(xs) = 0, (3.2)
where the superscript k means the kth layer of the laminated actuator/sensor; h(k), σ(k)a and σ
(k)s
denote the thickness and the axial stress of the kth layer of the actuator or the sensor, respectively.Integration of equation (3.2) gives
7∑k=1
σ(k)a (xa)h(k) = −
∫ xa
−aτa(ξa) dξa and
7∑k=1
σ(k)s (xs)h(k) = −
∫ xs
−bτs(ξs) dξs. (3.3)
The axial strain of actuator and sensor, which should be the same in all constituent layers andbe continuous across their interfaces with the surrounding media, is denoted by εa and εs. Theconstitutive relations for each constituent layer of the actuator and the sensor are [26,27]
σ(k)a =
⎧⎨⎩
E(4)a εa − epEz, (k = 4)
E(k)a εa, otherwise
and σ(k)s = E(k)
s εs, (k = 1, 2, . . . , 7), (3.4)
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with Ez being the electric field in the PZT layer of the actuator and
E(k)a =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
c11 − c213
c33, k = 4
E(k)
1 − (ν(k))2 , otherwise
; E(k)s =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
c11 − c213
c33+
e2p
ke, k = 4
E(k)
1 − (ν(k))2 , otherwise
;
ep = e31 − e33c13
c33; ke = k33 + e2
33c33
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
, (3.5)
where cij, eij and kij are the elastic, piezoelectric and dielectric constants of PZT-4, respectively;E(k) is the Young modulus and ν(k) is the Poisson’s ratio of the kth layer.
Substitution of equation (3.4) into equation (3.3) gives the axial strains of the actuator and thesensor as
εa(xa) = − 1Ka
∫ xa
−aτa(ξa) dξa + ephPZT
KaEz and εs(xs) = − 1
Ks
∫ xs
−bτs(ξs) dξs, (3.6)
with
Ka =7∑
k=1
E(k)a h(k) and Ks =
7∑k=1
E(k)s h(k). (3.7)
The second relation of equation (3.6), together with vanishing electric displacement Dz(xs) =epεs(xs) + keE′
z(xs) = 0 [27] in the PZT-4 layer of the sensor, gives the ratio of the output voltage ofthe sensor to the input voltage of the actuator as [27]
Uoutput
Uinput= (1/2b)
∫b−b [−E′
z(xs)hPZT] dxs
−EzhPZT= ep
keKsEz
12b
∫ b
−bdxs
∫ xs
−bτs(ξs) dξs, (3.8)
where E′z(xs) is the distributed electric field in the PZT-4 layer (sandwiched by two electrodes) of
the sensor.
(b) Analysis of the surrounding mediaLet ε(x; ξ ) denote the normal strain in the x-direction (figure 1) at point (x, 0) (i.e. the interfacebetween the encapsulation and the epidermis) owing to a concentrated force of unit magnitudealong the positive x-axis (figure 1) applied at point (ξ , 0). It can be decomposed into two separateparts as follows (see appendix B)
ε(x; ξ ) = εencapepidermis(x; ξ ) + εfree-surface
dermis (x; ξ ), (3.9)
where the first term on its right-hand side denotes the normal strain owing to the same force in thecorresponding bimaterial composed of an encapsulation half-plane and an epidermis half-plane;and the second term accounts for the effects of both the free-surface of the encapsulation layerand the semi-infinite dermis. Generalization of the method of reverberation ray matrix [28–30]from elastodynamics to elastostatics gives these two terms as (see appendix B for details)
εencapepidermis(x; ξ ) = − 3
2π (Eencap + Eepidermis)1
x − ξ(3.10)
and
εfree-surfacedermis (x; ξ ) = − 3
2π (Eencap + Eepidermis)
∫+∞
0f (k) sin [k(x − ξ )] dk, (3.11)
with
f (k) = cT(k) d(k). (3.12)
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Here,c(k) = [cencap(k) 01×4 01×2]T
and cencap(k) = [e−kHencap Hencap e−kHencap 1 0]
⎫⎬⎭ (3.13)
andd(k) = [I10×10 + S(k)P(k)]−1s(k) − s(k), (3.14)
where ‘T’ denotes the transpose of a matrix or a vector, ‘I’ denotes the identity matrix and thescattering matrix S(k) is defined only in terms of material parameters as
S(k) =
⎡⎢⎢⎢⎣
Sfree-surface(k) 02×4 02×2
04×2 S(k; αencapepidermis) 04×2
04×2 04×4 Shalf-space(k; αepidermisdermis )
⎤⎥⎥⎥⎦ , (3.15)
with
S(k; α) =
⎡⎢⎢⎢⎣
−α 0 −1 + α 02αk −α 0 1 − α
−1 − α 0 α 00 1 + α −2αk α
⎤⎥⎥⎥⎦
Shalf−space(k; α) =
⎡⎢⎢⎢⎣
−α 02αk −α
−1 − α 00 1 + α
⎤⎥⎥⎥⎦
and Sfree-surface(k) =[−1 02k −1
]
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
, (3.16)
where
αencapepidermis = Eencap − Eepidermis
Eencap + Eepidermisand α
epidermisdermis = Eepidermis − Edermis
Eepidermis + Edermis, (3.17)
are the first Dundurs’ parameter [31] for the encapsulation/epidermis and epidermis/dermisinterfaces, respectively. The phase matrix P(k) in equation (3.14) is defined only in terms ofgeometric parameters as
P(k) =[
P(k; Hencap) 04×4 04×204×4 P(k; Hepidermis) 04×2
], (3.18)
with
P(k; H) =
⎡⎢⎢⎢⎣
0 0 e−kH He−kH
0 0 0 −e−kH
e−kH He−kH 0 00 −e−kH 0 0
⎤⎥⎥⎥⎦ . (3.19)
The source vector s(k) in equation (3.14) is defined as
s(k) = [01×2 sencapepidermis(k) 01×4]T
and sencapepidermis(k) = [1 −k 1 −k]
⎫⎬⎭ . (3.20)
The normal strains in the x-direction (figure 1) of surrounding media along the encapsulation–epidermis interface at the locations of the actuator and the sensor are denoted by εi−a and εi−s,respectively, and are obtained via the principle of superposition as
εi−a(xa) = −∫ a
−aτa(ξa)ε(xa; ξa) dξa −
∫ b
−bτs(ξs)ε(xa; ξs + l) dξs
and εi−s(xs) = −∫ a
−aτa (ξa) ε (xs; ξa − l) dξa −
∫ b
−bτs(ξs)ε(xs; ξs) dξs
⎫⎪⎪⎪⎬⎪⎪⎪⎭
. (3.21)
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(c) Interactions among the actuator, sensor and surrounding mediaThe continuity of axial strains requires εa = εi–a and εs = εi–s, which leads to the following coupledsingular integral equations in dimensionless form
λa
∫Xa
−1Ta(Ξa) dΞa +
∫ 1
−1
Ta(Ξa)Xa − Ξa
dΞa +∫ 1
−1Ta(Ξa) dΞa
∫+∞
0f(
Θ
a
)sin [(Xa − Ξa)Θ] dΘ
+ Ka
Ks
∫ 1
−1
Ts(Ξs)[(aXa − l)/b] − Ξs
dΞs + Ka
Ks
∫ 1
−1Ts(Ξs) dΞs
∫+∞
0f(
Θ
b
)
× sin[(
aXa − lb
− Ξs
)Θ
]dΘ = 1 (−1 < Xa < 1), (3.22)
and
λs
∫Xs
−1Ts(Ξs) dΞs +
∫ 1
−1
Ts(Ξs)Xs − Ξs
dΞs +∫ 1
−1Ts(Ξs) dΞs
∫+∞
0f(
Θ
b
)sin [(Xs − Ξs)Θ] dΘ
+ Ks
Ka
∫ 1
−1
Ta(Ξa)[(bXs + l)/a] − Ξa
dΞa + Ks
Ka
∫ 1
−1Ta(Ξa) dΞa
∫+∞
0f(
Θ
a
)
× sin[(
bXs + la
− Ξa
)Θ
]dΘ = 0 (−1 < Xs < 1) (3.23)
where Xa = xa/a, Xs = xs/b, function f is given by equation (3.12), and the other involveddimensionless functions and parameters are defined as
Ta(Ξa) = 1λa
ahPZT
τa(aΞa)epEz
and Ts(Ξs) = 1λs
bhPZT
τs(bΞs)epEz
, (3.24)
and
λa = 2πa(Eencap + Eepidermis)
3Kaand λs = 2πb(Eencap + Eepidermis)
3Ks. (3.25)
The solution of the coupled singular integral equations (3.22) and (3.23) has a square-rootsingularity at Ξ = ±1. Therefore, the shear force can be generally expressed in terms of theexpansion of Chebyshev polynomials of the first kind as follows [26]
Ta(Ξa) = 1√1 − Ξ2
a
+∞∑n=1
CanTn(Ξa) and Ts(Ξs) = 1√
1 − Ξ2s
+∞∑n=1
CsnTn(Ξs), (3.26)
where Tn is the nth-order Chebyshev polynomial of the first kind, and Can and Cs
n are thecoefficients to be determined. Equation (3.26) satisfies the force equilibrium conditions in equation(3.1) and also the boundary conditions for the axial force at two ends of the actuator or the sensoras follows
7∑k=1
σ(k)a (±a)h(k) = 0 and
7∑k=1
σ(k)s (±b)h(k) = 0, (3.27)
owing to the orthogonality of Chebyshev polynomials.Substitution of equation (3.26) into equations (3.22) and (3.23) leads to a set of linear
algebraic equations for determining the coefficients Can and Cs
n (see appendix C for details).Their dependence on the Young moduli Eepidermis and Edermis of the epidermis and dermis andthe thickness Hepidermis of epidermis is through three dimensionless parameters: Hepidermis/a,
αencapepidermis and α
epidermisdermis . The ratio of the output voltage of the sensor to the input voltage of the
actuator is related to the coefficient Cs1 by
Uoutput
Uinput= −π2
6
e2phPZT/ke
Ks
(Eencap + Eepidermis)b
KsCs
1. (3.28)
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0.8experimentsanalytical model
0.6
0.4
0.2U
outp
ut/U
inpu
t(10
–6)
0
1 2 3sensor number
4 5
Figure 2. The output voltage of several sensors, normalized by the input voltage of the actuator, is obtained from the analyticmodel and experiments. (Online version in colour.)
(d) Comparison with experimentsThe above-mentioned analytical model is validated by comparing with a model experimentwith a 150 µm thick, relatively stiff poly(dimethylsiloxane) (PDMS, Young’s modulus 1800 kPa)on a thick soft PDMS (Young’s modulus 10 kPa). Their large mismatch in Young’s moduli
gives αepidermisdermis ≈ 1 from equation (3.17). The CMS system, mounted on the surface of the stiff
PDMS, has a silicone encapsulation layer with the Young modulus Eencap = 60 kPa and thicknessHencap = 20 µm. The large elastic mismatch between the encapsulation layer and stiff PDMS givesα
encapepidermis ≈ 1. The CMS system consists of one actuator (length 2a = 140 µm) and five identical
sensors (length 2b = 60 µm), with the distance between the centres of the actuator and the sensorsl = 300, 900, 1500, 2100 and 2700 µm, respectively. The detailed structures of the actuator andsensors as well as their geometric and material parameters are given in appendix A. Figure 2shows the output voltage of all five sensors, normalized by the input voltage of the actuator. Thecomparison indicates a good agreement between the data from our validation experiment andthe prediction from our analytical model without any parameter fitting. For applications to thehuman skin, the comparison of experiments with the theory gives the real skin modulus.
The usefulness of the analytic model is to determine the Young modulus and thickness ofthe relatively stiff PDMS layer from the output voltage in the experiments by solving an inverseproblem. For a CMS device with encapsulation (with known material and geometric parameters),the normalized output voltage in equation (3.28) depends on the Young modulus and thickness ofthe stiff PDMS layer. Minimization of the difference in the output voltage between the analyticalmodel and experiments for all sensors gives the Young modulus 1600 kPa and thickness 140 µm,which are 11% and 6.7% less than their values in experiments. This error mainly results fromtwo orders of magnitude in elastic mismatch between the stiff PDMS layer (1800 kPa) and theencapsulation layer (60 kPa) and the soft PDMS (10 kPa). It is expected that this error will decreasesignificantly for measurement of skin modulus, because the Young moduli of epidermis (approx.102 kPa [24,32]), dermis (approx. 101 kPa [32]) and encapsulation layer (60 kPa) are all on the sameorder of magnitude.
4. A strategy to determine Young’s moduli of epidermis and dermisA strategy to simultaneously determine the Young moduli Eepidermis and Edermis and the thicknessHepidermis is proposed in this section. Figures 3–5 show the output voltage of the sensor (length
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0.14Hepidermis = 150 mm
Hepidermis = 120 mm
Hepidermis = 90 mm
Hepidermis = 60 mm
0.13
0.12
0.11
0.10
0.09
0.08
0.07
0.06
0.05100 120 140
Eepidermis (kPa)160 180 200
Uou
tput
/Uin
put(
10–6
)
Figure 3. The normalized output voltage of the sensor versus the Young modulus of epidermis for several values of epidermisthickness. Here, Edermis = 30 kPa. (Online version in colour.)
0.11
0.09
0.08
0.07
0.06
0.05
0.10
0.12
Uou
tput
/Uin
put(
10–6
)
10 20 30 40 50 60 70 80 90 100
Hepidermis = 150 mm
Hepidermis = 120 mm
Hepidermis = 90 mm
Hepidermis = 60 mm
Edermis (kPa)
Figure 4. The normalized output voltage of the sensor versus the Young modulus of dermis for several values of epidermisthickness. Here, Eepidermis = 150 kPa. (Online version in colour.)
2b = 60 µm) on the Young moduli Eepidermis of the epidermis and Edermis of the dermis, and thethickness Hepidermis of the epidermis layer, for the actuator length 2a = 140 µm and a distancel = 300 µm between the actuator and sensor. The encapsulation layer has Young’s modulusEencap = 60 kPa and thickness Hencap = 20 µm. The output voltage of the sensor, normalized by theinput voltage of the actuator, displays an approximately linear dependence on Eepidermis(figure 3).However, its dependence on Edermis and Hepidermis are non-monotonic (figures 4 and 5). Thesenon-monotonic dependences may create difficulties to uniquely determine Edermis and Hepidermis,which is common for all inverse problems.
The CMS system is redesigned in the following to scale its size proportionally as (2a,2b, l) = λ (140, 60, 300 µm), where λ is the scaling factor. For Eepidermis = 150 kPa, Edermis = 30 kPa,Eencap = 60 kPa and thickness Hencap = 20 µm, figure 6 shows the normalized output voltage ofthe sensor versus the scaling factor λ for several values of Hepidermis ranging from 50 to 200 µm.The output voltage of the sensor increases monotonically with λ either λ ≤ 1/5 or λ ≥ 5, whichcorrespond to small and large sizes, respectively, when compared with the thickness of theepidermis. For λ ≤ 1/5, the output voltage of the sensor is insensitive to the thickness of theepidermis, which suggests that the effects of both the thickness of the epidermis and the modulus
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0.14
0.12
0.10
0.08
0.06
0.040 50 100 150 200 250
Uou
tput
/Uin
put(
10–6
)
Hepidermis (mm)
Eepidermis = 200 kPaEepidermis = 150 kPaEepidermis = 100 kPa
Figure 5. The normalized output voltage of the sensor versus the thickness of the epidermis for several values of epidermismodulus. Here, Edermis = 30 kPa. (Online version in colour.)
1
10–1
10–1 1 10
10–2
10–3
Uou
tput
/Uin
put(
10–6
)
l
Hepidermis = 200 mm
Hepidermis = 150 mm
Hepidermis = 100 mm
Hepidermis = 50 mm
Figure 6. The normalized output voltage of the sensor versus the scaling factor λ for several values of epidermis thickness.Here, Eepidermis = 150 kPa and Edermis = 30 kPa. (Online version in colour.)
of dermis are negligible. These observations suggest the use of a CMS system consisting oftwo sets of actuator/sensors. One set has small-size actuator and sensors with (2asmall, 2bsmall,lsmall) approximately one-fifth of the representative thickness of the epidermis, to determineEepidermis independently. Another set has large-size actuator and sensors with (2alarge, 2blarge,llarge) approximately five times of the thickness of the epidermis, to determine Edermis andHepidermis simultaneously after Eepidermis is determined.
5. Concluding remarksAn analytic model is developed for use of a CMS system to determine the Young moduli ofthe skin, including the epidermis and the dermis, as well as the thickness of the epidermis. Thepiezoelectric thin-film actuator and sensors in the CMS system interact with the human skin aswell as the encapsulation layer through the interfacial shear stress distribution over the surfaceof the actuator and sensors. The analytical model agrees well with the in vitro experiments fora bilayer structure of PDMS, without any parameter fitting. For evaluation of human skin, theanalytical model suggests a design for a CMS system that consists of two sets of actuator and
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sensors. A set of small-size actuator and sensors determines the Young modulus of epidermis,whereas a set of large-size actuator and sensors determine the thickness of the epidermis and theYoung modulus of dermis.
Ethics. The research work did not involve active collection of human data or any other ethical issues.Data accessibility. The datasets supporting this article are already available in the article itself.Author’s contributions. J.Y. carried out the analytic modelling, analysed the data and drafted the manuscript; C.D.carried out the experiments, participated in data analysis and revised the manuscript; Y.S. participated inthe analytical modelling and discussions; Y.M. participated in data analysis and discussions; X.F. and J.A.R.analysed the data and revised the manuscript; Y.H. designed the study, analysed the data and finalized themanuscript. All authors gave final approval for publication.Competing interests. The authors declare that they have no competing interests.Funding. J.Y. acknowledges the support from the National Natural Science Foundation of China (grant no.11402133). X.F. acknowledges the support from the National Basic Research Programme of China (grantno. 2015CB351900) and the National Natural Science Foundation of China (grant no. 11320101001). Y.H.acknowledges the support from the US National Science Foundation (grant nos. DMR-1121262, CMMI-1534120, CMMI-1300846 and CMMI-1400169) and from the US National Institutes of Health (grant no.R01EB019337).Acknowledgements. The authors thank the reviewers for their helpful comments and suggestions.
Appendix AThe actuator or sensor in the CMS system [24] consist of seven layers
PI (h1 = 2.4µm, E1 = 2.5 GPa, ν1 = 0.34)
Au (h2 = 300 nm, E2 = 78 GPa, ν2 = 0.44)
Cr (h3 = 10 nm, E3 = 279 GPa, ν3 = 0.21)
PZT (h4 = 500 nm)
Pt (h5 = 300 nm, E5 = 168 GPa, ν5 = 0.38)
Ti (h6 = 20nm, E6 = 110 GPa, ν6 = 0.34)
and PI (h7 = 1.2µm, E7 = 2.5GPa, ν7 = 0.34),
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(A 1)
where hk, Ek and νk denote the thickness, Young’s modulus and Poisson’s ratio of the kth elasticlayer, respectively. The polyimide (PI) layer on the Au-electrode side is in direct contact with theskin.
The PZT layer (layer 4) in the actuator or sensor is transversely isotropic with its polingdirection along the layer thickness. Their elastic, piezoelectric and dielectric coefficients are [33]
c11 = 139 GPa, c12 = 77.8 GPa, c13 = 74.3 GPa,
c33 = 113 GPa, c44 = 25.6 GPa, c66 = (c11 − c12)2
;
e31 = −6.98 C m−2, e33 = 13.8 C m−2, e15 = 13.4 C m−2
and k11 = 6.0 × 10−9 C2 (Nm2)−1, k33 = 5.47 × 10−9 C2 (Nm2)−1.
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(A 2)
Appendix BThe fundamental solution for the layered half-plane owing to a horizontal unit point-force atthe interface is obtained below, following the method of reverberation ray matrix [28–30]. The
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1x
z12
z23
z34
z32
x x
z21x x
H12 = Hencap
H23 = Hepidermis
half-plane
E23 = Eepidermis
E34 = Edermis
encapsulation
epidermis
dermis
E12 = Eencap
2
3
4
Figure 7. Dual coordinates for the encapsulation, epidermis and dermis.
general solution for the displacements and stresses in an incompressible medium under plane-strain condition can be expressed as
u1 = ∂ϕ1
∂x+ z
∂ϕ2
∂x, u3 = ∂ϕ1
∂z+ z
∂ϕ2
∂z− ϕ2
and σ31 = 2E3
(∂2ϕ1
∂x∂z+ z
∂2ϕ2
∂x∂z
), σ33 = 2E
3
(∂2ϕ1
∂z2 + z∂2ϕ2
∂z2 − ∂ϕ2
∂z
)⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭
, (B 1)
where E is Young’s modulus, and ϕ1 and ϕ2 are two harmonic functions that satisfy
∂2ϕ1
∂x2 + ∂2ϕ1
∂z2 = 0 and∂2ϕ2
∂x2 + ∂2ϕ2
∂z2 = 0. (B 2)
The application of the Fourier transform
f (k) =∫+∞
−∞f (x) e−ikx dx, (B 3)
to equations (B 1) and (B 2) gives the general solutions of displacements and stresses in thetransformed domain as
u1(z; Aξ , Dξ ) = ik[(A1 e|k|z + D1 e−|k|z) + z(A2 e|k|z + D2 e−|k|z)]
u3(z; Aξ , Dξ ) = |k|(A1 e|k|z − D1 e−|k|z) − (A2 e|k|z + D2 e−|k|z)
+ |k|z(A2 e|k|z − D2 e−|k|z)
σ31(z; Aξ , Dξ ) = 2E3
ik|k|[(A1 e|k|z − D1 e−|k|z) + z(A2 e|k|z − D2 e−|k|z)]
σ33(z; Aξ , Dξ ) = 2E3
[k2(A1 e|k|z + D1 e−|k|z) + k2z(A2 e|k|z + D2 e−|k|z)
− |k|(A2 e|k|z − D2 e−|k|z)]
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
for ξ = 1, 2, (B 4)
where |k| denotes the absolute value of k, and Aξ = 0 for the semi-infinite medium shown infigure 7, which also shows dual local coordinate systems, distinguished by the superscripts, foreach layer. For convenience, we label the modulus (and thickness) of the encapsulation layer,epidermis and dermis by the superscripts 12, 23 and 34, respectively. The general solutions for
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the encapsulation layer and epidermis in dual local coordinates can be expressed as
u1(zIJ ; AIJξ , DIJ
ξ )
u3(zIJ ; AIJξ , DIJ
ξ )
σ31(zIJ ; AIJξ , DIJ
ξ )
σ33(zIJ ; AIJξ , DIJ
ξ )
or
u1(zJI; AJIξ , DJI
ξ )
u3(zJI; AJIξ , DJI
ξ )
σ31(zJI; AJIξ , DJI
ξ )
σ33(zJI; AJIξ , DJI
ξ )
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
for IJ = 12, 23, (B 5)
and that for dermis in local coordinates as
u1(z34; 0, D34ξ ), u3(z34; 0, D34
ξ ), σ31(z34; 0, D34ξ ) and σ33(z34; 0, D34
ξ ). (B 6)
The two sets of solutions in equation (B 5) should satisfy the following relations
u1(zIJ; AIJξ , DIJ
ξ ) = u1(HIJ − zIJ ; AJIξ , DJI
ξ )
u3(zIJ; AIJξ , DIJ
ξ ) + u3(HIJ − zIJ ; AJIξ , DJI
ξ ) = 0
σ31(zIJ ; AIJξ , DIJ
ξ ) + σ31(HIJ − zIJ ; AJIξ , DJI
ξ ) = 0
σ33(zIJ ; AIJξ , DIJ
ξ ) = σ33(HIJ − zIJ ; AJIξ , DJI
ξ )
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
for IJ = 12, 23, (B 7)
which leads to ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
AIJ1
AIJ2
AJI1
AJI2
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭
= P(HIJ)
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
DIJ1
DIJ2
DJI1
DJI2
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭
for IJ = 12, 23, (B 8)
with
P(HIJ) =
⎡⎢⎢⎢⎢⎢⎣
0 0 e−|k|HIJHIJ e−|k|HIJ
0 0 0 −e−|k|HIJ
e−|k|HIJHIJ e−|k|HIJ
0 0
0 −e−|k|HIJ0 0
⎤⎥⎥⎥⎥⎥⎦ . (B 9)
The traction-free surface of encapsulation layer and the perfectly bonded interfaces betweenadjacent layers require
σ31(0; A12ξ , D12
ξ ) = 0
and σ33(0; A12ξ , D12
ξ ) = 0,
⎫⎬⎭ (B 10)
u1(0; A23ξ , D23
ξ ) = u1(0; A21ξ , D21
ξ )
u3(0; A23ξ , D23
ξ ) + u3(0; A21ξ , D21
ξ ) = 0
σ31(0; A23ξ , D23
ξ ) + σ31(0; A21ξ , D21
ξ ) + 1 = 0
and σ33(0; A23ξ , D23
ξ ) = σ33(0; A21ξ , D21
ξ ),
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(B 11)
andu1(0; 0, D34
ξ ) = u1(0; A32ξ , D32
ξ )
u3(0; 0, D34ξ ) + u3(0; A32
ξ , D32ξ ) = 0
σ31(0; 0, D34ξ ) + σ31(0; A32
ξ , D32ξ ) + 1 = 0
and σ33(0; 0, D34ξ ) = σ33(0; A32
ξ , D32ξ ).
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
(B 12)
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Equations (B 10), (B 11) and (B 12) can be rewritten as
S1
{A12
1
A122
}+{
D121
D122
}= 0, (B 13)
S2
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
A211
A212
A231
A232
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
+
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
D211
D212
D231
D232
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
= 3E12 + E23
12ik|k|
⎧⎪⎪⎪⎨⎪⎪⎪⎩
1−|k|
1−|k|
⎫⎪⎪⎪⎬⎪⎪⎪⎭
(B 14)
and S3
{A32
1
A322
}+
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
D321
D322
D341
D342
⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭
= 0, (B 15)
where
S1 =[
−1 02|k| −1
], (B 16)
S2 =
⎡⎢⎢⎢⎣
−α2 0 −1 + α2 02α2|k| −α2 0 1 − α2
−1 − α2 0 α2 00 1 + α2 −2α2|k| α2
⎤⎥⎥⎥⎦ (B 17)
and S3 =
⎡⎢⎢⎢⎣
−α3 02α3|k| −α3
−1 − α3 00 1 + α3
⎤⎥⎥⎥⎦ , (B 18)
with
α2 = E12 − E23
E12 + E23 and α3 = E23 − E34
E23 + E34 . (B 19)
Equations (B 13), (B 14) and (B 15), together with equation (B 8), yield
D10×1 = 3E12 + E23
12ik|k| (I10×10 + S10×8P8×10)−1s10×1
and A8×1 = P8×10D10×1
⎫⎪⎬⎪⎭ , (B 20)
where I10×10 is the identity matrix
D10×1 = {D121 D12
2 D211 D21
2 D231 D23
2 D321 D32
2 D341 D34
2 }T
and A8×1 = {A121 A12
2 A211 A21
2 A231 A23
2 A321 A32
2 }T
⎫⎬⎭ , (B 21)
P8×10 and S10×8 are the phase matrix and scattering matrix given by
P8×10 =[
P(H12) 04×4 04×2
04×4 P(H23) 04×2
](B 22)
and
S10×8 =
⎡⎢⎢⎣
S1 02×4 02×2
04×2 S2 04×2
04×2 04×4 S3
⎤⎥⎥⎦ , (B 23)
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respectively, and s10×1 is the source vector
s10×1 = {0 0 1 −|k| 1 −|k| 0 0 0 0}T. (B 24)
By the inverse Fourier transform
f (x) = 12π
∫+∞
−∞f (k) eikx dk, (B 25)
the normal strain in the x-direction along the encapsulation/epidermis interface can be derivedin the physical domain from equations (B 4), (B 8) and (B 20) as
ε11(x) = ∂
∂x
[1
2π
∫+∞
−∞u1(0; A21
ξ , D21ξ ) eikx dk
]
= − 14π i
3E12 + E23
∫+∞
−∞k|k| {e
−|k|H12H12 e−|k|H12
1 01×7}(I10×10 + S10×8P8×10)−1s10×1 eikx dk
= − 12π i
3E12 + E23
∫+∞
0{e−kH12
H12 e−kH121 01×7}(I10×10 + S10×8P8×10)−1s10×1sin (kx) dk,
(B 26)
Equation (B 26) can be further decomposed into two separate parts as
ε11(x) = εbi-material11 (x) + εc
11(x), (B 27)
where εbi-material11 denotes the normal strain in the x-direction along the encapsulation/epidermis
interface owing to the same force in the corresponding bi-material composed of an encapsulationhalf-plane and an epidermis half-plane defined as
εbi-material11 (x) = − 1
4π i3
E12 + E23
∫+∞
−∞k|k| {e
−|k|H12H12 e−|k|H12
1 01×7}s10×1 eikx dk
= − 14π i
3E12 + E23
∫+∞
−∞k|k| eikx dk = − 3
2π (E12 + E23)1x
, (B 28)
and εc11 is the complementary part defined as
εc11(x) = − 1
4π i3
E12 + E23
×∫+∞
−∞k|k| {e
−|k|H12H12 e−|k|H12
1 01×7}[(I10×10 + S10×8P8×10)−1 − I10×10]s10×1 eikx dk
= − 12π
3E12 + E23
×∫+∞
0{e−kH12
H12 e−kH121 01×7}[(I10×10 + S10×8P8×10)−1 − I10×10]s10×1sin (kx) dk.
(B 29)
Appendix CThe infinite series in equation (3.26) for the normalized shear stress is truncated to the Nth
term when substituted into equations (3.22) and (3.23). Multiplying√
1 − X2aUm−1(Xa) and√
1 − X2s Um−1(Xs) on both sides of equations (3.22) and (3.23), and then integration from −1 to
+1 with respect to Xa and Xs, respectively, give a set of linear algebraic equations for Can and Cs
n
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asN∑
n=1
(Λa−amn Ca
n + Λa−smn Cs
n) = − 1π
δm1
andN∑
n=1
(Λs−amn Ca
n + Λs−smn Cs
n) = 0,
⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭
(C 1)
where Un is the nth-order Chebyshev polynomial of the second kind, δij is the Kronecker delta,and
Λa−amn = δmn − λagmn + cos
(m − n)π2
∫+∞
0
2mΘ
Jm(Θ)Jn(Θ)f(
Θ
a
)dΘ
Λa−smn = Ka
Ks
∫+∞
0
2mbaΘ
Jm
(aΘb
)Jn(Θ)
[f(
Θ
b
)+ 1
]cos
[(m − n)π
2− lΘ
b
]dΘ
Λs−smn = δmn − λsgmn + cos
(m − n)π2
∫+∞
0
2mΘ
Jm(Θ)Jn(Θ)f(
Θ
b
)dΘ
and Λs−amn = Ks
Ka
∫+∞
0
2mabΘ
Jm
(bΘa
)Jn(Θ)
[f(
Θ
a
)+ 1
]cos
[(m − n)π
2+ lΘ
a
]dΘ
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
, (C 2)
with Jn being the nth-order Bessel function of the first kind. The function gmn in equation (C 2) isdefined as
gmn =
⎧⎪⎪⎨⎪⎪⎩
4m[(−1)m−n + 1]
π2[(m − n)2 − 1][(m + n)2 − 1], if (m − n)2 �= 1;
0, if (m − n)2 = 1
. (C 3)
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