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Research ArticleElectrical Impedance Tomography-Based
AbdominalSubcutaneous Fat Estimation Method Using Deep Learning
Kyounghun Lee ,1 Minha Yoo,2 Ariungerel Jargal ,3 and Hyeuknam
Kwon 4
1Center for Mathematical Analysis and Computation, Yonsei
University, Seoul 03722, Republic of Korea2National Institute for
Mathematical Science, Daejeon 34047, Republic of Korea3Department
of Computational Science and Engineering, Yonsei University, Seoul
03722, Republic of Korea4College of Science and Technology, Yonsei
University, Wonju 26493, Republic of Korea
Correspondence should be addressed to Hyeuknam Kwon;
[email protected]
Received 13 January 2020; Revised 5 April 2020; Accepted 30
April 2020; Published 11 June 2020
Academic Editor: Liangjiang Wang
Copyright © 2020 Kyounghun Lee et al. This is an open access
article distributed under the Creative Commons Attribution
License,which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly
cited.
This paper proposes a deep learning method based on electrical
impedance tomography (EIT) to estimate the thickness ofabdominal
subcutaneous fat. EIT for evaluating the thickness of abdominal
subcutaneous fat is an absolute imaging problemthat aims at
reconstructing conductivity distributions from current-to-voltage
data. Existing reconstruction methods based onEIT have difficulty
handling the inherent drawbacks of strong nonlinearity and severe
ill-posedness of EIT; hence, absoluteimaging may not be possible
using linearized methods. To handle nonlinearity and ill-posedness,
we propose a deep learningmethod that finds useful solutions within
a restricted admissible set by accounting for prior information
regarding abdominalanatomy. We determined that a specially designed
training dataset used during the deep learning process
significantly reducesill-posedness in the absolute EIT problem. In
the preprocessing stage, we normalize current-voltage data to
alleviate the effectsof electrodeposition and body geometry by
exploiting knowledge regarding electrode positions and body
geometry. Theperformance of the proposed method is demonstrated
through numerical simulations and phantom experiments using a
10channel EIT system and a human-like domain.
1. Introduction
Abdominal obesity is closely linked to metabolic syndromeand
cardiovascular diseases [1–3]. As a major health indica-tor, it is
desirable to estimate the regional distribution ofabdominal fats,
such as subcutaneous and visceral fats. Com-puted tomography (CT)
and magnetic resonance imagingcan quantitatively estimate the
distribution of abdominalfat [4]. However, these methods are
expensive and unsuitablefor daily use. Furthermore, CT has
associated safety issuesbased on radiation exposure [5]. Therefore,
there is a growingdemand for a cheaper and safer abdominal fat
evaluationmethod that is practical for continuous self-monitoring
totrack body fat status as part of a daily routine.
Electrical impedance tomography (EIT) [6–8] may be thetop
candidate for meeting this demand based on its low eco-nomic
burden, continuous self-monitoring capabilities, andsuitability for
daily routines. EIT aims at visualizing the dis-
tributions of electrical conductivity inside the human body.Such
distributions can be used to evaluate the thickness ofsubcutaneous
fat because the electrical properties of adiposetissue are
significantly different from those of other tissues[9–11]. EIT uses
a number of electrodes (typically 8 to 32)attached to the surface
of the body. Current-voltage dataare acquired by applying
alternating currents over variousfrequencies (from tens of
kilohertz to megahertz) and mea-suring voltages through the
attached electrodes. These volt-ages reflect internal conductivity
distributions. Here, theamount of current injected to human body is
less than orequal to 10mA at the current excitation frequency 100
kHz,and the safe range of current depends on the excitation
fre-quency [12]. Conductivity distributions are recovered fromthe
current-voltage data using reconstruction algorithms. Itis
theoretically guaranteed that a conductivity distributioncan be
uniquely identified based on (infinite) current-voltagedata
[13–15]. The EIT imaging reconstruction problem of
HindawiComputational and Mathematical Methods in MedicineVolume
2020, Article ID 9657372, 14
pageshttps://doi.org/10.1155/2020/9657372
https://orcid.org/0000-0002-8520-8999https://orcid.org/0000-0003-3676-6086https://orcid.org/0000-0002-4362-2585https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/9657372
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recovering conductivity distributions inside the abdomen,which
is referred to as the absolute imaging problem [16],is highly
nonlinear and severely ill-posed. Despite over30 years of
development of EIT reconstruction methods,performance for absolute
imaging is still insufficient forclinical applications, although
difference imaging has beensuccessful in some applications as a
contrast imaging method[6, 17]. There have been numerous studies on
EIT recon-struction algorithms, such as backprojection [18],
NOSER[19], GREIT [20], the D-bar method [21], factorizationmethod
[22], and regularized least squares method [23].Representative
methods for solving the absolute imagingproblem are the regularized
least squares method [6, 19, 24]and D-bar method [25–27]. The
regularized least squaresmethod is based on the minimization
problem argmin γ1/2kFðγÞ −Vk2 + RegðγÞ for recovering a
conductivity distribu-tion γ from given data V, where F is a
forward operator thatmaps data from γ to data in Vði:e:,F : γ⟶VÞ,
where RegðγÞ is a regularization term and k · k is the Euclidean
norm.These methods handle ill-posedness by forcing the mini-mizer
to have a desired property that is determined basedon prior
knowledge regarding γ and incorporated in RegðγÞ.However, this
approach fails to produce useful images forabsolute abdominal
imaging with regularization, includingL2 and L1 regularization, and
variation [6] (see Section 2.3).The D-bar method performs nonlinear
direct reconstruction,but reconstruction can be inaccurate when
data are measuredfor only a small portion of the boundary [28].
When describing the forward operator F, most conven-tional
methods use a physics-based model. Specifically, ageneralized
Laplace equation representing electric potentialdistributions at
low frequencies is typically developed fromMaxwell’s equation [6,
7]. Physics-based models not onlydepend on the relationships
between conductivity distribu-tions and current-voltage data, but
also on geometrical fac-tors, body shape, and electrode positions
[29]. Therefore,errors or uncertainty in geometry factors can
easily causethe reconstruction process to misinterpret the
underlyingcurrent-voltage data, thereby compromising image
recon-struction results. In fact, the influence of geometrical
factorsis so severe that regularization in physics-based models
isnot sufficient for absolute imaging [27, 30–32]. To handlethis
undesirable influence, it would be advantageous to incor-porate
geometrical factors in the regularization term Regð·Þ.However, this
is a difficult proposition for physics-basedmodels based on the
difficulty of explicitly describing geo-metrical influences on data
during regularization.
To avoid using physics-based models, which are thefundamental
cause of many of the difficulties in absoluteimaging, we propose
using a deep learning technique thatincorporates a data-based
model. Recently, deep learningmethods have been actively applied to
EIT [33]. Suchmethods can be divided into two main types: methods
thatmap images to images and methods that map data to images.The
former type enhances the resolution of relatively low-resolution
images generated by conventional methods [27,34]. The latter type
directly learns mappings from measureddata to images [35, 36]. In
this study, we developed a method
of the second type by adopting a multilayer perceptron(MLP),
which is one of the deep-learning techniques. MLPsuse fully
connected layers, meaning each node in each inte-rior layer is
connected to all nodes in the next layer [37]. Thisfully connected
structure is suitable for EIT because bound-ary voltage data are
entangled with the global structures ofconductivity distributions
[7].
Furthermore, we combine an MLP with a special datanormalization
technique to reduce inherent geometricalinfluences. We establish an
operator G : V⟶ γ that canbe considered as a backward operator
corresponding to theforward operator F by using a training dataset
consisting ofgeometrical dependency-reduced voltage data (Section
3.2).Specifically, we normalize the measured voltage V to mini-mize
geometrical dependency and apply this normalizedvoltage to the MLP
using a normalization map (Section3.1). When generating a training
dataset D, we use simplifiedconductivity distributions by layering
imaging domains ofuniform thickness (Section 3.3). This
simplification makesthe estimation of the thickness of subcutaneous
fat much eas-ier and reduces the ill-posedness of the absolute
imagingproblem by reducing the number of unknown variables.
This remainder of this paper is organized as follows. InSection
2, we present preliminary information, includingthe electrical
properties of the abdomen, data acquisitionmethods, and
conventional methods. The proposed methodis introduced in Section
3, which has three subsections focus-ing on data normalization, the
MLP, and training datasets. InSection 4, we present numerical
simulation results. Finally,we conclude this paper in Section
5.
2. Preliminary Study: Conductivity and Data
2.1. Electrical Properties of the Abdomen. The abdomen
hasdifferent electrical properties related to different organsand
can be roughly divided into four regions: subcutaneousfat (just
below the skin), abdominal muscle, visceral fat (fatthat surrounds
the organs), and organ tissue, as shown inFigure 1. Fat and muscle
tissues have very distinct conduc-tivity spectra over the
frequencies plotted in Figure 2 [38],which reveal that the
conductivity of muscle tissue is sixtimes greater than that of fat.
Let γf and γm denote the con-ductivity values of fat and muscle,
respectively. Then, weassume that
γf < γm ≤ 10γf : ð1Þ
These distinct electrical properties of organs in the abdo-men
motivate the use of impedance data and EIT techniquesfor the
estimation of subcutaneous fat thickness by distin-guishing fat
from muscle.
2.2. Data Acquisition. Let an imaging object occupy a two-
orthree-dimensional space Ω bounded by its surface ∂Ω.
Theconductivity in Ω at an angular frequency ω and positionr = ðx,
yÞ or ðx, y, zÞ is denoted by γðrÞ. A set of surface elec-trodes
attached to ∂Ω apply currents and measure corre-sponding voltages.
When applying a sinusoidal current
2 Computational and Mathematical Methods in Medicine
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between a chosen pair of electrodes at an angular frequencyω,
the induced voltage in the body can be expressed as UðrÞsin ðωt +
θðω, rÞÞ, where θ indicates the phase angle. Then,the corresponding
time-harmonic potential uðrÞ =UðrÞ expðθðω, rÞ
ffiffiffiffiffiffi−1p Þ is governed by
∇· γ∇uγð Þ = 0 inΩ,n ⋅ γ u∇γð Þ = g on ∂Ω,
(ð2Þ
where n is the outward unit normal vector corresponding to∂Ω and
g is the boundary current density on ∂Ω induced bythe applied
current.
Let ðε1, ε2,⋯, εEÞ denote the attached surface electrodes.We
sequentially apply M different currents using chosenelectrode pairs
fðε1+ , ε1−Þ, ðε2+ , ε2−Þ,⋯, ðεM+, εM−Þg, where1±,⋯,M± ∈ f1, 2,⋯,
Eg. Let uj denote the induced potentialin (2) corresponding to the
jth applied current, where asinusoidal current of I mA at an
angular frequency ω isapplied through the electrode pair ðεj+ ,
εj−Þ. By denoting
(a) (b)
Figure 1: Abdominal CT image (a) and corresponding segmented
image (b) separated into three main regions (subcutaneous fat
coloredblue, muscle colored red, and visceral fat colored yellow)
with bone (white) and other tissues (gray).
102 103 104 105 106
Frequency (Hz)
0
0.1
0.2
0.3
0.4
0.5
0.6Conductivity (S/m)
FatMuscle
Figure 2: Conductivity values of subcutaneous fat (blue) and
muscle (red) tissues over the frequency range of 100Hz to 1MHz
[38].
3Computational and Mathematical Methods in Medicine
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the corresponding Neumann data as gj, we haveÐε j+gjds =
I = −Ðε jgjds, where gj is approximately zero on ∂ΩðEj+ ∪
Ej−Þ, where ds is the surface element. To estimate a
conduc-tivity distribution, we measure the voltage
differencebetween the ith pair of electrodes subject to the jth
appliedcurrent as
V j,i = uγj εi+ − u
γj
�� ��εi− , ð3Þfor j, i = 1, 2⋯ ,M. We do not use Vj,i for ðj, iÞ
as fεj+ , εj−g∩ fεi+ , εi−g ≠ 0 based on the uncertainty caused by
skin-electrode contact impedances [7, 39, 40].
The voltage V j,i reflects the conductivity distribution
γaccording to the following relation:
V j,i ≈1I
ðΩ
γ rð Þ∇uγj rð Þ ⋅ ∇uγi rð Þdr, ð4Þ
where dr is the area element. The voltage V j,i heavily
dependson the body geometryΩ and electrode positions ðε1,
ε2,⋯,εEÞ,which are difficult to acquire in practice. To minimize
thedependency on body geometry and electrode positions, wefix the
positions of the electrodes on a curved plate, as shownin Figure 3.
The shape of the curved plate is predeterminedand designed to fit
the human abdomen. By concatenatingall voltages V j,i in order, we
generate a vector V of V j,i valuesas follows:
V = V j1,i1,V j2,i2,⋯,V jM∗ ,iM∗h iT
, ð5Þ
where ðj1, i1Þ, ðj2, i2Þ,⋯, ðjM∗, iM∗Þ are the ordered
indexpairs of fðj, iÞ: fεj+ , εj−g ∩ fεi+ , εi−g = 0g and M∗ is the
num-ber of voltages in V. Then, V is referred to as a vector
ofcurrent-voltage data.
2.3. Conventional Method: Sensitivity Approach. The mostwidely
used EIT image reconstruction method is the sensitiv-ity method
[19], which operates based on the voltage-conductivity relationship
in (4). This method requires a dis-cretized imaging domain Ω with L
subregions (mostly trian-gular) Ω1,Ω2,⋯,ΩL, which can be defined
as
Ω = ∪Lℓ=1Ωℓ: ð6Þ
Assuming that γ is constant in each subregion Ω, thevoltage in
(4) can be expressed approximately as a linear sys-temV = Sγγ for γ
= ½γ ∣Ω1, γ ∣Ω2,⋯,γ ∣ΩL�T Where Sγ is anM∗ × L sensitivity matrix
defined as ðSγÞα,β = 1/I
ÐΩℓ∇uγj ðrÞ ·
∇uγj ðrÞdr with α = ðj, iÞ, β = ℓ. γ can be derived from V
bysolving the linear system V = Sγγ. However, the matrix Sγis
ill-conditioned. Therefore, to recover γ from V, the regu-larized
least squares method [6] is used as follows:
V↦ γ≔ argminγ
12 Sγγ‐V�� ��2 + λRReg γð Þ, ð7Þ
where k⋅k is the standard Euclidean norm, Reg is a
regulari-zation operator, and λR > 0 is a regularization
parameter.Tikhonov and total variation regularizations are also
widelyused [19, 40]. However, the regularized least squares
methodfor absolute EIT suffers from the fundamental difficulty
inhandling the ill-conditioned matrix Sγ based on theunknown γ and
forward modelling errors.
3. Method: Absolute EIT Reconstruction UsingDeep Learning
In this section, we discuss the proposed absolute
imagereconstruction algorithm for estimating subcutaneous
fatthickness. We develop a map G from V to γ based on deep-learning
techniques combined with the special data normal-ization method
introduced in [11]. The map G is defined bytwo other maps, namely,
the data normalization map Ψand conductivity reconstruction map Ξ,
as follows:
G = Ξ ∘Ψ: ð8Þ
In Section 3.1, we presented a data normalization map Ψfrom the
data V to V̂, which minimizes forward modellingerror as
follows:
Ψ : V↦ V̂: ð9Þ
In Sections 3.2 and 3.3, for reconstructing the conductiv-ity γ
from the normalized data V̂, we introduced a map Ξ,which is defined
as
Ξ : V̂↦ γ, ð10Þ
based on deep learning techniques.
Electrode array
Electrode array
Figure 3: Left image shows an illustrative human with an
electrodearray attached to their abdomen in 3D. The right image
shows across section of the left image with the electrode array
depicted inblack. In the right image, blue, red, yellow, white, and
gray colorsrepresent subcutaneous fat, muscle, visceral fat, bone,
and othertissues, respectively.
4 Computational and Mathematical Methods in Medicine
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3.1. Normalization of Current-Voltage Data. The reconstruc-tion
of the conductivity distribution γ from the data V is verysensitive
to forward modelling errors caused by the inaccu-racy of electrode
positions and geometrical uncertainty [7].To alleviate such forward
modelling errors, we normalizethe data V based on information
regarding boundary geom-etry. The map Ψ is designed to minimize the
geometrydependency of the data V j,i, which are components of V,
asfollows ([11], the equation (2.18)):
Ψ Vð Þ = V̂, where V̂j,i ≔Sj,iVj,i
≈ÐΩ∇vj rð Þ ⋅ ∇vi rð ÞdrÐ
Ωγ rð Þ∇uγj rð Þ ⋅ ∇uγi rð Þdr
,
ð11Þ
and Sj,i = 1/IÐΩ∇vjðrÞ · ∇viðrÞdr, where vj is the solution
to
vj = 0 in Ω with the boundary condition n ⋅ ∇vj = gj on ∂Ω.Here,
V̂j,i can be considered as a weighted harmonic averageof
conductivity whose weight depends on the distribution of∇uγj ·
∇u
γi . It should be noted that if the conductivity distribu-
tion γ is homogeneous and represented as a constant, thenuγj ðrÞ
= vjðrÞ/γðrÞ and V̂j,i = γ for all i, j, regardless of the
elec-trode positions and boundary geometry.
3.2. Multilayer Perceptron. We reconstruct the
conductivitydistribution γ from the normalized data V̂ using an
MLP,which is a deep-learning technique that can capture nonlin-ear
relationships between input and output data [41, 42].An MLP can
serve as a tool for creating a representationfunction based on a
given set of credible pairs of input andoutput data, which is
referred to as a training dataset. This
representation functionality can be considered as an
inversesolver for EIT. We recover the conductivity γ from the
nor-malized voltage data V by constructing a mapping Ξ : V̂⟶ γ
using an MLP. The map Ξ is constructed by composit-ing multiple
linear and nonlinear functions called activationfunctions. Below,
we provide details regarding how Ξ is con-structed from linear and
nonlinear functions.
An MLP consists of several layers of nodes or neurons, asshown
in Figure 4. To apply an MLP to EIT, we define thenodes in the
input layer as the normalized voltages in the vec-tor V and the
nodes in the output layer as the conductivityvalues γ for each
subregion Ω bin (6). The hidden layers,which are neither input nor
output layers, are used to extractcomplex features from the
relationships between the conduc-tivity and voltage data. Let an
MLP contain J layers and letthe jth layer contain Nj nodes. Since
the nodes in the firstlayer (input layer) are measured voltage data
and the nodesin the last layer (output layer) are the conductivity
valuesfrom subregions, we have N1 =M∗ and NJ = L. The
repre-sentation function Ξ in the MLP has the following form
ofsuccessive compositions:
Ξ V̂� �
= hJ−1∘⋯∘h2 ∘ h1 V̂� �
, ð12Þ
where hj : ℝN j ⟶ℝN j+1 is given by
hj oj� �
= oj+11 , oj+12 ,⋯,o
j+1N J+1
h iTwith oj+1m = ξ 〠
N j
n=1Wjm,n o
jn
0@
1A
ð13Þ
1260
512256
128 64 32 15Input Output
Input Output
V(∈R1260) 𝛾(∈R15)
Structure of hidden layers
𝛾(1), 𝛾(2), … , 𝛾(NT) V(1), V(2), … ,V(NT)
Specially designed abdomen model only for training
Solve (2) and from (3)
Forward simulationV
Compute (8)
Normalization
Normalized current-voltage data
Training data generation
1.5
1
0.5
00 100 200 300 400 500 600 700 800 9001000
1.6
1.5
1.5
1.5
1.5
1
0.5
0 0 100 200 300 400 500 600 700 800 900 1000
1000
1000
1260
512256
128 64 32 15Input OutputV(∈R1260) 𝛾(∈R15)
𝛾(1), 𝛾(2), … , 𝛾(NT) V(1), V(2), … ,V(NT)
Solvevv (2) and from (3)
ForFF ward simulationV
Compute (8)
Normalization
Training data generation
.5
1
.5
00 100 200 300 400 500 600 700 800 9001000
1.6
1.5
1.5
1.5
1.5
1
0.5
0 0 100 200 300 400 500 600 700 800 900 1000
1000
1000
1000
1000
⌃ ⌃ ⌃
⌃
Figure 4: Schematic diagram of the MLP used in this study. To
apply the MLP, we require a training dataset, as shown in the blue
box. In thered box, we illustrate the details of the MLP structure.
The numbers in the black box indicate numbers of nodes. The inputs
are obtained fromthe training dataset and the outputs are
conductivity values.
5Computational and Mathematical Methods in Medicine
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for oj ∈ℝN j . Here, Wjm,n is the weight connecting the mthnode
(neuron) in the ðj + 1Þth layer to the nth node (neuron)in the jth
layer and ξ is an activation function. It shouldbe noted that o1
are the input data, meaning o1 = V̂ andoN j are the output data,
meaning oNJ = γ. In this study, arectified linear unit ξ ðxÞ = J
max ðx, 0Þ was used as theactivation function. It should be noted
that Ξ is determined
by the weights W ≔ ffW1m,ngm=1,⋯,N2n=1,⋯,N1 ,
fW2m,ngm=1,⋯,N3n=1,⋯,N2 ,⋯,
fWJ−1m,ngm=1,⋯,N Jn=1,⋯,N J−1g. Therefore, we denote ΞWðV̂Þ≔
ΞðV̂Þ.To determine W, we minimize the function
W ≔ arg minW
〠NT
k=1ΞW V̂
kð Þ� � − γ kð Þ������2, ð14Þ
for a given training dataset ðγð1Þ, V̂ð1ÞÞðγð2Þ, V̂ð2ÞÞ,⋯, ðγðNT
Þ,V̂ðNT ÞÞ, where NT is the number of training data. As shownin
(14), the set of weights W, meaning the map Ξ is deter-mined by the
training dataset. In the next section, we will pres-ent the
training dataset adopted in this study.
3.3. Training Dataset from Numerical Simulations forConstructing
the Map Ξ. The map Ξ used in (12) and (13)is determined by W, which
is defined by the minimizationin (14). The result of the
minimization in (14) depends on
the training dataset fðγðiÞ, V̂ðiÞÞgNTi=1. Therefore, the map Ξ
isultimately determined by the training dataset. Consequently,the
design of the training dataset is very important. There-fore, in
this section, we present the details regarding howthe training data
were defined.
Generating training data requires solving equation (2) toderive
the voltage dataV that determine g and the conductiv-ity
distributionb γ. We adopted the body shape in (5) and thenormalized
voltage V̂ in (11) for a given domain Ω, the elec-trode positions
for the domain Ω from two-dimensionalabdomen CT axial images, and
considered a case with 10electrodes on the central front part of
the abdomen. Accord-ing to [11], it is acceptable to consider a
limited regionaround the electrode array, as shown in Figure 5,
because
measured voltages are rarely affected by the conductivity
farfrom the electrodes. In the restricted region, we use a
specialtype of internal domain that enables us to estimate the
thick-ness of abdominal fat more easily. We propose dividing
theimaging domain into L disjoint layers Ω1,⋯,ΩL with thick-nesses
of d0 such that
Ωℓ ≔ r ∈Ω : d0 ℓ − 1ð Þ < dist r, ∂Ωð Þ < d0ℓf g andΩL≔Ω \
∪L−1ℓ=1Ωℓ,
ð15Þ
for ℓ = 1, 2,⋯, L − 1, where dist (r, ∂Ω) denotes the dis-tance
between r and ∂Ω. In this study, we used 15 thin layers(i.e., L =
15). By using the layers Ω, we can simply divide thedomainΩ into
three regions of subcutaneous fat, muscle, andother tissues, whose
conductivity values are denoted as γf ,γm, and γr , respectively.
Then, the conductivity distribu-tion can be expressed as
γ rð Þ =γf , r ∈ ∪1 ≤ ℓ ≤ ℓfΩℓ,γm, r ∈ ∪ℓ f ≤ ℓ ≤ ℓmΩℓ,
γr , otherwise,
8>><>>:
ð16Þ
where ℓf and ℓm are indices for the subregions of subcuta-neous
fat and muscle, respectively. It should be noted thatℓf < ℓm
because the subcutaneous fat is the outermostregion. Therefore, the
thicknesses of the subcutaneous fatand muscle regions can be easily
estimated as d0ℓf andd0ðℓm − ℓf Þ, respectively.
In this study, we tested all possible values of ℓf and ℓmwhile
maintaining at least one layer for each region (i.e., min-imum of
ℓf = 1, maximum of ℓf = 13, minimum of ℓm = 2,and maximum of ℓm =
14). Therefore, we testing 91 differentpartitions for the three
regions. Regarding the conductivityvalues for subcutaneous fat γf ,
muscle γm, and other tissuesγr , we assigned conductivity values
ranging from 1 to 10 witha step size of 0.5 according to (1).
Specifically, we testedγf = 1, γm = 2:0, 2:5, 3:0,⋯, 10:0, and γr =
1:5, 2:0, 2:5,⋯,9:5 satisfying γf < γr < γm ≤ 10γf .
Therefore, we tested 17
Electrode array
Region of interest
(a)
𝛺1𝛺2𝛺3
𝛺L
…
(b)
Figure 5: Region of interest in the imaging domain (a) and
subregions of layers (b).
6 Computational and Mathematical Methods in Medicine
-
different values for γm and nine different values for γr .
Con-sidering the 91 different partitions for the three types
oforgans, a total of 13923ð= 91 × 17 × 9Þ different
conductivitydistributions were used for the training dataset.
To produce a normalized voltage V̂ with a given conduc-tivity
distribution γ in a given domain with a given electrodearray, we
used 45 different current application patterns (withall possible
electrode pairs using 10 electrodes). For eachapplication, we
measured 28 voltages using all electrodepairs for which no current
was applied. We used a set ofpairs ðγ, V̂Þ of simplified
conductivity distributions and nor-malized data for the training
dataset.
By using the training dataset fðγðiÞ, V̂ðiÞÞgNTi=1, we
con-structed the representation function Ξ defined in (12)
usingTensorFlow [43] with five hidden layers ðJ = 7Þ. We set
the
numbers of nodes in each hidden layer as ðN2,N3,N4,N5,N6Þ =
ð512, 256, 128, 64, 32Þ.
4. Results
In this section, we present numerical simulations to validatethe
proposed method. In Section 4.1, we present recon-structed images
of the human abdominal model to determinesubcutaneous fat
thickness, as well as conductivity values fordifferent fat
thicknesses and body shapes. To calculate thepercentage error of
thickness estimation, in Section 4.2, wepresent numerical
simulations with various fat thicknessesin a circular domain.
Additionally, numerical experimentsin Section 4.3 demonstrate the
robustness of the proposedreconstruction algorithm in a nonabdomen
domain withrandom data and nonabdomen domain with regular data.
(i) Training image generation: generate γðkÞ (specially designed
abdomen model used solely for training)(ii) Data acquisition:
obtain VðkÞ from (2) and (3)(iii) Normalization: obtain V̂ðkÞ from
(11)(iv) Minimization: find the minimizer W of (14)(v) Construction
of Ξ: compute Ξ from (13) and (12)
Algorithm 1: Construction of Ξ.
2
4
6
8
10
(i)
(ii)
(iii)
(a) (b) (c)
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
2
4
6
8
10
-4
-2
0
2
4
6
-6
-4
-2
0
2
4
6
1.2
0.8
0.6
0.4
0.2
1
Figure 6: Numerical simulations for testing the proposed
reconstruction algorithm with different subcutaneous fat and muscle
thicknessesand body shapes. (i) Thick subcutaneous fat. (ii) Thin
subcutaneous fat. (iii) Different body shape. (a) True conductivity
distributions. (b)Reconstructed images generated by the proposed
method. (c) Reconstructed images generated by the regularized least
squares method.
7Computational and Mathematical Methods in Medicine
-
Finally, in Section 4.4, we present a numerical validation
ofdata normalization.
We present the image reconstruction algorithm below.To test the
proposed algorithm, we normalized the measureddata V to obtain V̂
and inserted it into the function Ξ, result-ing in the image γ =
ΞðV̂Þ.
4.1. Image Reconstruction. This section presents the
imagereconstruction results of the proposed method in compari-son
to those of the conventional method (regularized least
squares method), which was originally presented in (7). Thefirst
test image contains variations in the thicknesses of sub-cutaneous
fat and muscle. For the second test image, wechanged the body
shape. To generate an internal conductiv-ity distribution of the
abdomen for our simulations, we usedCT images and assigned
conductivity values to each organsatisfying (1), as shown in Figure
6(a). To generate thecurrent-voltage data V for (5), we applied
45ð= 10 × 9 ÷ 2Þcurrents to all possible pairs of electrodes. For
each appli-cation, we measured 28 voltages between the
remaining
# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
25 26 27 28
3 4 5 6 7 8 9 3 4 5 6 7 8 3 4 5 6 7 3 4 5 6 3 4 5 3 4 34 5 6 7 8
9 10 5 6 7 8 9 10 6 7 8 9 10 7 8 9 10 8 9 10 9 10 10
1
2
Injectpair
Measurepair
Figure 7: Electrode pairs for current application and voltage
measurement when current is applied through the first and second
electrodes.
0 3Depth (cm)
–4
2
5
7
Cond
uctiv
ity (S
/m)
True 𝛾 ProposedConventional
12
2.1
(a)
True 𝛾 ProposedConventional
0 3
Depth (cm)
0
2
5
7
12
Cond
uctiv
ity (S
/m)
–41.2
(b)
Figure 8: Profiles corresponding to the first and second rows of
images in Figure 6. Each profile line begins at the top center of
thecorresponding image and moves to a vertical depth of up to 3 cm.
In both plots, gray represents true conductivity values, and blue
and redrepresent the reconstructed conductivity values generated by
the proposed and conventional methods, respectively. (a) Thick fat
casecorresponding to the first row in Figure 6. (b) Thin fat case
corresponding to the second row in Figure 6.
7
6
5
4
3
2
1
(a)
0.3 2.1 3.9Thickness of subcutaneous fat (cm)
0
3.5
7
Thic
knes
s err
or (%
)
(b)
0.3 2.1 3.9Thickness of subcutaneous fat (cm)
7
17.5
28
Cond
uctiv
ity er
ror (
%)
(c)
Figure 9: (a) One representative circular model for the test
process among 13 models with subcutaneous fat thicknesses ranging
from 0.3 to3.9 cm. (b) Percentage errors of thickness estimation
using the proposed method. (c) Percentage errors of conductivity
estimation using theproposed method.
8 Computational and Mathematical Methods in Medicine
-
electrode pairs, where no current was applied (as shown inFigure
7).
A total of 1260ð= 45 × 28Þ voltages ðM∗ = 1260Þ wereused to
reconstruct the conductivity distributions. Thecurrent-voltage
dataVwas obtained by solving the governingequation (2) using the
finite element method. In Figure 6, we
present reconstructed images and ground-truth images. InFigure
8, we present profiles for ease of comparison.
In Figure 6, we present the results for varying subcutane-ous
fat and muscle thicknesses. The different types of testdomains are
arranged in row (i) for thick subcutaneous fat,row (ii) for thin
subcutaneous fat, and row (iii) for a different
2
4
6
8
10
(a)
2
4
6
8
10
(b)
2
4
6
8
10
(c)
2
4
6
8
10
(d)
0 200 400 600 800 1000 1200 14000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(e)
2
4
6
8
10
(f)
Figure 10: (a, c) True conductivity distributions. (b, d)
Corresponding image reconstruction results generated by the
proposed method. (e)Voltage data generated based on random numbers
drawn from a Gaussian distribution. (f) Image reconstruction
results generated by theproposed method.
9Computational and Mathematical Methods in Medicine
-
body shape based on a different CT image. For comparison,we
present images of (a) the true conductivity distributionand the
reconstructed conductivity distributions generatedby (b) the
proposed method and (c) conventional method.We use the same color
bar for the true conductivity distribu-tions and the images
reconstructed by the proposed method,and use an adjusted color bar
to improve image contrast forthe images reconstructed by the
regularized least squaresmethod. If we use the same color bar as
the true images forthe images reconstructed by the regularized
least squaresmethod, then the resulting images are almost
homogeneous(i.e., no distinct image contrast). The images
reconstructedby the proposed method contain distinct borders
betweensubcutaneous fat and muscle, whereas the images
recon-structed by the regularized least squares method fail to
dividesubcutaneous fat and muscle.
For ease of observing subcutaneous fat estimations, inFigure 8,
we present profiles for the images in Figure 6.The profiles
represent regions from the top center of eachimage to a vertical
depth of up to 3 cm from each image.In Figure 8, we present both
(a) thick fat and (b) thin fat casescorresponding to the first and
second rows in Figure 6. In thecase with thick fat, the proposed
method succeeds in captur-ing the border between fat and muscle,
whereas the conven-tional method cannot identify fat thickness and
conductivityvalues. Furthermore, the proposed method can
accuratelyreconstruct the conductivity value of fat. When the fat
isthin, the proposed method is still able to estimate fat
thick-ness and conductivity values, whereas the conventionalmethod
cannot determine fat thickness and conductivityvalues.
4.2. Thickness Estimation. In this section, we present the
per-centage errors (relative errors of subcutaneous fat thicknessas
percentages) for estimating subcutaneous fat thicknessand
conductivity values using the proposed method (Section3). To derive
quantitative results, we used a circular model,rather than a body
shape (for both training and testing), witha radius of 10 cm and
various subcutaneous fat thicknessesranging from 0.3 cm to 3.9 cm
with a step size of 0.3 cm, asshown in Figure 9(a). The estimations
of fat thickness werederived from the reconstructed concentric
circles (layers)by measuring the length from the boundary to the
borderof each layer, where the conductivity values change
abruptly.The results of estimating subcutaneous fat thickness
arevalues corresponding to 13 thickness classes (possible
fatthickness classes for the training model): 0.3, 0.6, 0.9,
1.2,1.5, 1.8, 2.1, 2.4, 2.7, 3.0, 3.3, 3.6, and 3.90 cm. Because
thetesting model used the same variations in subcutaneous
fatthickness as the training model, the errors are discretely
dis-tributed and can be equal to zero (Figure 9(b)). The
conduc-tivity values of fat were also estimated and the
correspondingpercentage errors are presented in Figure 9(c). The
errors ofestimating thickness and conductivity values are always
lessthan 7% and 28%, respectively, for subcutaneous fat
thick-nesses ranging from 0.3 to 3.9 cm.
4.3. Robustness. The goal of this section is to demonstrate
therobustness of the proposed method for nonabdominal data.
Based on the results presented in this section, we can con-clude
that the proposed reconstruction algorithm does notartificially
generate human abdomen images from nonab-dominal data. For
verification, we tested two types of data:(1) impedance data from
abdominal shape domains withnonabdominal conductivity distributions
and (2) data con-sisting of random numbers.
We use the same abdominal domain shape and electrodealignment as
those used in Section 4.1 and shown inFigure 6(i). The first
impedance data we tested came from anabdominal shape domain with
various conductivity anoma-lies. The background conductivity value
was two, the upperanomalous conductivity value was seven, and the
lower anom-alous conductivity value was five, as shown in Figure
10(a).Next, we tested impedance data from an abdominal shapedomain
with random conductivity distributions drawn froma Gaussian
distribution, as shown in Figure 10(c). The recon-struction results
in Figures 10(b), 10(d), and 10(f) do notexhibit a fat-muscle
structure and show almost entirely back-ground conductivity
values.
4.4. Data Normalization. In this section, we present evidenceof
the benefits of using the normalized data V in (11), ratherthan
using the data V in (5) with the same current measure-ment patterns
discussed in Section 4.1. To this end, we com-pare the results of
MLPs (Section 3.2) different geometricalshapes for the training
process (circular model) and testingprocess trained using V̂ andV.
For the purpose of comparinggeometrical influences, we used
(elliptical model). In thetraining process, we used a circular
model to create MLPmaps Ξnorm and Ξorig from V̂ and V,
respectively, to γ usingthe network in Figure 4. For the circular
model, we main-tained a radius of 10 cm and generated concentric
circlesfor subcutaneous fat, muscle, and interior tissues. The
firstouter layer of the model was subcutaneous fat with
0 2.1 3Depth (cm)
0
1
5
7
Cond
uctiv
ity (S
/m)
True 𝛾 𝛾 recontructed by orig𝛾 recontructed by norm
Figure 11: True and reconstructed conductivity values near
theboundary and up to a depth of 3 cm. The gray line represents
trueconductivity values. The red and blue lines
representreconstructed conductivity values generated from Ξorig and
Ξnormusing V and V̂, respectively. The results of using Ξorig fail
tocapture the border between the subcutaneous fat and muscle.
10 Computational and Mathematical Methods in Medicine
-
thicknesses ranging from 0.3 cm to 3.9 cm. The second layerof
the model was muscle with various thicknesses that satis-fied the
condition of the total thickness of fat and musclebeing equal to
4.2 cm. We set the conductivity value of thesubcutaneous fat to 1
S/m and tested various conductivityvalues for the muscle and
interior tissues ranging from 2 to10 S/m and from 1.5 to 9.5 S/m,
respectively, satisfyingthe condition that the conductivity value
of the muscle wasalways greater than that of the interior tissues.
According tothe settings described above, the total number of
trainingdata was 13923. For testing, we used an elliptical model
witha major axis of 15 cm and a minor axis of 9 cm and a
subcu-taneous fat thickness of 2.1 cm. The other settings were
thesame as those used for the training data. We applied eachMLP map
(Ξnorm and Ξorig) to the elliptical model. For easeof comparison,
Figure 11 presents conductivity profilesderived from the
reconstructed images based on Ξnorm andΞorig. The results of using
Ξnorm reveal accurate subcutaneousfat thicknesses, while those of
using Ξorig are inaccurate, asshown in Figure 11.
4.5. Phantom Experiments. In this section, we present phan-tom
experiments to validate the proposed method. We useda specially
designed phantom with a boundary shape repre-senting the human
abdomen and attached 10 electrodes tothe front of the phantom, as
shown in Figure 12(a). To gen-erate conductivity distributions
inside the phantom, weplaced a toroidal agar fabricated from
gelatin away from theboundary of the phantom at a uniform distance
from theboundary in all directions. We used a saline solution
(0.1%NaCl) to fill the two regions separated by the agar,
namely,the near boundary and the middle of the phantom. Here,the
agar represents the muscle layer and the regions sepa-rated by the
agar represent the regions of subcutaneous fatand the interior
organs. The electrical conductivity valuesof the agar and saline
water are 9.2 S/m and 0.2 S/m, respec-tively. The thickness of the
agar is 3.2 cm and the distancefrom the boundary is 1 cm, meaning
the thicknesses of themuscle and fat layers are 3.2 cm and 1 cm,
respectively. We
used a Sciospec 16 channel EIT system to apply 45 currentsand
measured 28 voltages for each current application, asdescribed in
Section 4.1. The amount of current appliedwas 8mA (as peak
amplitude) with an excitation frequencyof 100 kHz. We applied the
proposed method and conven-tional method to the measured data from
the phantom forimage reconstruction. To consider a more obese
abdomencase, we reduced the thickness of the agar to 1 cm by
trim-ming the outer section of the agar, resulting in a
distancefrom the agar to the boundary of 3.2 cm, as shown inFigure
12(b). In Figure 13, we present the image reconstruc-tion results
for the phantom experiments. In the case with athin fat layer (1
cm), the conductivity distribution of the pro-posed method abruptly
changes at the borders, whereas theconductivity distribution of the
conventional method doesnot reveal a clear border between fat and
muscle. In the casewith a thick fat layer (2 cm), the reconstructed
conductivityvalues in the fat region are constant for the proposed
method,but the conventional method yields irregular
reconstructedconductivity values in the fat region. These results
demon-strate that both the thicknesses and conductivity
distribu-tions of fat layers can be more clearly identified by
theproposed method compared to the conventional method.However, the
reconstructed conductivity values at the fatregion are
overestimated.
5. Conclusions and Discussion
We proposed an absolute EIT reconstruction method forabdominal
fat estimation using an MLP, which is a deep-learning technique.
The absolute EIT problem is nonlinear,ill-posed, and severely
affected by forward modelling errorsstemming from uncertainty in
electrode positions and bodygeometry. We adapted an MLP to capture
the nonlinearrelationships between current-voltage data and
conductivitydistributions. To alleviate forward modelling errors,
we nor-malized the current-voltage data based on information
regard-ing electrode positions and body geometry. When
performingreconstruction, we separated the problem domain into
sub-layers of uniform thickness to facilitate the estimation of
1.0 cm3.2 cm
(a)
2.0 cm2.2 cm
(b)
Figure 12: Abdomen-shaped phantom with agars of different
thicknesses. (a) 3.2 cm of the thickness of the agar and 1 cm of
the distance tothe boundary. (b) 2.2 cm of the thickness of the
agar and 2 cm of the distance to the boundary.
11Computational and Mathematical Methods in Medicine
-
abdominal fat thickness. This specially designed separationof
the problem domain significantly reduced the numberof unknown
variables, thereby reducing the ill-posednessof the absolute EIT
problem. We validated the proposedmethod through numerical
simulations and phantom exper-
iments using 10 channel EIT systems with a human-likedomain and
varying thicknesses of fat and muscle.
Based on the presence of errors in the data V, a widerdomain
should be considered for stable reconstruction ofthe conductivity
distribution γ as follows:
200
150
100
50
–50
–100
–150
–200
0
(a)
200
150
100
50
–50
–100
–150
–200
0
(b)
104.210Depth (cm)
–200
–100
0
100
200
Cond
uctiv
ity (S
/m)
ProposedConventional
(c)
200
150
100
50
–50
–100
–150
–200
0
(d)
200
150
100
50
–50
–100
–150
–200
0
(e)
ProposedConventional
104.220Depth (cm)
–200
–100
0
100
200
Cond
uctiv
ity (S
/m)
(f)
Figure 13: Reconstructed images from phantom experiments using
the proposed and conventional methods. Reconstructed images for
the1 cm fat layer generated by the (a) proposed method and (b)
conventional method. (c) Profile of the reconstructed conductivity
values alongthe depth direction. For the 3.2 cm fat layer,
reconstructed images generated by the (d) proposed method and (e)
conventional method.
12 Computational and Mathematical Methods in Medicine
-
ϒε ≔ γ : Sγγ −V�� �� < ε , ð17Þ
where ε > 0 is the error tolerance. Based on the
ill-posednature of EIT, small errors in the data can result in
abruptchanges in outputs ðdiamðϒεÞ≫ 1Þ. The regularizationsin (7)
can be considered as an attempt to restrict the widedomain Yε by
using prior knowledge regarding sparsity(L1), smoothness (L2), and
sharpness (total variation), butthese regularizations cannot
completely eliminate infeasiblesolutions, meaning they cannot
guarantee stable absoluteEIT reconstruction. In the proposed deep
learning method,the restriction of Yε can be achieved easily to
reject infeasiblesolutions by designing suitable training data.
Specifically, oneshould only select feasible conductivity
distributions for thetraining set to satisfy the desired properties
for restriction.
The fully connected nature of MLP layers may be redun-dant
because boundary voltage is insensitive to local pertur-bations in
conductivity, meaning the entanglement betweenboundary voltages and
conductivity is weak at regions farfrom electrodes.
One could use partially connected layers. This method
istypically referred to as a convolutional neural network [44].The
use of partially connected layers reduces the computa-tional cost
of the training process because it requires fewerweights to be
optimized.
In this study, we assumed that the thickness of subcuta-neous
fat was uniform when constructing training data. Thismodel not only
makes estimation of the thickness of subcuta-neous fat easier, but
also makes the problem less ill-posed byreducing the number of
unknown variables. However, theproposed method may be less accurate
when the thicknessof subcutaneous fat is nonuniform.
Data Availability
TheMatlab data used to support the findings of this study
areavailable from the corresponding author upon request.
Disclosure
A preliminary version of this manuscript was presented inthe
form of an abstract at the Chinese Society of Industryand Applied
Mathematics 15th Annual Meeting in Qingdaoin 2017. Since then, this
research has been updated andextended with new results.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
K.L. was supported by the National Research Foundationof Korea
under grant number NRF-2015R1A5A1009350.M.Y., K.L., and A.J. were
supported by the NationalResearch Foundation of Korea under grant
number NRF-2017R1E1A1A03070653.
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14 Computational and Mathematical Methods in Medicine
https://itis.swiss/database/https://www.tensorflow.org/https://www.tensorflow.org/
Electrical Impedance Tomography-Based Abdominal Subcutaneous Fat
Estimation Method Using Deep Learning1. Introduction2. Preliminary
Study: Conductivity and Data2.1. Electrical Properties of the
Abdomen2.2. Data Acquisition2.3. Conventional Method: Sensitivity
Approach
3. Method: Absolute EIT Reconstruction Using Deep Learning3.1.
Normalization of Current-Voltage Data3.2. Multilayer Perceptron3.3.
Training Dataset from Numerical Simulations for Constructing the
Map Ξ
4. Results4.1. Image Reconstruction4.2. Thickness Estimation4.3.
Robustness4.4. Data Normalization4.5. Phantom Experiments
5. Conclusions and DiscussionData
AvailabilityDisclosureConflicts of InterestAcknowledgments
