Spatiotemporal Regularization for Inverse ECG Modeling

Abstract

Advanced sensing such as the wearable sensor network provides an unprecedented opportunity to capture a wealth of information pertinent to space-time electrical activity of the heart, and facilitate the inverse electrocardiographic (ECG) modeling with the readily available data of body surface potential mapping. However, it is often challenging to derive heart-surface potentials from body-surface measurements, which is called the “inverse ECG problem”. Traditional regression is not concerned about spatiotemporal dynamic variables in complex geometries, and tends to be limited in the ability to handle high-dimensional spatiotemporal data for solving the inverse ECG problem. This paper presents a comparison study of regularization methods in the performance to achieve robust solutions of the inverse ECG problem. We first introduce the forward and inverse ECG problems. Second, we propose two spatiotemporal regularization (STRE) models to increase the robustness of inverse ECG modeling. Finally, case studies are conducted on the two-sphere geometry, as well as a real-world torso-heart geometry to evaluate the performance of different regularization methods. Experimental results show that STRE models effectively tackle the ill-conditioned inverse ECG problem and yield 56.3% and 67.3% performance improvement compared to the traditional Tikhonov regularization in the two-sphere and the torso-heart geometries, respectively. The spatiotemporal regularization methodology is shown to have strong potential to achieve robust solutions for high-dimensional predictive modeling in the inverse ECG problem.

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Spatiotemporal regularization for inverse ECG
modeling
Bing Yao & Hui Yang
To cite this article: Bing Yao & Hui Yang (2020): Spatiotemporal regularization for
inverse ECG modeling, IISE Transactions on Healthcare Systems Engineering, DOI:
10.1080/24725579.2020.1823531
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Spatiotemporal regularization for inverse ECG modeling
Bing Yaoand Hui Yang
Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA, USA
ABSTRACT
Advanced sensing such as the wearable sensor network provides an unprecedented opportunity
to capture a wealth of information pertinent to space-time electrical activity of the heart, and
facilitate the inverse electrocardiographic (ECG) modeling with the readily available data of body
surface potential mapping. However, it is often challenging to derive heart-surface potentials from
body-surface measurements, which is called the “inverse ECG problem.”Traditional regression is
not concerned about spatiotemporal dynamic variables in complex geometries, and tends to be
limited in the ability to handle high-dimensional spatiotemporal data for solving the inverse ECG
problem. This paper presents a comparison study of regularization methods in the performance to
achieve robust solutions of the inverse ECG problem. We first introduce the forward and inverse
ECG problems. Second, we propose two spatiotemporal regularization (STRE) models to increase
the robustness of inverse ECG modeling. Finally, case studies are conducted on the two-sphere
geometry, as well as a real-world torso-heart geometry to evaluate the performance of different
regularization methods. Experimental results show that STRE models effectively tackle the ill-condi-
tioned inverse ECG problem and yield 56.3% and 67.3% performance improvement compared to
the traditional Tikhonov regularization in the two-sphere and the torso-heart geometries, respect-
ively. The spatiotemporal regularization methodology is shown to have strong potential to achieve
robust solutions for high-dimensional predictive modeling in the inverse ECG problem.
KEYWORDS
Inverse ECG problem;
spatiotemporal data;
Tikhonov regularization;
body surface potential
mapping; wearable
sensor network
1. Introduction
Modern healthcare systems are increasingly investing in
advanced sensing and imaging to facilitate the effective
modeling, monitoring, and management of complex dynam-
ics in the patients’health conditions. For example, body-
area sensor network helps capture multi-directional informa-
tion pertinent to the heart electrical activity. The 12-lead
electrography (ECG) provides 12 directional views of the
cardiac electrodynamics, i.e. 12 ECG time series (Yang et al.,
2012; Yang & Leonelli, 2016). Such ECG time series enable
medical scientists to investigate cardiac electrical activity and
further identify heart diseases by checking waveform abnor-
malities (Penzel et al., 2016; Yang et al., 2013). For example,
the patterns of ST depression/elevation, significant Q waves,
or inverted T-waves in ECG cycles often indicate different
stages in the progression of myocardial infarction.
Advanced sensing and imaging bring a data-rich environ-
ment and provide an unprecedented opportunity to investi-
gate and further optimize medical diagnostics and treatment
for smart and personalized health (Lee et al., 2018; Lin et al.,
2018; Si et al., 2017). Realizing the full potential of sensing
data depends to a great extent on advanced analytical and pre-
dictive methods, which are challenged by the complex geom-
etry and high-dimensional data structure. For example, ECG
time series is a projected view of the space-time cardiac
dynamics, and the electrical activity in the heart is observed
with noise and interference when propagating from the heart
to the body. This projection diminishes important space-time
characteristics pertinent to the cardiac electrodynamics. In
addition, electric potentials are distributed on the complexly-
shaped heart surface, and are dynamically evolving over time.
Such spatiotemporal structure poses significant challenges in
the predictive modeling of heart-surface signals from body-
surface ECG data (which is also called the inverse ECG prob-
lem (Oster et al., 1997; Rudy, 2009)).
The inverse ECG problem is ill-conditioned (Rudy &
Burnes, 1999; Wang et al., 2010), its solution is sensitive to
measurement noise and model uncertainties. Achieving a
robust solution to inverse ECG problem requires the inte-
gration of statistical regularization with the physics model
governing the cardiac electrodynamics. This paper presents
a comparison study of regularization methods in the per-
formance to achieve robust solutions of the inverse ECG
model. We first introduce the forward and inverse ECG
problems. Second, we propose two spatiotemporal regular-
ization (STRE) models to increase the robustness in the
inverse ECG solution. Finally, case studies in both a simu-
lated two-sphere geometry and a real-world torso-heart
geometry are conducted to evaluate the performance of dif-
ferent regularization models.
The remainder of this paper is organized as follows:
Section 2 presents the literature review of ECG sensing.
CONTACT Hui Yang huy25@psu.edu Department of Industrial and Manufacturing Engineering, The Pennsylvania State University, University Park, PA,
16802 USA.
School of Industrial Engineering and Management Oklahoma State University, Stillwater, OK, USA.
ß2020 “IISE”
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING
https://doi.org/10.1080/24725579.2020.1823531

Section 3 introduces the inverse ECG problem and different
regularization methods. We propose two spatiotemporal
regularization (STRE) methods in Section 4.Section 5 shows
the comparison study of different regularization methods in
the two-sphere geometry and the torso-heart geometry.
Section 6 concludes the present investigation.
2. ECG sensing
Electrocardiography (ECG) is a noninvasive procedure to
record cardiac electrical signals, which is commonly used to
assist physicians in the assessment of heart health condi-
tions. Effective ECG monitoring systems are critical to inter-
preting cardiac rhythm, identifying heart abnormalities, and
further facilitate the allocation, planning, and execution of
timely medical treatments. Figure 1(a) shows the conven-
tional 12-lead ECG system that requires the strategic place-
ment of ten electrodes, four on the limbs to record both
bipolar leads (I, II, III) and unipolar leads (aVR, aVL, aVF),
and six on the chest to record precordial leads (V1-V6). The
12-lead ECG system generates twelve ECG time series and
provides twelve directional views of the cardiac
electrodynamics.
The EASI ECG system (Dower et al., 1988) is a quasi-
orthogonal system that derives the standard 12-lead ECG
from only five electrodes (see Figure 1(b)). Electrode E is on
the sternum, electrode S is at the sternal manubrium, and
electrodes A and I are at the left and right mid-auxiliary
lines, respectively. The fifth is a ground electrode and is
placed on one of the other clavicles. Moreover, 3-lead vec-
torcardiogram (VCG) system investigates the electrical activ-
ity of the heart along three orthogonal X, Y, Z planes of the
body, i.e. frontal, transverse, and sagittal (Dawson et al.,
2009; Edenbrandt & Pahlm, 1988). As shown in Figure 1(c),
VCG loops consist of 3-D recurring, near-periodic patterns
of the cardiac electrodynamics. Each cardiac cycle consists
of three loops describing electrical activity corresponding to
P, QRS, and T waves. The VCG analysis is widely used in
the literature to investigate different types of heart diseases,
including congenital heart disease (Braunwald et al., 1955),
bundle branch block (Villongco et al., 2014), and myocardial
infarction (Yang et al., 2013).
Conventional ECG systems place ECG sensors at a very
limited number of locations over the thorax and provide the
cardiac electrical information with low spatial resolution. It
has been suggested that high-resolution ECG mapping is
more conducive to the diagnostic assessment of heart dis-
eases such as acute cardiac ischemia (Herring & Paterson,
2006). ECG sensors placed at different locations on the torso
surface respond to the cardiac electrodynamics differently.
Researchers have developed the body surface potential map-
ping (BSPM) to provide a comprehensive 3D picture
describing the electrical activity on the body surface pro-
jected from the heart using a large number (32-231) of ECG
sensors (Bond et al., 2010; Lacroix et al., 1991; Rudy &
Burnes, 1999). High-resolution BSPMs provide richer car-
diac information than traditional ECG systems, and have
been used to detect different heart diseases such as acute
cardiac ischemia (Kornreich et al., 1993) and atrial fibrilla-
tion (Bonizzi et al., 2010).
However, ECG images describe the space-time distribu-
tion of body surface signals, which is a projected view of
spatiotemporal cardiac electrodynamics. Important spatial
information pertinent to the cardiac electrical activity is
diminished and blurred when propagating from the heart to
the body surface. For example, although BSPM achieves the
early detection of myocardial infarction (McMechan et al.,
1995), it is limited in the ability to characterize the extent
and location of 3D infarct on the heart surface (Zarychta
et al., 2007). Clinicians call for the estimation of heart-sur-
face electric potentials from body-surface signals for a better
investigation and characterization of heart patho-
logical behaviors.
3. Inverse ECG problem
3.1. Forward and inverse ECG problems
Cardiac electrical signal is initiated by the sinoatrial (SA)
node, i.e. the pacemaker of the heart, and then propagates
Figure 1. The illustration of (a) 12-lead ECG, (b) EASI ECG, and (c) 3-lead VCG systems.
2 B. YAO AND H. YANG

through the right and left atria toward the atrioventricular
node (AVN). The electric impulse further travels through
the bundle of His and Purkinje fibers, and enter the left and
right ventricles, which completes the cardiac cycle. As shown
in Figure 2, the forward ECG problem denotes the predic-
tion of electric potentials on the body surface based on the
excitation and propagation of space-time electrodynamics in
the heart. The objective of the inverse ECG problem is to
estimate cardiac electrical sources from electrical signals (e.g.
BSPMs) measured on the body surface.
Both forward and inverse ECG modeling require the for-
mulation of cardiac electrical sources. Two source models are
generally utilized, namely “activation-based”source model
and “potential-based”source model. In the activation-based
model, the source is defined by a moving set of current
dipoles that are aligned along the activation wavefront. The
cardiac electrodynamics is characterized and parameterized
by the arrival timing of the depolarization wavefront at each
location in the heart (Han et al., 2008). However, waveform
parameterization is generally nonlinear in the unknown acti-
vation time, which leads to the inverse problem that requires
non-convex and nonlinear least square minimization (Erem
et al., 2014). In order to address this problem and increase the
computation efficiency, a variety of methods have been devel-
oped in the literature such as the modified Multiple Signal
Classification algorithm (Huiskamp & Greensite, 1997), elec-
trophysiological propagation-based parameter initialization
(Van Dam et al., 2009), and cardiac electrical sparse imaging
technique (Yu et al., 2015).
The potential-based inverse formulation consists of trans-
membrane potential (TMP)-based and epicardial potential-
based models. In TMP source formulation, the 3D myocar-
dium is modeled as a continuum formed by interpenetrating
spaces: the intracellular and extracellular domains. The TMP
is then defined as /m¼/i/e, i.e. the difference between
intracellular potential /iand extracellular potential /e(He
et al., 2003; Tilg et al., 2002; Wang et al., 2010). In epicardial
potential-based formulation, the potential distribution on
the heart surface represents the electric source of the inverse
modeling, which is also called the heart-surface potential
(HSP)-based inverse ECG model. Instead of modeling the
entire myocardium, the HSP-based model only requires to
model the epicardial surface using boundary element
method (BEM) (Barr et al., 1977; Brooks et al., 1999; Ghosh
& Rudy, 2009; Wang et al., 2011; Yao et al., 2016). Cheng
et al. (2003) conducted a systematic comparison between the
two source models (i.e. activation-based and potential-based
source models), and no significant differences were found
between the reconstructed heart-surface potentials. The acti-
vation-based model requires nonconvex and nonlinear opti-
mization, and is more computationally demanding. Thus, in
the rest of this paper, we will focus on the potential-based
model, especially the HSP-based inverse ECG problem.
Denoting electric potential distributions on the heart and
body surfaces as /Hand /Brespectively, the relationship
between /Hand /Bis established as
/B¼RBH/H(1)
where RBH is the transfer matrix, the calculation of which
will be detailed in section 3.2. Studying the forward ECG
problem facilitates optimizing ECG measurements (i.e. loca-
tions of electrodes) (Chen & Yang, 2016), investigating how
electrophysiological properties (e.g. geometry, anisotropy,
and conductivity) of various tissues impact the correspond-
ing ECG signals (Geneser et al., 2008), and validating results
from the inverse ECG model (Yao et al., 2016). The inverse
ECG model is conducive to the investigation of cardiac
pathological activities and will further facilitate timely detec-
tion and effective treatments for heart diseases.
3.2. Derivation of transfer matrix R
BH
The transfer matrix RBH is derived from physics-based prin-
ciples (i.e. divergence theorem and Green’s theorem).
Divergence theorem states that if Eis a vector field and is
continuously differentiable in the 3D space enclosed by V
R3with a piecewise-smooth boundary S, then
ðS
ðEnÞdS ¼ðV
ðr  EÞdV (2)
where the unit vector nis orthogonal to surface Sand
points outward. If wand /are twice differentiable in V,
and define the vector field E¼/rwwr/, followed by
Eq. (2), we have
ðS
ð/rwwr/ÞndS ¼ðV
ð/r2wwr2/ÞdV (3)
In the human body, the torso is modeled as a volume con-
ductor and the heart is the bioelectric source (see Figure 3).
The boundary of the torso-heart geometry Sis formed by
body surface S
B
and heart surface S
H
. Let w¼1=ð4prÞand /
be the electric potentials inside the human body. Given the
fact that no bioelectrical sources exist outside S
H
(i.e.
r2/¼0), and electric field outside S
B
(i.e. in the air) is negli-
gible (i.e. r/¼0onS
B
), the value of electric potential /oat
an arbitrary location ofollowed by Eq. (3) is then derived as
/o¼ 1
4pðSH
/H
rn
r3dSH1
4pðSH
r/Hn
rdSH
þ1
4pðSB
/B
rn
r3dSB(4)
Thus, electric potentials /Hon the heart surface and
potential distribution /Bon the body surface is expressed as
Figure 2. The illustration of forward and inverse ECG problems.
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING 3

/B¼ 1
4pðSH
/HdXBH 1
4pðSH
r/Hn
rdSHþ1
4pðSB
/BdXBB
/H¼ 1
4pðSH
/HdXHH 1
4pðSH
r/Hn
rdSHþ1
4pðSB
/BdXHB
(5)
where dXij represents the solid angle subtended at a location
on surface S
i
by a mesh element on surface S
j
(see Figure
3(b)). BEM (Barr et al., 1977; Henneberg & Plonsey, 1993)
is implemented to solve the above integrals, in which S
B
and
S
H
are discretized into triangle meshes. The complex inte-
gration is then approximated by the sum of a series of
numerical integration over mesh elements. Thus, each term
in Eq. (5) is discretized as
ABB/BþABH /HþMBH NH¼0
AHB/BþAHH /HþMHH NH¼0(6)
where the coefficient matrices, M’s and A’s depend on the
body-heart geometry, and NHis the discretized normal
component of r/H(Barr et al., 1977). Rearranging the two
equations in Eq. (6), the transfer matrix is expressed as
RBH ¼ðABB MBHM1
HHAHBÞðMBHM1
HHAHH ABHÞ1(7)
which is then incorporated to estimate heart-surface poten-
tials /Hfrom the body-surface measurements, /B, in the
inverse ECG modeling.
However, matrix RBH is generally ill-posed with a large
condition number (i.e. condðRBH ¼jjRBHjjjjR1
BHjjÞ), resulting
in the ill-conditioned inverse ECG problem that is sensitive
to measurement noise (Ghosh & Rudy, 2009; Yao & Yang,
2016). Considering a small change d/Bin /B, the resulted
variation d/Hof estimated heart-surface potential /His
expressed as d/H=/H’condðRBH Þd/B=/B:The large con-
dition number condðRBH Þposes a significant challenge on
achieving robust solutions to the inverse ECG problem. In
addition, simplified assumptions are made when applying
the physics-based models (e.g. the human torso is modeled
as a homogeneous volume conductor and random fluctua-
tions in the electrophysiological property are neglected).
Such assumptions may not hold true in real-world situations
and may introduce uncertainties when predicting /H:
Therefore, achieving a robust solution of the inverse ECG
problem calls for the integration of ideal physics-based mod-
els with statistical regularization.
3.3. Regularization methods in the inverse ECG problem
As aforementioned, inverse ECG modeling is ill-conditioned,
and the resulted solution is unstable and sensitive to meas-
urement noise and uncertainties. It is necessary to add pen-
alty constraints (i.e. regularization) to stabilize the model
and obtain robust solutions. A variety of regularization
methods have been developed in the early literature among
which Tikhonov regularization is the most widely used to
penalize the L
2
norm of the inverse solution including zero-
order (Tikh_0th) and first-order (Tikh_1st). The Tikh_0th
method shrinks the unreliable components in the magnitude
of heart-surface potential and improves overall regularity of
the inverse solution (Dawoud et al., 2008; Rudy & Burnes,
1999; Wang et al., 2011). Higher-order Tikhonov regulariza-
tion focuses on penalizing the higher-order derivatives and
increasing the spatial smoothness of the inverse solution
(Throne & Olson, 2000). Various existing literature also
focused on L
1
-norm regularization, i.e. penalizing the L
1
-
norm of the inverse solution, including L
1
-norm zero-order
(L1_0th) (Wang et al., 2011) and first-order regularization
(L1_1st) (Ghosh & Rudy, 2009; Shou et al., 2011).
Moreover, Rahimi et al. (2013) generalized the spatial regu-
larization with L
p
-norm penalty (1 <p<2) to achieve a
balanced inverse solution from the L
2
- and L
1
-norm meth-
ods. In traditional regularization models, the inverse ECG
problem is solved individually at each time point, but did
not account for the temporal correlation in the evolving
dynamics of electric potentials.
The human heart is a space-time system with cardiac
electrodynamics that is varying in both space and time.
Robust inverse ECG modeling requires accounting for both
the spatial and temporal correlations among electric poten-
tials. Various regularization techniques have been proposed
in the literature to address the space-time structure of car-
diac electrical signals (Brooks et al., 1999; Greensite, 2003;
Ritter et al., 2015; Wang et al., 2010; Yu et al., 2015).
However, many of the methods require significant computa-
tional effort (e.g. large-scale matrix inversion (Brooks et al.,
1999)), restricted assumption of heart-surface electric poten-
tials (Messnarz et al., 2004), or estimation of high-dimen-
sional tissue property (Wang et al., 2010). Thus, generalized
spatiotemporal regularization methods are needed to
account for both the spatial and temporal correlations in a
Figure 3. (a) A cross-section of the human body from the NIH Visible Human Project (Ackerman, 1998). (b) A cross-section of the torso-heart geometry. The heart
enclosed by surface S
H
is the bioelectric source. The thorax bounded between S
B
and S
H
is modeled as a volume conductor.
4 B. YAO AND H. YANG

computation-efficient way, and further improve the solution
regularities in inverse ECG modeling.
This paper proposes two spatiotemporal regularization
models (i.e. STRE_L1 and STRE_L2) and presents a com-
parison study of different regularization methods in the per-
formance to achieve robust solutions of the inverse ECG
problem. Specifically, we propose to address the spatial
structure of the heart-surface potential distribution using a
spatial Laplacian defined on the irregular triangle mesh.
STRE_L1 model penalizes the L_1 norm of the spatial
Laplacian to achieve an estimated distribution that is spa-
tially piecewise-constant with few discontinuities. STRE_L2
model penalizes the L_2 norm of the spatial Laplacian to
increase the spatial smoothness and stability of heart-surface
signals. In addition to the spatial regularization, we propose
to increase the temporal regularity by constraining the tem-
poral difference within a narrow time window. We further
propose two optimization algorithms (i.e. the augmented
iterative algorithm with lagged diffusivity and the dipole
multiplicative update algorithm) to efficiently solve the two
spatiotemporal regularization models.
4. Research methodology
4.1. Spatial and temporal regularization
The electric potentials are spatially distributed on the com-
plex torso-heart geometry, and are highly correlated among
the adjacent areas. The torso-heart geometry is generally dis-
cretized into irregular triangle meshes through the BEM
method. The complex structure of corresponding potential
distributions is addressed by the spatial Laplacian defined
on irregular triangle meshes (Huiskamp, 1991; Yao &
Yang, 2016).
As shown in Figure 4, the pairwise node distance is not a
constant in the 3D triangle mesh as opposed to the regular
lattice with orthogonal and regular grids. Traditional finite
difference method is not applicable to derive the spatial
Laplacian on the irregular triangle mesh. Here, we estimate
the spatial Laplacian through the interpolation. Let /i
denote the field value at node i,r
ij
denote the distance
between node iand node j, and 
ribe the average distance
between node iand its neighbors, i.e.

ri¼Pj:i$jrij
ni
(8)
where i$jindicates node iand node jare neighbors, and
n
i
is the number of neighbors of node i. Then, the field
value /j0at location j0as shown in Figure 4 is expressed as
/j0¼/iþ
ri
rij
ð/j/iÞ(9)
Hence, the spatial Laplacian at node ifor the field value /
on a 3D triangle mesh is estimated as
Ds¼4

r2
i
1
niX
ni
j¼1
/j0/i
!
¼4

riX
ni
j¼1
ð1
ni
/j
rij
1
rij
!
/iÞ(10)
where 1
rij ¼1
niPj:i$j1
rij :The matrix element of D
s
is therefore
defined as
Dij ¼
4
ri
1
ri

,ifi¼j
4
ri
1
ni
1
rij
,ifi6¼ j,i$j
0, otherwise
8
>
>
>
>
<
>
>
>
>
:
(11)
Finally, the spatial regularization at node ion a 3D triangle
mesh is defined as
ðDsð/ÞÞi¼X
j:i$j
Dij/j(12)
In addition, potential distributions on the heart and body
surfaces are dynamically changing over time, which are
denoted by /Hðs,tÞand /Bðs,tÞrespectively. Note that s
denotes the spatial location on the body or heart surface,
and tdenotes a specific time point. In addition to spatial
correlations, the temporal correlation needs to be addressed
as well for robust inverse ECG solutions. Here, we define
the temporal regularization as
X
T
t¼1X
s¼tþw=2
s¼tw=2
j/Hðs,tÞ/Hðs,sÞj2
2(13)
where Tdenotes total time index, jj
2represents L
2
norm, and
wis a time window. Note that wis often selected as a small
number, which is due to the fact that temporal correlation is
stronger for closer time points, and potential distributions tend
to be very different at two time indices that are far away.
4.2. Spatiotemporal regularization (STRE) models with
L1 or L2 penalization
Integrating the transfer matrix RBH with spatial and tem-
poral regularization, two spatiotemporal regularization mod-
els, i.e. STRE with L1 norm (STRE_L1) and STRE with L2
norm (STRE_L2), will be investigated. STRE_L1 is formu-
lated as
Figure 4. An illustration of the irregular triangle mesh.
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING 5

min
/Hðs,tÞ
J1¼X
T
t¼1
fj/Bðs,tÞRBH/Hðs,tÞj2
2
þk2
tX
s¼tþw=2
s¼tw=2
j/Hðs,tÞ/Hðs,sÞj2
2gþk2
sjDs/Hðs,tÞj1
(14)
STRE_L2 is formulated as
min
/Hðs,tÞ
J2¼X
T
t¼1
fj/Bðs,tÞRBH/Hðs,tÞj2
2
þk2
tX
s¼tþw=2
s¼tw=2
j/Hðs,tÞ/Hðs,sÞj2
2gþk2
sjDs/Hðs,tÞj2
2
(15)
where k
s
and k
t
denote the spatial and temporal regulariza-
tion parameters respectively, and jj
prepresents L1 (L2)
norm when p¼1ðp¼2Þ:Regularization parameters k
s
and
k
t
can be chosen by cross validation or L-curve method
(Hansen & O’Leary, 1993).
STRE_L1 model penalizes the L
1
norm of the spatial
Laplacian of heart-surface potentials. L
1
regularization
encourages the sparsity of the spatial Laplacian, and achieves
a solution of heart-surface potential /Hthat is spatially
piecewise constant with few discontinuities. STRE_L2 model
penalizes L
2
norm of Ds/H:L
2
regularization shrinks unreli-
able or redundant components of the spatial Laplacian, and
achieves the estimation of /Hthat is spatially smoothed and
reliable. The performance of STRE_L1 and STRE_L2 will be
evaluated in the case studies in section 5.
Both STRE models involve spatial and temporal correla-
tions, and it is nontrivial to obtain the analytical solution to
both models. Two different numerical algorithms are pro-
posed to solve STRE_L1 and STRE_L2, respectively.
4.2.1. Augmented iterative algorithm with lagged diffusiv-
ity to solve the STRE_L1 model
L
1
regularization model has been well-known to be computa-
tionally demanding due to the non-differentiability of the L
1
penalty function. We propose the augmented iteration algo-
rithm with lagged diffusivity to solve the STRE_L1 model.
Specifically, an augmented forward model is defined as follows:
UB¼ZBHUH(16)
where UB¼½/0
B1,:::,/0
Bt,:::,/0
BT 0,UH¼½/0
H1,:::,/0
Ht,:::,/0
HT0,
which concatenate potential distributions over different time
points, and ZBH ¼INRBH is a block diagonal matrix consist-
ing of RBH and has the form of
ZBH ¼
RBH 0::: 0
0RBH ::: 0
.
.
..
.
...
..
.
.
00::: RBH
2
6
6
6
4
3
7
7
7
5
Hence, the objective function of STRE_L1 is reformulated as
min
UH
JðUHÞ¼fjUBZBH UHj2
2þk2
sjLUHj2
1þk2
tX
w=2
l¼1
jAlUHj2
2g
(17)
where L¼INDsis the augmented matrix for spatial
Laplacian which is defined the same way as ZBH , and Alis
the augmented matrix for the temporal regularization. Note
that ldenotes the time lag. For example, matrix A1has the
form of
A1¼
ININ
ININ
..
...
.
ININ
2
6
6
6
4
3
7
7
7
5
Matrix A2has the form of
A2¼
IN0IN
IN0IN
..
...
...
.
IN0IN
2
6
6
6
4
3
7
7
7
5
The iterative algorithm of lagged diffusivity (Ghosh &
Rudy, 2009) is then implemented to solve the augmented
objective function in Eq. (17), as shown in Table 1. Solving
the STRE_L1 model requires the computation of the
inverse of a matrix with dimensionality of ðNTÞðN
TÞat each iteration, which can be solved using the Block-
Jacobi-Iteration (see details in the Appendix). The compu-
tational effort will increase significantly with the increase of
Nor T.
4.2.2. Dipole multiplicative algorithm to solve the
STRE_L2 model
STRE_L2 model also incorporates the space-time structure
of potential distributions, whose analytical solution is
Table 1. The augmented iterative algorithm with lagged diffusivity to solve STRE_L1 model.
1: Set constants k
s
,k
t
and w.
2: Initialization:
^
Uð0Þ
H¼ðZ0
BHZBH þk2
sL0Lþk2
tXw=2
l¼1A0
lAlÞ1Z0
BHUB
3: Repeat
^
UðkÞ
H¼ðZ0
BHZBH þk2
sL0WðkÞ
gLþk2
tXw=2
l¼1A0
lAlÞ1Z0
BHUBð18Þ
WðkÞ
g¼1
2diag½1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jL^
Uðk1Þ
Hj2
1þg
q
where g¼105is a small positive number to guarantee that the denominator is nonzero.
4: until convergence
6 B. YAO AND H. YANG

difficult to reach. Iterative algorithms (e.g. multiplicative-
update (MU) algorithm) are generally utilized to solve com-
plex objective functions. Traditional MU algorithms require
the model solution to be nonnegative. However, electric
potentials in real-world situations can be positive or nega-
tive. Hence, traditional MU methods are not applicable to
solve the inverse ECG problem.
Yao and Yang (2016) developed the algorithm of dipole
multiplicative update (DMU) to solve STRE_L2, which is
inspired by the dipole assumption in physics. In the DMU
method, the electric potenital /His split into the positive
part /þ
Hand negative part /
H, which are defined as /þ
H¼
maxf0, /Hgand /
H¼maxf0, /Hg:Hence, /Hcan be
written as /H¼/þ
H/
H:Table 2 shows the updating rules
for both /þ
Hand /
H(see details of DMU algorithm in Yao
and Yang (2016) and Yao et al. (2018)).
5. Experimental studies
The STRE_L1 and STRE_L2 models are then implemented
to estimate the spatiotemporal electric potentials on the
heart surface from the body-surface measurements (i.e.
BSPM). The model performance of STRE_L1 and STRE_L2
is evaluated in both a simulated two-sphere geometry and a
real-world torso-heart geometry (as shown in Figure 5)by
the relative error (RE):
RE ¼Ps,tjj^
/Hðs,tÞ/Hðs,tÞjj
Ps,tjj/Hðs,tÞjj (19)
where ^
/Hðs,tÞand /Hðs,tÞdenote the estimated and refer-
ence potential distribution on the heart surface. The model
performance of both STRE models is benchmarked with
regularization methods commonly used in current practice,
i.e. zero-order Tikhonov (Tikh_0th), truncated singular
value decomposition (TSVD) (please see the appendix for
more details of TSVD), first-order L1-norm (L1_1st), and
first-order Tikhonov (Tikh_1st). Note that in L1_1st and
Tikh_1st, the first-order gradient operator denoting the nor-
mal-derivative of heart-surface potential distribution defined
in (Ghosh & Rudy, 2009). We also investigate the perform-
ance improvement of the two STRE models over the L1 and
L2 methods only with the spatial regularization defined in
Eq. (12), i.e. spatial L1-norm (L1-spatial), and spatial
Tikhonov (Tikh-spatial). Specifically, the objective function
of L1-spatial is
min
/Hðs,tÞ
Jð/Hðs,tÞÞ¼X
T
t¼1
fj/Bðs,tÞRBH/Hðs,tÞj2
2
þk2
sjDs/Hðs,tÞj1g
The objective function of Tikh-spatial is
min
/Hðs,tÞ
Jð/Hðs,tÞÞ¼X
T
t¼1
fj/Bðs,tÞRBH/Hðs,tÞj2
2
þk2
sjDs/Hðs,tÞj2g
5.1. Simulation studies on the two-sphere geometry
As shown in Figure 5(a), the two-sphere geometry consists
of two concentric spheres. Each sphere is discretized into a
triangle mesh with 184 nodes and 364 triangles. A time-
varying dipole of the electric current pðtÞ¼
½pxðtÞ,pyðtÞ,pzðtÞ is placed at the geometry center, where
pxðtÞ¼10ð0:9þejt1:4jÞcos ð2pðt1:48ÞÞ,pyðtÞ¼2ð1:1
þejt1:6jÞcos ð2pðtþ1ÞÞ,pzðtÞ¼ð1þetÞcos ð2pðt1:2ÞÞ:
Because of the perfect symmetry of the two-sphere geom-
etry, we can analytically calculate the electric potentials gen-
erated by the dipole pðtÞ, which are distributed on the inner
and outer sphere as
/Hðs,tÞ¼ 1
4pr
pðtÞrHðsÞ
r2
BrH
2rH
rB
þð
rB
rH
Þ2

(20)
/Bðs,tÞ¼ 3
4pr
pðtÞrBðsÞ
r3
B
(21)
where rH¼1:0 and rB¼1:5 are the radii of the inner and
outer spheres, rHðsÞand rBðsÞdenote the distance vectors
starting from the geometry center to the two spheres, and
r¼1 represent the electric conductivity within the two-
sphere geometry. STRE_L1 and STRE_L2 are implemented
to estimate electric potentials on the inner sphere /Hðs,tÞ
from /Bðs,tÞon the outer sphere. The regularization param-
eters k
s
and k
t
are selected as 0.015 and 1.15 respectively for
STRE_L2, which are 0.5 and 1.0 for STRE_L1 according to
the L-curve method. The time window is selected as w¼4
in both methods. In the simulation study, Gaussian noise
Table 2. The dipole multiplicative update algorithm to solve STRE_L2 model.
1: Set constants k
s
,k
t
and w.
2: Initialization: f/þ
Hgand f/
Hg(positive random matrices.
3: Repeat
4: for t¼1, :::,Tdo
ð/þ
H,tÞi ðA/
H,tÞiþBiþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ððA/
H,tÞiþBiÞ2þ4ðAþ/þ
H,tÞiðA/þ
H,tÞi
qð2Aþ/þ
H,tÞi
ð/þ
H,tÞið/
H,tÞi ðA/þ
H,tÞiBiþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ððA/þ
H,tÞiBiÞ2þ4ðAþ/
H,tÞiðA/
H,tÞi
qð2Aþ/
H,tÞi
ð/
H,tÞi
where
A¼RT
BHRBH þk2
sDT
sDsþ2k2
twIB ¼/T
BðtÞRBH þ2k2
tXt1
s¼tw=2/T
HðsÞþ2k2
tXtþw=2
s¼tþ1/T
HðsÞ
5: end for
6: until convergence
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING 7

with mean zero and different levels of variance r2
(i.e.
r¼0:1, 0:2, 0:3) are added to /Bðs,tÞto test the model
performance.
Figure 6 compares the RE’s between STRE_L1, STRE_L2,
and the traditional regularization methods, i.e. Tikh_0th,
TSVD, L1_1st, Tikh_1st, L1_spatial, and Tikh_spatial. In the
present investigation, the experiment with different noise
levels (i.e. r¼0:1, 0:2, 0:3) is replicated 20 times, and the
resulted RE in the bar chart is shown with ± one standard
deviation. According to Figure 6, the RE monotonically
increases for all the methods when the noise level r
increases from 0 to 0.3. Specifically, the RE derived from
STRE_L1 and STRE_L2 increases from 0.0664 to
(0.0702 ± 0.0017) and from 0.0664 to (0.0706 ± 0.0019),
respectively, both of which are significantly smaller than the
RE’s from Tikh_0th (i.e. from 0.1455 to 0.1626 ± 0.0016),
TSVD (i.e. from 0.1398 to 0.1752 ± 0.0057), L1_1st (i.e. from
0.1026 to 0.1086 ± 0.0014), and Tikh_1st (from 0.1025 to
0.1084 ± 0.00087). Specifically, STRE_L2 method achieves a
56.3% performance improvement on average over the com-
monly used Tikh_0th method. Furthermore, note that when
there is no temporal noise in /B, the two STRE methods
yield similar performance (i.e. RE ¼0.0664) compared to
L1_spatial and Tikh_patial. As the noise level increases, the
superiority of STRE_L1 and STRE_L2 becomes prominent.
For example, when r¼0:3, STRE_L1 and STRE_L2 yield
the RE of 0.0702 ± 0.0017 and 0.0706 ± 0.0019, achieving
22.4% and 20.9% reduction in RE compared to L1_spatial
(0.0905 ± 0.0027) and Tikh_patial (0.0892 ± 0.0027),
respectively.
Figure 7(a) shows the true distribution of the inner-
sphere electric potentials calculated from Eq. (19). Note that
the inner-sphere potential mapping is time-varying, and
Figure 7 only shows the potential distribution at t¼50ms:
Figure 7(b) shows the estimated potential distributions by
STRE_L1, STRE_L2, and traditional regularization methods
(i.e. Tikh_0th, TSVD, L1_1st, Tikh_1st, L1_spatial, and
Tikh_spatial) when no noise is added in the outer-sphere
potential mapping /Bðs,tÞ:Figure 7(c) shows the estimated
potential distributions by the different methods with a noise
level of r¼0:3in/Bðs,tÞ:Both STRE models work better
in preserving the color scale and patterns of the true refer-
ence mapping compared to the traditional regularization
methods under different noise conditions.
5.2. Experimental studies on the torso-heart geometry
We further evaluate the performance of STRE_L1 and
STRE_L2 models, and make comparisons with traditional
regularization methods in the real-world torso-heart geom-
etry as shown in Figure 5(b). The heart surface is triangu-
lated with 960 triangles and 482 nodes, and the body surface
mesh consists of 677 triangles and 352 nodes. Note that the
heart surface potentials are simulated by solving the mono-
domain reaction-diffusion model (Aliev & Panfilov, 1996).
The STRE_L1 and STRE_L2 models are implemented to
estimate heart-surface potentials /Hðs,tÞfrom potential
mapping on the body surface /Bðs,tÞ:The regularization
parameters k
s
and k
t
are selected as 0.7 and 0.2 respectively
for STRE_L2, which are 6 and 0.015 for STRE_L1 according
Figure 5. (a) Two-sphere geometry; (b) real-world torso-heart geometry.
Figure 6. Performance comparison of relative errors (REs) in the two-sphere geometry between STRE_L1, STRE_L2, and traditional regularization methods, i.e.
Tikh_0th, TSVD, L1_1st, Tikh_1st, L1_spatial, and Tikh_spatial.
8 B. YAO AND H. YANG

to the L-curve method. The time window is selected as
w¼4 in both methods. Similarly, Gaussian noise with mean
zero and different levels of variance r2
(i.e.
r¼0:005, 0:01, 0:05) are added to /Bðs,tÞto test the per-
formance of different regularization models.
Figure 8 shows the performance comparison of RE
between STRE_L1, STRE_L2, and the traditional regulariza-
tion methods (i.e. Tikh_0th, TSVD, L1_1st, Tikh_1st,
L1_spatial, and Tikh_spatial) in the real-world torso-heart
geometry. In the present investigation, the experiment with
different noise (i.e. r¼0:005, 0:01, 0:05) is replicated 20
times, and the resulted bar-chart of RE is shown with ± one
standard deviation. Note that the RE monotonically
increases for all methods when the noise level rincreases
from 0 to 0.05. Specifically, the RE derived from STRE_L1
and STRE_L2 increases from 0.0895 to (0.2111 ± 0.0058) and
from 0.0894 to (0.2100 ± 0.0043) respectively, which are
much smaller than the RE’s from Tikh_0th (from 0.3454 to
0.4478 ± 0.0027), TSVD (from 0.346 to 0.458 ± 0.001), L1_1st
(from 0.6665 to 1.8165 ± 0.0983), Tikh_1st (from 0.5710 to
1.6234 ± 0.0553), L1_spatial (from 0.0895 to 0.2136 ± 0.0037),
and Tikh_spatial (from 0.0894 to 0.2118 ± 0.0061).
Specifically, STRE_L2 method achieves a 67.3% performance
improvement on average compared to the commonly used
Tikh_0th method. It also may be noted that the estimation
error (i.e. RE) in this real geometry study increases dramat-
ically when adding extra noise to the body surface data
/Bðs,tÞ, as compared to the two-sphere geometry. This is
due to the fact that inverse ECG modeling tends to be more
sensitive to noise with real-world torso-heart geometry
which is more complex and irregular.
Figure 9(a) shows the true potential distribution on the
heart surface. Note that the heart-surface potential mapping
is time-varying, and Figure 9 only shows the mapping at
t¼200ms:Figure 9(b) shows the estimated potential distri-
butions by STRE_L1, STRE_L2, and the traditional regular-
ization methods (i.e. Tikh_0th, TSVD, L1_1st, Tikh_1st,
L1_spatial, and Tikh_spatial) when there is no noise added
to the body-surface potential mapping /Bðs,tÞ:Figure 9(c)
shows the estimations by the 8 regularization methods with
a noise level of r¼0:005 in the body-surface measure-
ments /Bðs,tÞ:Notably, the color scale and patterns of the
true reference mapping are better preserved by the two
STRE methods under different noise conditions compared
to the traditional regularization approaches.
Note that the number of time points in the data depends
on the sampling frequency of the ECG signals. In our
experiment, there are 301 and 501 time points in the two-
sphere and torso-heart case studies, respectively. We further
investigate how the sampling frequency impacts the RE in
the inverse ECG solution. Specifically, we reduce the sam-
pling frequency by half. In other words, there are 151 and
251 time points in the data for the two-sphere and torso-
heart case studies, respectively. The optimal temporal regu-
larization parameter k
t
becomes 0.5 for the two-sphere
geometry and 0.09 for the torso-heart geometry in the
STRE_L2 method. Table 3 shows the corresponding relative
error solved by STRE_L2 for each case. Note that the opti-
mal value for k
t
decreases when the sampling frequency is
reduced and the resulted RE increases slightly for all the
noise conditions in both the simulation and real-geometry
case studies. This is due to the fact that the temporal correl-
ation between two successive signals becomes weaker when
the time interval between those two points is wider, which
is reflected in the decrease of the temporal regularization
parameter and the increase of estimation error.
As shown in Figures 6 and 8, both STRE_L1 and
STRE_L2 achieve superior performance in estimating the
Figure 7. (a) Reference potential distribution on the inner sphere at t¼50ms;Estimated potential distributions on the inner sphere by STRE_L1, STRE_L2, and
traditional regularization methods (i.e. Tikh_0th, TSVD, L1_1st, Tikh_1st, L1_spatial, and Tikh_spatial) at t¼50ms (b) when there is no noise in /B(c) with a noise
level of r¼0:3in/B:
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING 9

time-varying cardiac potentials in both the two-sphere
geometry and the torso-heart geometry. Figure 10 further
compares the computation time of the two algorithms.
Note that when the data become noisier, it will take more
iterations for both algorithms to converge. Therefore, as
the noise level rincreases, the computation time for both
methods increases. Notably, STRE_L2 method requires less
computation time in comparison with STRE_L1 under dif-
ferent noise conditions in both geometries. On average,
solving STRE_L2 is 78.9% and 80.0% faster than solving
STRE_L1 in the two-sphere geometry and the torso-heart
geometry, respectively. This is due to the fact that the
augmented iteration algorithm to solve STRE_L1 requires
computing the inverse of a matrix with a dimensionality
of ðNTÞðNTÞat each iteration, whose computation
effort will increase dramatically as Nor Tbecomes large.
On the other hand, the DMU method in STRE_L2
improves the estimation at each iteration without the cal-
culation of any matrix inversion, which is more computa-
tionally efficient than STRE_L1. Therefore, STRE_L2
should be more preferable when solving the inverse
ECG problem.
Figure 8. Performance comparison of relative errors (REs) in the torso-heart geometry between STRE_L1, STRE_L2, and traditional regularization methods, i.e.
Tikh_0th, TSVD, L1_1st, Tikh_1st, L1_spatial, and Tikh_spatial.
Figure 9. (a) Reference potential distribution on the heart surface at t¼200ms;Estimated potential distributions on the heart surface by STRE_L1, STRE_L2, and
traditional regularization methods (i.e. Tikh_0th, TSVD, L1_1st, Tikh_1st, L1_spatial, and Tikh_spatial) at t¼200ms when there is no noise in /B(c) with a noise
level of r¼0:005 in /B:
Table 3. Relative error (i.e. RE) solved by STRE_L2 with different number of time points in the data.
Num. of time points r¼0:1r¼0:2r¼0:3
Two-sphere 301 0:06765:11040:068361:31030:070661:9103
151 0:067668:31040:06961:91030:07161:6103
r¼0:005 r¼0:01 r¼0:05
Torso-heart 501 0:107269:11030:138362:21030:191463:5103
251 0:11466:41030:139565:21030:201164:2103
10 B. YAO AND H. YANG

6. Conclusions
This paper presents a comparison study of spatiotemporal
L1 and L2 regularization methods, as well as other trad-
itional regularization approaches in the performance to
achieve robust solutions of the inverse ECG problem. We
first introduce the forward and inverse ECG problems, and
different traditional regularization methods. Second, we
propose two spatiotemporal regularization models (i.e.
STRE_L1, STRE_L2) to cope with the spatiotemporal data
structure and further increase the robustness in inverse
ECG modeling. Finally, we validate and evaluate the per-
formance of STRE_L1 and STRE_L2 on the two-sphere
geometry, as well as a real-world torso-heart geometry.
Experimental results show that both STRE models effect-
ively tackle the ill-conditioned inverse ECG problem and
yield a better estimation of the heart-surface potentials
compared to the traditional regularization methods (i.e.
Tikh_0th, TSVD, L1_1st, Tikh_1st, L1_spatial, and
Tikh_spatial). Specifically, the STRE_L2 method achieves
56.3% and 67.3% performance improvement compared to
the traditional Tikhonov regularization in the two-sphere
geometry and the torso-heart geometry, respectively.
Moreover, STRE_L2 is more computationally efficient than
STRE_L2. Specifically, STRE_L2 method is 78.9% and
80.0% faster compared to the STRE_L1 method in the two-
sphere geometry and the torso-heart geometry, respectively.
Therefore, STRE_L2 demonstrates stronger potential to pro-
vide robust solutions of the inverse ECG problem and help
medical scientists to investigate the cardiac activity and a
vast array of heart diseases.
Funding
The authors of this work would like to acknowledge the NSF I/UCRC
Center for Healthcare Organization Transformation (CHOT), NSF I/
UCRC award #1624727 for funding this research.
ORCID
Hui Yang http://orcid.org/0000-0001-5997-6823
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Xi¼C1
ii ðYiX
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CijXjÞ
4: end for
5: until convergence
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING 11

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Appendix A
A.1. Block Jacobi iteration
The algorithm shown in Table 1 to solve the STRE_L1 model requires
the computation of the inverse of a matrix with a dimension of ðN
TÞðNTÞat each iteration (see Eq. (18)), which can be solved by
the method of Block Jacobi Iteration. At iteration k, Eq. (18) can be
rewritten as
ðZ0
BHZBH þk2
sL0WðkÞ
gLþk2
tX
w=2
l¼1
A0
lAlÞ^
UðkÞ
H¼Z0
BHUB(A1)
which can be denoted as 
C
X¼
Yfor simplicity, and the vectors 
X
and 
Ydenote ^
UðkÞ
Hand Z0
BHUB, respectively. Note that matrix 
Chas
the following block structure:

C¼
C11 C12 ::: C1T
C21 C22 ::: CT2
.
.
..
.
..
.
..
.
.
CT1CT2::: CTT
2
6
6
6
4
3
7
7
7
5
where each of the block element C
ij
is a matrix with the size NN.
The augmented vectors 
Xand 
Yare further divided as 
X¼
½X0
1,X0
2,:::,X0
T0and 
Y¼½Y0
1,Y0
2,:::,Y0
T0, respectively. The i-th block
equation of Eq. (A1) is written as
CiiXiþX
j6¼i
CijXj¼Yi(A2)
Hence, Xican be solved as
Xi¼C1
ii ðYiX
j6¼i
CijXjÞ(A3)
Based on Eq. (A3), we have the algorithm of Block Jacobi Iteration in
Table 4 to solve the inversion of big matrix.
A.2. Truncated singular value decomposition (TSVD)
The objective function for the TSVD is
min
/H
jRBH/H/Bj2(A4)
The least-square solution of the above objective function is
/H¼ðR0
BHRBH Þ1R0
BH/B(A5)
which is unstable because matrix RBH is ill-posed. Denote the singular
value decomposition of RBH as:
RBH ¼USV0¼X
N
i¼1
riuiv0
i(A6)
where uiand viare orthonormal vectors, and r
i
’s denote the diagonal
elements of S, and satisfy r1r2::: rN:From Eqs. (A5) and
(A6), we have
/H¼X
N
i¼1
u0
i/B
ri
vi(A7)
In order to improve the stability of the inverse solution in the presence
of noise, the singular values from rkþ1to r
N
are set to zero, and the
corresponding robust solution from TSVD is
/H¼X
k
i¼1
u0
i/B
ri
vi(A8)
where kserves as the regularization parameter, and can be determined
by the method of cross validation.
IISE TRANSACTIONS ON HEALTHCARE SYSTEMS ENGINEERING 13