Unipola Elec og am Eigen alue Dis ibu ion Analysis o he Iden i ica ion o
A ial Fib osis
Jenni e Riccio1, Sa a Roche 2, Lau a Ma ´
ınez-Ma eu2, Alejand o Alcaine3,1, Ja ie Saiz2,
Juan Pablo Ma ´
ınez1,3, Pablo Laguna1,3
1BSICoS, I3A, IIS A ag´
on, Uni e sidad de Za agoza, Za agoza, Spain
2Ci2B, Uni e si a Poli `
ecnica de Val`
encia, Valencia, Spain
3CIBER en Bioingenie ´
ıa, Bioma e iales y Nanomedicina (CIBER-BBN), Spain
Abs ac
A ial ib osis plays an impo an ole in he pa hogene-
sis o a ial ib illa ion (AF). Low bipola elec og ams (b-
EGMs) peak- o-peak ol age a eas indica e sca issue and
a e conside ed a ge s o AF subs a e abla ion. Howe e ,
his app oach igno es he spa io empo al in o ma ion em-
bedded in he signal and he dependence o b-EGMs on
ca he e o ien a ion. This wo k p oposes an app oach o
de ec ib osis based on he eigen alue dominance a io
(EIGDR) in an ensemble (clique) o unipola elec og ams
(u-EGMs). A 2-D issue wi h a cen al ci cula pa ch o
ib osis has been simula ed using he Cou emanche cellu-
la model. Maps o EIGDR ha e been compu ed using wo
sizes o elec ode cliques, om he o iginal u-EGMs wi hin
he ensemble o a e a ime alignmen o hese signals.
Pe o mance o each map in de ec ing ib osis has been
e alua ed using ecei e ope a ing cha ac e is ic cu es
and de ec ion accu acy. Bes esul s achie e an a ea un-
de he cu e (AUC) o 0.98 and an accu acy (ACC) o 1
when we use as ma ke he gain in eigen alue dominance
p oduced by he ensemble alignmen .
1. In oduc ion
A ial ib osis ep esen s a s uc u al abno mali y o
he a ium, which al e s he elec ical conduc ion and ex-
ci abili y o he issue. Fib oblas s p oli e a ion and hei
sec e ion o ex acellula ma ix p o eins, such as collagen,
cha ac e ize ib o ic issue [1]. These p o eins a e mainly
in ol ed in he epa a i e p ocess o eplace damaged my-
oca dial pa enchyma [2].
A ial ib osis is obse ed o be closely ela ed o a ial
ib illa ion (AF), e en i hei causal ela ionship is s ill
challenging [3]. On he one hand, an ex ensi e ib o ic
p ocess in he a ium can p omo e pe sis en AF [3]; on
he o he hand, s uc u al a ial emodeling ound in AF
p oduces ib osis and al e s issue unc ion [2]. Elec o-
physiologically, a ial ib osis p oduces low- ol age in ac-
a diac elec og ams (EGMs), which can be iden i ied us-
ing elec oana omical mapping (EAM) [4]. Peak- o-peak
bipola ol age maps can be cons uc ed h ough da a ob-
ained du ing subs a e mapping [5], being bipola ol age
an in e es ing ma ke du ing sinus hy hm (SR) as well as
in AF. Low- ol age a eas a e ypically de ined as hose
wi h peak- o-peak bipola ol age below 0.5 mV du ing
SR. Howe e , some d awbacks o his p ocedu e should be
pu in e idence. Fi s , peak- o-peak ol age measu e does
no p o ide in o ma ion abou mo phological ea u es o
empo al end embedded in he signal; he e o e, ol age
h esholding does no ake in o accoun he p esence o un-
de lying abno mali ies in he a ia. Second, he me hod-
ology o de ine low ol age a eas has no been s anda d-
ised [6]. Thi d, spa ial he e ogenei y is no accoun ed o .
Fou h, low bipola ol age can also be in luenced by o he
ac o s han ib osis, such as ac i a ion di ec ion, elec ode
size, in e elec ode dis ance and il e ing [4], [6], as well
as by echnical p oblems such as poo elec ode con ac in
ana omically di icul si es (e.g. he pulmona y eins) o
i s ins abili y in ime.
In o de o o e come hese limi a ions, in his wo k we
p opose he dominan - o- emaining eigen alue dominance
a io (EIGDR) o unipola EGMs (u-EGMs) as a measu e
o he ol age wa e on oughness and co ela e i wi h
he p esence o ib osis. We compu e maps o eigen alues
a ios conside ing wo di e en elec ode clique a ange-
men s (ECA) and we e alua ed he abili y o each map o
de ec a ib osis pa ch in he con ex o a simula ion s udy.
Maps ha e been c ea ed om he whole leng h o u-EGMs,
using 0oand 45oca he e o ien a ions wi h espec o he
wa e on p opaga ion.
2. Ma e ials
A 2-D a ial issue o 4x4 cm o hexahed ic elemen s
simula ed wi h 100 µm esolu ion using he Cou emanche
Compu ing in Ca diology 2020; Vol 47 Page 1 ISSN: 2325-887X DOI: 10.22489/CinC.2020.434
cellula model [7] has been used. Conduc ion he e ogene-
i y induced by ch onic AF in le a ium has been consid-
e ed in he model. The simula ed issue includes a ci cula
pa ch o di use ib osis ha ing a diame e o 2 cm, whe e
20% nodes ha e andomly been assigned he Malecka
model o ib oblas s [8] and a conduc i i y educ ion o
30%. Le xk(n)be he u-EGMs compu ed wi h a sampling
equency o 1 kHz a a high-densi y mul i-elec ode a ay
(MEA) o 15×15 elec odes, k∈ {1,...,225}, loca ed
a si es (i, j), i, j ∈ {1, .., 15}. The MEA has an in e -
elec ode dis ance d= 2 mm, is cen e ed in he issue slice
and loca ed a 1 mm dis ance om he issue su ace. Each
simula ed u-EGM is 500 ms long and con ains a single ac-
i a ion (depola iza ion plus epola iza ion) co esponding
o one sinus bea .
3. Me hods
3.1. Eigen alue analysis
In his wo k, we p opose and assess EIGDR om u-
EGMs a each 3×3 and 2×2 clique om he MEA, as
ib osis ma ke s. Eigen alues we e ob ained om he
spa ial co a iance ma ix o bo h o iginal and aligned u-
EGMs wi hin clique ensembles. Signals ha e been aligned
as p oposed in [9], acco ding o he maximum c oss-
co ela ion wi h espec o he highes ampli ude u-EGM.
Fo each elec odes clique, he a io Ro he dominan - o-
emaining eigen alues, compu ed as:
R=λ1
PK
k=2 λk
,(1)
whe e Kis he numbe o u-EGMs xk(n)in he clique
ensemble (4 o 9 in his s udy), is es ima ed o quan i y
EIGDR. Fo a heo e ical analysis o EIGDR, he model
xk(n) = αks(n−τk) + k(n) + k(n)(2)
is conside ed o he u-EGMs xk(n)a he k h elec ode,
whe e s(n)is he clean and space in a ian u-EGM in he
case o a plane wa e p opaga ion, τkis he delay o he k h
u-EGM s(n−τk)wi h espec o he ime e e ence in he
ensemble, αkis a pa ame e accoun ing o u-EGM ampli-
ude (dec eased a ib osis, αk<1, wi h espec o no mal
issue, αk= 1), k(n) ep esen s he u-EGM ib o ic com-
ponen (absen in no mal issue), and k(n)is he noise
componen o he k h u-EGM. The ene gy o s(n)is de-
no ed by Es, and Es0 ep esen s he ene gy o i s de i a i e
s0(n). The delays τka e cha ac e ized by hei a iance,
β2σ2
θ, wi h β > 1 in ib osis, a ac o accoun ing in e sely
o ib osis gene a ed speed educ ion ela i e o no mal
issue whe e β=1. αkis modelled as andom a iable
wi h mean E[αk] = αand a iance σ2
α. Noise is con-
side ed o be ze o-mean, Gaussian, whi e and unco ela ed
wi h τkand k(n), wi h a iance σ2
. The a iance o he
ze o-mean ib o ic componen ac oss clique elec odes is
deno ed by σ2
.
Fou u-EGMs scena ios o non-aligned (NA) and
aligned (A) u-EGMs a non- ib o ic (NF) and ib o ic (F)
a eas a e conside ed and hei app oxima e heo e ical
eigen alues a e de i ed acco ding o he p ocedu e p e-
sen ed in [10]. Table 1 shows he eigen alues λkand
EIGDR o he ou scena ios.
Fib osis ma ke s:
•F om he de i a ions in equa ions p esen ed in Table 1,
i is clea ha R>RF, as esul o h ee concomi an e -
ec s appea ing simul aneously a ib osis: a) highe mo -
phology dispe sion, σ2
, b) lowe ampli ude, α < 1, and c)
la ge delays esul ing in la ge misalignmen dispe sion
β > 1. Then Ris p oposed as one ma ke o ib osis
de ec ion.
•Simila ly, o he same wo i s easons, i also esul s
ha RA>RA
F, sugges ing RAas o he ib osis de ec ion
ma ke .
•Analyzing he a io ∆RFbe ween EIGDR in non-
ib o ic a eas wi h espec o ib o ic ones, ep esen ing he
eigen alue concen a ion los by ib osis, and hen a mea-
su e o he sepa abili y powe o RFas a ma ke o ib o-
sis, we ob ain o misaligned u-EGMs, σ2
θ>0:
∆RF=R
RF
≈β2σ2
θEs0+N(σ2
+σ2
)
(α2+σ2
α)
σ2
θEs0+Nσ2
.(3)
To e alua e how ∆RF a ies wi h he le el o misalign-
men σ2
θwe pe o m he de i a i e, ob aining:
∂∆RF
∂σ2
θ
≈
−Es0Nσ2
+σ2
1−β2α2+σ2
α
α2+σ2
α(σ2
θEs0+Nσ2
)2.
(4)
The e m 1−β2α2+σ2
α is ypically >0since in i-
b osis βcan ge alues up o 2 and αup o 1/8 [11], esul -
ing ha ∂∆RF
∂σ2
θ
<0, jus i ying he ad an age o alignmen ,
since he highe he misalignmen σ2
θ, he lowe he sepa a-
bili y capaci y o RF o disc imina e be ween ib osis and
non- ib osis, sugges ing ha RA
Fis be e sui ed ma ke
han RF.
•Al e na i ely, he a io ∆RAbe ween EIGDR be o e
and a e alignmen , ep esen ing he gain in eigen alue
concen a ion p oduced by he ensemble alignmen , is con-
side ed. In he case o ib osis, i is eached he alue:
∆RA=RA
F
RF
≈
Esβ2σ2
θα2+σ2
αEs0+N(σ2
+σ2
)
N(σ2
+σ2
)(Es−β2σ2
θEs0)
.
(5)
To e alua e how ∆RA a ies wi h he le el o ib osis σ2
we also pe o m he de i a i e, ob aining:
Page 2
u-EGM model λkEIGDR
NA, NF xk(n) = s(n−τk) + k(n)λk≈


(Es−σ2
θEs0)K/N +σ2
, k = 1;
σ2
θEs0K/N +σ2
, k = 2;
σ2
, k = 3,...,K,
R ≈ Es−σ2
θEs0
σ2
θEs0+Nσ2
.
A, NF xk(n) = s(n) + k(n)λk≈EsK/N +σ2
, k = 1;
σ2
, k = 2,...,K, RA≈Es
Nσ2
.
NA, F xk(n)=αks(n−τk)+ k(n)+ k(n)λk≈




α2+σ2
α(Es−β2σ2
θEs0)K/N +σ2
+σ2
, k = 1;
α2+σ2
αβ2σ2
θEs0K/N +σ2
+σ2
, k = 2;
σ2
+σ2
, k = 3, ., K
RF≈Es−β2σ2
θEs0
β2σ2
θEs0+
N(σ2
+σ2
)
(α2+σ2
α)
A, F xk(n) = αks(n) + k(n) + k(n)λk≈(α2+σ2
αEsK/N +σ2
+σ2
, k = 1;
σ2
+σ2
, k = 2,...,K, RA
F≈Es
N(σ2
+σ2
)
(α2+σ2
α)
.
Table 1: Models o non-aligned (NA) and aligned (A) u-EGMs a non- ib o ic (NF) and ib o ic (F) a eas, wi h hei
espec i e eigen alues λkand eigen alue dominance a ios EIGDR compu ed ollowing he p ocedu e p esen ed in [10]
∂∆RA
∂σ2
≈−EsNβ2σ2
θα2+σ2
αEs0
N(σ2
+σ2
)2(Es−β2σ2
θEs0)
.(6)
Since o small delays τk,Esβ2σ2
θEs0i esul s ha
∂∆RA
∂σ2
<0, making his ma ke ∆RAbecoming smalle
he highe he ib o ic componen σ2
, jus i ying o conside
i as a po en ial ib osis ma ke . Also, ∂∆RA
∂α2>0, so he
la ge he ib osis, ( educed α2), he u he ge s ∆RA e-
duced. Howe e , ∂∆RA
∂β2>0, and since βinc eases in
ib osis, i esul s in a coun e ac ing e ec o he ma ke
sensi i i y o ib osis. Since ib osis e ec s on u-EGM am-
pli ude, α2, and mo phology, σ2
, a e much mo e ma ked
han on conduc ion eloci y, β2, [11], i is expec ed ha he
i s wo endencies domina e, making he ma ke ∆RA o
educe wi h ib osis. Simula ion and eal expe imen s can
elucida e he beha iou in p ac ice.
3.2. Assessmen o R,RAand ∆RA o i-
b osis de ec ion
Maps o R,RAand ∆RAha e been c ea ed p ocess-
ing he comple e MEA wi h he elec ode lines in wo o i-
en a ions wi h espec o he wa e o m p opaga ion di ec-
ion, pa allel (0o) and oblique (45o) and wo ECA (3×3
and 2×2). The 3×3ECA p o ides one EIGDR o each
squa ed g oup o nine elec odes wi h diagonal e ices a
(i, j), and (i+2, j+2), i, j ∈ {1,...,13}, gi ing a o al o
13 ×13 pixel maps o ma ke s R,RAo ∆RA. The 2×2
ECA p o ides alues a each squa ed g oup o ou elec-
odes wi h diagonal e ices a (i, j)and (i+ 1, j + 1),
esul ing in maps o 14 ×14 pixels, i, j ∈ {1,...,14}.
Recei e ope a ing cha ac e is ic (ROC) cu es ha e been
used o e alua e he ma ke s abili y in disc imina ing i-
b o ic om non- ib o ic a eas. Fo ha pu pose, a g ound
u h mask has been c ea ed labelling hese a eas. Cliques
lying in he issue in e ace, i.e., in he line sepa a ing he
ib o ic pa ch om non- ib o ic issue, we e no conside ed
in he e alua ion. Fo each map, h esholds o ib osis
iden i ica ion ha e been a ied o compu e he ROC cu e.
The a ea unde he cu e (AUC) and he maximum accu-
acy (ACC) ha e been compu ed o each map, as a mea-
su e o he abili y o he ma ke o de ec ib osis.
AUC/ACC
Clique Angle R RA∆RA
3×30o0.88/0.86 0.96/0.95 0.97/0.95
45o0.77/0.75 0.96/0.93 0.98/1
2×20o0.84/0.80 0.85/0.79 0.72/0.70
45o0.76/0.71 0.96/0.93 0.94/0.89
Table 2: AUC and ACC o he h ee di e en me ics
4. Resul s
Figu e 1 shows he mean and s anda d de ia ion (SD)
o each s udied ma ke R,RAand ∆RAa ib o ic and
heal hy simula ed issues, using 2×2 o 3×3 cliques. The
h ee ma ke s we e signi ican ly lowe in he heal hy han
in he ib o ic issue (Wilcoxon ank-sum es , p < 0.05).
RAp esen s he highe a io o ib o ic o heal hy is-
sue γ= 2.66 (2.51) o 3×3 (2×2) cliques, being ∆RA
he ma ke wi h mo e signi ican di e ences be ween bo h
ypes o issue (p < 0.05).
Table 2 con ains AUC and ACC o he ma ke s consid-
e ed. Resul s show highe alues when ime alignmen o
u-EGMs is pe o med, especially when MEA has an o ien-
a ion o 45owi h espec o he wa e on di ec ion. The
bes disc imina ion powe is ob ained by ∆RA, wi h AUC
= 0.97 (0.98) o pa allel (diagonal, whe e ACC = 1) o i-
en a ion, using 3×3ECA. Figu e 2 shows he maps o R,
RAand ∆RAob ained wi h 3×3 ECA and wi h pa allel
ca he e o ien a ion. In he lowe panels, he de ec ed i-
b o ic a eas a e shown, using he h esholds ha maximize
he de ec ion accu acy o each ma ke .
5. Discussion and Conclusions
In his wo k, he eigen alue dominance a ios, EIGDR,
o u-EGMs ha e been p oposed and e alua ed using a sim-
Page 3
3x3 2x2
0
5
10
15
20 F
NF
3x3 2x2
0
100
200
300
400 F
NF
3x3 2x2
0
5
10
15
20
25 F
NF
= 1.72
= 1.56 = 2.51
= 2.66
= 1.49 = 1.55
R RA∆RA
Figu e 1: EIGDR ma ke s (R,RAand ∆RA) mean and
SD om he wo cliques, 3×3 and 2×2, conside ing bo h
MEA o ien a ions, 0oand 45o.γis he ac o ela ing he
mean om ib o ic o non- ib o ic a eas o each ma ke
and clique size.
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
3
4
5
6
7
8
9
10
11
12
13
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
3
4
5
6
7
8
9
10
11
12
13
40
60
80
100
120
140
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
3
4
5
6
7
8
9
10
11
12
13
8
10
12
14
16
18
R RA∆RA
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10111213
0
1
2
3
4
5
6
7
8
9
10
11
12
13
0
0.2
0.4
0.6
0.8
1
Figu e 2: Top panels: maps o R,RAand ∆RA om
3×3 cliques o ca he e o ien a ion o 0o. Ci cle encom-
passes ib o ic a ea. Lowe panels: de ec ed ib o ic a eas
(b own), using he h esholds ha maximize de ec ion ac-
cu acy o each ma ke . Blue (b own) colo inside he ci cle
deno es FN (TP), while ou side deno es TN (FP) de ec ion,
espec i ely.
ula ed a ial issue including a eas wi h ib osis and o he s
o no mal issue, in o de o disc imina e hem. In clini-
cal se ing, bipola ol age is commonly used as a su o-
ga e o a ial ib osis, bu he phenomenon is much mo e
complex and ol age canno be conside ed as a subs i-
u e o ib osis as assessed by MRI. He e, h ee di e en
EIGDR ma ke s ha e been s udied, e alua ing hei abili y
in wo possible ECA o wo ca he e - o-wa e on o ien a-
ions and conside ing depola iza ion and epola iza ion o
he u-EGMs. The p oposed ma ke s a e good candida es
o de ec ing ib o ic issue. Resul s in e ms o AUC and
ACC con i m he hypo hesis ha educing misalignmen is
bene icial and sugges ∆RA, ep esen ing he eigen alue
dominance gain by alignmen , as he be e sui ed o i-
b o ic a eas iden i ica ion, especially when he 3×3ECA
is used. Howe e , hese esul s need o be complemen ed
wi h o he simula ion con igu a ions, such as smalle i-
b o ic a eas and pa chy ib osis, as well as wi h eal da a,
whe e di e en alues o he elec odes size can be es ed
and he quali y o issue-elec ode con ac could also be
conside ed.
Acknowledgmen s
Funding comes om EU P og amme H2020 unde he
Ma ie Sklodowska-Cu ie G an No 766082 (MY-ATRIA),
Gobie no de A ag´
on (BSICoS G oup T39-20R) co unded
by FEDER 2014-2020 “Building Eu ope om A agon”,
ellowship ACIF/2018/174 om Gene ali a Valenciana,
and PID2019-104881RB-I00 om MICINN, Spain.
Re e ences
[1] Tzeis S, As es as D, Va das P. A ial ib osis: ansla ional
conside a ions o he managemen o AF pa ien s. AER
Jou nal 2019; 8(1):37–41.
[2] Bu s ein B, Na el S. A ial ib osis: mechanisms
and clinical ele ance in a ial ib illa ion. JACC 2008;
51(8):802–809.
[3] Pla ono PG. A ial ib osis: an obliga o y componen o
a hy hmia mechanisms in a ial ib illa ion?. J Ge ia Ca -
diol. 2017; 14(4):233–237.
[4] Rod ´
ıguez-Ma˜
ne o M e al. Valida ing le a ial low ol age
a eas du ing a ial ib illa ion and a ial lu e using mul i-
elec ode au oma ed elec oana omic mapping. JACC: Clin-
ical Elec ophysiology 2018; 4(12):1541-1552.
[5] Bhak a D, Mille JM. P inciples o elec oana omic map-
ping. Indian Pacing Elec ophysiol J. 2008; 8(1):32-50.
[6] Sim I, Bishop M, O’Neill M, Williams SE. Le a ial ol -
age mapping: de ining and a ge ing he a ial ib illa ion
subs a e. J In e Ca d Elec ophysiol 2019; 56:213-227.
[7] Cou emanche M, Rami ez RJ, Na el S. Ionic mechanisms
unde lying human a ial ac ion po en ial p ope ies: in-
sigh s om a ma hema ical model. Am. J. Physiol. 1998;
275(1), H301-21.
[8] Malecka MM, G eens ein JL, Giles WR, T ayano a NA.
Elec o onic coupling be ween human a ial myocy es and
ib oblas s al e s myocy e exci abili y and epola iza ion.
Biophys. J. 2009; 97:2179–2190.
[9] S¨
o nmo L, Laguna P. Bioelec ical Signal P ocessing in
Ca diac and Neu ological Applica ions, Else ie (Aca-
demic P ess), Ams e dam, 2005.
[10] Laguna P e al. Eigen alue-based ime delay es ima ion
o epe i i e biomedical signals. Digi al Signal P ocessing
2018; 75:107-119.
[11] Vigmond E e al. Pe cola ion as a mechanism o explain
a ial ac iona ed elec og ams and een y in a ib o-
sis model based on imaging da a. Hea Rhy hm 2016;
13(7):1536-1543.
Add ess o co espondence:
Jenni e Riccio
C/ Ma iano Esquillo s/n, Edi icio I+D+i, Lab 6.1.01
50018 Za agoza, Spain.
jen iccio@uniza .es
Page 4