Monitoring blood potassium concentration in hemodialysis patients by quantifying T-wave morphology dynamics
Abstract
We investigated the ability of time-warping-based ECG-derived markers of T-wave morphology changes in time (dw) and amplitude (da), as well as their non-linear components (dNLw and dNLa), and the heart rate corrected counterpart (dw,c), to monitor potassium concentration ([K+]) changes (Δ[K+]) in end-stage renal disease (ESRD) patients undergoing hemodialysis (HD). We compared the performance of the proposed time-warping markers, together with other previously proposed [K+] markers, such as T-wave width (Tw) and T-wave slope-to-amplitude ratio (TS/A), when computed from standard ECG leads as well as from principal component analysis (PCA)-based leads. 48-hour ECG recordings and a set of hourly-collected blood samples from 29 ESRD-HD patients were acquired. Values of dw, da, dNLw, dNLa and dw,c were calculated by comparing the morphology of the mean warped T-waves (MWTWs) derived at each hour along the HD with that from a reference MWTW, measured at the end of the HD. From the same MWTWs Tw and TS/A were also extracted. Similarly, Δ[K+] was calculated as the difference between the [K+] values at each hour and the [K+] reference level at the end of the HD session. We found that dw and dw,c showed higher correlation coefficients with Δ[K+] than TS/A—Spearman’s (ρ) and Pearson’s (r)—and Tw—Spearman’s (ρ)—in both SL and PCA approaches being the intra-patient median ρ≥0.82 and r≥0.87 in SL and ρ≥0.82 and r≥0.89 in PCA respectively. Our findings would point at dw and dw,c as the most suitable surrogate of Δ[K+], suggesting that they could be potentially useful for non-invasive monitoring of ESRD-HD patients in hospital, as well as in ambulatory settings. Therefore, the tracking of T-wave morphology variations by means of time-warping analysis could improve continuous and remote [K+] monitoring of ESRD-HD patients and flagging risk of [K+]-related cardiovascular events. Palmieri, F.; Gomis, P.; Ferreira, D.; Ruiz, J.E.; Bergasa, B.; Martín-Yebra, A.; Bukhari, H.A.; Pueyo, E.; Martínez, J.P.; Ramírez, J.; Laguna, P.
Full text
ͷ
ǤǣȋͰͱͲͳʹ͵Ȍ
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Ch onic kidney disease (CKD) is de ined as he p esence o kidney damage, pe sis ing o 3 mon hs o mo e,
i espec i e o he cause1. I ep esen s a s a e o p og essi e loss o kidney unc ion ul ima ely esul ing in need
o enal eplacemen he apy such as hemodialysis (HD) o ansplan a ion. The de elopmen o CKD and i s
p og ession o his e minal s age, called end-s age enal disease (ESRD), emains a signi ican sou ce o educed
quali y o li e and p ema u e mo ali y2. In pa icula , sudden ca diac dea h (SCD) ep esen s an impo an cause
o dea h in ESRD-HD pa ien s3. Va ious isk ac o s may be esponsible o SCD in his pa ien popula ion,
including le en icula hype ophy and ib osis, diso de ed bone-mine al me abolism, HD-induced changes
in elec oly e, and luid and acid-base s a us, which may lead o elec oca diog aphic (ECG) abno mali ies and
en icula a hy hmia3,4.
Recen s udies ha e shown ha blood po assium concen a ions (
[
K+
]
) ou side he physiological in e al a e
associa ed wi h inc eased mo ali y isk5. In heal hy condi ions, he main enance o
[
K+
]
homeos asis is ensu ed
by no mal enal ac i i y6. Howe e , ESRD-HD pa ien s su e om
[
K+
]
imbalance, leading o a high incidence
ͷCen e de Rece ca en Enginye ia Biomèdica, Uni e si a Poli ècnica de Ca alunya, Ba celona, Spain. CIBER en
ÀǡȋǦȌǡǡǤLabo a o ios Rubió, Cas ellbisbal,
Ba celona, Spain. ͺValencian In e na ional Uni e si y, Valencia, Spain. ͻNeph ology Depa men , Hospi al
ÀǡǡǤͼǡ ǡ×ǡǡ
ǡ Ǥ ͽWilliam Ha ey Resea ch Ins i u e, Queen Ma y Uni e si y o London, London, UK. *email:
ƪǤǤ
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o a hy hmic e en s. The p o-a hy hmic consequences o
[
K+
]
imbalance can be explained conside ing ha
po assium cu en s a e in ol ed in he epola iza ion p ocess o he ca diac ac ion po en ial (AP), de e mining
memb ane po en ial and e ac o iness o he myoca dium7. The e o e, e en modes de ia ions o
[
K+
]
om i s
no mal ange (hypokalemia i
[
K+
]
< 3.5 mmol/L o hype kalemia i
[
K+
]
> 5 mmol/L) may lead o hospi alisa-
ion o dea h in ESRD-HD pa ien s8. E alua ion o
[
K+
]
le els is cu en ly based on blood samples ha equi e
u he analyses in he labo a o y, limi ing con inuous moni o ing. Non-in asi e ma ke s able o ack a ia ions
in
[
K+
]
le els a e he e o e needed.
The elec oca diog am (ECG) is a non-in asi e, easily accessible, and inexpensi e p ac ice ha e lec s he
elec ical ac i i y o he hea . In pa icula , he T-wa e e lec s he spa io- empo al epola iza ion o he en-
icle, and i s analysis has been used o measu e he ulne abili y o a pa ien o en icula a hy hmias9. This
ac is o pa icula in e es because T wa es a e equen ly al e ed in ESRD-HD pa ien s4. The QT in e al is he
s anda d index o en icula epola iza ion, and i has been p oposed o moni o ESRD-HD pa ien s10. Howe e ,
he e ec s o HD on QT in e al, and i s co ec ed e sion QTc, a e s ill con o e sial, since se e al s udies11
epo ed a p olonga ion du ing he HD sessions, bu o he s epo ed opposi e end o e en no changes a all12.
This mo i a es he analysis o he o e all T-wa e mo phology as a po en ial po assium le el ma ke .
Di e en T-wa e mo phology ma ke s ha e been p e iously epo ed o be co ela ed wi h
[
K+
]
, such as he
T-wa e igh slope13, he wid h o he T-wa e (
Tw
)14, he T-wa e slope- o-ampli ude a io (
T
S
/
A)15, and a mo -
phology combina ion sco e, which in eg a es ea u es like T-wa e asymme y, la ness and no ching16. Howe e ,
hese ma ke s ely on speci ic local ea u es o he T-wa e a he han in he o e all T-wa e mo phology, which
may ha e a s onge po en ial in ollowing [K+
]
han indices based on local ea u es.
A ecen s udy epo ed a ime-wa ping based me hodology o quan i y changes in he o e all T-wa e
mo phology17. Six indices we e p oposed, du
w and d
a
, e lec ing mo phological a ia ions in ime and ampli ude,
espec i ely, as well as hei non-linea e sion, dNL
w and dNL
a as epo ed in17 and wo no el ma ke s de i ed om
du
w and named d
w
and dw
,
c . The main goal o his s udy is o in es iga e he po en ial o hese ma ke s in moni-
o ing bo h hypo- and hype kalemia e en s excluding he a iabili y due o he hea a e (HR) and o compa e
hei pe o mance agains
Tw
and
T
S
/
A in s anda d single-lead app oach and by applying p incipal componen
analysis (PCA) as mul ilead space educ ion echnique. Howe e , some o he abo e men ioned indices may no
be obus enough o ou pu pose. I is he case o d
w
which does no p o ide in o ma ion abou he di ec ion
o he T-wa e mo phological a ia ion (i.e. i he e is s e ching o sho ening) and has been ound o be co -
ela ed wi h HR. The e o e, we ha e adap ed he o iginal me hodology17 o accoun o hypo- and hype kalemia,
and we p opose a new ma ke ha is independen o HR, hus o e ing a mo e p ecise
[
K+
]
moni o ing ool
o a hy hmic isk s a i ica ion in ESRD-HD pa ien s. P elimina y esul s ex ac ed om a smalle subse o
pa ien s ha e been p esen ed a Compu ing in Ca diology con e ence18,19 while he elec ophysiological basis
was s udied in Bukha i e al.20.
The no el ies o he p esen s udy wi h espec o he s a e-o - he-a a e: (1) he usage o T-wa e ime wa p-
ing analysis o non-in asi e
[
K+
]
moni o ing, oge he wi h he de elopmen o a HR co ec ion ool o he
ime-wa ping ma ke , dw
,
c ; (2) he p oposal o a PCA spa ial ans o ma ion lead o ma ke ex ac ion and (3)
he alida ion o he p oposed ma ke s in compa ison wi h p e iously published bioma ke s (
Tw
and
T
S
/
A ) and
wi h hei ex ac ion om s anda d leads.
Ǥ The s udy popula ion included 29 pa ien s om he Neph ology wa d om Hospi al
Clínico Uni e si a io Lozano Blesa (Za agoza, Spain). Inclusion c i e ia we e (i) 18-yea -old (o olde ), (ii) ha -
ing a diagnosed ESRD pa hology and (iii) unde going HD a leas h ee imes pe week (wi h enous o cannula
access). Table1 shows he popula ion cha ac e is ics. The s udy p o ocol was app o ed by he A agon’s esea ch
e hics commi ee (CEICA, e . PI18/003) and all pa ien s and/o hei legal gua dians signed in o med consen .
All he p ocedu es and all he me hods we e pe o med in acco dance wi h he Helsinki Decla a ion. The da a-
base collec ion is s ill ongoing, wi h he cu en size signi ican enough o a pilo s udy21,22.
Ǥ Gene al in o ma ion. Sex, age, concomi an he apies (e.g. assump ion o an i-a hy h-
mic d ugs), kidney disease e iology and HD ea men ela ed in o ma ion we e collec ed o each en olled
pa ien , as de ailed in Table1.
Blood sample analysis. Fo each pa ien , six blood samples we e aken and analysed du ing he HD session: he
i s one a he HD onse and he nex h ee, e e y subsequen hou (Fig.1, h
0
o h
3
in ed). The 5- h blood sam-
ple was collec ed a he end o he HD (minu e 215- h o 245- h, depending on he HD session du a ion) while
he 6- h blood sample was aken a e 48 h, immedia ely be o e he nex HD session. Po assium, magnesium,
calcium, u ea, c ea inine, bica bona e and pH we e measu ed om each blood es . Blood po assium concen a-
ions alues o each blood es a e gi en in Table2.
ECG measu emen s. A 48h, s anda d 12-lead ECG Hol e eco ding, (H12+, Mo a a Ins umen s, Milwaukee,
WI, USA, sampling equency o 1 kHz, ampli ude esolu ion o 3.75
μ
V), was ob ained o each en olled pa ien ,
s a ing he acquisi ion 5 min be o e he HD onse (Fig.1, blue line). The block diag am p esen ed in Fig.2a
desc ibes he main s eps o he whole da a p ocessing implemen ed in his wo k.
ǦǤ ECG il e ing. Hol e ECG signals con ain baseline d i and o he noises, such as
powe -line and muscula ac i i y (Fig.2a). The e o e, an ini ial p e-p ocessing is needed o imp o e he signal-
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o-noise a io (SNR) and enable ECG wa e o m analysis. Fi s , baseline wande was emo ed wi h a high-pass,
o wa d-backwa d 6- h o de Bu e wo h il e wi h 0.5 Hz cu -o equency23 (Fig.2a). Then, esidual noise
ou o he T-wa e band was emo ed wi h a 6- h o de low-pass Bu e wo h il e wi h 40 Hz cu -o equency.
ECG wa e o m de ec ion and delinea ion. A wa ele -based single-lead me hod24 was applied o de ec QRS
complexes and hen delinea e T-wa e onse s and ends in each o he 12 leads. The wa ele ans o m (WT)
decomposes he signal in he ime-scale domain, allowing i s ep esen a ion a di e en esolu ions. I is, he e-
o e, a sui able ool o analyze ECG signals, which con ain pa e ns wi h di e en equency con en (QRS com-
plexes, P and T-wa es).
Single-lead delinea ion. The disc e e dyadic WT is implemen ed in such a way ha i keeps empo al esolu ion
a di e en scales. The de ec ion o he iducial poin s is ca ied ou ac oss he adequa e WT scales, a ending
o he dominan equency componen s o each ECG wa e: Q,R,S wa es co espond o a simul aneous e ec in
scales 21–22 , while he T and P wa es a ec mainly scales 2
4
o 25 , see24 o de ails. ECG wa e peaks co espond
o ze o c ossings in he WT, and ECG maximum slopes co espond o WT’s maxima and minima. Depending
on he numbe and pola i y o he slopes ound, a wa e mo phology is assigned and bounda ies a e loca ed using
h eshold-based c i e ia. The onse (end) o a wa e occu s be o e (a e ) he i s (las ) signi ican slope associ-
a ed wi h he wa e24.
Selec ion ules o mul i-lead delinea ion. To ob ain mul ilead peak loca ions, a median pos -p ocessing selec-
ion ule o e he single-lead-based de ec ed loca ions is used. The pos -p ocessing ules o bounda ies consis
o o de ing he single-lead anno a ions and selec ing as he onse (end) o a wa e he i s (las ) anno a ion
whose k nea es neighbou s lay wi hin a
δ
ms in e al24,25.
Single-lead analysis. Fi s , we pe o med he analysis using he single-lead ECG, aking he T-wa es om leads
V3 o V6, as used in a p e ious s udy26 o
[
K+
]
es ima ion, and lead II being he mos widely used in pa ien
moni o ing27. These T wa es we e u he delinea ed by using he abo e mo ioned delinea o 24 and he bio-
ma ke es ima ion is pe o med as desc ibed below in sec ion named “Time wa ping analysis”.
Spa ial lead educ ion by p incipal componen analysis. Nex , a spa ial lead educ ion by P incipal Componen
Analysis (PCA) was made since i was ound o be a obus spa ial ans o ma ion o emphasize wa e o m SNR28.
In his wo k PCA was spa ially applied o he 8 independen leads, lea ned o e he T-wa e segmen o mainly
emphasize his wa e o m, and esul ing in 8 p incipal componen s (PCs) o ans o med leads. The coe icien s
de ining he PCA ans o ma ion we e ob ained om he eigen ec o s o he 8×
8
in e lead au o-co ela ion
ma ix compu ed o e he T-wa es in a 10-min wide window a he end o he HD session. The co ec deline-
a ion o T-wa es is c ucial o emphasize only T-wa e ene gy con en . The i s PCA, deno ed as PC1, was used
o he subsequen ECG analysis, as i is he ans o med lead whe e he T-wa es ha e maximal ene gy, and hus,
maximal SNR o mo phological cha ac e isa ion28,29. PC1 was u he delinea ed by applying again24, and each
T-wa e was u he low-pass il e ed a 20 Hz using a 12- h o de Bu e wo h il e o es ic shape analysis o
he dominan band o he T-wa e so emo ing emaining noisy componen s ha could s ill co up he T-wa e
shape analysis.
ǦǤ Two-minu e ECG segmen s, cen ed on he 5- h and 35- h minu es o each a ail-
able hou , we e analysed. The window du a ion was sho enough o hold he assump ion o s abili y o bo h
[
K+
]
and HR alues. Figu e5a shows he a e age RR in e al o each selec ed i- h 2-min segmen s o a gi en
pa ien along he ECG eco ding. While he blood samples (pu ple diamonds) we e collec ed each hou du ing
he HD, he wa ping pa ame e s we e compu ed e e y hal an hou o ge a mo e de ailed iew o e ime.
Fo each i- h 2-min segmen , a mean wa ped T-wa e (MWTW) was compu ed. Fi s o all, he p edominan
T-wa e pola i y (e.g. upwa d, downwa d e c) wi hin a gi en window, was de ined as ha ha ing he highes
numbe o occu ences. This pola i y change can be physiological o induced by delinea o oscilla ion when by-
phasic o egula T-wa es appea s almos indis inguishable. A T-wa e was conside ed o ha e in e ed pola i y
i he magni ude o i s peak had nega i e sign and ice- e sa. Only hose T-wa es ha ing he same pola i y as
ECG acquisi ion
0565 125 185 215
245
HD Pos HD
2880
ℎ1
ℎ0ℎ2ℎ3ℎ4ℎ5
Time (min)
Figu e1. Diag am o he s udy p o ocol: h
0
o h
5
a e he ime poin s (in minu es) o blood sample ex ac ion.
h
4
is aken a he end o he HD (minu e 215- h o 245- h, depending on he HD du a ion).
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O iginal ECG
ECG il a ion PCA
Time-wa ping analysisECG p e-p ocessing
(a)
(b)
Figu e2. Analysis s ages pe o med in his s udy. In panel(a) is he low cha showing he ECG p ocessing
s eps o T-wa e ime-wa ping ma ke s ex ac ion. The analysis s a s wi h he o iginal ECG, ollowed by a
il e ing s ep be o e spa ial PCA analysis, o conclude wi h ma ke s compu a ion. Panel(b) shows an example
o he linea and nonlinea ime-wa ping ma ke s o he same pa ien as in Fig.5a. In pa icula , subpanel (i)
shows bo h he e e ence (blue) and he i- h MWTW ( ed) while subpanel (ii) shows he wa ping unc ion ( ed
do ed line) ha op imally ela es he e e ence and s udied MWTWs. Subpanel (iii) shows he MWTWs a e
wa ping and subpanel (i ) a e he no malized e e ence and wa ped MWTWs.
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he p edominan one we e conside ed in he ollowing s eps. Then, all hese selec ed T-wa es we e aligned wi h
espec o hei g a i y cen e and used o compu e an ini ial MWTW
17. Finally, all he T-wa es we e checked o
ind and disca d he ou lie s, de ined as ha ing T-wa e du a ion ou side he ange
T
dmi±1.5 ×σdi cen e ed a he
i- h ensemble T-wa e mean,
T
dmi , and bounded unc ion o he T-wa e du a ion s anda d de ia ion σdi . Among
he emaining, only hose T-wa es highly co ela ed (Pea son’s co ela ion coe icien > 0.98) wi h he p e ious
ini ial MWTW we e used o ecalcula e he inal MWTW. The MWTW a he end o he HD ea men was aken
as he e e ence, gi en ha i is he ime when he pa ien (a) is supposed o ha e eco e ed he no mal
[
K+
]
le el
and (b) was discha ged om hospi al, being an app op ia e e e ence o ou -o -hospi al ambula o y moni o ing.
Since hype kalemia has been epo ed o cause T-wa e in e sions30, any MWTWs wi h nega i e-pola i y
was in e ed be o e pe o ming he wa ping wi h he e e ence MWTW. P e ious o wa ping, he wo MWTWs
we e aligned wi h espec o hei g a i y cen e , so ha only changes in he T-wa e mo phology, and no hose
associa ed wi h hei ela i e delay, we e quan i ied by he wa ping algo i hm.
Fo compa ison pu poses, bo h
Tw
14 and
T
S
/
A15 we e ex ac ed om each MWTW and hei pe o mance,
wi h espec o T-wa e ime-wa ping based bioma ke s in moni o ing
[
K+
]
, was assessed. This wo k pe o m
a clinical s udy ollowing p e ious analysis es ing he ma ke by elec ophysiological simula ions as epo ed
in Bukha i e al.20.
Table 1. Cha ac e is ics o he s udy popula ion. Values a e exp essed as numbe (
%
) o ca ego ical a iables,
and median (IQR) o con inuous a iables.
(N = 29)
Age (yea s)
7
5
(
12
)
Gende (male) 20
(
70%
)
An i-a hy hmic d ugs (yes) 9
(
31%
)
Implan ed pace-make (yes) 1
(
3%
)
Time unde HD ea men (mon hs) 15
(
59
)
HD session du a ion
210 min 3
(
10%
)
240 min 26
(
90%
)
Kidney disease e iology
Diabe es melli us 17
(
59%
)
In e s i ial neph i is 2
(
7%
)
Glome uloneph i is 2
(
7%
)
Tube ous scle osis 1
(
3%
)
Polycys ic kidney 1
(
3%
)
Cance 1
(
3%
)
Unknown 5
(
18%
)
HD liquid composi ion
Po assium (1.5 mmol/L) 21
(
72%
)
Po assium (3 mmol/L) 5
(
17%
)
Po assium (dec easing) 3
(
11%
)
Calcium (2.5 mg/dL) 21
(
72%
)
Calcium (3 mg/dL) 8
(
28%
)
HD echniques
Con en ional 18
(
62%
)
Online 8
(
28%
)
Ace a e- ee bio il a ion wi h dec eas-
ing in a-HD [K+]3
(
10%
)
Table 2. Blood po assium concen a ion
[
K+
]
alues (in mmol/L) a each blood ex ac ion du ing he HD (
h
0
o h
4
) and HR (bea s/min). Spea man’s (
ρ
) and Pea son’s ( ) in a-pa ien co ela ion coe icien s be ween
[
K+
]
and RR. Values a e exp essed as median (IQR).
h
0
h
1
h
2
h3h
4
ρ
(K
+
)5.0 (1.4) 3.8 (1.1) 3.6
(0.8) 3.4 (0.7) 3.3 (0.6) 0.10 (1.35) 0.09 (1.45)
HR 81 (28) 76 (28) 80 (23) 80 (17) 80 (25)
ͼ
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T-wa e ime wa ping. The me hod he e applied was o iginally p oposed by Ramí ez e al.17. Le
i( i)=[ i( i(1)),..., i( i(Ni))]
T
be he MWTW o a gi en i- h segmen , and ( )=[ ( (1),..., ( (N ))]
T
he e e ence MWTW, whe e
i=[ i(1),..., i(Ni)]
T
and =[ (1),..., (N )]
T
wi h N
i
and N being he
o al T-wa e du a ion, in samples, o i and espec i ely. Figu e2b illus a es he wa ping me hod applied
be ween one o he i- h MWTW ( ed) and he e e ence MWTW (blue). Le
γ
i(
)
be he wa ping unc ion ha
ela es and i , such ha he composi ion
(
i◦
γ
i)( ) deno es he e-pa ame iza ion o ime domain wa ping
o i( i) using
γ
i(
)
, i.e.
(
i◦
γ
i)( ) ep esen s he ampli ude alues o i( i) i i s empo al ec o was . The
squa e- oo slope unc ion (SRSF) was p oposed ins ead o he o iginal T-wa es31,32 o ind he op imal wa ping
unc ion. This was applied by pe o ming ime-wa ping on he SRSFs o he T-wa es, p e en ing he “pinching
e ec ” in cases when T-wa e ampli udes di e 33. This ans o ma ion is de ined as:
The op imal wa ping unc ion is he one ha minimizes he ampli ude di e ence be ween he SRSF o (
)
and i(
γ
i( )
)
32:
The dynamic p og amming algo i hm was used o ob ain he solu ion o his op imisa ion p oblem34. Fig-
u e2b(ii) shows he op imal wa ping unc ion be ween he wo wa es in Fig.2b(i). The wa ped T-wa e, i(γ ∗
i
( )
)
is shown in Fig.2b(iii), oge he wi h he e e ence T-wa e, (
)
.
Time wa ping bioma ke s. The index du
w (co esponding o he index deno ed as d
w
in17), shown as he yellow
a ea in Fig.2b(ii), quan i ies he amoun o wa ping needed o op imally align he wo T-wa es, and is de ined as
he a e age o he absolu e di e ence alue be ween γ∗
i
(
)
and :
The o iginal de ini ion o du
w(i
)
17 was modi ied he e o allow he ma ke o be signed, he e o e dis inguishing
T-wa e widenings om na owings. This signed dw(i
)
was de ined as:
whe e sd
(
i
)
was used o accoun o he sign o he dw(i
)
and i was compu ed as:
wi h Nu
being he se o T-wa e up-slope samples. A posi i e sign means ha he i( i) has o be widened o i
he (
)
and ice- e sa o a nega i e sign.
A e applying ime wa ping be ween bo h MWTWs, he ampli ude di e ence be ween (
)
and i(γ ∗
i
( )
)
is quan i ied as he a ea con ained be ween (
)
and i(γ ∗
i
( )
)
, no malized by he L2-no m o (
)
:
whe e sa(i)=
N
n
=1( i(γ ∗
i
( )) − ( )
)
is used o accoun o he da(i
)
sign es ima ion.
Bo h dw(i
)
and da(i
)
inco po a e in o ma ion om he linea and non-linea di e ences be ween bo h T-wa es
in ime and ampli ude domain, espec i ely. The non-linea componen s can be quan i ied as in17:
whe e
γ
∗
i
,
l(
)
(g een line in Fig.2b(ii)) is he bes linea i ing o
γ
∗
i
(
)
acco ding o he leas absolu e esidual
c i e ion35. The pa ame e dNL
w(i
)
quan i ies he non-linea wa ping by compu ing he a ea o he dashed magen a
egion be ween
γ
∗(
)
and
γ
∗
i
,
l(
)
(in Fig.2b(ii)). Finally, he ma ke dNL
a(i
)
quan i ies he esidual in o ma ion
in ampli ude domain a e no malising MWTWs (Fig.2b(i )).
(1)
q ( )=sign˙
( )
˙
( )
.
(2)
γ∗
i =a g min
γi( )
q −q[ i◦γi]
=a g min
γ
i( )
q −q iγi ˙γi( )
.
(3)
d
u
w(i)=
1
N
N
n=1
|γ∗
i( (n)) − (n)|.
(4)
dw(i)=sd(i)
|sd(i)|1
N
N
n=1
|γ∗
i (n)− (n)|
.
(5)
sd(i)=
n∈Nu
(γ ∗
i (n)− (n))+
n/∈Nu
( (n)−γ∗
i (n)).
(6)
da(i)=
sa(i)
sa(i)
i(γ ∗
i( )) − ( )
( )
×100
.
(7)
dNL
w(i)=1
N
N
n=1
|γ∗
i( (n)) −γ∗
i,l( (n))|
.
(8)
dNL
a(i)=
( )
( )−
i(γ ∗
i( ))
i(γ ∗
i
( ))
×100
.
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Hea - a e-co ec ed T-wa e wa ping. I is well known ha T-wa e du a ion and QT in e al a e s ongly
dependen on HR36. Al hough aligning he T-wa es acco ding o hei g a i y cen e educes mos o he
dependence o dw(i) on HR, he e may s ill be some esidual dependence in T-wa e mo phology ha should be
compensa ed o (e.g. see Fig.5a a ound hou s h = 9, 12 and 43). We assume ha dw(i
)
, as o iginally p oposed
in (4), can be modelled as he sum o wo componen s:
whe e dw
,
HR(i
)
is he HR dependen componen and dw
,
c(i
)
is he non-HR dependen componen accoun ing
o (
K+ ) induced a ia ions and possibly o he s no HR ela ed.
To es ima e he co ec ed componen dw
,
c(i
)
we depa om he li e a u e, whe e se e al o mulae o HR-
dependency co ec ion o epola iza ion ela ed ime in e als, like he QT in e al, ha e been de eloped37–40,
including a a ie y o app oaches (e.g. linea , hype bolic, exponen ial models e c.) being in es iga ed and es ed
in iew o he complex ela ionship be ween QT in e al and HR39. To de i e a co ec ion o mula and es ima e
dw
,
c(i
)
, we s a ed om a linea app oxima ion o a hype bolic model unde small RR changes, de i ed simila ly
o he QT in e al co ec ion (QTc)38,39,
Le ’s call
R
R he e e ence RR in e al associa ed o a e e ence hea bea and
R
R
i
he one o he i- h RR in e al
om one bea a he i- h segmen , hen
As he
Q
T
i
−
Q
T di e ence, also dw(i
)
is a measu e o wid h change be ween he e e ence and he cu en i- h
mean T-wa es om hei espec i e obse a ions ime windows, hen i is possible o ex end p e ious ela ion
in (11) o dw(i
)
ob aining he HR ela ed componen
By subs i u ing (12) in (9) we ob ain
The alue dw
,
c(i
)
can be assumed o be non-ze o mean, and unco ela ed o HR, ha is:
wi h dw
,
c(i
)
ze o mean and unco ela ed o HR. Then, dw(i
)
becomes:
whe e he pa ame e s b,
β
and
α
, once join ly es ima ed (i.e.
ˆ
b
,
ˆ
β
and ˆ
α
) can be used o de i e
ˆ
dw
,
c(i
)
as:
No e ha ,
ˆ
β
and ˆ
α
canno be assessed om (15) wi h a di ec ly leas squa e i ing, since he DC componen b in
(15) la gely a ec s he esul s. Ra he , i is possible o join ly es ima e
ˆ
b,
ˆ
β and ˆ
α
, and hen use he esul s in (16).
This es ima e can be u he app oxima ed linea ly o small RR changes. Deno ing RR
(
i
)
=RRi−RR ,
R
R
i
can be exp essed as
R
Ri=RR +RR
(
i
)
and by eplacing his in he igh side o (12):
Ope a ing on he e ms and unde he assump ion ha RR
(
i
)
RR ,
(
RR(i)
RR
)
1
and by using he Taylo ’s
se ies expansion, we ha e
Subs i u ing (18) in (13):
whe e b,
α
,
β
and
(
RR )
(
α−1
)
a e cons an alues; hen placing:
he ac ual dw(i
)
dependency wi h RR will be:
(9)
dw(i)=dw
,
c(i)+dw
,
HR(i)
.
(10)
QT =β(RR)α.
(11)
Q
Ti−QT =β
(RRi)α−(RR )α
.
(12)
d
w,HR(i)=β
(RRi)α−(RR )α
.
(13)
d
w(i)=dw,c(i)+β
(RRi)α−(RR )α
.
(14)
dw
,
c(i)=b+dw
,
c(i)
,
(15)
dw(i)=b+dw,c(i)+β
(RRi)α−(RR )α
,
(16)
ˆ
d
w,c(i)=dw(i)−ˆ
β(RRi)ˆα−(RR )ˆα
.
(17)
dw,HR(i)=β
(RR +RR(i))α−(RR )α
.
(18)
(RR +RR(i))α−(RR )α≃αRR(i)(RR )(α−1).
(19)
dw(i)≃b+dw,c(i)+αβRR(i)(RR )(α−1),
(20)
αβ(RR )(α−1)=c
,
(21)
dw(i)≃b+dw
,
c(i)+cRR(i)
.
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F om he geome ical poin o iew, b and c can be es ima ed as he ze o-c ossing and he slope, espec i ely, o
he leas -squa es line i o he dw(i
)
alues (in a RR
(
i
)
s. dw(i
)
g aph). Then, he dw(i
)
componen ha does
no dependen on he RR, meaning i is assumed no co ela ed, can assessed as:
whe e ˆc is he es ima ed slope om he Hol e eco ding, see Fig.3,
ˆ
dw
,
c(i
)
is hen he co ec ed es ima ed o
dw
,
c(i
)
, wi h
R
R
i
and
R
R he mean RR in e al om he i- h s udied segmen and he e e ence windows espec-
i ely and ˆc pa ame e is es ima ed o e e y pa ien du ing he ime cou se o he Hol e eco ding. When he
linea app oxima ion p esen ed abo e canno be assumed,
ˆ
b
,
ˆ
β
, and ˆ
α
can be join ly es ima ed om he model
in (15), and use he (16) as he co ec ed es ima e.
An example o he es ima ed
ˆ
dw
,
c(i
)
is gi en in Fig.5a whe e bo h
ˆ
dw
,
c(i
)
and dw(i
)
whe e displayed. No ice
how he p oposed co ec ion o mula emo ed he HR-dependency, o example a ound h = 12.
Table3 p o ides an o e iew o he mo phology ma ke s s udied in his wo k.
[K+]Ǥ The p oposed bioma ke s ha e been compa ed wi h he
ela i e a ia ions in
[
K+
]
(deno ed as [K+](h
)
) wi h espec o a e e ence
[
K+
]
ha was aken a he end o
he HD:
(22)
ˆ
d
w
,
c(i)=dw(i)−ˆcRR(i)=dw(i)−ˆc(RRi−RR )
.
Table 3. T-wa e mo phology ma ke s o
[
K+
]
moni o ing. *du
w co espond o he ma ke deno ed as d
w
in17,
while he e. d
w
is ese ed o he newly in oduced signed e sion.
Ma ke s Desc ip ion
O iginal ma ke s om17
d
u
w
*Time-domain changes be ween
The e e ence and he i- h MWTW (ms).
dN
L
w
Nonlinea componen o he
Time-domain changes be ween
The e e ence and he i- h MWTW (ms)
da
Rela i e ampli ude changes be ween
The e e ence and he i- h MWTW (%)
dN
L
a
Rela i e nonlinea ampli ude changes
A e no malising he e e ence
And he i- h MWTW (%)
Speci ically p oposed in his wo k
dw
Signed e sion o he
P e iously p oposed
d
u
w
* (ms)
d
w
,
cHea a e co ec ed e sion o
dw
(ms)
(b)(a)
Figu e3. Sca e plo showing he alues o bo h d
w
panel (a) and
ˆ
dw
,
c panel(b) wi h espec o
RR
o a gi en
pa ien in PCA app oach. Spea man’s co ela ion coe icien s (
ρ
) and p- alues o bo h d
w
and
ˆ
dw
,
c a e shown on
op o each panel, while he leas -squa e i ing eg ession lines a e plo ed in ed.
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being
[
K+]h he concen a ion a he h- h hou du ing he HD and
[
K+] he concen a ion a he end o he
ea men . An example o he [K+](h
)
e olu ion is shown in Fig.5a (pu ple diamonds).
Ǥ Resul s a e p esen ed as median and in e qua ile ange (IQR). Spea man ank co -
ela ion coe icien (
ρ
) and Pea son co ela ion ( ) we e used o co ela ion analysis be ween [K+
]
and he
p oposed bioma ke , gi ing in o ma ion abou bo h he mono onic ela ion and he s eng h o he associa ion
be ween he ime wa ping based bioma ke s and
[
K+
]
changes and hen p o iding a mo e comple e cha ac e i-
sa ion. The a e age du a ion o he ECG eco dings was 44 h mainly due o elec ode de achmen o ea ly ba -
e y exhaus ion. Fo his eason, co ela ion coe icien s we e compu ed using he i s i e alues o [K+](h
)
h oughou he HD and he wa ping ma ke s e alua ed a he co esponding i- h segmen poin s (
h=(i−1)
/2
whe e i=1, 3, 5, 7, 9 o i=1, 3, 5, 7, 8 depending on he HD du a ion). All s a is ical analyses we e pe o med
using MATLAB e sion R2018b.
In his s udy, ECG signals and
[
K+
]
om 29 ESRD-HD pa ien s we e in es iga ed. An example o d
w
and
ˆ
dw
,
c ime
e olu ion o a pa icula pa ien , in PCA app oach, was p o ided in Fig.3.
RR
was ep esen ed on he x-axis in
bo h panels, while d
w
and
ˆ
dw
,
c we e shown on he y-axis in panel (a) and panel (b), espec i ely. The leas -squa e
i ing line ( ed line) was depic ed in bo h panels. Spea man’s co ela ion coe icien s (
ρ
) and p- alues we e also
showed in each panel. High and signi ican co ela ion (
ρ=−0.9
0
and p- alue
<
0.00
1
) was ound be ween
RR
and d
w
. Howe e , a e co ec ing o he HR-dependency, ρ=0.0
3
and p- alue =0.7
6
.
Co ela ion be ween
[
K+
]
and mean HR exp essed in bea s pe minu e (bpm) ha e also been compu ed and
he esul s a e p esen ed in Table2, wi h a Spea man’s co ela ion coe icien median (IQR) alues o 0.10 (1.35),
and a median p- alue o p=0.33. These alues we e 0.09 (1.45), p = 0.22 o Pea son’s co ela ion coe icien .
Table4 shows he in a-pa ien Spea man’s (
ρ
) and Pea son’s ( ) co ela ion coe icien s compu ed be ween he
ela i e a ia ions in
[
K+
]
(deno ed as [K+
]
) wi h espec o a e e ence
[
K+
]
ha was aken a he end o he
HD and he ime-wa ping pa ame e s. In bo h single-lead and PCA app oaches, he highes median Spea man’s
and Pea son’s co ela ion coe icien s we e ound o du
w , d
w
and dw
,
c being ρ≥0.8
2
and
≥0.8
6
o single-lead
analysis and ρ≥0.8
2
and
≥0.8
9
in PCA.
Boxplo s in Fig.4 show he dis ibu ions o [K+
]
and he p oposed PCA-based ime-wa ping desc ip o s
du ing HD. Figu e5b shows he a e age ime e olu ion o PCA-based du
w , d
w
,
ˆ
dw
,
c and dNL
w in he s udied popula-
ion along he moni o ing pe iod, while he e olu ion o d
a
and dNL
a is shown Fig.5c.
Repola iza ion abno mali ies play a undamen al ole in he genesis o a hy hmic e en s and he isk inc eases
in pa ien s a ESRD wi h imbalance in
[
K+
]
41. In his wo k, wo p e iously epo ed po assium es ima o s,
Tw
14 and
T
S
/
A15, ou wa ping-based ECG-de i ed bioma ke s o
[
K+
]
moni o ing p oposed in Ramí ez e al.17,
du
w , d
a
, dNL
w , dNL
a , and he he e p oposed modi ied e sions d
w
and dw
,
c we e es ed as bloodless indices o
[
K+
]
a ia ions in ESRD-HD pa ien s compu ed om s anda d leads as well as in a PCA-de i ed lead. The mos
p omising esul s in e ms o co ela ion we e ob ained o ma ke s du
w , d
w
, and dw
,
c , leading o he highes
median in a-pa ien ρ≥0.8
2
and
≥0.8
7
in single-lead and ρ≥0.8
2
and
≥0.8
9
in PCA lead espec i ely,
e idencing high mono onic and linea associa ion wi h
[
K+
]
and making hem a p omising non-in asi e indices
o blood
[
K+
]
moni o ing.
The signed bioma ke d
w
ollowed a simila ime-cou se as he unsigned du
w du ing he whole moni o ing
pe iod, showing a simila dis ibu ion in Fig.5a, as a esul o he ac ha he sign compu ed as in (5) is posi-
i e in oughly all he pa ien s. Tha can be explained by he ac ha he T-wa e mo phology in hype kalemia
is usually mo e peaked and sho e in ime han a T-wa e om egula
[
K+
]
concen a ions, as happens a he
end o HD, whe e he e e ence has been aken42,43. The e o e, all he o he MWTWs needed o be sh unk in
ampli ude and widened in ime du a ion du ing he wa ping p ocedu e o i he e e ence one, and his is gi en
by a posi i e signed d
w
. Howe e , o he ex e nal ac o s, such as he po assium emo al a es44 o he dialysa e
po assium le el45,46, migh also ha e played a ole in al e ing en icula epola iza ion ac i i y.
The wa ping algo i hm is applied o e he MWTWs compu ed om di e en obse ing windows wi h di e -
en HRs, as is e iden in Fig.5a. The e o e, a co ec ed e sion o he d
w
, de i ed simila ly o he QT co ec ion
o mula38,39, was p oposed since he HR in luences his ma ke as poin ed ou in Ramí ez e al.17, and can be
obse ed in Fig.5a as an example a ound hou s h
=
9, 12 and 43. A la ge numbe o models ha e been p oposed
o he compu a ion o QTc alues independen o HR37–40. Howe e , a p e ious s udy38 ound ha he linea
eg ession model i s be e han any o he model o he ela ionship be ween QT and he RR in e als. Also,
o small RR a ia ions, in sec ion “Hea - a e-co ec ed T-wa e wa ping” i is shown ha hype bolic QT o
RR dependency becomes linea . The e o e, we used a linea model o de i e an HR-co ec ed index, dw
,
c . This
app oach was used o es ima e he d
w
componen s ic ly ela ed o
[
K+
]
emo ing i s ela ion wi h HR as showed
in Fig.3, whe e he HR-dependency, clea ly isible in panel (a), was cancelled a e he co ec ion, panel (b).
Compa ing he esul s o du
w , d
w
and
ˆ
dw
,
c , all o hem ha e p o ed o be highly co ela ed wi h
[
K+
]
a ia ions.
Howe e , i is impo an o emembe ha du
w (and so d
w
) is biased by he HR e ec s as p e iously desc ibed17,
while
ˆ
dw
,
c is no longe dependen on i , possibly being esponsible o he lowe IQR in he co ela ion, 0.25,
as compa ed o 0.35 and 0.36 o d
w
and du
w , espec i ely (see Table4, PCA column). I should also be no ed
ha he small di e ences be ween he
ρ
and compu ed o
ˆ
dw
,
c and d
w
could be due o he low HR a ia ions
(23)
[K+]
(
h
)
=
(
[K+]
h
−[K+]
)