A icle
Red blood cell linge ing modula es hema oc i
dis ibu ion in he mic oci cula ion
Yazdan Rashidi,
1,
*G e a Simiona o,
1,2
Qi Zhou,
3
Thomas John,
1
Alexande Kihm,
1
Mohammed Bendaoud,
1,4,5
Timm K €
uge ,
3
Miguel O. Be nabeu,
6,7
La s Kaes ne ,
1,8
Ma hias W. Laschke,
2
Michael D. Menge ,
2
Ch is ian Wagne ,
1,9
and Alexis Da as
1,
*
1
Expe imen al Physics, Saa land Uni e si y, Saa b uecken, Ge many;
2
Ins i u e o Clinical and Expe imen al Su ge y, Saa land Uni e si y,
Hombu g, Ge many;
3
School o Enginee ing, Ins i u e o Mul iscale The mo luids, Uni e si y o Edinbu gh, Edinbu gh, Uni ed Kingdom;
4
Uni e si 
e G enoble Alpes, CNRS, LIPhy, G enoble, F ance;
5
LaMCScI, Facul y o Sciences, Mohammed V Uni e si y o Raba , Raba ,
Mo occo;
6
Cen e o Medical In o ma ics, Ushe Ins i u e, Uni e si y o Edinbu gh, Edinbu gh, Uni ed Kingdom;
7
The Bayes Cen e, Uni e si y
o Edinbu gh, Edinbu gh, Uni ed Kingdom;
8
Theo e ical Medicine and Biosciences, Saa land Uni e si y, Hombu g, Ge many; and
9
Physics
and Ma e ials Science Resea ch Uni , Uni e si y o Luxembou g, Luxembou g, Luxembou g
ABSTRACT The dis ibu ion o ed blood cells (RBCs) in he mic oci cula ion de e mines he oxygen deli e y and solu e ans-
po o issues. This p ocess elies on he pa i ioning o RBCs a successi e bi u ca ions h oughou he mic o ascula ne wo k,
and i has been known since he las cen u y ha RBCs pa i ion disp opo iona ely o he ac ional blood low a e, he e o e
leading o he e ogenei y o he hema oc i (i.e., olume ac ion o RBCs in blood) in mic o essels. Usually, downs eam o a
mic o ascula bi u ca ion, he essel b anch wi h a highe ac ion o blood low ecei es an e en highe ac ion o RBC lux.
Howe e , bo h empo al and ime-a e age de ia ions om his phase-sepa a ion law ha e been obse ed in ecen s udies.
He e, we quan i y how he mic oscopic beha io o RBC linge ing (i.e., RBCs empo a ily esiding nea he bi u ca ion apex
wi h diminished eloci y) in luences hei pa i ioning, h ough combined in i o expe imen s and in silico simula ions. We de el-
oped an app oach o quan i y he cell linge ing a highly con ined capilla y-le el bi u ca ions and demons a e ha i co ela es
wi h de ia ions o he phase-sepa a ion p ocess om es ablished empi ical p edic ions by P ies e al. Fu he mo e, we shed ligh
on how he bi u ca ion geome y and cell memb ane igidi y can a ec he linge ing beha io o RBCs; e.g., igid cells end o
linge less han so e ones. Taken oge he , RBC linge ing is an impo an mechanism ha should be conside ed when s udying
how abno mal RBC igidi y in diseases such as mala ia and sickle-cell disease could hinde he mic oci cula o y blood low o
how he ascula ne wo ks a e al e ed unde pa hological condi ions (e.g., h ombosis, umo s, aneu ysm).
INTRODUCTION
Pa i ioning o ed blood cells (RBCs) h ough bi u ca ions o
blood essels de e mines how he oxygen is deli e ed o is-
sues and o gans. Ea ly s udies ha e shown ha his pa i ion-
ing isa complex phenomenon(1–4) and, since hen, i has been
an a ea o in ensi e esea ch, a ac ing a wide ange o expe -
imen al (5–13) and nume ical (14–20)s udies.Mo eaccu-
a ely, essels wi h a highe low a e end o collec an e en
highe p opo ion o RBCs. This ac is known as he
Zwei ach-Fung e ec and leads o he e ogenei y in he hema -
oc i be ween a ious essels (2). Unco e ing he mechanisms
o such a pa i ioning beha io is c ucial o unde s and no
only he anspo o oxygen and solu es ac oss ascula ne -
wo ks bu also he de elopmen o emodeling o he ne wo ks
hemsel es, as RBC dynamics we e ecen ly disco e ed o
Submi ed Augus 31, 2022, and accep ed o publica ion Ma ch 13, 2023.
*Co espondence: yazdan. ashid[email p o ec ed] o alexis.da as@
uni-saa land.de
Yazdan Rashidi, G e a Simiona o, and Qi Zhou con ibu ed equally o his
wo k.
Edi o : Timo Be z.
SIGNIFICANCE We demons a e in i o ha he linge ing beha io o ed blood cells (RBCs) a he apex o bi u ca ions
modula es hei pa i ioning in he mic oci cula ion. The in luence o his mechanism opens new ways o unde s anding how
al e ed RBC p ope ies in pa hologies can hinde he mic o ascula blood low, o in luence he de elopmen o emodeling
o he ascula ne wo ks. Fo ins ance, we show ha igid RBCs linge less a ypical Y-shaped bi u ca ions and mani es a
dis ibu ion pa e n close o he well-known Zwei ach-Fung e ec . We also highligh ce ain p ope ies o he bi u ca ion
geome y ha con ibu e o he linge ing in ensi y o RBCs.
1526 Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023
h ps://doi.o g/10.1016/j.bpj.2023.03.020
Ó2023 Biophysical Socie y.
This is an open access a icle unde he CC BY-NC-ND license (h p://
c ea i ecommons.o g/licenses/by-nc-nd/4.0/).
unde pin essel emodeling, p esumably by media ing he
wall shea s ess di e ence be ween neighbo ing b anches
ia he ou e o e ec i e blood iscosi y (21).
An empi ical model ha e ec i ely ecapi ula es he
ime-a e age beha io o he Zwei ach-Fung e ec was
de eloped by P ies e al. in he 1990s (22–24) and has
been widely employed by he esea ch communi y o
s udying mic o ascula blood low. Howe e , ecen
in i o and in silico s udies demons a ed ha no able de i-
a ions om his empi ical model can a ise o a ious ea-
sons (5,7,8,10,25,26). In pa icula , i has been shown ha
RBC pa i ioning a bi u ca ions in he smalles essels o
he mic o ascula ne wo k end o de ia e om he empi -
ical p edic ions mo e se e ely (11,12,21,27,28). On he
o he hand, i s es ablished h ough nume ical simula ions
(25) and subsequen ly alida ed by in i o expe imen s
(29), an in iguing beha io o RBCs in he mic o ascula-
u e was highligh ed: hey linge a he apex o bi u ca ions
(i.e., he cells empo a ily esiding nea he b anching poin
o essels wi h diminished eloci y) o ce ain pe iods o
ime, consequen ly modi ying he dynamics o cells en e ing
downs eam essels and he cha ac e is ic in e -cell dis-
ances ea u ing in e mi en oids.
P e iously, Bagchi e al. (25) ex ensi ely cha ac e ized he
ime-dependen co ela ion be ween single-cell linge ing
imes and geome y o he ne wo k, bu he causal ela ion-
ship be ween he linge ing pa e n and he a e age beha io
o RBC dis ibu ion h ough he bi u ca ion was no explic-
i ly in es iga ed. Thei ollowing wo k (28) u he combined
a con inuum model wi h he cellula simula ion o dis inguish
he e ec s o plasma skimming and cell sc eening. Mo e
ecen ly, Bagchi e al. epo ed how changes in he ac ion
o linge ing popula ion be ween igidi ied and heal hy cells
a e ela ed o de ia ions om iden i y o he RBC low a e
and o e all blood low a e (30). No wi hs anding he p og-
ess abo e, a mechanis ic unde s anding o he de ia ion
om iden i y due o he Zwei ach-Fung e ec as well as
he baseline beha io o how such de ia ions a e associa ed
wi h he p opo ion o absolu e esidence ime o linge ing
cells is ye o be ob ained.
In he p esen s udy, we sys ema ically analyze how he
linge ing o RBCs in luences hei pa i ioning a highly
con ined capilla y-le el bi u ca ions in he mic oci cula ion
(whe e he cells li e ally squeeze hemsel es h ough and
negligible cell- ee laye exis s con a y o p io s udies; c .
(22–24)). Th ough ca e ully designed in i o expe imen s
co obo a ed by complemen a y in silico simula ions on ime-
scales longe han he linge ing ime and ele an o physio-
logical condi ions, we show ha global de ia ions om he
empi ical model by P ies e al. (24) a e s ongly co ela ed
wi h no malized a e age du a ion o cell linge ing e en s.
We u he e eal how linge ing modula es RBCs pa i ioning,
wi h a p ima y ocus on mechanis ic unde s anding o he de-
ia ion om he baseline beha io o Zwei ach-Fung e ec as
pos ula ed by he empi ical model (24). The asymme ic d i e
om RBC linge ing is ound o ei he ein o ce o e e he
Zwei ach-Fung e ec . In addi ion, gi en ha RBCs usually
exhibi la ge de o ma ions when linge ing, we also in es iga e
nume ically how cells wi h inc eased s i ness di e om
heal hy ones in hei linge ing and pa i ioning beha io . Ou
esul s demons a e ha RBC linge ing is a cause o s a is ical
de ia ion om he Zwei ach-Fung pa i ioning and p o ide
u he e idence ha pa hologically igidi ied cells can impai
he dis ibu ion o RBCs ac oss he mic oci cula ion (31–38)
due o al e ed linge ing p ope ies.
MATERIALS AND METHODS
In i o expe imen s
Pe missions
Expe imen s we e pe o med in Sy ian golden hams e s acco ding o he
Ge man legisla ion on p o ec ion o animals and we e app o ed by he local
go e nmen al animal p o ec ion commi ee (pe mission numbe 25/2018).
Hams e s we e main ained on a s anda d 12/12 h day/nigh cycle and wa e
and ood we e p o ided ad libi um. De ailed me hods a e desc ibed below
o each ype o in es iga ion.
Animal p epa a ion and mic oscopy
Hams e s wi h an age o 5–7 weeks, weighing 55–70 g, we e used o he
implan a ion o a do sal skin old chambe (39). The su ge y was pe o med
unde deep anes hesia (150 mg/kg ke amin (Se umwe ke Be nbu g AG)/
0.25 mg/kg domi o (O ion Pha ma) in ape i oneal), wi h in aope a i e
pain medica ion by ca p o en (10 mg/kg, Zoe is, subcu aneous). A p e i-
ously desc ibed p o ocol (39) was ollowed. B ie ly, he back o he ham-
s e s was sha ed and a i anium chambe consis ing o wo ames was
implan ed on he li ed do sal skin old. On he ame wi h a ci cula obse -
a ion window (diame e o 10 mm) he cu is, subcu is, and e ac o muscle
we e emo ed o expose he s ia ed skin muscle o la e obse a ion o he
mic oci cula ion. The window was closed wi h a co e glass ha was
ixed wi h a snap ing. An implan ed do sal skin old chambe is depic ed
in Fig. 1 a. Animals we e allowed o eco e o 72 h a e he p ocedu e.
Hams e s we e anes he ized as desc ibed abo e be o e in a i al mic o-
scopy. A olume o 100 mL o he luo escen plasma ma ke luo escein
iso hiocyana e (FITC)-labeled (5%, 150 kDa, Sigma-Ald ich) was injec ed
e o-o bi ally and he animals we e ixed on a Plexiglas s age, as illus a ed
in Fig. 1 b. Se e al capilla y bi u ca ions in di e en a eas o he chambe
window we e chosen o epi luo escence mic oscopy (Axio Examine A1,
Zeiss). FITC-labeled dex an was exci ed wi h a 480-nm ligh -emi ing
diode (LED) (Colib i 7, Zeiss), and image con as was addi ionally
enhanced by simul aneously using ansmi ed blue ligh , which is abso bed
by hemoglobin, making RBCs appea da ke . Imaging was pe o med wi h
20(LD A-Plan, NA ¼0.35, Zeiss), 50(LD EC Epiplan-Neo lua , NA ¼
0.55, Zeiss), o 100(LD C Epiplan-Neo lua 100,NA¼0.75, Zeiss)
long-dis ance objec i es. Video acquisi ion was ca ied ou wi h a digi al
came a (Hamama su O ca Flash 4.0, C13440) using he so wa e ZEN
3.1 Blue (Zeiss). Fo each ideo, 2000 ames we e eco ded a di e en
a es acco ding o low eloci y (100–170 Hz). A 2 2 binning was applied
du ing acquisi ion. An example o an ob ained image is gi en in Fig. 1 c.
Image analysis and linge ing quan i ica ion
Despi e he animal being ixed on he Plexiglas s age, i s b ea hing and/o
muscles mo emen caused sligh ansla ions o he mic oscopic ield o
iew in he image sequence (Fig. S1 a). To de e mine he mo ion o he
RBCs in he essels, we i s co ec hose ansla ions by ansla ing
each image by he maximum o i s 2D co ela ion wi h he i s image in
RBC linge ing and hema oc i dis ibu ion
Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023 1527
he sequence. This p ocess c ea es a mo ie whe e he essels and su ound-
ing issues a e s ill ixed (see Fig. S1 b). A Gaussian il e (wi h s anda d
de ia ion o wo pixels) is hen applied o despeckel he images, and a
mask is d awn a ound he essels. The c ea ion o his mask is acili a ed
by also conside ing he image showing he s anda d de ia ion o each pixel
in ensi y along ime, which is highe in he essels wi h ansien (da k)
RBCs and o e s a guide o he eye o he mask d awing (see Fig. S1 c).
The diame e s Do he a ious essels a e compu ed as wice he a e age
dis ance om he skele on pixels o he masks o he bo de o he essels,
using s anda d Ma lab unc ions bwmo ph and bwdis (see Fig. S1 c). The
image se ies is hen in e ed and bina ized inside he mask a ea wi h a
h eshold based on a use -de ined ac ion o he O su h eshold (40). The
O su h esholding me hod was used because he a e age in ensi y o images
is usually sligh ly licke ing. Howe e , he au oma ed h eshold algo i hm
o e es ima ed he h eshold equi ed o de ec he whole RBCs; hence,
we implemen ed a use -de ined a io o adjus i (see Fig. S1 d).
The hema oc i Hwas calcula ed based on he bina ized image: H¼
4s
VRBC
ARBCpR , whe e 4sis he measu ed a e age a ea ac ion o RBCs in he
essel, VRBC ¼56:5mm3and ARBC a e he cell olume and he su ace
a ea (measu ed empi ically in each essel o a selec ion o single cells)
o he RBCs espec i ely, and R is he essel adius. This co ec ion,
simila o he app oach in p e ious wo k (7), akes in o accoun he de o -
ma ion o he cells in he plane while a oiding any coun ing mis ake ha
close RBCs (o he segmen a ion algo i hm) could induce (see Suppo ing
ma e ial o mo e de ails). This co ec ion is alida ed using 2D p ojec ion
slices o he nume ical simula ion, whe e he hema oc i alue is known as
simula ion inpu ; he e o o he alida ion is below 1%. Fo de ec ed
RBCs, he s anda d Ma lab unc ion Weigh edCen oid was used o
calcula e hei cen e o mass. In his unc ion, he cen e o mass is calcu-
la ed based on he loca ion and in ensi y ~
R¼PN
i¼1PN
j¼1½xi;yjIðxi;yjÞ=
PN
i¼1PN
j¼1Iðxi;yjÞ, whe e xand ya e he coo dina es o each pixel and
Iðx;yÞ e e s o he in ensi y o he pixel a posi ion ½x;y.
A acking algo i hm was hen used o de e mine he eloci y o he de-
ec ed RBCs (41). The a ia ions o ligh in ensi y along he essels, in
conjunc ion wi h he collisions and ansien agg ega ions o RBCs, did
no allow us o eliably ollow all RBCs along hei en i e ajec o ies. How-
e e , as we ocused on bi u ca ions whe e he mo he essel ðMÞand he
wo daugh e essels ha e a hema oc i low enough o dis inc single
RBCs, his acking allowed us o ex ac he spa ial dis ibu ion o he
RBC eloci y (i.e., we pe o med pa icle acking elocime y (PTV)),
on which we pe o med ou u he analyses. In his a icle, we will e e
o he wo daugh e essels as main daugh e ðMDÞand seconda y daugh e
ðSDÞ. By de ini ion, MD is he daugh e essel wi h highe ac ional blood
low a e and SD is he one wi h lowe alue.
To de ec he linge ing o RBCs, we applied a me hod as in ou p io
wo k (29) by de ining a ci cle wi h a diame e o 6mm (i.e., he main
diame e o a hams e RBC) su ounding he bi u ca ion apex (Fig. 3 a).
The cells we e conside ed o linge i hei cen e o mass was loca ed in
his ci cle and a he same ime had a eloci y lowe han he local minimum
de ec ed in he p obabili y densi y unc ion (PDF) o a collec i e da ase o
RBC eloci ies (Fig. 3 b). The eloci y PDF displayed in Fig. 3 bwas ob-
ained by conside ing he M essel and he bi u ca ion a ea. In his s udy,
low and cell s a is ics we e cha ac e ized wi hin speci ic ime in e als
(10–20 s), longe han he ansi ime o indi idual RBCs bu s ill sho
enough so ha he low is quasi-s eady (i.e., no sys ema ic end and/o sig-
ni ican change in he olume low a es can be iden i ied). Wi hin such in-
e als, sel -consis en obse ables could be well de ined.
Nume ical simula ions
Complemen a y simula ions o ime-dependen RBC low in he ep esen-
a i e capilla y bi u ca ions we e un wi h HemeLB (h ps://gi hub.com/
hemelb-codes/hemelb) using he imme sed-bounda y-la ice-Bol zmann
me hod ollowing ou p e ious app oach (21). Fi s , h ee-dimensional
(3D) luminal su ace models we e econs uc ed om bina y masks o
he capilla y bi u ca ions using open-sou ce so wa e PolNe (42), assuming
ci cula c oss sec ions o a ying diame e along each essel. Then he low
domain enclosed by he luminal su ace was uni o mly disc e ized in o cu-
bic la ices o ine oxel size Dx¼0:25 mm, aimed a esol ing cell dy-
namics wi h high esolu ion a he bi u ca ion apex. To ini ialize he
simula ion, in low/ou low bounda y condi ions based on expe imen al
ime-a e age olume low a es we e imposed a he end o he M essel
and he MD b anch (simila ly de ined as abo e o he expe imen s) o
each bi u ca ion, and a e e ence p essu e o he SD b anch. No-slip
bounda y condi ions we e imposed a he essel wall. The physical leng h
co esponding o each simula ion ime s ep was D ¼1:04 106s.
The simula ion o each bi u ca ion was ini ialized wi h (RBC- ee)
plasma low. Once he plasma low became con e ged, RBCs we e andomly
inse ed om he M essel h ough a cylind ical low inle wi h cons an
eeding hema oc i ðHFÞas measu ed om expe imen s (see Suppo ing ma-
e ial sec ion ‘‘simula ion hema oc i and RBC ini ializa ion’’). When RBCs
eached he end o MD o SD, hey we e emo ed om he simula ion
domain. Each RBC was modeled as a cy osol- illed capsule wi h an iso opic
and hype elas ic memb ane consis ing o 5120 iangula ace s. The me-
chanical p ope ies o he RBCs we e go e ned by elas ic moduli go e ning
di e en ene gy con ibu ions, such as shea ing and bending o he mem-
b ane. The cy osol was ea ed as a New onian luid wi h he same iscosi y
as he suspending medium (i.e., blood plasma). The iscosi y o he RBC
memb ane i sel was no conside ed. To p obe he e ec o con inemen ,
wo di e en RBC diame e s we e conside ed (D bc ¼6mmco esponding
o no mal hams e RBC size and D bc ¼8mmco esponding o la ge cells
o simila size o a human RBC). Two di e en memb ane s i ness le els
(s ain modulus ks¼5106;5105N=m) we e conside ed o ep e-
sen ing a no mal and a ha dened cell, espec i ely. Fo cell-cell and cell-
wall in e ac ions, epulsi e po en ials in e sely decaying wi h he dis ance
we e implemen ed. High-enough esolu ion was used o ensu e he cell-
cell/cell-wall dis ance did no all below one g id poin .
RESULTS AND DISCUSSION
De e mina ion o low a es and baseline
Zwei ach-Fung pa i ioning
To de e mine how he linge ing o RBCs in luences hei
pa i ioning, we used he empi ical model o P ies e al.
abc
FIGURE 1 Expe imen al me hods. (a) Do sal
skin old chambe implan ed on he back o a ham-
s e . (b) Anes he ized hams e placed unde nea h
he objec i e o an epi luo escence mic oscope. (c)
An exempla mic o ascula ne wo k imaged by
luo escence mic oscopy. The dyed plasma appea s
b igh ; RBCs in capilla ies (see a ows) and he su -
ounding issues a e da k. To see his igu e in colo ,
go online.
Rashidi e al.
1528 Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023
o he Zwei ach-Fung e ec (24) as a baseline. Fo his pu -
pose, we i s de e mined he blood low a e Qin each
essel. Since he essels we e smalle han he main adius
o he RBCs, we assumed a plug low. We also assumed ha
he di e ence be ween he ube and discha ge hema oc i s
was negligible in ou obse a ions. The low a es wi hin
all bi u ca ion segmen s we e he e o e ini ially assessed
om he eloci ies (V) and diame e (D) measu emen s o
he indi idual segmen s
CQD¼CVDpD
22
;(1)
simila ly o p e ious s udies (22,29,43). Mass conse a ion
s ipula es ha he o al olume low a e om he M essel
QMis di ided in o he daugh e b anches and sa is y QM¼
QMD þQSD. The subsc ip s MD and SD e e o he main
daugh e and seconda y daugh e , espec i ely. To obey
his conse a ion law and minimize expe imen al e o s,
alues ob ained om Eq. 1 we e co ec ed ollowing he
no maliza ion p ocess desc ibed by P ies e al. in p e ious
simila measu emen s (22). Simila ly, he measu ed e y h o-
cy e low a es Ei¼HiQi,Hibeing he local hema oc i ,
we e also co ec ed o obey EM¼EMD þESD. The empi -
ical law o he Zwei ach-Fung e ec de eloped by P ies
e al. (22–24) desc ibes he ela ion be ween ac ional
blood low, FQb, and he ac ional e y h ocy e lux, FQe,
in he mic o ascula bi u ca ion. The ac ional blood lows
FQb o bo h daugh e essels we e calcula ed by di iding
he low a e Qin he espec i e daugh e by he low a e
in he M essel. The ac ional e y h ocy e low a es FQe
we e compu ed as he a io o he co ec ed e y h ocy es
low a es (e.g., EMD=EM o he MD). Wi h hese de ini-
ions, he model om P ies e al. (22–24) s a es
whe e logi ðxÞ¼lnðx=ð1xÞÞ and A,B,X0a e ela ed o
he bi u ca ion geome y and eeding hema oc i in he M:
whe e H e e s o he measu ed hema oc i , calcula ed om
he a e age a ea ac ion o RBCs in he mask du ing he
expe imen (23,24). The subsc ip s indica e he essel o
which he quan i y is conside ed.
Due o he he e ogeneous blood low and di e se geome-
ies in he mic o ascula u e, he FQeand FQbda a ex-
ac ed in i o a e na u ally sca e ed (22,23), which may
also apply o egula in i o ascula ne wo ks (10,11).
Howe e , P ies e al. de i ed hei empi ical model o
obus single-bi u ca ion p edic abili y (see Fig. 4 in
Re . (22) and Fig. 2 in Re . (23)) wi h in i o da a ob ained
h ough a classic p o ocol ha enabled con inual a ia ion o
he hema oc i s in indi idual bi u ca ions.
In he p esen s udy, we se ou o in es iga e why indi-
idual bi u ca ion cases (a widely adop ed app oach (5–8,
16–18) o s udy he RBC pa i ioning beha io ) would
nicely sca e a ound his empi ical limi as de ined by
Eq. 2, bea ing in mind ha he e ec s o low a io, essel
geome y, and eeding hema oc i we e al eady accoun ed
o in he o iginal model. To do his expe imen ally, we
conside a se o bi u ca ions selec ed in i o o examine
how he RBC pa i ioning in hem de ia es om he base-
line spli (as pos ula ed by Eq. 2) and u he co ela e he
de ia ion wi h expe imen al ac o s no conside ed by he
empi ical model. In pa icula , we ocus on he linge ing
beha io , which appea s a po en ial mechanism o such
de ia ions in highly con ined capilla ies. Indeed, he ini ial
s udy o P ies e al. analyzed a e iola bi u ca ions wi h
cha ac e is ic diame e s abou 8mm o la ge (22); i.e.,
la ge han he main diame e o hams e RBCs. In hei
s udies o bi u ca ions wi h lowe con inemen le el, he
linge ing beha io (i any) was p obably no playing a
signi ican ole in he pa i ioning o cells. In he case p e-
sen ed he e, s udied essel diame e s ange be ween 3.5
and 5.6 mm; i.e., conside ably smalle han he diame e
o he hams e RBCs. Such high con inemen would a o
AMD ¼13:29 D2
MDD2
SD 1
D2
MDD2
SD þ1ð1HMÞDM;ASD ¼AMD;B¼1þ6:981HM
DM;X0¼0:9641HM
DM;
(3)
FQe¼
8
>
>
>
>
<
>
>
>
>
:
0;i FQb<X
0
1;i FQb>1X0
logi 1AþBlogi FQbX0
12X0;o he wise
(2)
RBC linge ing and hema oc i dis ibu ion
Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023 1529
he linge ing o RBCs as he cells canno easily a oid in e -
ac ing wi h he essel wall a he bi u ca ion.
Linge ing and pa i ioning
Global desc ip ion
Expe imen s. Fig. 2 a–cshows h ee ep esen a i e bi u -
ca ions ( wo Y shaped and one T shaped) whe e all equi ed
pa ame e s could be ex ac ed om he image sequences. A
cha ac e is ic mo ie is p o ided as Video S1. We calcula ed
he empi ical p edic ions by Eq. 2 o each bi u ca ion (nine
in o al) and de ined he de ia ions om he model as
dZF ¼DFQeEX DFQeZF. Al e na i e de ini ions o
he de ia ion (e.g., in signed o m DZF) a e discussed in
he Suppo ing ma e ial (Fig. S7). We no ed DFQe¼
FQeðMDÞFQeðSDÞ, he di e ence o ac ional e y h ocy e
low a e, and he exp essions EX and ZF e e o he expe -
imen al da a and he empi ical p edic ion, espec i ely. The
ac ional blood low a e FQbis gi en by he expe imen al
measu emen s, and he p edic ed ac ional e y h ocy e low
a e FQZF
eis compu ed h ough Eq. 2 (see Fig. 4 a o a
schema ic).
To quan i y he linge ing o RBCs a each bi u ca ion, we
i s de e mined he a e age du a ion o he linge ing e en s
C D. The o al linge ing ime was conside ed as he sum o
he du a ion o all de ec ed linge ing e en s. We hen de e -
mined he a io o his o al linge ing ime o he numbe o
he RBCs as he cha ac e is ic a e age linge ing ime C D.
Since his ime is highly co ela ed wi h he low a e and
he ansi ime o an RBC h ough he bi u ca ion (i.e., he
ed ci cle de ined in Fig. 3 a), we no malized i by s¼
R=CVMD. The adius R¼3mm is he adius o he ed ci cle
de ined in Fig. 3 aand VM he a e age eloci y in he M essel.
abc
ed
FIGURE 2 Rep esen a i e bi u ca ions om expe imen s ha a e simula ed. (a–c) Th ee bi u ca ions selec ed om he expe imen s he ea e e e ed o as
BIF-a, BIF-b, and BIF-c, espec i ely. The M essel, he MD, and SD a e anno a ed wi h a ows indica ing he low di ec ion. The plasma was s ained wi h
luo escen dye (b igh in he images). Da k a eas in he essels indica e he RBCs. The bo de o he masks used o analyze he bi u ca ions a e depic edin
colo ed symbols. Consis en colo s and symbols a e used o hese bi u ca ions h oughou he a icle. (d– ) Simula ed RBC low in he econs uc ed domain
o c opped bi u ca ions, espec i ely (a–c) as cha ac e ized expe imen ally. To see his igu e in colo , go online.
FIGURE 3 RBC eloci y dis ibu ions in he
expe imen and simula ion. (a) Spa ial dis ibu ion
o RBC eloci ies ob ained h ough pa icle acking
in expe imen o BIF-b. Linge ing is conside ed
o occu i a cell is inside he ed ci cle (6mm,
size o a ypical hams e RBC) wi h a eloci y lowe
han he h eshold eloci y, which is de e mined as a
local minimum in he PDF o eloci ies ob ained
om he M essel and he bi u ca ion a ea
(M þb ), as shown in (b). Fo compa ison, he e-
loci y PDFs o he MD and SD b anches and he
M essel a e also shown. (c) Co esponding colo
map o RBC eloci ies measu ed along simula ed
cell ajec o ies in BIF-b. (d) Veloci y PDFs
measu ed in di e en essel b anches o he simu-
la ed BIF-b as o he expe imen al da a in (b). To
see his igu e in colo , go online.
Rashidi e al.
1530 Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023

The ime sis hen a cha ac e is ic ad ec ion ime ha a cell
would need o go h ough he linge ing de ec ion a ea (whe e
he cell eloci ies diminish; Fig. 3 b). We de ined he a io o
hese wo cha ac e is ic imes as Pel¼C D= s. The a io Pel
de ines he equi alen o a P
ecle numbe , since i compa es
wo cha ac e is ic imes o di e en anspo modes a he
bi u ca ion (linge ing and ad ec ion). As shown in Fig. 4 c
o nine di e en bi u ca ions (see expe imen al de ails in
Table S1), he de ia ion dZF om he Zwei ach-Fung p edic-
ion Eq. 2 s ongly co ela es wi h Pel(Pea son co ela ion co-
e icien ¼0:74,p<0:05).
In e es ingly, we obse ed in some case ha he de ia ion
om he Zwei ach-Fung e ec caused by he linge ing can
e e he pa i ioning. Indeed, in he case o he ( ed)
cha ac e is ic bi u ca ion highligh ed in Fig. 2 a, he MD
(wi h highe FQb) is he b anch wi h lowe FQe.Thisis
o en e e ed o in he li e a u e as e e se pa i ioning
(7,10,11,26). In ou da a, we only obse ed such e e se pa -
i ioning o he highes Pelz1:75.
Simula ions. The h ee ep esen a i e bi u ca ions om ex-
pe imen s in Fig. 2 a–cwe e also simula ed unde equi alen
low condi ions and RBC olume ac ions (see de ails in
Table S1), wi h he linge ing phenomenon o RBCs success-
ully ep oduced (Fig. 2 d– ). These bi u ca ions we e selec ed
o nume ical sc u iny because hei expe imen al coun e -
pa s oughly co e he en i e ange o p ima y obse ables
(e.g., Pel,dZF,dp ). A compila ion o anima ion ideos gene -
a ed om nume ical simula ions a e p o ided as Video S2.To
quan i y he linge ing beha io , an equi alen p ocedu e o he
expe imen al da a analysis was applied o analyze he simula-
ion da a oo. Exempla ajec o ies o RBCs nea he bi u ca-
ion apex om simula ions o a ying eeding hema oc i s a e
shown in Fig. S3.
In line wi h he expe imen al PTV measu emen s, he cell
eloci ies diminished as he RBCs app oached he bi u ca-
ion apex (Fig. 3 c). A e en e ing one o he daugh e
b anches (MD o SD), he ins an aneous eloci y o RBCs
would inc ease again. These pa e ns we e well cap u ed
by he eloci y PDFs compiled o indi idual essel
b anches in each bi u ca ion (Fig. 3 d). Quan i ica ion o
linge ing e en s based on he simula ed RBC eloci ies e-
ealed ha he pe cen ages o linge ing cells o he h ee bi-
u ca ions we e 63%, 85%, and 4%, espec i ely. Fo
ha dened RBCs wi h a memb ane s ain modulus 10 imes
as la ge, he a ios became 23%, 77%, and 42%. No e ha
he p opo ion o linge ing cells does no unequi ocally
de e mine Pel; a he , he a e age linge ing ime domina es,
conside ed as he sum o ime du a ion (moni o ed om all
de ec ed linge ing e en s) di ided by he o al numbe o
cell ansi s. In o he wo ds, a highe p opo ion o linge ing
cells does no necessa ily lead o a highe P
ecle numbe .
Fo ins ance, a lowe po ion o cells linge s in BIF-a han
in BIF-b, bu hey linge o longe ime compa ed wi h hei
ad ec ion ime, he e o e causing a la ge Pel.
The co ela ion be ween he linge ing in ensi y o RBCs
and hei pa i ioning a he bi u ca ions, namely Peland
dZF, was e alua ed simila ly o he expe imen al da a
(Fig. 4 d). A s ong associa ion was ound, indica ing a linea
inc ease o dZF agains Pel(Pea son- co ela ion coe icien
0.71, p<0:05). Ne e heless, he absolu e magni ude o dZF
and Pelwe e subs an ially smalle (al hough in a p opo ional
manne ) han hei expe imen al coun e pa in Fig. 4 c.The
FIGURE 4 Expe imen al and simula ion esul s
o RBC pa i ioning e sus linge ing. (a) Compa i-
son o expe imen al RBC pa i ioning agains he
empi ical p edic ions o he Zwei ach-Fung e ec
( o BIF-a as in Fig. 2 a). The axes FQeand FQb
a e o he ac ional RBC lux and ac ional blood
low, espec i ely. The de ia ion om he empi ical
p edic ion was calcula ed as dZF ¼DFQeEX 
DFQeZF.(b) The ba s show expe imen al esul s
agains he lines, ep esen ing empi ical p edic ions
by Eq. 2. The da a he e a e o he h ee cha ac e -
is ic bi u ca ions in Fig. 2 a–c, wi h consis en colo s
and symbols. Dashed lines and hollow symbols he e
e e o he SD essel wi h lowe ac ional low a e
FQb. No e he in e sion o he Zwei ach-Fung e ec
in BIF-a. (c) Expe imen al de ia ion dZF om Eq. 2
as a unc ion o he linge ing P
ecle numbe Pel. The
poin s wi h e o ba s a e expe imen al da a, and he
solid line shows linea eg ession i ing o he da a
poin s, wi h he shaded a ea indica ing 95% con i-
dence in e al p edic ion om he i . All da a ga h-
e ed om nine bi u ca ions (including he h ee
bi u ca ions BIF-a, BIF-b, and BIF-c in Fig. 2 a–c)
a e included in his g aph. The Pea son- co ela ion
coe icien (n ¼9) is ¼0.74 (p¼0.02). E o ba s in (a–c) a e compu ed by e o p opaga ion om he unde lying expe imen al measu emen s. (d) Co -
ela ion be ween dZF and Pelin simula ions (n ¼12). The c osses a e simula ion da a and he solid line shows linea eg ession i ing o he da a poin s. The
Pea son- co ela ion coe icien is ¼0.71 (p¼0.014). To see his igu e in colo , go online.
RBC linge ing and hema oc i dis ibu ion
Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023 1531
highes P
ecle numbe was only Pel¼0:65 in he simula-
ions compa ed wi h a maximum alue o Pel¼1:75 in
he expe imen s.This disc epancy may ha e a isen om simu-
la ion con igu a ions ha a e di e en om in i o expe i-
men s; e.g., absence o endo helial su ace laye (ESL;
oughly 0.4–0.5 mm in hickness) and RBC glycocalyx in
he nume ical model. Indeed, al hough a epulsi e cell-wall
po en ial was implemen ed, i is possible ha he ESL coa ed
by cilia ed s uc u es in e ac s wi h he RBCs in mo e complex
ways (20,44,45). Fo ins ance, i may con ibu e o enhanced
con inemen o gi e ise o an a ac ion o ce. Because hese
ad anced cell-wall in e ac ions ha e no been de ini i ely
cha ac e ized in he li e a u e, wi hou a angible physical
model, we could no cap u e such e ec s in ou simula ions.
In any case, he co ela ion be ween he linge ing P
ecle num-
be Peland he de ia ion om he P ies model was quali a-
i ely simila be ween he expe imen and he simula ion.
We no e ha no e e se pa i ioning (o in e sion o he
Zwei ach-Fung e ec ) wasobse edin ou simula ions, which
is in ag eemen wi h he ac ha e e se pa i ioning in expe -
imen s only occu ed o bi u ca ions wi h Pel>1:75.
Linge ing asymme y in expe imen s
Fo insigh s in o how he linge ing beha io could modi y
RBC pa i ioning a he bi u ca ion, we u he examined
i s symme y be ween he wo daugh e b anches. Indeed,
as can be seen om Video S1, he RBCs end o en e he
lowe daugh e b anch ðMDÞalmos only i he e is ano he
cell al eady linge ing a he bi u ca ion apex. Due o cell
collisions a he apex, i was no possible o p ecisely de e -
mine he p opo ion o linge ing cells en e ing each
daugh e me ely by analyzing hei ajec o ies. The e o e,
we calcula ed he a e age ad ec ion ime ao he cells be-
ween he (blue) mask in bi u ca ion and he (magen a)
mask in he daugh e (Fig. 5 a). This ad ec ion ime was
de e mined by calcula ing he con olu ion o he empo al
hema oc i in he daugh e mask and ha in he bi u ca ion
mask (see Fig. 5 aand Fig. S2 b). We hen checked i he e
was a linge ing e en a he bi u ca ion (also moni o ed in a
hin mask) a a ime aea lie o each cell de ec ion.
We ealized ha , o such so ing, some linge ing e en s a
he bi u ca ion could no be a ibu ed o any ad ec ed cell
in o he daugh e s. Indeed, he ad ec ion ime o each cell
can a y sligh ly, depending on i s in e ac ions wi h o he
cells nea he apex. The e o e, a e being empo ally ans-
la ed by a, he cell de ec ion can be loca ed close o he
linge ing e en bu no exac ly wi hin i s du a ion. Fo
mo e accu a e cha ac e iza ion, we he e o e measu ed he
p oximi y ime ibe ween he closes linge ing e en and
he ansi ime (de ec ion ime in he daugh e minus a)
o each cell (Fig. 5 b). Cells wi h a p oximi y ime i¼0
can hen be de e mined as linge ing. We u he de ined
dp ¼jpSD pMDjas he di e ence be ween he p opo ion
o linge ing cells in he wo daugh e b anches (Fig. 5 ).
FIGURE 5 Asymme y in RBC linge ing a he bi u ca ion. (a) Example o mask se used o de e mine he ad ec ion ime ao RBCs om he bi u ca ion
(blue mask) o he SD b anch (magen a mask) ia con olu ion o he hema oc i in bo h masks (see main ex and Fig. S2). Equi alen analysis is pe o med
independen ly o he MD b anch. (b) Ex ac ion o he p oximi y ime io each cell. The hema oc i a ia ion o e ime, ex ac ed om he daugh e ’s mask,
is i s ansla ed by he a e age ad ec ion ime a(see Fig. S2). Then, o each peak o his hema oc i along ime, a p oximi y ime iis ob ained as he
empo al dis ance om he closes linge ing e en . This p ocess is pe o med independen ly o he wo daugh e s. (c–e) CDF o he p oximi y ime o
linge ing e en s o bo h daugh e s o he h ee cha ac e is ic bi u ca ions. When a signi ican di e ence (conside able dp ) is obse ed in he expe imen s,
highe p opo ion o linge ing cells (and a close p oximi y wi h linge ing e en s) is usually ob ained in he daugh e b anch wi h he lowe ac ional RBC
lux FQe.( ) The co ela ion be ween expe imen al dp (di e ence in he p opo ion o linge ing cells in SD and MD calcula ed om CDFs a i¼0) and
Pel. The Pea son- co ela ion coe icien is 0.59 ðp¼0:09Þ. To see his igu e in colo , go online. E o ba s a e compu ed by e o p opaga ion om he
expe imen al unce ain ies o pSD and pMD.
Rashidi e al.
1532 Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023
Because cells wi h a small p oximi y ime imigh also ha e
been linge ing o migh ha e in e ac ed wi h ano he cell
linge ing a he apex, we used he di e ence be ween he i s
wo poin s o he cumula i e densi y unc ion (CDF) dis ibu-
ion o i(Fig. 5 c–e) as e o ba s o pSD and pMD. As can be
seen in Fig. 5 c–e, he p oximi y ime idis ibu ion and he
p opo ion o cells iden i ied as linge ing a e dis inc o
he wo daugh e s unde la ge Pel. Fo he expe imen ally
obse ed cases as highligh ed in Fig. 5 , he asymme y in
he p opo ion o linge ing cells is weakly co ela ed wi h
he linge ing in ensi y Pel. In simula ion, bo h dp and Pel
o he simula ed bi u ca ions ( h ee in o al) a e smalle
han hei expe imen al coun e pa bu in a p opo ional
manne wi h quali a i e ag eemen (see Fig. S2 c).
O igin o linge ing in expe imen s
To unde s and wha causes he cells o linge , we iden i ied
geome ical ea u es o he bi u ca ions co ela ing wi h he
linge ing P
ecle numbe Pel. Due o he de ini ion o he
linge ing e en s as a signi ican educ ion o he RBC eloc-
i y a he bi u ca ion, i is likely ha cells linge a he s agna-
ion poin ; i.e., he in e sec ion o he bi u ca ion’s wall and
he low di ide plane. No e ha a s agna ion poin is consid-
e ed because he 3D essels a e p ojec ed in he mid- essel
plane. A his posi ion, he cells will also expe ience he
smalles d ag o ce, likely no o push he cell owa d any
daugh e essel. The posi ion o he s agna ion poin depends
on he ac ional low a e o he daugh e b anches (46). The
dis ance o he s agna ion poin om he cen e line Ls, which
is depic ed in he Fig. 6 a, can be app oxima ed h ough
QMD
QSD
¼
1þ2
pa csinLs
RMþ2Ls
pR2
Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
ML2
s
q
12
pa csinLs
RM2Ls
pR2
Mffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2
ML2
s
q;(4)
whe e RMis he adius o he M b anch (see Suppo ing ma e-
ial o u he jus i ica ion). Fo he si ua ions whe e nume i-
cal simula ions we e compa ed wi h he expe imen al da a,
posi ions o he s agna ion poin de e mined his way we e
loca ed wi hin one pixel o dis ance om he posi ion indi-
ca ed by he s eamlines (see Fig. S4). This app oxima ion
can hen be conside ed o be consis en wi h he a ailable nu-
me ical da a. Nume ical simula ions also showed ha he shi
o he s agna ion poin due o s udied RBC ansi s we e negli-
gible compa ed wi h o he expe imen al unce ain ies (see
Fig. S5).
Since he cells a he s agna ion poin expe ience he small-
es d ag o ce, one can assume ha he domina ing o ces on
he cells a ise om hei in e ac ion wi h he endo helial laye .
In u n, his in e ac ion p obably depends on he geome y o
he wall a his loca ion, whichcan mos ly be desc ibed h ough
i s cu a u e. To de e mine he cu a u e o each bi u ca ion, a
ci cle was i ed on he neighbo hood o he s agna ion poin
FIGURE 6 Po en ial o igin o RBC linge ing. (a) Schema ic highligh ing he low spli and he s agna ion poin a bi u ca ions. Flow in he mo he b anch
spli s in o lows QMD and QSD in he MD and SD b anches, espec i ely. The s agna ion poin is indica ed by he g een poin , whose dis ance om he apex Ls
is highligh ed wi h he a ow and scale symbol. The illed ci cle shows an idealized c oss sec ion o he mo he essel wi h simpli ied and la low sepa a ix.
(b) Co ela ion o he linge ing P
ecle numbe Pelwi h he cu a u e zo he bi u ca ion a he s agna ion poin . (c) Co ela ion be ween Peland he low a e
a io be ween he SD and MD QSD=QMD.(d) Co ela ion be ween Peland he dis ance be ween he s agna ion poin and he bi u ca ion apex LS.(e) Co -
ela ion be ween LSand he cu a u e a he s agna ion poin z.In(b)–(e), he poin s a e expe imen al da a, he solid lines show linea eg ession i ing o he
da a poin s, and he shaded a eas indica e he 95% con idence in e al p edic ion om each i . To see his igu e in colo , go online. E o ba s in (b–e) a e
compu ed by e o p opaga ion om he unde lying expe imen al measu emen s.
RBC linge ing and hema oc i dis ibu ion
Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023 1533
(see Fig. S4). The cu a u e is de ined as he in e se o he
adius o his ci cle. To es ima e he measu emen unce ain y,
we used mul iple i edci cles, each calcula ed using a di e en
leng h o he neighbo hood (leng hs be ween 4and 8mm,
cen e ed on he s agna ion poin ). The mean alue o he cu a-
u es is aken as he bes app oxima ion, whe eas he s anda d
de ia ion gi es he e o ba s. The esul s showed ha he cu -
a u e o he bi u ca ion a he s agna ion poin co ela es wi h
Pel( ¼0:59 and p¼0:09) (see Fig. 6 b). In e es ingly,
when es ing co ela ions wi h u he possibly in luencing pa-
ame e s, we ound ha he linge ing P
ecle numbe Pelalso
co ela es wi h he a io o he low a es be ween he SD and
MD QSD=QMD (Fig. 6 c). Since his pa ame e de e mines
he dis ance LSbe ween he apex o he bi u ca ion and he
s agna ion poin h ough Eq. 4, his pa ame e LSalso co e-
la es wi h Pel(Fig. 6 d). Fu he mo e, he bi u ca ions whe e
he s agna ion poin is u he om he apex gene ally ha e a
lowe cu a u e a he s agna ion poin , as shown by Fig. 6 e.
We he e o e hypo hesize ha he low a e a io de e mines
whe e he cells can linge , whe eas he cu a u e o he endo-
helial laye a his linge ing posi ion de e mines he in ensi y
o he linge ing. The linge ing P
ecle numbe would hen be
de e mined by an in e play be ween global pa ame e s o he
bi u ca ion ðQSD =QMDÞand local geome ies o he endo he-
lial laye ( he cu a u e a he s agna ion poin , z). When i
comes o he expe imen al co ela ions, he esul ha a lowe
coe icien is ound be ween Peland z han Peland QSD=QMD
likely a ises om he ac ha he de e mina ion o zalso elies
on he app oxima ion o LSand hen he measu emen o
QSD=QMD. The e o p opaga ion on hese successi e quan i-
ies hen likely dec eased a i icially he co ela ion be ween
Peland he successi ely de e mined pa ame e s LSand z.
Mo e geome ical pa ame e s, which ailed o demon-
s a e a signi ican co ela ion wi h Pel, a e included in
he Suppo ing ma e ial (see Fig. S6).
E ec o cell igidi y and hema oc i in
simula ions
Besides he cu a u e o he s agna ion poin a he apex,
o he ac o s may also a ec he in e ac ion be ween
he cells and he bi u ca ion, leading o ei he weakened
o enhanced RBC linge ing. Taking a ypical Y- ype
bi u ca ion o example (Fig. 7 a), h ee physiologically
ele an aspec s we e nume ically explo ed: he cell igidi y
ks, he eeding hema oc i HF, and he cell size DRBC
(D bc ¼6mm unless o he wise speci ied).
Inc eased cell igidi y (imi a ing diseased RBCs wi h
highe -le el s i ness) a he Y- ype bi u ca ion is ound o
in oduce s ong cell-wall and cell-cell s e ic epulsion a
he apex (see enla ged gaps nea he b anching poin in
Fig. 7 a), hus educing he o e all linge ing equency
and a e age du a ion o indi idual linge ing e en s as indi-
ca ed by a smalle linge ing P
ecle numbe Pel(BIF-a in
Fig. 7 b). The same is ound o BIF-b oo (Fig. 7 b).
Con a ily, o a T- ype bi u ca ion (less common bi u ca-
ion ype in mic o ascula ne wo ks), s i e RBCs end o
ge s uck a he apex due o cell collisions and can lead o
mo e in ense linge ing ins ead (see BIF-c in Fig. 7 b). The
dis inc e ec o RBC de o mabili y on i s linge ing
beha io a he Y- ype and T- ype bi u ca ions obse ed
he e is in line wi h a ecen epo on capilla y ascula
ne wo k (30).
On he o he hand, o iden ical low condi ions, an en-
iched hema oc i (HF¼33%, ela i e o he baseline
HF¼12%basedonexpe imen almeasu emen )in
he M b anch has a weakening e ec on RBC linge ing
a he bi u ca ion, whe eas a educed hema oc i
ðHF¼10%Þhas a s eng hening e ec compa ed wi h
he expe imen ally measu ed alue HF¼12% o he
same bi u ca ion (Fig. 7 c). The p obable eason behind
he dec eased Pelin his case is ha c owded cell a ic
educes he possibili y o indi idual RBCs esiding on
he apex o ex ended pe iods o ime as hey a e mo e
likely o be pushed o wa d by ollowing cells (‘‘he ding’’
e ec as epo ed in (47)). We also conside ed a di e en
RBC size gi en he a iabili y o cell mo phology
in i o, which is modeled h ough inc easing he de aul
cell diame e DRBC ¼6mm om DRBC ¼6mm o
DRBC ¼8mm while main aining he same HF(Fig. 7
c). The inc eased RBC size leads o highe le el o cell
con inemen in he capilla ies and educes he luc ua ion
ampli ude o RBC linge ing caused by hema oc i
al e a ion.
FIGURE 7 E ec o cell igidi y and eeding he-
ma oc i HFon he linge ing o RBCs in simula-
ions. (a) S e ic epulsion o ( op) no mal
(N) and (bo om) ha dened (H) RBCs a he apex
o a Y- ype bi u ca ion co esponding o BIF-a in
Fig. 2.(b)Pel e sus cell igidi y o no mal and
ha dened RBCs a BIF-a, BIF-b, and BIF-c’’ (see
Fig. 2). Solid lines a e o simula ions wi h HF
ma ching co esponding expe imen al condi ion,
whe eas dashed lines a e simula ions wi h ele a ed
hema oc i le els ( wice as high o BIF-a and
BIF-b, and h ee imes as high o BIF-c). (c)Pel
e sus eeding hema oc i HF o BIF-a wi h no mal RBCs (in de o mabili y) o wo di e en sizes D bc ¼6;8mm. To see his igu e in colo , go
online.
Rashidi e al.
1534 Biophysical Jou nal 122, 1526–1537, Ap il 18, 2023