

Ci a ion: Daud, M.U.; Abbas, G.;
A zaal, M.; Naz, M.Y.; Fa ima, N.G.;
Ghu a , A.; I an, M.; Mahnashi,
M.H.; Legu ko, S.; Pe ˚u, J.; e al.
Fini e Elemen Analysis o Sil e
Nano ods, Sphe es, Ellipsoids and
Co e–Shell S uc u es o
Hype he mia T ea men o Cance .
Ma e ials 2022,15, 1786. h ps://
doi.o g/10.3390/ma15051786
Academic Edi o s: We onika
K uszelnicka and
And zej Tompo owski
Recei ed: 21 Decembe 2021
Accep ed: 21 Feb ua y 2022
Published: 26 Feb ua y 2022
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Licensee MDPI, Basel, Swi ze land.
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condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
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ma e ials
A icle
Fini e Elemen Analysis o Sil e Nano ods, Sphe es, Ellipsoids
and Co e–Shell S uc u es o Hype he mia T ea men
o Cance
Muhammad Usama Daud 1, Ghulam Abbas 1,*, Muhammad A zaal 1,*, Muhammad Yasin Naz 2,
Nazma Gohe Fa ima 1, Abdul Ghu a 1, Muhammad I an 3, Ma e H. Mahnashi 4, S anislaw Legu ko 5,
Jana Pe ˚u 6, Jiˇ íK a och íl6and Usama Muhammad Niazi 7
1Depa men o Physics, Faisalabad Campus, Riphah In e na ional Uni e si y, Faisalabad 44000, Pakis an;
[email p o ec ed] (M.U.D.); [email p o ec ed] (N.G.F.);
[email p o ec ed] (A.G.)
2Depa men o Physics, Uni e si y o Ag icul u e, Faisalabad 38040, Pakis an; [email p o ec ed]
3Elec ical Enginee ing Depa men , College o Enginee ing, Naj an Uni e si y, Naj an 61441, Saudi A abia;
[email p o ec ed]
4Depa men o Pha maceu ical Chemis y, College o Pha macy, Naj an Uni e si y,
Naj an 11001, Saudi A abia; [email p o ec ed]
5Facul y o Mechanical Enginee ing, Poznan Uni e si y o Technology, 60-965 Poznan, Poland;
s anislaw[email p o ec ed]
6Facul y o Mechanical Enginee ing, VSB—Technical Uni e si y o Os a a, Po uba,
708 00 Os a a, Czech Republic; [email p o ec ed] (J.P.); [email p o ec ed] (J.K.)
7Depa men o Mechanical Enginee ing Technology, Na ional Skills Uni e si y, Islamabad 44000, Pakis an;
[email p o ec ed]
*Co espondence: [email p o ec ed] (G.A.); [email p o ec ed] (M.A.)
Abs ac :
The ini e elemen analysis echnique was used o in es iga e he sui abili y o sil e
nano ods, sphe es, ellipsoids and co e–shell s uc u es o he hype he mia ea men o cance . The
empe a u e o he sil e nanos uc u es was aised om 42 o 46
◦
C, in o de o kill he cance ous
cells. The ime aken by he nanos uc u es o a ain his empe a u e, wi h ex e nal sou ce hea ing,
was also es ima ed. The hea ans e module in COMSOL Mul iphysics was used o he ini e
elemen analysis o hype he mia, based on sil e nanos uc u es. The he mal esponse o di e en
shapes o sil e nanos uc u es was e alua ed by placing hem inside he sphe ical domain o he
umo issue. The p oposed geome ies we e hea ed a di e en ime in e als. Op imiza ion o he
geome ies was pe o med o achie e he bes ea men empe a u e. I was obse ed ha sil e
nano ods quickly a ain he desi ed empe a u e, as compa ed o o he shapes. The sil e nano ods
achie ed he highes empe a u e o 44.3
◦
C among all he analyzed geome ies. Mo eo e , he cen al
olume, used o iden i y he he mal esponse, was he maximum o he sil e nano-ellipsoids.
The mal equilib ium in he ea men egion was a ained a e 0.5
µs
o hea ing, which made hese
s uc u es sui able o hype he mia ea men .
Keywo ds:
COMSOL Mul iphysics; hype he mia; su ace coa ing; ini e elemen analyses;
sil e nanos uc u es
1. In oduc ion
Cance is a mul i ac o ial illness p oduced by a complex combina ion o he edi a y and
en i onmen al a iables [
1
–
3
]. T ea ing cance is he mos challenging heal h issue in he
21s cen u y [
4
–
7
]. The cells o cance a e ound o agg essi ely in ade o he bodily egions.
These cells ei he o m a umo oge he , o can dissemina e o he blood s eam o lymph
sys em [
8
,
9
]. These cells can me as asize o o he o gans and de elop new umo s in places
a away om he ini ial illness si e. The abili y o cance cells o me as asize o o he a eas
o he body is de e mined by a a ie y o ac o s, including blood low and he ype o cance
Ma e ials 2022,15, 1786. h ps://doi.o g/10.3390/ma15051786 h ps://www.mdpi.com/jou nal/ma e ials
Ma e ials 2022,15, 1786 2 o 13
cells, and also he ini ial loca ion o he cance [
8
–
10
]. The subs an ial ad ancemen s in
cance esea ch ha e esul ed in a be e unde s anding o cance a he gene ic, molecula ,
and cellula le els, allowing o no el he apeu ic a ge s and p ocedu es [
11
]. Cance
ea men s include chemo he apy, su ge y and adia ion he apy [
12
,
13
]. Ano he way o
ea ing cance is by using hype he mia, which in ol es hea ing he umo egion wi hou
damaging he no mal cells [
14
]. Using hype he mia, he empe a u e a he umo loca ion
is aised o a ce ain le el o kill he cance cells [
15
]. The a ge ed umo si es mus a ain a
empe a u e in he ange o 42–46
◦
C o hype he mia o be he apeu ically e ec i e in
cance he apy [
16
]. P o ein dena u a ion occu s when cells a e exposed o his empe a u e
ange, esul ing in a high ac ion o co-agg ega ed dena u ed p o eins [
15
]. Fu he mo e, a
high empe a u e in luences he cellula s uc u e unc ion and al e s in acellula p ocesses,
ul ima ely leading o cance mo ali y [16,17], as illus a ed in Figu e 1.
Ma e ials 2022, 15, x FOR PEER REVIEW 2 o 14
umo s in places a away om he ini ial illness si e. The abili y o cance cells o me as-
asize o o he a eas o he body is de e mined by a a ie y o ac o s, including blood low
and he ype o cance cells, and also he ini ial loca ion o he cance [8–10]. The subs an-
ial ad ancemen s in cance esea ch ha e esul ed in a be e unde s anding o cance a
he gene ic, molecula , and cellula le els, allowing o no el he apeu ic a ge s and p o-
cedu es [11]. Cance ea men s include chemo he apy, su ge y and adia ion he apy
[12,13]. Ano he way o ea ing cance is by using hype he mia, which in ol es hea ing
he umo egion wi hou damaging he no mal cells [14]. Using hype he mia, he em-
pe a u e a he umo loca ion is aised o a ce ain le el o kill he cance cells [15]. The
a ge ed umo si es mus a ain a empe a u e in he ange o 42–46 °C o hype he mia
o be he apeu ically e ec i e in cance he apy [16]. P o ein dena u a ion occu s when
cells a e exposed o his empe a u e ange, esul ing in a high ac ion o co-agg ega ed
dena u ed p o eins [15]. Fu he mo e, a high empe a u e in luences he cellula s uc u e
unc ion and al e s in acellula p ocesses, ul ima ely leading o cance mo ali y [16,17],
as illus a ed in Figu e 1.
Figu e 1. Mechanism o hype he mia he apy o cance cells.
Figu e 2 shows he ollowing wo examples o hype he mia-media ed cance ea -
men : pho o he mal he apy (PTT) and magne ic hype he mia he apy (MHT) [18–21].
Insu icien blood low, nou ishmen and oxygen supply a e es ablished inside he blood
essels in he umo en i onmen , due o he apid empe a u e change in malignan cells;
howe e , umo s a e mo e esis an o empe a u e luc ua ions [18,19]. Hype he mia has
been used in cance he apy o imp o e he he apeu ic e ec i eness.
Whole-body hype he mia, local hype he mia and egional hype he mia a e all
possible h ough hype he mia he apy [7]. Whole-body hype he mia in ol es hea ing
h ough an ex e nal hea sou ce, such as mic owa es o adio equencies, which may
ha e unheal hy side e ec s because o non-selec i e hea ing h ough non-selec i e p o-
cess [21]. Regional hype he mia hea s a la ge a ea o cells, such as a body ca i y, a limb
o an o gan. The egional pe usion echnique, o he con inuous hype he mic pe i oneal
pe usion (CHPP) s a egy, can be used o apply egional hype he mia. D awing blood
om he pa ien ’s body, hea ing i , and hen pumping i back in o he damaged o gan
cons i u es he egional pe usion app oach. The CHPP echnique is used o ea pe i o-
neal cance s, including p ima y pe i oneal meso helioma and s omach cance [22]. I p e-
sen in small a eas, he cance cells a e killed by in oducing hea ca ie s (Fe, Co, Ni, Ag,
Au, e c.) in o he body. This is e e ed o as local body hype he mia, whe e nanopa icles
a e used as hea suscep o s [23].
Figu e 1. Mechanism o hype he mia he apy o cance cells.
Figu e 2shows he ollowing wo examples o hype he mia-media ed cance ea -
men : pho o he mal he apy (PTT) and magne ic hype he mia he apy (MHT) [
18
–
21
].
Insu icien blood low, nou ishmen and oxygen supply a e es ablished inside he blood
essels in he umo en i onmen , due o he apid empe a u e change in malignan cells;
howe e , umo s a e mo e esis an o empe a u e luc ua ions [
18
,
19
]. Hype he mia has
been used in cance he apy o imp o e he he apeu ic e ec i eness.
Whole-body hype he mia, local hype he mia and egional hype he mia a e all
possible h ough hype he mia he apy [
7
]. Whole-body hype he mia in ol es hea ing
h ough an ex e nal hea sou ce, such as mic owa es o adio equencies, which may ha e
unheal hy side e ec s because o non-selec i e hea ing h ough non-selec i e p ocess [
21
].
Regional hype he mia hea s a la ge a ea o cells, such as a body ca i y, a limb o an o gan.
The egional pe usion echnique, o he con inuous hype he mic pe i oneal pe usion
(CHPP) s a egy, can be used o apply egional hype he mia. D awing blood om he
pa ien ’s body, hea ing i , and hen pumping i back in o he damaged o gan cons i u es
he egional pe usion app oach. The CHPP echnique is used o ea pe i oneal cance s,
including p ima y pe i oneal meso helioma and s omach cance [
22
]. I p esen in small
a eas, he cance cells a e killed by in oducing hea ca ie s (Fe, Co, Ni, Ag, Au, e c.) in o
he body. This is e e ed o as local body hype he mia, whe e nanopa icles a e used as
hea suscep o s [23].
Ma e ials 2022,15, 1786 3 o 13
Ma e ials 2022, 15, x FOR PEER REVIEW 3 o 14
Figu e 2. Schema ic o hype he mia u ilizing nanopa icles: (a) pho o he mal ea men wi h lase
i adia ion o he cance si e, and (b) magne ic hype he mia he apy wi h magne ic ield exposu e
[21].
The nanopa icles (NPs) can be hea ed h ough high-in ensi y ocused ul asounds,
magne ic hype he mia, mic owa e/ adio equencies and plasmonic pho o- he mal he -
apy. The hea ing me hod is selec ed by conside ing he gi en condi ions. Since all hype -
he mia ea men echniques ha e d awbacks, he e is a need o de elop new cance
ea men app oaches ha a e mo e e ec i e and less ha m ul o he heal hy cells. When
a magne ic ield, in al e na ing mode, is applied o he NPs in magne ic hype he mia,
elec omagne ic ene gy is ans o med in o hea . The hea ing o NPs is possible due o
h ee mechanisms, namely, ic ional hea ing in an aniso opic magne ic pa icle, eddy
cu en s ha ha e high elec ical conduc i i y [24], and hys e esis [25]. The e ec s o eddy
cu en s a e gene ally negligible, due o he small size o NPs. The hea ing p ocess de-
pends upon he shape, na u e, size and he mal cha ac e is ics o he issue, and he e-
quency and magni ude o he magne ic ield.
In designing NPs o p ac ical use, he majo issue is he biocompa ibiliza ion and
unc ionaliza ion o he su ace coa ing ma e ials, along wi h he selec ion o a sui able
co e. Noble me als, long-chain o ganic ligands, ino ganic polyme s and o ganic polyme s
a e some examples o su ace coa ings. The impo ance o such coa ings is ha hese a e
used o ancho he unc ional g oups, o example, bioma ke s, pep ides and an ibodies.
These coa ings a e also used o p e en clus e ing o NPs, due o in e ac ions be ween
pa icles, which e en ually p o ides s abili y o he colloidal solu ions p epa ed wi h NPs.
Fu he mo e, hese coa ings enhance he biocompa ibili y o NPs o p e en ing he leak-
age o oxic ions om he magne ic co e in o he biological sys em [26,27].
Due o hei s ong magne ic cha ac e is ics, ou s anding biocompa ibili y and low
cos , magne ic i on oxide nanopa icles a e he mos ex ensi ely employed magne ic na-
noma e ials. I on oxide nanopa icles a e being esea ched o hei applica ions in a a-
ie y o ields, including biomedical, en i onmen al science, sensing, elec onic de ices
and ene gy s o age [28,29]. Se e al esea che g oups ha e epo ed he po en ial uses o
i on oxide nanopa icles in hype he mia. Table 1 p o ides a compa ison o pas s udies
on di e en nanos uc u ed ma e ials o he ea men o cance .
Figu e 2.
Schema ic o hype he mia u ilizing nanopa icles: (
a
) pho o he mal ea men wi h
lase i adia ion o he cance si e, and (
b
) magne ic hype he mia he apy wi h magne ic ield
exposu e [21].
The nanopa icles (NPs) can be hea ed h ough high-in ensi y ocused ul asounds,
magne ic hype he mia, mic owa e/ adio equencies and plasmonic pho o- he mal he -
apy. The hea ing me hod is selec ed by conside ing he gi en condi ions. Since all hy-
pe he mia ea men echniques ha e d awbacks, he e is a need o de elop new cance
ea men app oaches ha a e mo e e ec i e and less ha m ul o he heal hy cells. When
a magne ic ield, in al e na ing mode, is applied o he NPs in magne ic hype he mia,
elec omagne ic ene gy is ans o med in o hea . The hea ing o NPs is possible due o h ee
mechanisms, namely, ic ional hea ing in an aniso opic magne ic pa icle, eddy cu en s
ha ha e high elec ical conduc i i y [
24
], and hys e esis [
25
]. The e ec s o eddy cu en s
a e gene ally negligible, due o he small size o NPs. The hea ing p ocess depends upon
he shape, na u e, size and he mal cha ac e is ics o he issue, and he equency and
magni ude o he magne ic ield.
In designing NPs o p ac ical use, he majo issue is he biocompa ibiliza ion and
unc ionaliza ion o he su ace coa ing ma e ials, along wi h he selec ion o a sui able
co e. Noble me als, long-chain o ganic ligands, ino ganic polyme s and o ganic polyme s
a e some examples o su ace coa ings. The impo ance o such coa ings is ha hese a e
used o ancho he unc ional g oups, o example, bioma ke s, pep ides and an ibodies.
These coa ings a e also used o p e en clus e ing o NPs, due o in e ac ions be ween
pa icles, which e en ually p o ides s abili y o he colloidal solu ions p epa ed wi h NPs.
Fu he mo e, hese coa ings enhance he biocompa ibili y o NPs o p e en ing he leakage
o oxic ions om he magne ic co e in o he biological sys em [26,27].
Due o hei s ong magne ic cha ac e is ics, ou s anding biocompa ibili y and low
cos , magne ic i on oxide nanopa icles a e he mos ex ensi ely employed magne ic nano-
ma e ials. I on oxide nanopa icles a e being esea ched o hei applica ions in a a ie y o
ields, including biomedical, en i onmen al science, sensing, elec onic de ices and ene gy
s o age [
28
,
29
]. Se e al esea che g oups ha e epo ed he po en ial uses o i on oxide
nanopa icles in hype he mia. Table 1p o ides a compa ison o pas s udies on di e en
nanos uc u ed ma e ials o he ea men o cance .
Ma e ials 2022,15, 1786 4 o 13
Table 1.
A compa ison o he pas s udies u ilizing nanos uc u ed ma e ials o he ea men
o cance .
S udy Type Pa icle Type and Size Coa ing Resul s and D awbacks Re e ence
Magne ic
Hype he mia
(Expe imen al)
3–4 nm o Fe2O3Dex an
Pos - ea men umo de elopmen
is slowed.
Induces inad e en MF in il a ion.
Jo dan e al. (1997) [30]
Pho o- he mal
hype he mia
( heo e ical)
Silica nano shell o 20 nm
adius Au coa ed
Sui able o ob ain he mal egime.
The e is inconside a ion o he ole o
blood pe usion a e and
me abolic hea .
Domb o sky e al. (2011) [31]
Theo e ical model Fe3O4o 0.9 mm
The app op ia e dose o nanopa icles
o hype he mia.
The blood essels close o he umo
educe he empe a u e achie ed in he
issue, whe eas he e a e no la ge
blood essels a ound he umo .
Pa el M. e al. (2009) [32]
Theo e ical +
expe imen al Fe3O4o 10 nm
The highes achie able empe a u e
depends on su ace- o- olume a io.
I is possible o desc ibe
pa ien -speci ic models.
Hen ich F., Rahn H. and
Odenbachs. (2015)
[33]
Expe imen al
Pho o he mal
hype he mia
Lipos AuNPs o 5–8 nm Gold coa ed Hyb id NPs (biodeg adable)
o ea men .
Rengan e al. (2015)
[34]
Theo e ical Fe3O4o 18 nm
They ound ha s eady s able
empe a u e achie ed a e 200 s was
a he cen e o he umo .
The empe a u e cu e declines wi h
inc easing dis ance om he cen e a
di e en exposu e imes.
Wu, L., Cheng, J., Liu, W., &
Chen, X. (2015)
[35]
Sil e nanopa icles (AgNPs) s and ou among he me allic nanoma e ials because
o hei uses in heal h-ca e p oduc s, ex iles, consume p oduc s, medical de ices and
biosensing, due o hei unique physical and chemical cha ac e is ics. AgNPs show good
he mal, op ical and elec ical p ope ies and ac i i y agains ungus, bac e ia, and e en
i uses [
36
–
38
]. Recen ly, AgNPs ha e piqued he in e es o esea che s in nanomedicine,
since mul iple s udies ha e shown ha hese NPs can gene a e an i umo al e ec s in
in i o
and
in i o
umo models, po en ially bene i ing a a ie y o onco he apy modali ies
and diagnos ic ools [
25
–
28
]. In addi ion o an ibac e ial p ope ies, AgNPs ha e unique
cy o oxic e ec s agains mammalian cells, making sil e -based nanopa icles po en ially
use ul in umo ea men . The he apeu ic e icacy o AgNPs is based on hei dis inc
way o inducing cell dea h in mammalian cells. Despi e he physical and chemical ea u es,
such as size, shape and he e ogenei y o he capping ma e ial, hei mechanism o ac ion o
p omo e cance cell dea h is qui e de e minis ic [
39
]. AgNPs a e assembled in endosomes
a e being aken up by endocy osis- ela ed p ocesses, and he o ganelles a e subsequen ly
guided o unde go lysosomal usion. The lysosomal acidic en i onmen causes an inc ease
in he elease o sil e ions om AgNPs, which hen unbalances cellula homeos asis and
leads o apop o ic cell dea h, depending on he biological aspec o he a ge ed cell [40].
C ys alline and/o amo phous co e–shell s uc u es ha e dis inc p ope ies, along
wi h hei medical applica ions. The e is g adual deposi ion, acco ding o he S öbe
me hod, o small gold colloids on o he su ace o co es [
41
]. The gold nanopa icles hen
g ow in numbe and consolida e by o ming an isola ed island, o c ea e an impe ec ,
une en coa ing, which e en ually o ms a con inuous comple e shell ha su ounds he
co e. Wu e al. [
35
] conduc ed a heo e ical s udy on magne i e nanopa icles o analyze he
empe a u e dis ibu ion in he umo . They concluded ha he cons an empe a u e was
ob ained a e 200 s, which is a e y long ime. The mo i a ion behind ou wo k was o
achie e he desi ed empe a u e in a sho e ime pe iod, by using AgNPs. A compa a i e
s udy on he hea gene a ion, using a single sil e nanopa icle, nanosphe e, nano od and
Ma e ials 2022,15, 1786 5 o 13
nano-ellipsoid, was conduc ed in he epo ed wo k. The olume o hese shapes is se as
he ollowing:
Vsphe e ∼
=V od ∼
=Vellipsoid i
.
e
.,
Vsphe e =4
3π 3=
33510.32
nm3
,
Vellipsoid =
4
3πabc =
33401.41
nm3
,
V od =4
3π 2( +h)=
33324.96
nm3
. The he mal e ec o hea
p opaga ion in he umo cell, as well as he spa ial– empo al dis ibu ions o empe a u e
du ing he ea men , a e desc ibed in his model. Fu he mo e, he e ec o di e en
coa ing ma e ials, such as gold (Au) and polyme (PEG), is analyzed. The hickness o
he shell is disco e ed o be an essen ial componen ha in luences he he mal esponse
h ough he he mal cha ac e is ics o he ma e ials u ilized in he ea men sys em. Finally,
a ying quan i ies o NPs a e a ached o he co e su ace o simula e an incomple e coa ing
su ace. The hea ans e module in COMSOL Mul iphysics is used o simula e he hea ing
p ocess o he nanos uc u es, using ini e elemen simula ions [42].
2. Me hods
A 0.5
µm
sphe ical domain o issue enclosed he nanopa icle. COMSOL Mul iphysics
was used o iden i y he spa ial and empo al dis ibu ion o empe a u e in his domain.
The majo goal o his esea ch was o examine he he mal esponse o li ing issue when
he hea ing sou ce (sil e nanos uc u e) was o a ious o ms, i.e., nanosphe e, nano od,
nano-ellipsoid and complex co e–shell s uc u e.
Due o di e ences in he s uc u e and unc ion o issues, balancing he mal ene gy in
di e en biological issues is a di icul p ocess. Ene gy balancing is a ec ed by he ela i e
ele ance o a hea ans e echnique, he ele an ime scale o he deposi ed ene gy, and
changes in he bounda y and ini ial condi ions [
43
]. In o de o simula e he undamen al
ea u es o he he mal s a e o he o ganism (o i s componen s), as well as he impac s o
he bounda y and ini ial condi ions, simpli ying assump ions a e equen ly equi ed. The
applica ion o he ene gy conse a ion law o a con ol olume is ypically he i s s ep in
he design o such models.
Qgain =Qs o age +Qloss +W(1)
whe e
Qgain
is he hea gained by he issue,
Qs o age
is he hea s o ed in he issue,
Qloss
is
hea loss by conduc ion and
W
is wo k conduc ed by he issue. Hea loss by conduc ion o
hea exchange wi h lowing luids, as well as he wo k conduc ed by he issue, balance ou
he hea s o age. Hea gain and hea ene gy s o age due o hea gene a ed by uni issue
segmen q( , ) can be ep esen ed as an in eg al o e he con ol olume, as ollows:
Qgain =Zq( , )dV (2)
Qs o age =Zρc∂
∂ T( , )dV (3)
When no ine ia is p esen , he hea conduc ion p ocess is o en ep esen ed using he
Fou ie law o bio-hea ans e p oblems.
Qconduc ion =−k
∆L
A (T1−T2),T1>T2(4)
whe e
k
is he he mal conduc i i y o issue, wi h dimensions [W/m*K]. I is based on
he biological ma e ials’ mic oscopic s uc u es. The di e en ial o m o he conduc ion
componen o hea lux is as ollows:
qcond.=−k ∂
∂xT (x, )(5)
The con ec ion e m is exp essed as ollows:
qcon .=hTw−T luid(6)

Ma e ials 2022,15, 1786 6 o 13
whe e his known as he coe icien o hea ans e , and i depends upon luid eloci y. The
hea ans e due o luid low should con ibu e o blood pe usion and low dis ibu ion
in he biological issue, due o he di e si y o li ing s uc u es. The low is p opo ional o
he di e ence in a e y (Ta o Ta) and enous (T en) blood empe a u e.
qb=ωbρbcb(Ta −T en)(7)
I is possible ha blood lows ex emely slowly in capilla y beds o achie e pe ec he -
mal equilib ium wi h he issues. The e o e, he hea lux ela ed o pe usion
(Equa ion (7))
can be app oxima ed as ollows:
qb=ωbρbcb(Ta −T )(8)
Unde hese condi ions, he ans e o ene gy by he s eam o blood h oughou he
en i e olume is as ollows:
Qb=Zωbρbcb[Ta ( , )−T ( , )]dV (9)
E en hough i is commonly obse ed in model o mula ions, he esul o Equa ion (9)
is no uncondi ional; in la ge a e ies, i would be in alid, owing o in ense blood mixing.
By analyzing he a ious hea ansmission modes, i is possible o c ea e a he mal ene gy
balance by igno ing he wo k conduc ed by he issue ac oss an a bi a y olume elemen ,
as ollows:
Zρ c ∂
∂ T ( , )dV =Z−k ∇T ( , )dV +Zωbρbcb[Ta ( , )−T ( , )]dV +ZQm( , )dV (10)
We ob ain, in a one-dimensional scena io wi h a homogenously dis ibu ed sou ce o
me abolic hea , Qm.
ρ c ∂
∂ T (x, )=k ∂2
∂x2T (x, )+ωbρbcb[Ta ( , )−T ( , )] +Qm(11)
This is he mos common o m o he hea ans e equa ion o li ing o ganisms,
also known as he bio-hea ans e equa ion. Th ough scaling and dimensional analysis,
we seek o examine i s componen s and assump ions, which may al e depending on he
condi ions imposed by he modelled i em, as well as he con ibu ion o he anspo
mechanism in ol ed. The empe a u e dis ibu ion in he umo cell can be modeled using
he Fou ie hea equa ion, as ollows:
ρCp∂T
∂ +ρCpu.∇T+∇.q=Q+Qbio (12)
whe e,
q=−k∇T(13)
Qbio =ρbCp,bwb(Tb−T)+Qme (14)
which esul s in he ollowing:
δ zρCp∂T
∂ +∇.(−k∇T)=ρbCbωb(Tb−T)+Qme +Qex (15)
whe e
Cp
is issue-speci ic hea capaci y a cons an p essu e,
ρ
is issue densi y,
k
is cell
he mal conduc i i y,
ρb
is blood densi y, which is 1000
kg/m3
,
Cb
is speci ic hea o blood,
which is 4180
J/(kg ∗K)
,
ωb
is blood pe usion a e o alue 0.0064 1
/s
,
Tb
is a e ial blood
empe a u e, which is app oxima ely equal o co e body empe a u e, i.e., 37
°C
[
44
], Tis
he local empe a u e, and
Q=
10
16 W/m3
is hea dissipa ed by he nanopa icles in he
olume o he cell [
45
],
Qme =
5790
W/m3
o cance ous cells [
46
–
50
] and
Qex
is he hea
Ma e ials 2022,15, 1786 7 o 13
gene a ed by loss powe s. Ini ially, he empe a u e o he issue was aken as no mal o
he human body (Ti=37 °C). Ti( , 0)=T0i,∂Ti( ,0)
∂ =0, while i=1, 2.
The geome y, which consis ed o he issue and nanopa icle, was disc e ized on all
domains by ee e ahed al elemen s, as shown in Figu e 3. The e we e 60,246 meshing
domain elemen s a e meshing he geome y, while he elemen al size was chosen as ine .
The ollowing bounda y condi ions we e applied o comple e he p ocedu e:
1. The umo cell ecei es he hea lux om he pa icle in i s en i e y, i.e., con inui y.
2. The empe a u e o he ou e su ace o he issue is main ained a T=T0=37 °C.
Ma e ials 2022, 15, x FOR PEER REVIEW 7 o 14
whe e 𝐶 is issue-speci ic hea capaci y a cons an p essu e, 𝜌 is issue densi y, 𝑘 is
cell he mal conduc i i y, 𝜌 is blood densi y, which is 1000 kg/m, 𝐶 is speci ic hea
o blood, which is 4180 𝐽/(kg ∗ 𝐾), 𝜔 is blood pe usion a e o alue 0.0064 1/𝑠, 𝑇 is
a e ial blood empe a u e, which is app oxima ely equal o co e body empe a u e, i.e.,
37 ℃ [44], T is he local empe a u e, and 𝑄 =10 (W/m) is hea dissipa ed by he na-
nopa icles in he olume o he cell [45], 𝑄 =5790 W/m o cance ous cells [46–50]
and 𝑄 is he hea gene a ed by loss powe s. Ini ially, he empe a u e o he issue was
aken as no mal o he human body (𝑇=37 ℃). 𝑇(𝑟,0)=𝑇,(,)
 =0,while 𝑖=1,2.
The geome y, which consis ed o he issue and nanopa icle, was disc e ized on all
domains by ee e ahed al elemen s, as shown in Figu e 3. The e we e 60,246 meshing
domain elemen s a e meshing he geome y, while he elemen al size was chosen as
ine . The ollowing bounda y condi ions we e applied o comple e he p ocedu e:
1. The umo cell ecei es he hea lux om he pa icle in i s en i e y, i.e., con inui y.
2. The empe a u e o he ou e su ace o he issue is main ained a 𝑇=𝑇=37 ℃.
Figu e 3. The disc e ized geome y o issue and nanopa icle.
Sil e nanos uc u es o di e en shapes (sphe e, od and ellipsoid) we e analyzed,
as shown in Figu e 4. Fi s ly, he sphe ical o m o sil e , wi h a adius o 20 nm, was
analyzed. The olume o he pa icle was kep he same as he o he shapes by compu ing
he dimensions o he od and ellipsoid. Fo he nano od, he leng h o he cylinde was
aken as 𝐿 =73 nm, adius o he hemisphe ical caps 𝑅 =𝑅 =11 nm, and he di-
mensions o he ellipsoid we e aken as 12 −15 − 44.3 nm. To s udy he coa ing e ec ,
he hickness o he Au and PEG polyme coa ing was se as 5, 10, 20, 30 and 40 nm. The
co e–shell s uc u e, wi h a adius o 20 nm, was chosen o he pu pose o analyzing he
he mal e olu ion o he p oposed s uc u e. Th ee di e en coa ing su aces (one sphe -
ical and wo ellipsoidal) we e also simula ed on he sil e nanopa icle co e o compa e
he he mal esponses o he ellipsoidal and sphe ical su ace coa ings. A sil e nanopa -
icle, which p o ided a co e wi h a adius o 20 nm, was coa ed wi h gold. The simula ion
g ow h o gold nanopa icles on he co e o AgNP can be obse ed in Figu e 5. The he -
mal p ope ies o a ious ma e ials used in his wo k a e gi en in Table 2.
Figu e 3. The disc e ized geome y o issue and nanopa icle.
Sil e nanos uc u es o di e en shapes (sphe e, od and ellipsoid) we e analyzed,
as shown in Figu e 4. Fi s ly, he sphe ical o m o sil e , wi h a adius o 20 nm, was
analyzed. The olume o he pa icle was kep he same as he o he shapes by compu ing
he dimensions o he od and ellipsoid. Fo he nano od, he leng h o he cylinde was
aken as
Lcyl =
73
nm
, adius o he hemisphe ical caps
Rcyl =Rcap =
11
nm
, and he
dimensions o he ellipsoid we e aken as 12
−
15
−
44.3
nm
. To s udy he coa ing e ec ,
he hickness o he Au and PEG polyme coa ing was se as 5, 10, 20, 30 and 40 nm. The
co e–shell s uc u e, wi h a adius o 20 nm, was chosen o he pu pose o analyzing he
he mal e olu ion o he p oposed s uc u e. Th ee di e en coa ing su aces (one sphe ical
and wo ellipsoidal) we e also simula ed on he sil e nanopa icle co e o compa e he
he mal esponses o he ellipsoidal and sphe ical su ace coa ings. A sil e nanopa icle,
which p o ided a co e wi h a adius o 20 nm, was coa ed wi h gold. The simula ion
g ow h o gold nanopa icles on he co e o AgNP can be obse ed in Figu e 5. The he mal
p ope ies o a ious ma e ials used in his wo k a e gi en in Table 2.
Ma e ials 2022, 15, x FOR PEER REVIEW 8 o 14
Figu e 4. Di e en shapes used in simula ions: (a) nano-ellipsoid, (b) nano od and (c) nano-
sphe e.
Figu e 5. The co e–shell s uc u e simula ed in COMSOL Mul iphysics: (a) naked co e wi h adius
o 20 nm, wi h AgNPs wi h 4 nm adius a ached o he co e su ace, (b) co e (0 NP) and (c–e) 10, 40
and 70 nanopa icles a ached o co e, ( ) comple e shell (100 NPs).
Table 2. The mal esponse o sou ce ma e ials used in his wo k.
The mal
Conduc i i y
[W/m*k] Mass Densi y [kg/m3]
Speci ic Hea Ca-
paci y [J/kg*K]
Tissue [46] 0.512 1000 3800
Tumo [47] 71 21,500 132
Gold [48] 317 19,300 129
Sil e [49] 429 10,500 235
Polyme [48] 0.2 1000 1000
3. Resul s and Discussion
To in es iga e he po en ial use o sil e nanos uc u es, wi h di e en shapes, in hy-
pe he mia, a nano od, nano-ellipsoid and nanosphe e we e placed in a umo wi h a
sphe ical shape and a 500 nm adius. Thei hea ing e ec was simula ed o kill he umo
cells. The i s simula ion showed he maximum empe a u e o all geome ies in he is-
sue, when hea ed using an ex e nal sou ce. The hea ing o he nanopa icles caused he
empe a u e and he mal equilib ium o he umo al cell o change o e ime, as obse ed
in Figu e 6. The empe a u es a ained by using a ious simula ed o ms di e ed no ably.
The a ia ion in empe a u es can be explained by he di e ence in he p opo iona e su -
ace o each o m, gi en ha he he mal esponse is de e mined om he hea gene a ion
mul iplied by he olume and su ace (𝑄=10 (W/m). The maximum empe a u e, a -
ained using he nanosphe e, was 43 ℃. Simila ly, he nano-ellipsoid and nano od e-
sul ed in maximum a ainable empe a u es o 43.1 ℃ and 44.3 ℃, espec i ely. These
empe a u e alues we e aken om he cen e o he pa icles, which hen p opaga ed
in o he su ounding medium, as shown in Figu e 7. The he mal ield dis ibu ion o an
ellipse is la ge han ha o a sphe e o a od.
Figu e 4. Di e en shapes used in simula ions: (a) nano-ellipsoid, (b) nano od and (c) nano-sphe e.
Ma e ials 2022,15, 1786 8 o 13
Ma e ials 2022, 15, x FOR PEER REVIEW 8 o 14
Figu e 4. Di e en shapes used in simula ions: (a) nano-ellipsoid, (b) nano od and (c) nano-
sphe e.
Figu e 5. The co e–shell s uc u e simula ed in COMSOL Mul iphysics: (a) naked co e wi h adius
o 20 nm, wi h AgNPs wi h 4 nm adius a ached o he co e su ace, (b) co e (0 NP) and (c–e) 10, 40
and 70 nanopa icles a ached o co e, ( ) comple e shell (100 NPs).
Table 2. The mal esponse o sou ce ma e ials used in his wo k.
The mal
Conduc i i y
[W/m*k] Mass Densi y [kg/m3]
Speci ic Hea Ca-
paci y [J/kg*K]
Tissue [46] 0.512 1000 3800
Tumo [47] 71 21,500 132
Gold [48] 317 19,300 129
Sil e [49] 429 10,500 235
Polyme [48] 0.2 1000 1000
3. Resul s and Discussion
To in es iga e he po en ial use o sil e nanos uc u es, wi h di e en shapes, in hy-
pe he mia, a nano od, nano-ellipsoid and nanosphe e we e placed in a umo wi h a
sphe ical shape and a 500 nm adius. Thei hea ing e ec was simula ed o kill he umo
cells. The i s simula ion showed he maximum empe a u e o all geome ies in he is-
sue, when hea ed using an ex e nal sou ce. The hea ing o he nanopa icles caused he
empe a u e and he mal equilib ium o he umo al cell o change o e ime, as obse ed
in Figu e 6. The empe a u es a ained by using a ious simula ed o ms di e ed no ably.
The a ia ion in empe a u es can be explained by he di e ence in he p opo iona e su -
ace o each o m, gi en ha he he mal esponse is de e mined om he hea gene a ion
mul iplied by he olume and su ace (𝑄=10 (W/m). The maximum empe a u e, a -
ained using he nanosphe e, was 43 ℃. Simila ly, he nano-ellipsoid and nano od e-
sul ed in maximum a ainable empe a u es o 43.1 ℃ and 44.3 ℃, espec i ely. These
empe a u e alues we e aken om he cen e o he pa icles, which hen p opaga ed
in o he su ounding medium, as shown in Figu e 7. The he mal ield dis ibu ion o an
ellipse is la ge han ha o a sphe e o a od.
Figu e 5.
The co e–shell s uc u e simula ed in COMSOL Mul iphysics: (
a
) naked co e wi h adius o
20 nm, wi h AgNPs wi h 4 nm adius a ached o he co e su ace, (
b
) co e (0 NP) and (
c
–
e
) 10, 40 and
70 nanopa icles a ached o co e, ( ) comple e shell (100 NPs).
Table 2. The mal esponse o sou ce ma e ials used in his wo k.
The mal Conduc i i y
[W/m*k]
Mass Densi y
[kg/m3]
Speci ic Hea
Capaci y [J/kg*K]
Tissue [46] 0.512 1000 3800
Tumo [47] 71 21,500 132
Gold [48] 317 19,300 129
Sil e [49] 429 10,500 235
Polyme [48] 0.2 1000 1000
3. Resul s and Discussion
To in es iga e he po en ial use o sil e nanos uc u es, wi h di e en shapes, in
hype he mia, a nano od, nano-ellipsoid and nanosphe e we e placed in a umo wi h a
sphe ical shape and a 500 nm adius. Thei hea ing e ec was simula ed o kill he umo
cells. The i s simula ion showed he maximum empe a u e o all geome ies in he
issue, when hea ed using an ex e nal sou ce. The hea ing o he nanopa icles caused he
empe a u e and he mal equilib ium o he umo al cell o change o e ime, as obse ed in
Figu e 6. The empe a u es a ained by using a ious simula ed o ms di e ed no ably. The
a ia ion in empe a u es can be explained by he di e ence in he p opo iona e su ace
o each o m, gi en ha he he mal esponse is de e mined om he hea gene a ion
mul iplied by he olume and su ace (
Q=
10
8W/m2
. The maximum empe a u e,
a ained using he nanosphe e, was 43
°C
. Simila ly, he nano-ellipsoid and nano od
esul ed in maximum a ainable empe a u es o 43.1
°C
and 44.3
°C
, espec i ely. These
empe a u e alues we e aken om he cen e o he pa icles, which hen p opaga ed in o
he su ounding medium, as shown in Figu e 7. The he mal ield dis ibu ion o an ellipse
is la ge han ha o a sphe e o a od.
Ma e ials 2022, 15, x FOR PEER REVIEW 9 o 14
Figu e 6. 2D spa ial empe a u e dis ibu ions in umo al cell using (a) nano-ellipsoid, (b) nano od
and (c) nanosphe e. The empe a u e dis ibu ions we e ob ained a e 0.5 µm o hea ing.
A unc ional su ace is p o ided by he NPs coa ed wi h a ious ma e ials in hype -
he mia applica ions. The in luence o he coa ing su ace on he mal dissipa ion in he
su ounding medium was in es iga ed using simula ions. The sil e magne ic co e was
used as a hea sou ce, wi h a polyme - o gold-like shell. The he mal conduc i i y coe i-
cien s o he wo simula ed shell ma e ials esul ed in opposing hea dissipa ion e ec s
in ela ion o he shell hickness. The he mal conduc i i y o gold dec eased, while i
inc eased o he polyme .
Figu e 7. The ime-dependen and adial– empo al dis ibu ion o h ee di e en shapes: (a) em-
pe a u e e olu ion a he cen e o umo (x = 0) and (b) he adial dis ibu ion a e 3 µs o hea ing
p ocess.
The hype he mia p ocess can be ela ed o he unc ionali y o co e–shell s uc u es.
To achie e he maximum empe a u e, a sphe e, wi h a adius o 20 nm, was chosen o
he assigned nanopa icles. In his simula ion, he co e o AgNP was co e ed by he shell
o gold o PEG polyme , and hen he empe a u e induced by he co e was analyzed.
Di e en alues o shell hickness we e conside ed, and he e ec o he empe a u e in-
duced by he co e o he nanopa icle is gi en in Figu e 8. The empe a u e alls wi h an
inc ease in he hickness o he Au shell, and ises wi h an inc ease in he hickness o he
PEG polyme shell. As he he mal conduc i i y o gold is high, i apidly ans e s he
hea o i s su oundings. While he polyme , which has low he mal conduc i i y, p e-
se es mo e hea inside he pa icle, which esul s in a ise in empe a u e. The e ec and
in luence o he hea di ec ly depend on he conduc i i y. These indings clea ly show he
possibili y o con olling he empe a u e by changing he hickness o he shell. In
Figu e 6.
2D spa ial empe a u e dis ibu ions in umo al cell using (
a
) nano-ellipsoid, (
b
) nano od
and (c) nanosphe e. The empe a u e dis ibu ions we e ob ained a e 0.5 µm o hea ing.
Ma e ials 2022,15, 1786 9 o 13
A unc ional su ace is p o ided by he NPs coa ed wi h a ious ma e ials in hy-
pe he mia applica ions. The in luence o he coa ing su ace on he mal dissipa ion in
he su ounding medium was in es iga ed using simula ions. The sil e magne ic co e
was used as a hea sou ce, wi h a polyme - o gold-like shell. The he mal conduc i i y
coe icien s o he wo simula ed shell ma e ials esul ed in opposing hea dissipa ion
e ec s in ela ion o he shell hickness. The he mal conduc i i y o gold dec eased, while
i inc eased o he polyme .
Ma e ials 2022, 15, x FOR PEER REVIEW 9 o 14
Figu e 6. 2D spa ial empe a u e dis ibu ions in umo al cell using (a) nano-ellipsoid, (b) nano od
and (c) nanosphe e. The empe a u e dis ibu ions we e ob ained a e 0.5 µm o hea ing.
A unc ional su ace is p o ided by he NPs coa ed wi h a ious ma e ials in hype -
he mia applica ions. The in luence o he coa ing su ace on he mal dissipa ion in he
su ounding medium was in es iga ed using simula ions. The sil e magne ic co e was
used as a hea sou ce, wi h a polyme - o gold-like shell. The he mal conduc i i y coe i-
cien s o he wo simula ed shell ma e ials esul ed in opposing hea dissipa ion e ec s
in ela ion o he shell hickness. The he mal conduc i i y o gold dec eased, while i
inc eased o he polyme .
Figu e 7. The ime-dependen and adial– empo al dis ibu ion o h ee di e en shapes: (a) em-
pe a u e e olu ion a he cen e o umo (x = 0) and (b) he adial dis ibu ion a e 3 µs o hea ing
p ocess.
The hype he mia p ocess can be ela ed o he unc ionali y o co e–shell s uc u es.
To achie e he maximum empe a u e, a sphe e, wi h a adius o 20 nm, was chosen o
he assigned nanopa icles. In his simula ion, he co e o AgNP was co e ed by he shell
o gold o PEG polyme , and hen he empe a u e induced by he co e was analyzed.
Di e en alues o shell hickness we e conside ed, and he e ec o he empe a u e in-
duced by he co e o he nanopa icle is gi en in Figu e 8. The empe a u e alls wi h an
inc ease in he hickness o he Au shell, and ises wi h an inc ease in he hickness o he
PEG polyme shell. As he he mal conduc i i y o gold is high, i apidly ans e s he
hea o i s su oundings. While he polyme , which has low he mal conduc i i y, p e-
se es mo e hea inside he pa icle, which esul s in a ise in empe a u e. The e ec and
in luence o he hea di ec ly depend on he conduc i i y. These indings clea ly show he
possibili y o con olling he empe a u e by changing he hickness o he shell. In
Figu e 7.
The ime-dependen and adial– empo al dis ibu ion o h ee di e en shapes: (
a
) empe a-
u e e olu ion a he cen e o umo (x = 0) and (
b
) he adial dis ibu ion a e 3
µ
s o hea ing p ocess.
The hype he mia p ocess can be ela ed o he unc ionali y o co e–shell s uc u es.
To achie e he maximum empe a u e, a sphe e, wi h a adius o 20 nm, was chosen o
he assigned nanopa icles. In his simula ion, he co e o AgNP was co e ed by he shell
o gold o PEG polyme , and hen he empe a u e induced by he co e was analyzed.
Di e en alues o shell hickness we e conside ed, and he e ec o he empe a u e
induced by he co e o he nanopa icle is gi en in Figu e 8. The empe a u e alls wi h an
inc ease in he hickness o he Au shell, and ises wi h an inc ease in he hickness o he
PEG polyme shell. As he he mal conduc i i y o gold is high, i apidly ans e s he hea
o i s su oundings. While he polyme , which has low he mal conduc i i y, p ese es
mo e hea inside he pa icle, which esul s in a ise in empe a u e. The e ec and in luence
o he hea di ec ly depend on he conduc i i y. These indings clea ly show he possibili y
o con olling he empe a u e by changing he hickness o he shell. In addi ion, he
ma e ial and shell hickness de e mine he equi ed empe a u e o hype he mia, which
is dependen on he speci ic loca ion o use in he human body.
Ma e ials 2022, 15, x FOR PEER REVIEW 10 o 14
addi ion, he ma e ial and shell hickness de e mine he equi ed empe a u e o hype -
he mia, which is dependen on he speci ic loca ion o use in he human body.
Figu e 8. (a) The co e–shell s uc u e, (b) maximum empe a u e ob ained by coa ing o gold and
PEG polyme shell o hickness 5, 10, 20, 30 and 40 nm.
The shape o he coa ing su ace plays an impo an ole in he unc ionali y o
nanos uc u es. To s udy he e ec o he coa ing su ace, wo o ms o coa ing su aces,
namely, sphe ical (30 nm adius) and wo ellipsoids ( 𝑒𝑙𝑙𝑖𝑝𝑠𝑜𝑖𝑑:25−25−
43.2 nm,𝑒𝑙𝑙𝑖𝑝𝑠𝑜𝑖𝑑:22− 25 − 49 nm), we e conside ed. F om Figu e 9, i is clea ha he e
is no majo di e ence in he empe a u e o he coa ing su aces, which shows ha aniso -
opy o coa ing su aces is no impo an o he hype he mia p ocess, since gold nano-
pa icles g ow and o m an isola ed island, o c ea e an incomple e i egula coa ing,
which is hen ans o med in o a comple e shell o co e he co e. Fo his pu pose, small
AuNPs, wi h a 4 nm adius, we e a ached o he co e su ace o AgNPs, wi h a adius o
20 nm, which was hen embedded in o issue wi h a 0.5 µm adius. The olume co e age
a io o AuNPs, compa ed o he olume o he ull shell, was de e mined o desc ibe he
empe a u e p o ile o incomple ely co e ed nanopa icles wi h a ying amoun s o na-
nopa icles a ached, as shown in Figu e 10. I is obse ed ha he maximum empe a u e
a he cen e o he nanopa icle was 42.3 ℃ o he naked co e, while he e was a g adual
dec ease in empe a u e wi h an inc ease in he amoun o su ace coa ing. The minimum
empe a u e a he cen e o he nanopa icle was 39.9 ℃ when a comple e shell was
o med.
Figu e 9. Radial empe a u e dis ibu ions o di e en coa ed shapes.
Figu e 8.
(
a
) The co e–shell s uc u e, (
b
) maximum empe a u e ob ained by coa ing o gold and
PEG polyme shell o hickness 5, 10, 20, 30 and 40 nm.
The shape o he coa ing su ace plays an impo an ole in he unc ionali y o nanos-
uc u es. To s udy he e ec o he coa ing su ace, wo o ms o coa ing su aces, namely,
sphe ical (30 nm adius) and wo ellipsoids (
ellipsoid1
: 25
−
25
−
43.2
nm
,
ellipsoid2
: