ma hema ics
A icle
Analysis o he Pa ame ic Co ela ion in Ma hema ical
Modeling o In Vi o Glioblas oma E olu ion Using Copulas
Jacobo Ayensa-Jiménez 1,2 , Ma ina Pé ez-Aliaca 1,2 , Teodo a Randelo ic 2,3 , José An onio Sanz-He e a 4,
Mohamed H. Doweida 1,2,5 and Manuel Dobla é 1,2,3,5,*


Ci a ion: Ayensa-Jiménez, J.; Pé ez-
Aliaca , M.; Randelo ic, T.; Sanz-
He e a, J.A.; Doweida , M.H.; Dobla é,
M. Analysis o he Pa ame ic
Co ela ion in Ma hema ical Modeling
o In Vi o Glioblas oma E olu ion
Using Copulas. Ma hema ics 2021,9,
27. h ps://dx.doi.o g/10.3390/ma h
9010027
Recei ed: 24 No embe 2020
Accep ed: 21 Decembe 2020
Published: 24 Decembe 2020
Publishe ’s No e: MDPI s ays neu-
al wi h ega d o ju isdic ional claims
in published maps and ins i u ional
a ilia ions.
Copy igh : © 2020 by he au ho s. Li-
censee MDPI, Basel, Swi ze land. This
a icle is an open access a icle dis ibu ed
unde he e ms and condi ions o he
C ea i e Commons A ibu ion (CC BY)
license (h ps://c ea i ecommons.o g/
licenses/by/4.0/).
1
Mechanical Enginee ing Depa men , School o Enginee ing and A chi ec u e (EINA), Uni e si y o Za agoza,
50018 Za agoza, Spain; jacoboaj@uniza .es (J.A.-J.); 722195@uniza .es (M.P.-A.); mohamed@uniza .es (M.H.D.)
2A agon Ins i u e o Enginee ing Resea ch (I3A), Uni e si y o Za agoza, 50018 Za agoza, Spain;
753388@uniza .es
3A agón Ins i u e o Heal h Resea ch (IIS A agón), 50009 Za agoza, Spain
4Depa men o Mechanics o Con inuous Media and Theo y o S uc u es, School o Enginee ing,
Uni e si y o Se ille, 41092 Se illa, Spain; [email p o ec ed]
5Cen o de In es igación Biomédica en Red en Bioingenie ía, Bioma e iales y Nanomedicina (CIBER-BBN),
50018 Za agoza, Spain
*Co espondence: mdobla e@uniza .es
Abs ac :
Modeling and simula ion a e essen ial ools o be e unde s anding complex biological
p ocesses, such as cance e olu ion. Howe e , he esul ing ma hema ical models a e o en highly
non-linea and include many pa ame e s, which, in many cases, a e di icul o es ima e and p esen
s ong co ela ions. The e o e, a p ope pa ame ic analysis is manda o y. Following a p e ious
wo k in which we modeled he in i o e olu ion o Glioblas oma Mul i o me (GBM) unde hypoxic
condi ions, we analyze and sol e he e he p oblem ound o pa ame ic co ela ion. Wi h his aim,
we de elop a me hodology based on copulas o app oxima e he mul idimensional p obabili y densi y
unc ion o he co ela ed pa ame e s. Once he model is de ined, we analyze he expe imen al
se ing o op imize he u ili y o each con igu a ion in e ms o ga he ed in o ma ion. We p o e ha
expe imen al con igu a ions wi h oxygen g adien and high cell concen a ion ha e he highes u ili y
when we wan o sepa a e co ela ed e ec s in ou expe imen al design. We demons a e ha copulas
a e an adequa e ool o analyze highly-co ela ed mul ipa ame ic ma hema ical models such as
hose appea ing in Biology, wi h he added alue o p o iding key in o ma ion o he op imal design
o expe imen s, educing ime and cos in in i o and in i o expe imen al campaigns, like hose
equi ed in mic o luidic models o GBM e olu ion.
Keywo ds: copulas; design o expe imen s; glioblas oma mul i o me; ma hema ical modelling
MSC: 62H20; 62K05; 62P10
1. In oduc ion
Biological p ocesses usually in ol e se e al cell popula ions in e ac ing in a complex,
dynamic, and mul iple in e ac i e mic o-en i onmen [
1
]. Unde s anding hese in e ac ions
be ween cells and mic oen i onmen is c ucial in many physiological and pa hological
p ocesses [
2
]. Howe e , p og essing in his unde s anding wi h only in i o expe imen s
is di icul . Despi e hem being mo e ealis ic, isola ing e ec s o achie ing pa icula
condi ions is complex in such expe imen s due o echnical and/o e hical easons.
In i o expe imen s pe mi be e con ol o he a iables, while educing cos ly and
e hically-ques ioned animal assays. None heless, he p edic i e powe o cu en ly a ail-
able in i o models is s ill poo due o he s ong di icul ies ha we ace in ep oducing
he s uc u e and dis ibu ion o he di e en cell popula ions as well as he pa icula
en i onmen al condi ions in which cells li e, adap and eac (e.g., h ee-dimensionali y) [
3
].
Mic o luidics is a new in i o echnique ha allows mo e p ecise ep oduc ions o he
Ma hema ics 2021,9, 27. h ps://dx.doi.o g/10.3390/ma h9010027 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2021,9, 27 2 o 22
mic oen i onmen and cell dis ibu ion [
4
,
5
], including h ee-dimensionali y, hus making
in i o es s much close o he ac ual in i o condi ions. This pe mi s, o example, a mo e
eliable and e icien d ug es ing [6,7].
Finally, ma hema ical models allow o sepa a e and quan i y he e ec s o each mech-
anism o pa ame e , as well as o p edic he ou come in “wha i ” si ua ions, which a e
some imes impossible o achie e in in i o o in i o expe imen s [
8
,
9
]. Ne e heless, hese
models a e mos ly non-linea , in ol e highly-coupled mul iphysic in e ac ions, and in-
clude many pa ame e s. In many occasions, hose pa ame e s a e di icul o measu e
and ha e s ong hidden co ela ions. Mo eo e , i is usual o ha e a lack o da a bo h
o quan i ica ion and alida ion o he pa ame e s and esul s [
10
].
The e o e, hey a e
i ed only o he esul s a ailable, which usually co espond o e y speci ic condi ions.
This may lead o i ial conclusions ha could ha e been di ec ly de i ed om he model
assump ions, making he esul s only use ul o hose pa icula expe imen s, wi h he
ob ained conclusions impossible o gene alize.
In a p e ious pape [
11
], we add essed his pa ame ic analysis in a pa icula
p oblem— he ma hema ical modeling o he in i o (using mic o luidic de ices) e o-
lu ion o glioblas oma mul i o me (GBM), he mos agg essi e and le hal among p ima y
glioma umo s [
12
]. In Re . [
11
], we p esen ed a gene al amewo k in which he main cell
p ocesses in ol ed (p oli e a ion, chemo axis, andom mig a ion, apop osis, and nec osis),
in esponse o changes in he oxygen concen a ion, we e ma hema ically o mula ed.
We hen analyzed h ee di e en expe imen al con igu a ions, ep oducing he main GBM
mig a o y s uc u es (pseudopalisade and nec o ic co e o ma ion). An ex ensi e analysis
o all model pa ame e s was pe o med, bo h om li e a u e and by i ing he associa ed
in silico esul s wi h hose de i ed om he expe imen s. As main esul s o ha wo k,
we iden i ied a unique se o pa ame e s able o accu a ely ep oduce he quan i a i e
esul s o he h ee case-s udies. Howe e , we also ound wo model limi a ions: (i) he
sensi i i y analysis showed ha he model is s ongly a ec ed by small a ia ions in he
oxygen cell consump ion and di usion and (ii) a s ong co ela ion was ound be ween he
pa ame e s associa ed wi h hose wo mechanisms.
The objec i e o he p esen wo k is o p esen he possibili ies in his con ex o -
e ed by a me hodology ha is able o sepa a e he co ela ed e ec s ound in ha s udy,
and o ge a mo e accu a e and eliable ep esen a ion o he expe imen al esul s in he
pa ame ic space. Wi h ha pu pose, we app oxima e he mul idimensional p obabili y
densi y unc ion o he pa ame e s by means o app op ia e copulas. Copulas allow con-
side ing sepa a ely he ma ginal dis ibu ions and he dependence be ween a iables in
mul i a ia e s a is ical p oblems, including hose wi h high co ela ion. This pe mi s using
gene al models o he ma ginal dis ibu ions, while he a iable dependence model can
be di e en [
13
]. Copulas a e oday used in a wide ange o a eas in Economic sciences
and Enginee ing. The mos ecen models ha e been success ully applied in po olio
managemen and op imiza ion [
14
], ac ua ial analysis [
15
], quan i a i e inance and isk
heo y [
16
,
17
]. A pa icula ly ho opic is he s udy o clima e-agen ime se ies [
18
,
19
], hy-
d ology [
20
,
21
] and wea he and clima e esea ch [
22
,
23
]. Some e o s ha e been made in
anspo a ion esea ch [
24
] and a ic policy [
25
]. Recen ly, copulas ha e been success ully
applied in eliabili y analysis in ci il [
26
], mechanical and s uc u al [
27
], o sho e [
28
] and
so wa e [
29
] enginee ing. In Biology, copulas ha e been used in he ield o gene ics [
30
]
o model gene dependencies.
Up o he au ho s’ knowledge, he e is no wo k using copulas o he pa ame ic
analysis o e olu ion p ocesses in Biology, whe e, as commen ed, many o he pa ame-
e s in ol ed a e unknown and uncon olled, and high co ela ions be ween pa ame e s
a e common. We p o e he e ha copulas a e an adequa e ool o imp o e he analysis
o highly-co ela ed mul ipa ame ic ma hema ical models such as hose appea ing in
Biology, wi h he added alue o p o iding key in o ma ion o he op imal design o new
expe imen s wi h he highes in o ma ion possible, hus educing ime and cos no only in
in i o expe imen s bu also in sca ce and cos ly in i o cases.
Ma hema ics 2021,9, 27 3 o 22
2. Ra ionale o he App oach
2.1. De e minis ic and S ochas ic Models
Le us suppose ha ou p oblem may be ep esen ed by he ollowing ma hema ical
ela ionship:
u=F(λ,θ), (1)
wi h
•u
(an
m
-dimensional ec o ) he ou pu a iable, ha is, he ou come o he expe i-
men s, ha we measu e.
•λ
he a iables which we can con ol when pe o ming he expe imen s (such as
en i onmen al a iables, geome ic pa ame e s, o bounda y condi ions).
•θ
he model pa ame e s, ha we canno con ol and whose alues mus be de e mined
(θ∈Ω, wi h Ω he pa ame ic space o dimension n).
•F
he ma hema ical model, ha ela es he expe imen al con igu a ion
λ
wi h he
ou pu a iables uin e ms o he se o pa ame e s θ.
In ela ion o he accu acy and p ecision o he model, i is possible o de ine h ee
le els o analysis: (1) he model is pe ec and he expe imen al measu es a e noise- ee;
(2) he model is pe ec and he expe imen al measu es a e noisy; and (3) he model is
no pe ec and he measu emen s a e noisy. Only he hi d case is, in gene al, ealis ic in
complex p oblems as he one he e analyzed.
In addi ion, i is di icul o de ine uni e sal alues o he pa ame e s in biological
p oblems, since hey a e highly-dependen on he pa icula expe imen al con ex .
As a consequence o all he p e ious obse a ions, i is mo e app op ia e o conside a
s ochas ic app oach, and e o mula e Equa ion (1) as:
U=F(λ,Θ), (2)
whe e Uand Θa e now andom ec o s o dimensions mand n espec i ely.
The p oposed app oach is he e o e sui able when he ollowing condi ions a e sa is-
ied:
•
Many coupled phenomena a e p esen , being di icul o design expe imen s able o
isola e each o hem (complexi y).
•
The measu emen space is la ge and i is possible o pe o m a su icien ly big numbe
o expe imen s N(da a a ailabili y).
F om a ma hema ical poin o iew, hese wo s a emen s may be e o mula ed as:
• The model Fincludes many pa ame e s (n1) and/o is non-sepa able.
The sepa abili y o a model is e alua ed by he possibili y o app oxima ing Fas:
F(λ,θ)≃FM(λ,θ) =
M
∑
i=1
n
∏
j=1
Fi,j(λ,θj). (3)
The lowe
M
, he easie o de ine a se o di e en expe imen al con igu a ions
S={λj}j=1, ..., k
o isola e each o he pa ame e s
θj
by sol ing sepa a ely each
equa ion
uj=FM(λ
,
θ)
. Al hough his sepa abili y de ini ion is no e y igo ous,
i is enligh ening enough o ou pu poses.
•
The dimension o he measu emen space is high (
m
1) and/o he sample size
is la ge enough (
N
1). Wi hou loss o gene ali y, we conside ha
m
is, ac ually,
he educed dimensionali y o he space o in o he wo ds ha all a iables o he
ambien space a e independen .
2.2. Case S udy: In Vi o GBM E olu ion
The e ha e been many a emp s o de elop ma hema ical models o desc ibe how
umo s g ow and espond o he apies [
10
,
31
]. In pa icula , in p e ious wo ks, we demon-
Ma hema ics 2021,9, 27 4 o 22
s a ed he possibili y o de eloping GBM pseudopalisades [
32
] and nec o ic co es [
33
]
in i o. Figu e 1illus a es one o such expe imen s in which a high densi y cell cul-
u e is exposed o oxygen low by wo la e al channels bu , due o sel -induced hypoxia,
he o ma ion o a nec o ic co e in he cen al pa o he chambe is obse ed.
Figu e 1. Fo ma ion o a nec o ic co e in he mic o luidic de ice.
One o he main p oblems in hese models is he lack o eliable alues o he many
pa ame e s in ol ed ha o ces many imes o ely on alues i ed om di e en si ua-
ions, leading some imes o un eliable conclusions. We ecen ly p oposed a ma hema ical
model o GBM in i o e olu ion [
11
], oge he wi h an ex ensi e pa ame e discussion.
This model enables he simula ion o di e en s ages o GBM e olu ion unde se e al
expe imen al condi ions, showing obus ness, while keeping a small unce ain y ange in
he esul s. I is es ablished in e ms o h ee ad ec ion- eac ion-di usion equa ions and
he associa ed pa ame e s ha a e exp essed as:
∂Ca
∂ =∂
∂xDa∂Ca
∂x−KaχO2
a(O2)χCa
a(Ca)Ca∂O2
∂x
+1
τa
βa(O2)Ga(Ca,Cd)Ca−1
τad
Sad(O2)Ca
(4)
∂Cd
∂ =1
τad
Sad(O2)Ca(5)
∂O2
∂ =DO2
∂2O2
∂x2−αaHa(O2)Ca. (6)
Equa ion (4) quan i ies he e olu ion o he cell no moxic pheno ype concen a ion,
Ca
, wi h h ee e ms: andom di usion, g ow h-dea h sou ce, and chemo axis. Equa ion
(5)
models he e olu ion o he nec o ic pheno ype concen a ion,
Cd
, which con ains only
he dead cells de i ed om he no moxic pheno ype. Finally, Equa ion (6) de ines he
O2
concen a ion e olu ion in he hyd ogel in which cells a e embedded, conside ing
bo h oxygen di usion and cell consump ion. Func ions
βa
,
Ga
,
χO2
a
,
χCa
a
,
Sad
and
Ha
a e
nonlinea co ec ions accoun ing o cell me abolic beha io :
χO2
a
de ines a chemo axis co ec ion accoun ing o he oxygen concen a ion. I has
been shown ha GBM cells p esen wha is called he go o g ow beha io [
34
]: cells spend
esou ces in p oli e a ing when hey a e enough oxygena ed and ac i a e mig a ion mecha-
nisms unde hypoxia condi ions, ha is, when he oxygen concen a ion is unde a ce ain
hypoxia h eshold OH
2. The e o e, we s a e:
Ma hema ics 2021,9, 27 5 o 22
χO2
a(O2) = 1−O2/OH
2i 0 ≤O2≤OH
2
0 i O2>OH
2.(7)
χCa
a
de ines a chemo axis co ec ion accoun ing o he cell concen a ion. We assume
ha cellula mo ili y is only possible when he cell concen a ion is below he sa u a ion
capaci y o he hyd ogel CM:
χCa
a(Ca) = 1−Ca/CMi 0 ≤Ca≤CM
0 i Ca>CM.(8)
βa
accoun s o he dependence o he p oli e a ion ac i i y on he oxygen concen a-
ion, in ag eemen wi h he go o g ow pa adigm [
34
]. Cell p oli e a ion dec eases when he
oxygen concen a ion is unde he hypoxia h eshold,
OH
2
, and is o ally inac i a ed unde
o al lack o oxygen:
βa(O2) = O2/OH
2i 0 ≤O2≤OH
2
1 i O2>OH
2.(9)
Ga
is a logis ic g ow h co ec ion accoun ing o space and nu ien s a ailabili y [
35
].
Cell p oli e a ion dec eases when he cell concen a ion app oaches he hyd ogel sa u a ion
capaci y, CM:
Ga(Ca,Cd) = 1−Ca+Cd
CM. (10)
Sad
is a dea h ac i a ion unc ion accoun ing o he oxygen concen a ion. Cell dea h
is a complex phenomenon ha can be due o wo di e en cell mechanisms, nec osis,
and apop osis [
36
,
37
]. Cell nec osis is highly dependen on he oxygen concen a ion,
while cell apop osis is no . The e o e, we ha e chosen a so ansi ion unc ion o
Sad
depending on wo pa ame e s—a loca ion pa ame e ,
OA
2
, iden i ying he anoxia oxygen
concen a ion and a sp ead pa ame e , ∆OA
2, associa ed wi h he dea h s ochas ic na u e:
Sad(O2) = 1
2 1− anh O2−OA
2
∆OA
2!!. (11)
Finally,
Ha
is he Michaelis-Men en co ec ion ac o in oxygen consump ion, ela ed o
he oxida i e phospho yla ion kine ics [
38
]. The consump ion a e is cons an o high
oxygen concen a ions, bu dec eases o ze o wi h a homog aphic shape. The alue o he
oxygen concen a ion o which he consump ion a e is hal ed is he so-called Michaelis-
Men en cons an , OM
2. The unc ion Hais hen s a ed as:
Ha(O2) = O2
OM
2+O2
. (12)
Equa ions (4)–(6) a e complemen ed wi h he bounda y and ini ial condi ions. Fo he
expe imen s ca ied ou in ou mic o luidic de ices, we assume o al impe meabili y
(Neumann bounda y condi ions) o he cell popula ions and a ixed alue o he oxygen
concen a ion a bo h sides o he channel (Di ichle bounda y condi ions). The e o e, i
L
is he chambe leng h, we may w i e:
∂Ca
∂x=0, x=0, L
∂Cd
∂x=0, x=0, L
O2=Ol
2,x=0
O2=O
2,x=L,
(13)

Ma hema ics 2021,9, 27 6 o 22
wi h Ol
2and O
2 he oxygen le els a he le and igh channels o he chip.
The ini ial oxygen concen a ion is assumed o be homogeneous o e he whole
chambe and equal o he maximum o bo h la e al oxygen concen a ions, ha is
O2(x
,
=
0) = O0
2=max(Ol
2,O
2).
The esul ing expe imen al pa ame ic space consis s, he e o e, o h ee pa ame e s,
co esponding o he concen a ion a he bounda ies o he chip, (
Ol
2
,
O
2
), and he ini ial
cell concen a ion, (C0), assumed cons an h oughou he chip. Tha is:
λ= [Ol
2,O
2,C0]. (14)
OH
2
,
OA
2
,
∆OA
2
and
OM
2
ha e a clea meaning in e ms o cell me abolism and a e
assumed o be known and cons an o all cell cul u es used in ou expe imen s, a leas om
an illus a i e poin o iew. Besides, al hough
CM
is e y dependen on he expe imen al
condi ions (hyd ogel mechanical p ope ies, nu ien s, ...), we shall assume i is cons an ,
o he sake o simplici y. The alues o hese pa ame e s we e aken om a p e ious
wo k [11].
P e ious esea ch in compu a ional biology has mainly ocused on he alue o he
pa ame e s o , in he bes case, in hei (indi idual) unce ain y. Howe e , in many cases,
he i ing p ocess is e y complex and he pa ame e s a e highly co ela ed due o, a leas ,
wo ac s:
•Samples a iabili y
: Di e en physical phenomena may ha e an inhe en co ela ion
suppo ed by physical conside a ions, being his co ela ion independen o he expe -
imen s pe o med o he model used. Fo example, when wo king wi h GBM cellula
models, cell mo ili y is induced by he andom mo ion inhe en o any cell and se e al
axis e ec s d i en by ex e nal physical o chemical s imuli. Ma hema ical pa ame e s
ela ed o hese phenomena (e.g., di usion and chemo axis coe icien s) appea ing
in he model equa ions will p esen , he e o e, a s ong co ela ion in he di e en
expe imen al samples.
•Model complexi y
: The non-sepa abili y o he model and/o he expe imen s does
no allow o isola e he pa icula mechanisms. Fo example, when wo king wi h
GBM cellula models, wi hou u he measu emen s o cell oxygen consump ion o
oxygen lux, i is impossible o es ablish i a lack o oxygen in a ce ain egion is due o
high cell consump ion o due o low oxygen di usion. The ma hema ical pa ame e s
ela ed o hese phenomena (e.g., oxygen di usion and cell oxygen consump ion
coe icien s) should p esen a s ong co ela ion, al hough his co ela ion does no
ha e a physical meaning, being inhe en o he model o o he expe imen al se -up.
Thanks o he lexibili y, po abili y, au oma ion, in eg a ion, and minia u iza ion o
he mic o luidic expe imen s, a huge amoun o da a may be gene a ed.
Acco dingly, his
ype o expe imen s is a pe ec domain o applica ion o he amewo k p esen ed he ein.
3. Me hods
3.1. Da a Gene a ion and Nume ical Solu ion
As he me hodology is based on he a ailabili y o su icien da a, he da a se used
o illus a ing he me hodology was gene a ed syn he ically using nume ical simula ion.
Fo his pu pose, he assumed alues o he pa ame e s we e ex ac ed om Re . [
11
] and
a da a se o “expe imen al” measu emen s was gene a ed by simula ion, using andomly
gene a ed bounda y and ini ial condi ions.
The summa y o he model pa ame e s is shown in Table 1, oge he wi h he alue
used o da a gene a ion.
Ma hema ics 2021,9, 27 7 o 22
Table 1. Model pa ame e s and alues used o da a gene a ion.
Pa ame e Symbol Value Used o Da a
Gene a ion [11]
No moxic cell di usion coe icien Da5×10−10 cm2/s
No moxic cell chemo axis coe icien Ka7.5 ×10−9cm2/mmHg·s
Oxygen di usion coe icien DO21×10−5cm2/s
Oxygen consump ion coe icien αa1×10−9mmHg·cm3/cell·s
G ow h cha ac e is ic ime τa200 h
Dea h cha ac e is ic ime τad 48 h
Hypoxia ac i a ion h eshold OH
27 mmHg
G ow h sa u a ion capaci y CM5×107cell/mL
Anoxia ac i a ion loca ion pa ame e
OA
21.8 mmHg
Anoxia ac i a ion sp ead pa ame e ∆OA
20.1 mmHg
Michaelis-Men en cons an OM
22.5 mmHg
Wi h espec o he simula ed i ual expe imen , we se a chip leng h o
L=
0.1
cm
,
a mesh size o
∆x=
0.0025
cm
and a ime s ep o
∆ =
1000 s.
N=
400 di e en ex-
pe imen s,
{λi}i=1, ..., 400
, we e simula ed a ying he bounda y condi ions: he le and
igh channel oxygen concen a ions we e se andomly be ween 0 and 7
mmHg
using wo
independen uni o m dis ibu ions while he ini ial oxygen concen a ion was se o he
maximum o bo h alues, as men ioned. The ini ial cell p o ile is supposed o be uni o m
and andomly sampled om a ecip ocal dis ibu ion ( o ake in o accoun bo h he expo-
nen ial and sa u a ed g ow h egimes) be ween 4
×
10
6
and 5
×
10
7cell/mL
. The nume ical
solu ions a e ob ained o
m=
8 d and he ou pu a iable associa ed o he expe imen
i
,
ui=us(x
,
m
;
λi)
, is he nume ical solu ion o he model equa ions ( he ma hema ical ap-
p oach and nume ical p ocedu es and algo i hms a e de ailed in Re . [
11
]), wi h bounda y
and ini ial condi ions de ined by
λ
, a ime
m
and a poin s gi en by he de ined mesh
x
. He e,
xj=j∆x
,
j=
1,
. . .
, 41. The compu ed da a we e all pe u bed wi h a uni o m
noise
ej=
0.2
×uj×V
wi h
V
a andom uni o m dis ibu ion
V∼ U[−
1, 1
]
. Consequen ly,
ui
j=us(xj, m,λi) + ei
j,j=1, . . . , 41 and i=1, . . . , 400.
Wi hin he amewo k p esen ed in Sec ion 2.1,
u=F(λ
,
θ)
a e he nume ical solu ions
ob ained, wi h
λ
he con ol pa ame e s,
θ
he unknown pa ame e s and
F
he ma hema ical
model p esen ed.
3.2. Copula-Based Pa ame ic Model Analysis
3.2.1. Concep o Copulas
In P obabili y and S a is ics, a copula is an
n
-mul i a ia e p obabili y dis ibu ion
unc ion
U
whose ma ginals,
Ui
, a e uni o m dis ibu ions on
[
0, 1
]
[
39
]. They we e in o-
duced by Skla in 1959 [
40
]. As he ma ginal dis ibu ions a e known, a copula desc ibing
he s uc u al dependence be ween a iables is enough o pe ec ly de ine he model.
Ma hema ical de ini ion.
As men ioned, a copula is a unc ion C:In→I, whe e I= [0; 1]such ha :
• Fo u1, . . . , un∈I, and i ui=0 o some 1 ≤i≤n:
C(u1, . . . , un) = 0. (15)
• Fo uj∈I, 1 ≤j≤n:
C(1, . . . , 1, uj, 1, . . . , 1) = uj. (16)
Ma hema ics 2021,9, 27 8 o 22
•C
is
n
-non dec easing, ha is, o each
B=∏n
i=1[xi
;
yi]⊂In
, he
C
- olume o
B
is
non-nega i e:
ZBdC(u) = ∑
z∈×n
i=1{xi;yi}
(−1)#{k:zk=xk}C(z)≥0. (17)
We can dis inguish be ween pa ame ic and non-pa ame ic copulas. In his wo k,
we use a hyb id app oach, as we i he ma ginal dis ibu ions by means o ke nel es i-
ma o s [
41
] o he p obabili y densi y unc ions and use a pa ame ic copula. Wi h his
app oach, he equi ed da a-se g ows as O(n)whe e nis he space dimension.
3.2.2. Fi ing and Model Valida ion
Le us suppose we ha e a da a-se o alues o di e en expe imen s,
λi
, cha ac-
e ized in e ms o a esul an mean alue
µi
and a co a iance ma ix
Σi
,
i=
1,
. . .
,
N
,
ob ained om di e en measu emen s associa ed o he con igu a ion
i
. As he assumed
model
F
is known, i is possible, o each piece o da a
ui
, o ob ain he se o pa ame e s
θi
which bes i s i .
In o de o a oid pa hological nume ical con e gence, we only ake in o accoun
hose se s o pa ame e s
θi
which lie inside he bibliog aphy anges conside ed in Re . [
11
],
ampli ied by 50% o a oid conside ing he pa ame e s bounds as de e minis ic alues,
ha , as shown in Table 2, a e e y la ge anges. The e o e, he esul ing in e als a e
[(
1
−
κ)xin
,
(
1
+κ)xsup]
, being
xin
and
xsup
he lowe and uppe bounds de ailed in Re . [
11
]
and
κ=
0.5, as summa ized in Table 2. As a esul o his p ocess, we ob ain a da ase wi h
n=
6 (numbe o pa ame e s),
N=
111 (da ase size) and
m=
41 (measu emen space
dimension), so we a e unde he scope o he p esen ed amewo k: N×mn>1.
Table 2. Pa ame e anges conside ed in he analysis.
Pa ame e Lowe Bound Uppe Bound Uni s
Da3.3 ×10−12 7.5 ×10−5cm2/s
Ka1×10−10 1.1 ×10−3cm2/mmHg·s
DO25×10−63×10−5cm2/s
αa5×10−10 1.1 ×10−6mmHg·cm3/cell·s
τa8 3000 h
τad 24 917 h
Once
θi
,
i=
1,
. . .
,
N
a e ob ained, he nex s ep is he adjus men o he ma ginal
dis ibu ions. The alues
θi
j
,
j=
1,
. . .
,
n
, a e used o i ing he ma ginal andom a iable
Θj
whose cumula i e dis ibu ion is assumed o be
Gj
. He e, we can ollow ei he a pa a-
me ic ( ha is,
Gj(x) = Gj(x
;
αj)
) o a non-pa ame ic app oach (which is he one ollowed
in his wo k). The alues θi
ja e he e o e ans o med in o uni o mly dis ibu ed ones ia
he s anda d ans o ma ion
yi
j=Gj(θi
j)
. As
yi
a e conside ed uni o mly dis ibu ed wi h a
join dependence, i is possible o i his s uc u al dependence using pa ame ic copulas.
To summa ize, he s eps o he aining p ocess a e:
1. P oblem minimiza ion o ob ain θi. We ha e o minimize he esidual unc ion Ri:
Ri(θ) = F(λi,θ)−µiT(Σi)−1F(λi,θ)−µi, (18)
whe e he Mahalanobis dis ance has been used o ake in o accoun he sample a i-
abili y. Assuming ha Σi=σi2I, Equa ion (18) can be ew i en as:
Ri(θ) = 1
σi2

F(λi,θ)−µi


2. (19)
Ma hema ics 2021,9, 27 9 o 22
2. Ke nel densi y es ima ion o he ma ginal dis ibu ions om he da a θi
j.
3. T ans o ma ion in o uni o mly dis ibu ed alues yi
j.
4. Copula i ing o he yda a o cap u e he join dependence.
The p esen ed sequence o s eps allows mo ing om a da ase
S={θi}i=1, ..., N
o a
p obabilis ic model o he andom ec o
Θ
( he ma ginal ke nel densi ies and he copula
pa ame e s encoding he s uc u al dependence), as i is he aim o s a is ical p ocedu es.
To a oid o e i ing, we ollow a ypical ain- es app oach: we di ide he da ase s
λi−ui
(whe e
ui
includes
µi
and
Σi
) in wo sepa a e subse s, one used o aining and he
o he used o es ing.
I we conside now he es da a-se , he p ocedu e is:
1. P oblem minimiza ion o ob ain θi.
2. Tes ing he s a is ical i ing:
•
Ma ginal i ing: q-q plo s, his og ams, empi ical cumula i e dis ibu ion unc-
ions (ecd ), boxplo s, pa ame ic o non-pa ame ic s a is ical es s [42].
•
Join 2 s. 2 co ela ions: co ela ions, sca e plo s, pa ame ic s a is ical es s o
co ela ions [42].
•
Whole join s uc u al dependence: mul i a ia e pa ame ic and non-pa ame ic
s a is ical es s [43].
3.2.3. Model Analysis and Pa ame e Es ima ion
Once he dis ibu ion o he andom ec o
Θ
is lea ned, he model is known om a
p obabilis ic poin o iew. The i s s aigh o wa d applica ion is pa ame e es ima ion
I is impo an o emphasize ha wi h “pa ame e es ima ion” we e e o he pa ame e s
o he ma hema ical model, no o he pa ame e s o he dis ibu ions used in he s a is ical
cha ac e iza ion (ac ually, he s a is ical cha ac e iza ion may be non-pa ame ic), ha may
be es ima ed ia common s a is ical in e ence echniques. A poin es ima e o he model
pa ame e s is gi en by:
ˆ
θ=P[Θ], (20)
whe e
P
is a cen al endency ope a o , o example, he expec a ion ope a o
E
, mini-
mizing he
L2
squa ed no m dispe sion (i s minimum is he a iance), o he geome ic
median ope a o
M
, minimizing he
L2
no m dispe sion (i s minimum is he mean absolu e
de ia ion).
Howe e , i is mo e in e es ing o pe o m a con idence egion es ima ion. As sug-
ges ed in Re . [
44
], in his wo k, we use he so-called Highes Densi y Regions (HDR)
because o hei easy in e p e a ion, s aigh o wa d gene aliza ion o mul i-dimensional
spaces and di ec compu a ion. Recall ha , unde some dis ibu ional assump ions
(e.g., no mali y assump ion), HDR compu a ion is educed o o he s anda d con idence
egion compu a ion echniques (e.g.,
χ2
quan ile ole ance ellipsoids). HDR compu a ion
enables eliable pa ame e es ima ion since, gi en a signi ican le el h eshold
α
, i is
possible o de ine an HDR egion in which he pa ame e s a e loca ed wi h a
p=
1
−α
p obabili y. This may be pe o med o single pa ame e s, o , in gene al,
k
- uples o pa am-
e e s.
This me hodology is also applicable o condi ional dis ibu ions. Le us suppose ha
we know he alue o a ce ain subse o pa ame e s
θ∗
and le us de ine
θ= (θ0
,
θ∗)
.
Knowing he dis ibu ion
Θ
, ha is ob ained a e he i ing- alida ion p ocedu e, i is
possible o de ine he condi ioned dis ibu ion o
Θ
gi en
Θ∗=θ∗
by i s densi y
0
de ined
in e ms o he densi y o θ:
0(θ0|θ∗) = (θ0,θ∗)
R (η,θ∗)dη, (21)
so all HDR compu a ions a e now applied o he dis ibu ion o
Θ
gi en
Θ∗=θ∗
by
eplacing by 0.
Ma hema ics 2021,9, 27 16 o 22
(a) Pseudopalisade expe imen .
(b) Nec o ic co e expe imen .
(c) Double pseudopalisade expe imen .
Figu e 8. Dis ibu ion o he measu ed a iable o in silico expe imen s.

Ma hema ics 2021,9, 27 17 o 22
4.5. Design o Expe imen s
In his sec ion, he aim is o de e mine he expe imen al con igu a ion wi h he highes
u ili y, ha is, o choose bo h igh and le oxygen low le els and he ini ial cell concen a-
ion o ge he maximum possible in o ma ion om he new expe imen . We ocus he e on
he e ec o coupling be ween pa ame e s and how i a ec s he u ili y in e p e a ion and
model pa ame e es ima ion.
Two di e en amilies o simula ions we e ca ied ou . In he i s one, only one
pa ame e dependence is analyzed a a ime, lea ing he es ixed a he alue se in
Sec ion 3.1. These igu es show con igu a ions whe e, i he es o he pa ame e s a e
assumed o be known, he unknown pa ame e will be es ima ed accu a ely. This is he case
in Figu e 9a,b. In he second amily, wo pa ame e dependencies a e analyzed. They a e
conside ed as bi a ia e dis ibu ions in o de o obse e he e ec ha he pa ame e
co ela ion has in cha ac e izing hese pa ame e s, ha is, how i modi ies he u ili y alues.
Figu e 9c, which belongs o his amily, illus a es expe imen al con igu a ions whe e he
wo-dimensional ec o will be es ima ed accu a ely.
In Figu e 9we compa e he iso-u ili y cu es when analyzing one o wo pa ame e
dependencies o he pai o pa ame e s ela ed o oxygen, changes in he cell popula ion
and cell mo ili y espec i ely. We assume o all igu es
C0=
5
×
10
7cell/mL
. In hese
igu es we can see he mos use ul expe imen s ( hose con igu a ions co esponding o he
highes u ili y alues) and hose ha lead o a poo adjus men o he model pa ame e s.
This analysis may be pe o med o di e en pa ame e combina ions, and o di e en
deg ees o knowledge. Fo ins ance, Table 3summa izes all possibili ies when explo ing
he ela ionship be ween
DO2
and
αa
, as we a e in e es ed in he es ima ion o hese wo
pa ame e s, bo h indi idually o join ly. The cases analyzed in his pape a e epo ed in
he hi d column.
Table 3.
Di e en possibili ies when explo ing he ela ionship be ween
DO2
and
αa
in he u ili y
compu a ion.
Pa ame e s o Be Es ima ed Known Pa ame e s Figu e
DO2None -
DO2Da,Ka,τa,τad -
DO2Da,Ka,τa,τad,αaFigu e 9a
αaNone -
αaDa,Ka,τa,τad -
αaDa,Ka,τa,τad,DO2Figu e 9b
DO2,αaNone -
DO2,αaDa,Ka,τa,τad Figu e 9c
Ma hema ics 2021,9, 27 18 o 22
(a)DO2.
(b)αa.
(c)(DO2,αa).
Figu e 9.
Iso-u ili y cu es o pa ame e s ela ed o oxygen o an ini ial concen a ion o
C0=
5×107cell/mL.
5. Discussion
The ain- es me hodology based on copulas ollowed in he i ing p ocess has shown
ha i is possible o es ablish a g ada ion in he s eng h o he pa ame e dependencies.
Figu e 6a illus a es he s eng h o his ela ionship, showing ha he e a e pai s o
phenomena di icul o isola e om he expe imen al and/o compu a ional poin s o
iew. Fo example, cell andom mo ili y and chemo axis mig a ion (
τ=
0.77). Bo h
phenomena ha e simila e ec s bu in he opposi e di ec ion. Thus, i is di icul o isola e
Ma hema ics 2021,9, 27 19 o 22
hei indi idual e ec on cell beha io i we ha e limi ed measu emen s a ailable on he
cell p o iles. I is hen only possible o e alua e, on he ou come, hei combined esul an
e ec , ha is, he a e age cell mo ili y. This analysis may be done o each pa ame e
couple, jus i ying he app oach adop ed in his wo k.
I is impo an o no e ha he high complexi y o biological sys ems, esul ing in
coupling be ween pai s o a iables, is mode a ed by he alues o he es , since he
bi a ia e andom dis ibu ions (shown o example in Figu e 3a) a e only a p ojec ion o
he whole 6-dimensional join dis ibu ion. Compa ing Figu e 3a,b, o example, we can
obse e he condi ioning e ec in loca ion, sp ead, and di ec ionali y o he dependency.
Once he p obabilis ic model is i ed, p edic ing he ac ual alue o he model pa am-
e e s is easily ca ied ou . As i is obse ed in Figu e 7, he no mali y assump ion o he
con idence egion es ima ion is no always a good s a ing hypo hesis. Fi s , i does no ake
in o accoun he complexi y o he ela ionship be ween he model pa ame e s (i.e., physical
phenomena) and may lead o non eliable alues (meaningless physical magni udes, such
as nega i e oxygen di usion). Secondly, i may mislead wi h espec o he unce ain y
ha we ac ually ha e o di e en signi ican le els. In any case, he con iden egion
es ima ion using HDR and a p ope p obabilis ic analysis a e e y in o ma i e abou he
deg ee o eliabili y o he ma hema ical model used o a biological explana ion. These
wo obse a ions become e en mo e e iden when he unce ain y o he model is educed,
as i can be seen when compa ing Figu e 7a,b: he chosen signi ican le el has a majo
impac on he con idence egion size and shape. In all he cases analyzed, his unce ain y
educ ion makes he con idence egion o concen a e a ound he pa ame e alues used in
he da a gene a ion p ocess.
Knowledge o he model pa ame e a ia ion ( om a p obabilis ic poin o iew)
allows o p edic he ou come o a gi en expe imen . This can be used no only o
model calib a ion and alida ion, bu o expe imen al planning (deciding he app op ia e
ma e ial, equipmen o accu acy o he measu ing de ices and echniques o be used).
Fo example, in Figu e 8, i may be seen ha he nec o ic co e expe imen equi es less
accu acy in he measu emen o he cell p o ile in he cen al pa o he chambe o pa am-
e e es ima ion, while he pseudopalisade expe imen equi es a measu emen echnique
able o de ec ex emely low ali e cell concen a ions. I can also be obse ed ha he
appea ance o signi ican ali e cells a he igh side o he chambe in he pseudopalisade
expe imen would no be explained by he model pa ame e a iabili y, bu a he by a
model limi a ion.
The p obabilis ic knowledge o he model can be u he exploi ed in expe imen al
planning and design by using BED heo y. In he analysis pe o med in his wo k, he e
a e se e al aspec s impo an o ema k. All g aphics showing he u ili y unc ion a e
symme ic wi h espec o he line
Ol=O
. This is cohe en wi h he symme ical con ig-
u a ion o he expe imen al se -up (geome y and p ope ies). The u ili y alue should
he e o e no be modi ied by lipping he bounda y condi ions. Besides, i can be seen ha
he le el cu es belonging o
DO2
and
αa
ha e simila shapes. This is due o he co ela ion
be ween pa ame e s, as i can be obse ed om he Kendall co ela ion coe icien
τ
o
each pai o a iables (Figu e 6b). The coe icien co esponding o
DO2
and
αa
is high and,
consequen ly, hey a e s ongly co ela ed, so he expe imen s needed o cha ac e ize he
alue o one o hem a e simila o he ones needed o cha ac e ize he o he .
Iso-u ili y cu es gi e us a pic u e ha may be in e p e ed biologically and is cohe en
wi h he di e en phenomena occu ing in he mic o luidic de ice. Howe e , he coupling
be ween hem makes his in e p e a ion di icul . In his wo k, he u ili y has been com-
pu ed o ou di e en ini ial cell concen a ions, anging om a low concen a ion C0=
1
×
10
6cell/mL
o he chip sa u a ion concen a ion
C0=CM=
5
×
10
7cell/mL
. The max-
imum u ili y is always eached o he highes ini ial concen a ion (5 ×107cell/mL).
A summa y o he analysis is p esen ed in Table 4, whe e he bes expe imen al con ig-
u a ion is p esen ed o each o he pa ame e s’ calib a ion, oge he wi h he maximum
u ili y alue.
Ma hema ics 2021,9, 27 20 o 22
Table 4. Mos use ul expe imen al con igu a ion o each o he pa ame e s’ e alua ion.
Pa ame e s o Be
Es ima ed
Uppe O2Concen a ion
[mmHg]
Lowe O2Concen a ion
[mmHg]
Maximum
U ili y Value
DO27 0 1.58
αa5 2 1.53
(DO2,αa)5 1 2.58
τa7 0 0.07
τad 7 0 1.29
(τa,τad)7 0 1.63
Da8 0 0.51
Ka7 1 0.35
(Da,Ka)8 0 0.49
Fo he analyzed amily o expe imen s, he mos use ul expe imen s a e always he
ones pe o med o high concen a ed cell cul u es. As mos phenomena a e ela ed o cell
concen a ions, he highe he concen a ion, he mo e quan i iable he di e en biological
mechanisms. Besides, i esul s clea ha con igu a ions wi h oxygen g adien a e mo e
use ul o accu a ely cha ac e izing he pa ame e s ela ed o oxygen (
DO2
,
αa
) and cell
mig a ion (
Da
,
Ka
), when he o he pa ame e s a e assumed o be known. Howe e , his
g adien has o be mode a e o a oid egions o o al no moxia o o al anoxia. When
he aim is o pe ec ly disc imina e be ween hei e ec s, so e g adien s a e gene ally
p e e ed (Figu e 9c, Table 4). Finally, o high ini ial cell concen a ions, g ow h and
dea h pa ame e s a e also well cha ac e ized unde g adien condi ions: we need o
induce localized hypoxic condi ions in o de o e alua e g ow h unde sa u a ion capaci y
and dea h.
6. Conclusions
Ma hema ical modeling o complex cell p ocesses is e y challenging due o i s in-
insic non-linea i y, highly-coupled mul iphysic in e ac ions, and he many co ela ed
pa ame e s which a e di icul o measu e o simply unknown. These pa ame e s a e mos
imes ob ained o a pa icula p oblem unde speci ic condi ions, leading in many cases o
conclusions, di ec ly de i ed om he modeling assump ions and he e o e p o iding li le
new in o ma ion. Also, hey a e di icul o gene alize.
As a esul , a p ope and ex ensi e pa ame ic analysis is manda o y. This should
include an ex ensi e and de ailed s udy o he alues epo ed in he bibliog aphy, a ca e ul
sensi i i y analysis and a su icien numbe o di e en expe imen s, no only o calib a ion
bu also o alida ion, a oiding pa ame e o e i ing.
This analysis, al hough i allows he iden i ica ion o he op imal se o pa ame e s,
is mos imes di icul o ex end o o he p oblems wi h easonable accu acy and he e o e
wi h a ce ain alida ion o i s ac ual physical cha ac e and i s alue ange. I is also
di icul o disc imina e be ween co ela ed pa ame e s associa ed o mechanisms ha
canno be isola ed in he expe imen s. Hence, we need addi ional in o ma ion bo h o
ge a be e disc imina ion be ween hem, and o iden i y he op imal condi ions o
addi ional expe imen s o p o ide he maximum in o ma ion possible in o de o ge
such disc imina ion.
We ha e p o ed he e ha copulas a e a simple and powe ul ool o de ec and
imp o e highly-co ela ed mul ipa ame ic ma hema ical models such as hose appea ing
in Biology, wi h he added alue o p o iding key in o ma ion o he op imal design o new
expe imen s wi h he highes in o ma ion possible o he p oblem in hands, hus educing
ime and cos no only in ou in i o expe imen s bu also in sca ce and cos ly in i o cases.
Ma hema ics 2021,9, 27 21 o 22
Au ho Con ibu ions:
Concep ualiza ion M.D., M.H.D. and J.A.-J.; Me hodology J.A.-J.; So -
wa e and alida ion J.A.-J., M.P.-A.; W i ing o iginal d a M.D., J.A.-J. and M.P.-A.; Figu es and
isualiza ion J.A.-J., M.P.-A. and T.R.; Supe ision M.D. and M.H.D.; P ojec adminis a ion and
unding M.D., M.H.D. and J.A.S.-H. All au ho s e iewed he manusc ip . All au ho s ha e ead and
ag eed o he published e sion o he manusc ip .
Funding:
The au ho s g a e ully acknowledge he inancial suppo om he Spanish Minis y o
Economy and Compe i i eness (MINECO) and FEDER, UE h ough he p ojec PGC2018-097257-
B-C31, he Spanish Minis y o Science and Inno a ion h ough he p ojec PID2019-106099RB-
C44/AEI/10.13039/501100011033, he Go e nmen o A agon (DGA) and he Cen o de In es iga-
cion Biomedica en Red en Bioingenie ia, Bioma e iales y Nanomedicina (CIBER-BBN). CIBER-BBN
is inanced by he Ins i u o de Salud Ca los III wi h assis ance om he Eu opean Regional De elop-
men Fund.
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : Da a and codes a ailable unde eques o he au ho s.
Con lic s o In e es :
The au ho s decla e no con lic o in e es . The unde s had no ole in he design
o he s udy; in he collec ion, analyses, o in e p e a ion o da a; in he w i ing o he manusc ip ,
o in he decision o publish he esul s.
Re e ences
1.
Quail, D.F.; Joyce, J.A. Mic oen i onmen al egula ion o umo p og ession and me as asis. Na . Med.
2013
,19, 1423. [C ossRe ]
[PubMed]
2. Hanahan, D.; Weinbe g, R.A. Hallma ks o cance : The nex gene a ion. Cell 2011,144, 646–674. [C ossRe ] [PubMed]
3.
Scannell, J.W.; Blanckley, A.; Boldon, H.; Wa ing on, B. Diagnosing he decline in pha maceu ical R & D e iciency. Na . Re .
D ug Disco . 2012,11, 191. [PubMed]
4.
Sackmann, E.K.; Ful on, A.L.; Beebe, D.J. The p esen and u u e ole o mic o luidics in biomedical esea ch. Na u e
2014
,
507, 181. [C ossRe ] [PubMed]
5. Bha ia, S.N.; Ingbe , D.E. Mic o luidic o gans-on-chips. Na . Bio echnol. 2014,32, 760. [C ossRe ] [PubMed]
6.
Boussommie -Calleja, A.; Li, R.; Chen, M.B.; Wong, S.C.; Kamm, R.D. Mic o luidics: A new ool o modeling cance –immune
in e ac ions. T ends Cance 2016,2, 6–19. [C ossRe ] [PubMed]
7.
Ze an onakis, I.K.; Hughes-Al o d, S.K.; Cha es , J.L.; Condeelis, J.S.; Ge le , F.B.; Kamm, R.D. Th ee-dimensional mic o luidic
model o umo cell in a asa ion and endo helial ba ie unc ion. P oc. Na l. Acad. Sci. USA
2012
,109, 13515–13520. [C ossRe ]
[PubMed]
8.
By ne, H.; Ala con, T.; Owen, M.; Webb, S.; Maini, P. Modelling aspec s o cance dynamics: A e iew. Philos. T ans. R. Soc. Lond.
A Ma h. Phys. Eng. Sci. 2006,364, 1563–1578. [C ossRe ]
9. Ki ano, H. Compu a ional sys ems biology. Na u e 2002,420, 206. [C ossRe ]
10.
Bea e , E.L.; Loweng ub, J.S.; F ieboes, H.B.; Chuang, Y.L.; Jin, F.; Wise, S.M.; Fe a i, M.; Agus, D.B.; C is ini, V. Mul ipa ame e
compu a ional modeling o umo in asion. Cance Res. 2009,69, 4493–4501. [C ossRe ]
11.
Ayensa-Jiménez, J.; Pé ez-Aliaca , M.; Randelo ic, T.; Oli án, S.; Fe nández, L.; Sanz-He e a, J.A.; Ochoa, I.; Doweida , M.H.;
Dobla é, M. Ma hema ical o mula ion and pa ame ic analysis o in i o cell models in mic o luidic de ices: Applica ion o
di e en s ages o glioblas oma e olu ion. Sci. Rep. 2020,10, 1–21. [C ossRe ] [PubMed]
12.
B a , D.J. Glioblas oma: Biology, gene ics, and beha io . In Ame ican Socie y o Clinical Oncology Educa ional Book; Ame ican
Socie y o Clinical Oncology: Alexand ia, VA, USA, 2012; pp. 102–107._am.2012.32.102. [C ossRe ]
13. Ang, A.; Chen, J. Asymme ic co ela ions o equi y po olios. J. Financ. Econ. 2002,63, 443–494. [C ossRe ]
14.
Boubake , H.; Sghaie , N. Po olio op imiza ion in he p esence o dependen inancial e u ns wi h long memo y: A copula
based app oach. J. Bank. Financ. 2013,37, 361–377. [C ossRe ]
15.
McNeil, A.; F ey, R.; Emb ech s, P. Quan i a i e Risk Managemen : Concep s, Techniques, and Tools; P ince on Uni e si y P ess:
P ince on, NJ, USA, 2017.
16. Kole, E.; Koedijk, K.; Ve beek, M. Selec ing copulas o isk managemen . J. Bank. Financ. 2007,31, 2405–2423. [C ossRe ]
17. Meucci, A. A new b eed o copulas o isk and po olio managemen . Risk 2011,24, 122–126.
18.
Sola i, S.; Losada, M. Non-s a iona y wa e heigh clima e modeling and simula ion. J. Geophys. Res. Ocean.
2011
,116. [C ossRe ]
19.
Munkhamma , J.; Widén, J. An au oco ela ion-based copula model o gene a ing ealis ic clea -sky index ime-se ies. Sol. Ene gy
2017,158, 9–19. [C ossRe ]
20.
A ya, F.K.; Zhang, L. Copula-based Ma ko p ocess o o ecas ing and analyzing isk o wa e quali y ime se ies. J. Hyd ol. Eng.
2017,22, 04017005. [C ossRe ]

Ma hema ics 2021,9, 27 22 o 22
21.
Laux, P.; Wagne , S.; Wagne , A.; Jacobei , J.; Ba dossy, A.; Kuns mann, H. Modelling daily p ecipi a ion ea u es in he Vol a
Basin o Wes A ica. In . J. Clima ol. A J. R. Me eo ol. Soc. 2009,29, 937–954. [C ossRe ]
22.
Schoelzel, C.; F iede ichs, P. Mul i a ia e non-no mally dis ibu ed andom a iables in clima e esea ch–in oduc ion o he
copula app oach. Nonlinea P ocess. Geophys. 2008,15, 761–772. [C ossRe ]
23.
Laux, P.; Vogl, S.; Qiu, W.; Knoche, H.R.; Kuns mann, H. Copula-based s a is ical e inemen o p ecipi a ion in RCM simula ions
o e complex e ain. Hyd ol. Ea h Sys . Sci. 2011,15, 2401–2419. [C ossRe ]
24.
Zou, Y.; Zhang, Y. A copula-based app oach o accommoda e he dependence among mic oscopic a ic a iables. T ansp. Res.
Pa C Eme g. Technol. 2016,70, 53–68. [C ossRe ]
25.
Spissu, E.; Pinja i, A.R.; Pendyala, R.M.; Bha , C.R. A copula-based join mul inomial disc e e–con inuous model o ehicle ype
choice and miles o a el. T anspo a ion 2009,36, 403–422. [C ossRe ]
26.
Kilgo e, R.T.; Thompson, D.B. Es ima ing join low p obabili ies a s eam con luences by using copulas. T ansp. Res. Rec.
2011
,
2262, 200–206. [C ossRe ]
27.
Ba oli, G.; Mannini, C.; Massai, T. Quasi-s a ic combina ion o wind loads: A copula-based app oach. J. Wind Eng. Ind. Ae odyn.
2011,99, 672–681. [C ossRe ]
28.
Dong, S.; Zhou, C.; Tao, S.S.; Xue, D.S. Bi a ia e Gumbel dis ibu ion based on Clay on Copula and i s applica ion in o sho e
pla o m design. Pe iod. Ocean Uni . China 2011,41, 117–120.
29.
Pham, H. Recen s udies in so wa e eliabili y enginee ing. In Handbook o Reliabili y Enginee ing; Sp inge : London, UK, 2003;
pp. 285–302.
30.
Kim, J.M.; Jung, Y.S.; Sungu , E.A.; Han, K.H.; Pa k, C.; Sohn, I. A copula me hod o modeling di ec ional dependence o genes.
BMC Bioin o m. 2008,9, 225. [C ossRe ]
31.
Kim, Y.; Jeon, H.; O hme , H. The ole o he umo mic oen i onmen in glioblas oma: A ma hema ical model. IEEE T ans.
Biomed. Eng. 2017,64, 519–527. [C ossRe ]
32.
Ayuso, J.M.; Monge, R.; Ma ínez-González, A.; Vi umb ales-Muñoz, M.; Llamaza es, G.A.; Be ganzo, J.; He nández-Laín, A.;
San ola ia, J.; Dobla é, M.; Hube , C.; e al. Glioblas oma on a mic o luidic chip: Gene a ing pseudopalisades and enhancing
agg essi eness h ough blood essel obs uc ion e en s. Neu o-Oncology 2017,19, 503–513. [C ossRe ]
33.
Ayuso, J.M.; Vi umb ales-Muñoz, M.; Lacue a, A.; Lanuza, P.M.; Checa-Cha a ia, E.; Bo ella, P.; Fe nández, E.; Dobla e, M.;
Allison, S.J.; Phillips, R.M.; e al. De elopmen and cha ac e iza ion o a mic o luidic model o he umou mic oen i onmen .
Sci. Rep. 2016,6, 36086. [C ossRe ]
34.
Ha ziki ou, H.; Basan a, D.; Simon, M.; Schalle , K.; Deu sch, A. ‘Go o g ow’: The key o he eme gence o in asion in umou
p og ession? Ma h. Med. Biol. A J. IMA 2012,29, 49–65. [C ossRe ] [PubMed]
35.
S ame , B.; Mayo , R. Mechanisms and in i o unc ions o con ac inhibi ion o locomo ion. Na . Re . Mol. Cell Biol.
2017
,18, 43.
[C ossRe ] [PubMed]
36.
Galluzzi, L.; Vi ale, I.; Aa onson, S.A.; Ab ams, J.M.; Adam, D.; Agos inis, P.; Alnem i, E.S.; Al ucci, L.; Amelio, I.; An-
d ews, D.W.; e al. Molecula mechanisms o cell dea h: Recommenda ions o he Nomencla u e Commi ee on Cell Dea h 2018.
Cell Dea h Di e . 2018,25, 486. [C ossRe ] [PubMed]
37. Sendoel, A.; Henga ne , M.O. Apop o ic cell dea h unde hypoxia. Physiology 2014,29, 168–176. [C ossRe ] [PubMed]
38.
Chance, B.; Williams, G.R. The espi a o y chain and oxida i e phospho yla ion. Ad . Enzymol. Rela . A eas Mol. Biol.
1956
,
17, 65–134.
39.
Jawo ski, P.; Du an e, F.; Hä dle, W.K.; Rychlik, T. Copula Theo y and I s Applica ions: P oceedings o he Wo kshop Held in Wa saw,
Poland, 25–26 Sep embe 2009; Sp inge : Be lin, Ge many, 2010; Volume 198.
40. Skla , M. Fonc ions de epa i ion an dimensions e leu s ma ges. Publ. Ins . S a is . Uni . Pa is 1959,8, 229–231.
41. Wand, M.P.; Jones, M.C. Ke nel Smoo hing; CRC P ess: Boca Ra on, FL, USA, 1994.
42. Ko egoda, N.T.; Rosso, R. Applied S a is ics o Ci il and En i onmen al Enginee s; Blackwell Malden: Malden, MA, USA, 2008.
43.
Fan, Y. Goodness-o - i es s o a mul i a ia e dis ibu ion by he empi ical cha ac e is ic unc ion. J. Mul i a . Anal.
1997
,
62, 36–63. [C ossRe ]
44. Hyndman, R.J. Compu ing and g aphing highes densi y egions. Am. S a . 1996,50, 120–126.
45. Fishe , R.A. The Design o Expe imen s; Oli e and Boyd: Edinbu gh/London, UK, 1937.
46. Chalone , K.; Ve dinelli, I. Bayesian Expe imen al Design: A Re iew. S a . Sci. 1995,10, 273–304. [C ossRe ]
47. Shannon, C.E. A ma hema ical heo y o communica ion. Bell Sys . Tech. J. 1948,27, 379–423. [C ossRe ]
48.
Ahmed, N.A.; Gokhale, D. En opy exp essions and hei es ima o s o mul i a ia e dis ibu ions. IEEE T ans. In . Theo y
1989
,
35, 688–692. [C ossRe ]
49. Dema a, S.; McNeil, A.J. The copula and ela ed copulas. In . S a . Re . 2005,73, 111–129. [C ossRe ]