RESEARCH Open Access
A lowg aph model o bladde ca cinoma
G ego io Rubio
1*
, Belén Ga cía-Mo a
1
, C is ina San ama ía
1
, José Luis Pon ones
2
F om 1s In e na ional Wo k-Con e ence on Bioin o ma ics and Biomedical Enginee ing-IWBBIO 2013
G anada, Spain. 18-20 Ma ch 2013
* Co espondence: g ubio@imm.
up .es
1
Ins i u o de Ma emá ica
Mul idisciplina , Uni e si a
Poli ècnica de València, España
Abs ac
Backg ound: Supe icial bladde cance has been he subjec o nume ous s udies
o many yea s, bu he e olu ion o he disease s ill emains no well unde s ood.
A e he umo has been su gically emo ed, i may eappea a a simila le el o
malignancy o p og ess o a highe le el. The p ocess may be easonably modeled
by means o a Ma ko p ocess. Howe e , in o de o mo e comple ely model he
e olu ion o he disease, his app oach is insu icien . The semi-Ma ko amewo k
allows a mo e ealis ic app oach, bu calcula ions become equen ly in ac able. In
his con ex , lowg aph models p o ide an e icien app oach o success ully manage
he e olu ion o supe icial bladde ca cinoma. Ou aim is o es his me hodology
in his pa icula case.
Resul s: We ha e buil a success ul model o a simple bu ep esen a i e case.
Conclusion: The lowg aph app oach is sui able o modeling o supe icial bladde
cance .
Backg ound
Bladde umo s a e a challenge in u ology. They pose an impo an public heal h p o-
blem because hey a e biologically e y agg essi e and a e highly p e alen in wes e n
coun ies. App oxima ely 75-85 % o pa ien s wi h newly diagnosed bladde ca cinoma
ha e non muscle-in asi e bladde ca cinoma (NMI-BC), which can be managed wi h
ansu e h al esec ion (TUR). TUR is a su gical endoscope echnique used o emo e
he mac oscopic umo om he in e io o he bladde . Howe e i has a no able
endency o ecu (30-85 %) and less equen ly o p og ess o muscle in asi e s ages
(10-20 %). The objec o his s udy is he NMI-BC, ha makes up 70 % o he o al
heal h ca e cos o his disease. A e iew abou he NMI-BC may be ound in [1].
Bio echnological ad ances ha e allowed us o use di e en he apeu ic p ocedu es
(su ge y, adio he apy, chemo he apy, immuno he apy) success ully bu s ill many
pa ien s su e an un a ou able ou come wi hou con ol o disease. In p ac ice u olo-
gis s ha e a se ious p oblem: some pa ien s wi h simila cha ac e is ics unde go di e -
en e olu ion. Consequen ly, his c ea es a p oblem as o he choice o ea men o be
applied. U ologis s need ools o accu a ely p edic he eal e olu ion o he disease,
ha help hem o imp o e ea men modali ies and ollow-up schemes o non-muscle
in asi e bladde cance pa ien s. In his ega d an impo an con ibu ion [2] appea ed
in Eu opean U ology, he o icial jou nal o he Eu opean Associa ion o U ology.
Rubio e al.Theo e ical Biology and Medical Modelling 2014, 11(Suppl 1):S3
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© 2014 Rubio e al.; licensee BioMed Cen al L d. This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License (h p://c ea i ecommons.o g/licenses/by/2.0), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in
any medium, p o ided he o iginal wo k is p ope ly ci ed. The C ea i e Commons Public Domain Dedica ion wai e (h p://
c ea i ecommons.o g/publicdomain/ze o/1.0/) applies o he da a made a ailable in his a icle, unless o he wise s a ed.
By means o looking up ables he p obabili y o ecu ence and p og ession o a
pa ien is p o ided. Howe e only ime o i s ecu ence is conside ed, and he analy-
sis is educed o he Cox p opo ional haza ds eg ession model. La e wo ks ha e s u-
died he model alida ion, inding some limi a ions [3].
Ou eam has been wo king wi h u ologis s om Uni e si y Hospi al La Fe o he las
en yea s. We ha e de eloped se e al models ying o cap u e di e en aspec s o he
disease e olu ion. Ou aim o he nea u u e is o de ec he mos ele an p edic i e
ac o s, and also o pe o m an accu a e model o he disease e olu ion. The i s objec-
i e includes in es iga ing a he gene ic and molecula le el, while he second one could
be achie ed wi h a sui able mul is a e model. While he p ocess may be easonably mod-
eled by means o a Ma ko p ocess, in o de o mo e comple ely model he e olu ion o
he disease his app oach is insu icien . Speci ically, i is possible ha ime spen in a
s a e in luences he u u e e olu ion o he p ocess, i.e., i no only depends on he cu -
en s a e. The semi-Ma ko amewo k allows a mo e ealis ic app oach, bu calcula-
ions become equen ly in ac able. In his con ex , lowg aph models p o ide an
e icien app oach o he analysis o ime- o-e en da a, since hei in oduc ion in his
ield a ew yea s ago [4]. The p esen wo k is a i s s ep in o de o explo e he e olu ion
o he ecu ence p og ession p ocess by means o his me hodology.
The pape is o ganized as ollows: i s we e iew a ew basic concep s o su i al
analysis, phase- ype dis ibu ions and E lang dis ibu ions, needed o build he model.
Then we p esen he essen ials o lowg aph models and impo an ea u es o ou
app oach. The sec ion ha ollows deals wi h a simple lowg aph model o he ecu -
ence-p og ession p ocess in NMI-BC, cons uc ed using a da abase om La Fe
Uni e si y Hospi al o Valencia (Spain). Finally, some conclusions a e discussed.
Su i al analysis and phase- ype dis ibu ions
Su i al analysis
Su i al analysis echniques deal wi h he analysis o da a aking imes om a well-
de ined ime-o igin un il he occu ence o some pa icula e en o end-poin .
Tosumma izesu i alda a he ea e wokey unc ions: he Su i al Func ion and
he Haza d Func ion. Le T be he andom a iable associa ed wi h he su i al ime
( ime un il he ocu ence o he e en ).
The Su i al Func ion is
S( )=P(T≥ )=1−F( )
whe e F( ) is he dis ibu ion unc ion o T. I exp esses he p obabili y ha an indi-
idual su i es om he ime o igin o some ime beyond .
The Haza d Func ion is gi en by
λ( ) = lim
 →0
P( ≤T< + |T≥ )
 ,
which exp esses he haza d a e o he ins an aneous e en a e.
In su i al analysis da a a e equen ly censo ed [5], which means ha he e en o
in e es has no been obse ed. The ollow-up ime o hose pa ien s mus be aken
in o accoun , because i in o ms us o he ac ha he indi idual has been ee o
e en un il he p esen momen . Fo ins ance we s a ed wi h 957 pa ien s, o whom
Rubio e al.Theo e ical Biology and Medical Modelling 2014, 11(Suppl 1):S3
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434 unde wen a ecu ence, 24 a p og ession, and 499 had censo ed imes, which
means ha a he ime o hei las check-up hey had no ecu ence o p og ession.
Phase- ype dis ibu ions
In o de o model li e imes, mix u es o dis ibu ion unc ions a e use ul. In his con-
ex phase- ype dis ibu ions [6] a e e y in e es ing, because o hei p ope ies and
hey p o ide compu a ions wi h manageable analy ical exp essions. Le us summa ize
he main concep s: he dis ibu ion F(-) on [0, ∞) is a phase- ype dis ibu ion (PH-
dis ibu ion) wi h ep esen a ion (a,T) i i is he dis ibu ion o he ime un il abso p-
ion in a Ma ko p ocess on he s a es {1, ...,m, m + 1} wi h gene a o
TT
0
00
,
and ini ial p obabili y ec o (a,a
m+1
) whe e ais a ow m- ec o .
The ma ix To o de mis non-singula wi h nega i e diagonal en ies and non-
nega i e o -diagonal en ies, T
0
is a column ma ix wi h nonnega i e en ies, and i
holds ha
−Te =T0,
whe e edeno es a column ec o wi h all componen s equal o one.
The dis ibu ion F(-) is gi en by
F( )=1−αexp(T )e, ≥0(1)
and he densi y ( )by
( )=αexp(T )T0.
The su i al unc ion is
S( )=αexp(T )e(2)
and he haza d unc ion is gi en by
h( )=αexp(T )T0
αexp(T )e.
Finally, he Laplace ans o m is
L(s)=αm+1 +α(sI −T)−1T0, o Re(s)>0. (3)
Phase- ype dis ibu ions a e a closed class o ini e mix u es, and o m a class
weakly dense in he class o gene al dis ibu ions de ined on he posi i e eal line.
A pa icula case o phase- ype dis ibu ion, ele an in ou app oach, is he E lang dis-
ibu ion. An E lang dis ibu ion E[ , l] has a ep esen a ion (a,T) as a phase- ype [7]:
α= (1, 0, ...,0)
1×
T=⎛
⎜
⎜
⎜
⎜
⎜
⎝
−λλ
−λλ
......
−λλ
−λ
⎞
⎟
⎟
⎟
⎟
⎟
⎠ ×
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A ini e mix u e o E langs dis ibu ions is he e o e a phase- ype dis ibu ion. We
a e in e es ed in he class o mix u es o h ee E lang dis ibu ions s udied in [8]. The
dis ibu ion unc ion o he elemen s in his class is gi en by he exp ession
G( )=p1F1( )+p2F2( )+p3F3( ), (4)
wi h p
1
+p
2
+p
3
=1,p
i
>0,i=1,2,3.
Le usdeno e he h eeE langsbyE[
1
,µ
1
], E[
1
,µ
1
], E[
1
,µ
1
], wi h µ
i
>0and
i
a
posi i e in ege , i= 1, 2, 3. In he pa icula case wi h
1
=1,
2
=3,
3
= 5 he ep e-
sen a ion as phase- ype dis ibu ion is (a,T) whe e
α=(p1p200p30000) (5)
T=
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
−μ100000000
0−μ2μ2000000
00−μ2μ200000
000−μ200000
0000−μ3μ3000
00000−μ3μ300
000000−μ3μ30
0000000−μ3μ3
00000000−μ3
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
(6)
The e sa ili y o dis ibu ions gi en in (4) le ha e us se e al op ions o i hem o
ou in e es dis ibu ions. The way we ied was o pe o m some expe imen al compu-
a ions, conside ing di e en alues o
1
,
2
and
3
. Mix u e (5) was explici ly gi en in
[8], and we ound i wo ked e y well.
Flowg aph models
A lowg aph model is a g aphical ep esen a ion o a mul is a e model ha consis s o
di ec ed line segmen s (b anches) connec ing he s a es, namely, a di ec ed g aph. The
b anches a e labeled wi h ansmi ances, ha a e he ansi ion p obabili y p
ij
om
s a e i o s a e jmul iplied by an in eg al ans o m G
ij
(s) o he ansi ion ime p ob-
abili y densi y unc ion (PDF). This ans o ma ion can be a cha ac e is ic unc ion
(CF), a momen gene a ing unc ion (MGF), a Laplace ans o m (LT), o e en an
empi ical ans o m [9][10]. Flowg aphs a e used o ep esen semi-Ma ko p ocesses,
gi en ha allowed wai ing ime dis ibu ions go beyond he exponen ial dis ibu ion
di ec ly linked o Ma ko p ocesses.
Fo ins ance, Figu e 1 shows he lowg aph o he h ee-s a e illness-dea h model ha
we will use in his pape , based on [11].
T ansmi ancesa ecombinedacco ding oasys ema icp ocedu e(see[12],sec ion
2.5), in o de o compu e he ans o ms o he ansi ions o in e es . Fo ins ance, he
ules pe aining o he g aph in he Figu e 1 a e he ollowing:
1) The ansmi ance o ansi ions in se ies is he p oduc o he se ies
ansmi ances.
2) The ansmi ance o ansi ions in pa allel is he sum o he pa allel
ansmi ances.
These ules a e applied la e in building he model.
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In o de o pe o m he model, he i s s ep is o selec a sui able dis ibu ion o
he wai ing ime in each ansi ion. Ou app oach will be o compu e he empi ical dis-
ibu ions (Kaplan-Meie [5]) and app oxima e hem using mix u es o E lang dis ibu-
ions. Speci ically we use he mix u e gi en by (5)-(6). No e ha he cumula i e
dis ibu ion unc ion is easily compu ed om exp ession (1). The pa ame e s p
i
and µ
i
a e calcula ed by minimizing
||Fij( )−Gij( )||,(7)
whe e F
ij
is he empi ical dis ibu ion o he ansi ion ij and G
ij
he mix u e dis i-
bu ion o he same ansi ion. Ini ial alues o he minimiza ion p ocess a e needed.
In o de o es ima e hese alues (and also o decide a sui able mix u e, in ou case
(5)-(6)) we use a non-nega i e leas squa es i (Lawson-Hanson algo i hm [13]).
Mo e p ecisely, he idea is he ollowing. Based on [8], we y se e al E lang dis ibu-
ions in exp ession (4). Gi en F
1
,F
2
and F
3
, and an empi ical dis ibu ion Fwe con-
side he sys em
F=p1F1+p2F2+p3F3
1=p1+p2+p3
which we i by non-nega i e leas squa es, o compu e p
1
,p
2
and p
3
. In his way we
ob ain easonable ini ial alues o he pa ame e s p
i
and µ
i
.
Once he pa ame ic dis ibu ions ha e been compu ed, he Laplace ans o ms a e
easily calcula ed om (3). Then we compu e he Laplace ans o m ele an o he
ansi ions o in e es , applying he abo e ules. The inal s ep is o in e hese ans-
o ms o ob ain PDFs, o which we use an in e sion algo i hm called EULER, de el-
oped by Aba e and Whi [14].
Figu e 1 Th ee-s a e illness-dea h model.
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Flowg aph models o s ochas ic ne wo ks we e in oduced by Bu le and Huzu ba-
za [4]. An accoun o he heo y de eloped up o 2005 may be ound in [12]. A ecen
con ibu ion p oposing a p ognos ic model is [15].
A lowg aph model o bladde ca cinoma
Da a
The da abase was ob ained om La Fe Uni e si y Hospi al o Valencia (Spain). I
eco ds clinical-pa hological in o ma ion om 957 pa ien s, ollowed be ween Janua y
1995 and Janua y 2010. The p ima y umo is a NMI-BC, which means ha i is ca e-
go ized as s age Ta o T1, acco ding o he Wo ld Heal h O ganiza ion (WHO) TNM
classi ica ion s aging sys em [16]. A e emo al o he umo by TUR, i may ecu a
a simila s age, which we call ecu ence; o i may p og ess o muscle in asi e s ages
T2, T3 o T4, which we call p og ession. The da a eco d se e al ecu ence imes.
This means ha some pa ien s ha e no ecu ence a all, some ha e one o mo e
ecu ences, and some ha e p og ession (di ec ly o a e some ecu ence). In ou
model we ha e conside ed p og ession and one ecu ence. As s a ed abo e, 434
pa ien s unde wen a ecu ence, 24 a p og ession, and 499 had censo ed imes. Then,
63 pa ien s we e los . F om he emaining 371 pa ien s, 17 unde wen a p og ession,
226 a ecu ence and imes o he emaining 128 pa ien s we e censo ed. A ull
desc ip ion o da a may be ound in [17].
Flowg aph model
Ou aim in his pape is o es he lowg aph me hodology in his pa icula p oblem,
andsowepe o m hesimplemodelo Figu e 1. In s a e 0 he pa ien is ee o dis-
ease, a e he TUR o he p ima y umo . S a e 1 is he i s ecu ence, and s a e 2 is
p og ession. Time is gi en in yea s.
By way o example, we a e going o model he o e all isk o p og ession. So we a e
in e es ed in inding he p obabili y dis ibu ion o ime o each s a e 2 o he i s
ime s a ing in s a e 0, i espec i e o he pa h ha was aken. Tha is o say, he i s
passage dis ibu ion o going om disease ee o muscle in asi e s ages. Bu he aim
o a mo e gene al lowg aph model o he ecu ence - p og ession p ocess would be
o p edic he isk o ecu ence o p og ession om any s a e.
Pa ame ic dis ibu ions o all ansi ions and hei Laplace ans o ms a e pe -
o med acco ding o he p ocedu e desc ibed abo e. Minimiza ion is ca ied ou by
means o he cons Op im unc ion, om he R S a s Package [18]. We use he eucli-
dean no m in (7). Empi ical and pa ame ic dis ibu ions o each ansi ion a e shown
in Figu es 2, 3 and 4.
Le us calcula e he i s passage dis ibu ion o going om s a e 0 o s a e 2. Fo his
we compu e he Laplace ans o m o he ime o p og ession. Applying he ules 1
and 2 abo e, i would be gi en by:
LT(s)=p01 p12LT01 (s)LT12(s)+p02LT02(s)
Howe e , i mus be aken in o accoun ha ou lowg aph is ac ually pa o a mo e
gene al g aph ha would model he disease p ocess, see Figu e 5. Passage om s a e 0
o s a e 2 is no ce ain o occu : a pa ien may only su e ecu ences, o e en no
ecu ence. The p obabili y o aking he conside ed pa h is p
01
p
12
+p
02
, and we mus
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Figu e 2 E lang mix u e (smoo h line) and empi ical dis ibu ion (s ep unc ion) o ansi ion 01.
Figu e 3 E lang mix u e (smoo h line) and empi ical dis ibu ion (s ep unc ion) o ansi ion 02.
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di ide he p eceding LT(s) by his p obabili y o ob ain he ue Laplace ans o m [12,
pag. 19]
LT(s)=p01 p12LT01 (s)LT12(s)+p02LT02(s)
p01 p12 +p02
Figu e 4 E lang mix u e (smoo h line) and empi ical dis ibu ion (s ep unc ion) o ansi ion 12.
Figu e 5 Recu ence - p og ession p ocess.
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P obabili ies p
ij
a e assigned om es ima ions based on ou da a. They simply consis
o he a ios be ween he numbe o p og essions o ecu ences and he numbe o
pa ien s who could unde go he ele an ansi ion. Calcula ions a e qui e sensi i e o
hese alues. We ied wi h he cu en and also p e ious da abase. The bes esul s
we e ob ained aking p
01
= 0.3967742, p
02
= 0.02507837 and p
12
= 0.03252033.
To eco e hePDFweusea a ian o hein e sion algo i hm EULER [15]. F om
his unc ion we ob ain he su i al unc ion (wi h ega d o p og ession), ha is
shown in Figu e 6, join ly wi h he empi ical su i al unc ion. The haza d unc ion
may be also easily compu ed. Thus we ha e a pa ame ic model o p edic he p ob-
abili y o being ee o p og ession a a gi en ime. The p ocedu e may be easily used
o de ine isk g oups, simply by calcula ing he su i al unc ions o pa ien s g ouped
acco ding o common cha ac e is ics. Then he moni o ing and ea men o pa ien s
can be adjus ed acco ding o hei isk.
All compu a ions we e made in R. Besides he men ioned packages, we also used he
expm [19], Ma ix [20] and su i al [21] packages.
Discussion
A pa ame ic app oach in he amewo k o lowg aph models in ol es explo ing pa a-
me ic models looking o he dis ibu ions ha ma ch he da a be e . In [12] his o-
g ams o sample wai ing imes a e sugges ed. In his pape we p opose a i ing
p ocedu e using mix u e o E lang dis ibu ions. Figu es 2, 3 and 4 show g aphically
ha he i ed pa ame ic dis ibu ions ma ch he empi ical dis ibu ions e y well.
Figu e 6 Su i al unc ion model (smoo h line) and empi ical su i al unc ion (s ep unc ion).
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