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Modélisation Conjointe de l'Évolution de La Fonction Rénale
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�1 �1
1 3
�1 �1
1
st3
rd
< 2.3 g.kg�1
< 2.3g.kg�1
t
T ⇤
i Ti, �i
Ti = min(T ⇤i , Ci)
�i =
⇢0 T ⇤
i > Ci
1
i T ⇤i Ci
T ⇤
T ⇤
T ⇤ R+ f(t) f(t) = lim�t!0+P (tT ⇤<t+�t)
�tf(t)�t t t+�t
T ⇤ F (t) = P{T ⇤ t}t F (t) =R t
0 f(u)duF (0) = 0 limt!+1 F (t) = 1
S(t) S(t) = P (T ⇤ > t) = 1 � F (t)t
tt +1 t 2 R+
�(t)
�(t) = lim
�t!0+
P (t T ⇤ < t+�t|T ⇤ � t)
�t=
f(t)
S(t)
�(t)�t t t+�tt
⇤(t) F (t) =R t0 �(u)du
T ⇤
tjtj
ˆS(tj) = ˆP (T ⇤ > tj) = ˆP (T ⇤ > tj |T ⇤ > tj�1)ˆP (T ⇤ > tj�1)
pj = P (T ⇤ > tj |T ⇤ > tj�1) j
p̂j = (nj � dj)/nj
dj tj nj
tj nj = nj�1 � cj�1 � dj�1 cj�1
tj�1 tj tpj t
ˆS(t) =Y
j:tjt
nj � djnj
=
Y
j:tjt
1� dj
nj
!
ˆS(t)
n
L(✓) =nY
i=1
li(Ti|✓)
li(Ti|✓) i Ti ✓
T ⇤i Ci
i
f(T ⇤i ) i �i = 1
S(Ci) �i = 0
n
L(✓) =nY
i=1
f(Ti)�iS(Ti)
1��i
�(t) = �0(t) exp(�X)
�(t) �0(t)�X
ln[� ln(
ˆS(t))]
u Z
T ⇤ t > u
f(t) = lim
�t!0+
P (t T ⇤ < t+�t)
�t
= lim
�t!0+
P (t T ⇤ < t+�t, T ⇤ > u)
�t
= lim
�t!0+
P (t T ⇤ < t+�t|T ⇤ > u)P (t > u)
�t
= P (t > u) lim
�t!0+
P (t T ⇤ < t+�t|T ⇤ > u)
�t
P (t > u) t
t = u lim�t!0+P (tT ⇤<t+�t|T ⇤>u)
�t t = uT ⇤
C C > u
P (t > C) = P (t > C, t > u)
= P (T > u)P (T > C|T > u)
P (T > u) 1
ere
P (T > C|T > u) eme
�(t,X, Z) = �0(t) exp(�X + �(t)Z)
= �0(t) exp(�X + 11Z2[0;u]�1z + 11Z2]u;T ]�2Z)
= �0(t) exp(�X + �1Z1 + �2Z2)
T �X �1 �2 Z
]0;u] ]u;T ]
n i i = 1, . . . , kYi = (Yi1, Yi2, . . . , Yini)
T ni i N =
Pki=1 ni
Yi = �Xi + ↵iZi + ✏i8>><
>>:
↵i ⇠ MVN (0,B)✏i ⇠ N (0,⌃i)
↵i ? ↵j 8↵i ? ✏i 8
i �
i i ↵i
B✏ij ⌃i
L(✓)
L(✓) =NY
i=1
Zf(Yi|↵i)f(↵i)d↵i
Yi = �Xi + ✏i✏i ⇠ N (0,⌃i)
L(✓) =NY
i=1
✓1
2⇡
◆ni/2
|Vi|�1/2exp
✓� 1
2
(Yi �Xi�)TV �1
i (Yi �Xi�)
◆
�
l(✓) = �1
2
NX
i=1
ni log(2⇡) + log |Vi|+ (Yi �Xi�)TV �1
i (Yi �Xi�)
Yi
Yi = f(Xi,�, Zi, ui) + ✏i
Yij j i
Yij =
8<
:
b01i + b11itij + ✏ij tij ⌧1b02i + b12itij + ✏ij ⌧1 < tij ⌧2b03i + b13itij + ✏ij tij > ⌧2
b0ki b1ki eme eme
⌧1 ⌧2✏ij ⇠ N (0,�2
) ✏ij ? ✏ij0 8 j, j0
tij = ⌧1 b01i + b11i⌧1 = b02i + b12i⌧1
) b02i = b01i + b11i⌧1 � b12i⌧1
tij = ⌧2 b02i + b12i⌧2 = b03i + b13i⌧2
) b03i = b02i + b12i⌧2 � b13i⌧2
) b03i = b01i + b11i⌧1 � b12i⌧1 + b12i⌧2 � b13i⌧2
Yij =
8<
:
b01i + b11itij + ✏ij tij ⌧1b01i + b11i⌧1 + b12i(tij � ⌧1) + ✏ij ⌧1 < tij ⌧2b01i + b11i⌧1 + b12i(⌧2 � ⌧1) + b13i(tij � ⌧2) + ✏ij tij > ⌧2
= 11tij2[0;⌧1[(b01i + b11itij)
+11tij2[⌧1;⌧2[(b01i + b11i⌧1 + b12i(tij � ⌧1))+11tij2[⌧2;+1[(b01i + b11i⌧1 + b12i(⌧2 � ⌧1) + b13i(tij � ⌧2))+✏ij
= b01i + b11imin(tij , ⌧1) + b12imin(max(tij , ⌧1)� ⌧1, ⌧2 � ⌧1) + b13imax(0, tij � ⌧2) + ✏ij
ˆY (t)
Yi Ti
f(Yi, Ti)
�(t|X1i;↵i) = �0(t) exp(�X1i + ⇣g(↵i; t))
�0(t)
� X1i
⇣⇣
g(↵i; t)
t E(Yi|↵i) = �Xi + ↵iZi
dY (t)dt
f(Yi, Ti) =
Zf(Yi|↵i)f(Ti|↵i)f(↵i)d↵i
L(✓;Y, T, �) =NY
i=1
Zf(Yi|↵i; ✓)�(Ti|↵i; ✓)
�S(Ti|↵i; ✓)f(↵i|✓)d↵i
f(Yi|↵i; ✓) E[Yi] = �X2i+↵iZi
Ini�2
↵i
�(Ti|↵i; ✓)�
S(Ti|↵i; ✓)f(↵i; ✓)
↵
↵ = 0.05
↵ = 0.05
B
↵ = 0.20
↵ = 0.05
↵ = 0.05
↵ = 0.20
↵ = 0.20
↵ = 0.05
(t) =
(140� (t))⇥ (t)
(t)
!⇥✓
1.041.23
◆
�1
�1
µ �1
1
�2
� �1
p
�1
� �1 < �1 �2
<
<
<�
<�
µ �1
<�
1
<�
<�
>
ln[� ln(
ˆS(t))]
95%)
� �1 < �1
� < <� < <
� µ �1 < µ �1
<1
� <� <
Temps depuis la greffe (en années)
log(−l
og(S
(t))
0 2 4 6 8 10 12 14 16 18 20
−8−6
−4−2
02
PgPr < 2.3 g.kg − 1PgPr ≥ 2.3 g.kg − 1
ln[� ln(
ˆS(t))]
< < �1
�
95%)
� <
� µ �1 < µ �1
<1
� <� �1 < �1
<� �1 < �1
�� �1 < �1
< �
95%)
� <
� µ �1 < µ �1
<1
� <� �1 < �1
<� �1 < �1
�� �1 < �1
< �1 ��1
� � �1
<
95%)
� <
� µ �1 < µ �1
<1
� <� �1 < �1
� �1 < �1
<� �1 < �1
�� �1 < �1
� �1 < �1
<<
� < <� < <
� µ �1 < µ �1
<1
� < <� <
1 3
2
�1
�1
b1i
b3i
b2i
<µ �1 <
< �1 <b0 <b1 <
b2 <b3 <
� �1 <b0 <b1 <
b2 <b3 <
< �1 �b0 <b1 <
b2 <b3 <
� �1 �b0 <b1 <
b2 <b3 <
<<
� < <� µ �1 < µ �1
<� <
ˆY (t)
Résidus
Fréquence
−50 0 50 100
01000
2000
3000
4000
5000
6000
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1.25 7
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Temps post−greffe (en années)
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l (m
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)
1.25 7
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0 50 100 150
−50
5
Prédictions
Rés
idus
sta
ndar
disé
s
0 5 10 15
3040
5060
70
PgPr < 2.3 g.Kg−1 et âge du donneur < 55 ans
Temps post−greffe (en années)
Cla
iranc
e de
la c
réat
inin
e (m
l.min−1
)
●
●
●
●
●●
●●
●●
●●
●
●
●
●
●
●
●
●
●
● ● ● ●●
● ●●
●
●
●
●
●
●
●
●
●
●
●
1.25 70.5 3.5
Modèle longitudinalModèle conjoint
0 5 10 15
3040
5060
PgPr ≥ 2.3 g.Kg−1 et âge du donneur < 55 ans
Temps post−greffe (en années)
Cla
iranc
e de
la c
réat
inin
e (m
l.min−1
)1.25 70.5 3.5
Modèle longitudinalModèle conjoint
0 5 10 15
2030
4050
60
PgPr < 2.3 g.Kg−1 et âge du donneur ≥ 55 ans
Temps post−greffe (en années)
Cla
iranc
e de
la c
réat
inin
e (m
l.min−1
)
1.25 70.5 3.5
Modèle longitudinalModèle conjoint
0 5 10 15
2030
4050
PgPr ≥ 2.3 g.Kg−1 et âge du donneur ≥ 55 ans
Temps post−greffe (en années)
Cla
iranc
e de
la c
réat
inin
e (m
l/min
)
1.25 70.5 3.5
Modèle longitudinalModèle conjoint
< µ �1 <
�1
2.3 g.kg�1
1 3
1 3
< 2.3 g.kg�1
< 2.3 g.kg�1
23
1
Effet aléatoire sur le niveau initial
Fréq
uenc
e
−40 −20 0 20 40
050
100
150
200
250
Effet aléatoire sur la pente 1
Fréq
uenc
e
−20 0 20 40
050
100
150
200
Effet aléatoire sur la pente 2
Fréq
uenc
e
−10 −5 0 5 10
010
020
030
0
Effet aléatoire sur la pente 3
Fréq
uenc
e
−10 −5 0 5
010
020
030
040
0
0 5 10 15
2030
4050
6070
Temps post−greffe (en années)
Cla
iranc
e de
la c
réat
inin
e (m
l.min−1
)
●
●
●
●
●●
●●
●●
●
●
●
●
●
●
●
●
●
● PgPr < 2.3 g.Kg−1 et âge du donneur < 55 ansPgPr ≥ 2.3 g.Kg−1 et âge du donneur < 55 ansPgPr < 2.3 g.Kg−1 et âge du donneur ≥ 55 ansPgPr ≥ 2.3 g.Kg−1 et âge du donneur ≥ 55 ans
< µ �1 <
Effet aléatoire sur le niveau initial
Fréq
uenc
e
−40 −20 0 20 40
050
100
150
200
250
Effet aléatoire sur la pente 1
Fréq
uenc
e
−20 0 20 40
050
100
150
200
Effet aléatoire sur la pente 2
Fréq
uenc
e
−10 −5 0 5 10
050
100
150
200
250
300
350
Effet aléatoire sur la pente 3
Fréq
uenc
e
−10 −5 0 5
010
020
030
040
0
0 5 10 15
1020
3040
5060
70
Temps post−greffe (en années)
Cla
iranc
e de
la c
réat
inin
e (m
l.min−1
) ●
●●●●● ● ● ●
●●
●●
●●
●●
●●
●
● PgPr < 2.3 g.Kg−1 et âge du donneur < 55 ansPgPr ≥ 2.3 g.Kg−1 et âge du donneur < 55 ansPgPr < 2.3 g.Kg−1 et âge du donneur ≥ 55 ansPgPr ≥ 2.3 g.Kg−1 et âge du donneur ≥ 55 ans
< µ �1 <
Effet aléatoire sur le niveau initial
Fréq
uenc
e
−20 0 20 40
050
100
150
Effet aléatoire sur la pente 1
Fréq
uenc
e
−40 −20 0 20 40 60
050
100
150
200
250
300
Effet aléatoire sur la pente 2
Fréq
uenc
e
−10 0 10 20
010
020
030
0
Effet aléatoire sur la pente 3
Fréq
uenc
e
−10 −5 0 5
050
100
150
200
