Quantitative modeling of metastasis: cancer at the organism scale

(1)

HAL Id: hal-03261886

https://hal.inria.fr/hal-03261886

Submitted on 16 Jun 2021

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Quantitative modeling of metastasis: cancer at the

organism scale

Sébastien Benzekry

To cite this version:

Sébastien Benzekry. Quantitative modeling of metastasis: cancer at the organism scale. Doctoral.

United States. 2021. �hal-03261886�

(2)

Quantitative modeling of metastasis: cancer at the organism scale

S. Benzekry

Head of the Inria-Inserm team COMPO

Marseille, France

(3)

MATHEMATICAL MODELING

J. Ciccolini

R. Fanciullino

A. Rodallec

Mechanistic

modeling

Pharmacometrics

Statistics

Data science

D. Barbolosi

S. Benzekry

D. Barbolosi, S. Benzekry

F. Gattacceca

S. Benzekry

PHARMACOLOGY

Experimental

J. Ciccolini

R. Fanciullino

B. Lacarelle

Clinical

MEDICINE

Pr L. Greillier Dr X. Muracciole

Pr S. Salas

(4)

Clinical problem

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g(t, v)

ﬂ

(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

g

(v)

= (

–

_≠

—

ln(v)) v

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

µ

V

p

(t)

“

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0

g

(t, V

0

)ﬂ(t, V

0

) = —(V

p

(t))

ﬂ

(0, v) = ﬂ

0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0 t œ]0, +Œ[, V œ]V

0

,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) = µV

p

(t)

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0

V

_œ]V

₀

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(

g

(t, V )

ﬂ

(t, V )) = 0 t œ]0, +Œ[, V œ]V

0

,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0

V

œ]V

0

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(g(v)ﬂ(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) = d(V

p

(t))

ﬂ

(0, v) = 0

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

8 >

<

>

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

d

(V

p

(t))

Ä

+

s

_V+Œ₀

d

(v)ﬂ(t, v)dv

ä

ﬂ

(0, v) = ﬂ

0

g

(V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

+

⁄

_+Œ V0

—

(V )

ﬂ

(t, V ) dV

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

=

µ

V

p

(t)

g

(V

0

)ﬂ(t, V

0

) =

d

(V

p

(t))

Å

+

⁄

+Œ V0

d

(v)ﬂ(t, v)dv

ã

4

(5)

Clinical problem

Early-stage breast cancer

• 94% of cases are local or regional at diagnosis but

30% will relapse

• Estimation of the metastatic risk is key to

individualize adjuvant

therapy

• Reduce the number of chemo cycles for patients with low risk

Brain metastases in non-small cell lung cancer (NSCLC)

• Decide whether to use

whole brain radiation therapy

or just

(6)

Objectives

• Use a

mechanistic model

to predict metastasis

• Combine with machine learning algorithm to

select features

(7)

Clinical problem

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g(t, v)

ﬂ

(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

g

(v)

= (

–

_≠

—

ln(v)) v

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

µ

V

p

(t)

“

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0

g

(t, V

0

)ﬂ(t, V

0

) = —(V

p

(t))

ﬂ

(0, v) = ﬂ

0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0 t œ]0, +Œ[, V œ]V

0

,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) = µV

p

(t)

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0

V

_œ]V

₀

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(

g

(t, V )

ﬂ

(t, V )) = 0 t œ]0, +Œ[, V œ]V

0

,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0

V

œ]V

0

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(g(v)ﬂ(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) = d(V

p

(t))

ﬂ

(0, v) = 0

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

8 >

<

>

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

d

(V

p

(t))

Ä

+

s

_V+Œ₀

d

(v)ﬂ(t, v)dv

ä

ﬂ

(0, v) = ﬂ

0

g

(V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

+

⁄

_+Œ V0

—

(V )

ﬂ

(t, V ) dV

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

=

µ

V

p

(t)

g

(V

0

)ﬂ(t, V

0

) =

d

(V

p

(t))

Å

+

⁄

+Œ V0

d

(v)ﬂ(t, v)dv

ã

4

(8)

Experimental

Clinical

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cells)

104 105 106 107 108 109 1010

Metastatic burden (cells)

104 105 106 107 108 109 1010 Orthotopic implantation Surgery (t=34) Pre-surgical PT Post-surgical MB Time (days) 0 10 20 30 40 50 60 70

Primary tumor size (cells)

104 105 106 107 108 109 1010

Metastatic burden (cells)

104 105 106 107 108 109 1010 Orthotopic implantation Surgery (t=34) Pre-surgical PT Post-surgical MB

SMARTc wet lab

collaboration with J. Ebos, Roswell Park Cancer Institute, Buffalo, NY

n >

400 animals

with different treatments

and schedules

Months 0 10 20 30 40 50 Metastases diameter (mm) 0 5 10 15 20 25 30 35 40 45 A Months 0 10 20 30 40 50

Primary tumor diameter (mm)

0 5 10 15 20 25 30 35 40 45 TKI CT1 mTKI CT2 TGI fit Data B Months 0 10 20 30 40 50

Number of visible metastases

0 2 4 6 8 10 12 14 16 18 20 C Months 0 10 20 30 40 50 Metastases diameter (mm) 0 5 10 15 20 25 30 35 40 45 Data

Gompertz growth prediction

D

1

• Number and sizes

of brain metastasis in individual NSCLC

patients (n=31)

• Databases of metastatic relapse in breast cancer patients with

no adjuvant therapy (n=642, p=21 and n=167, p=9)

n

p

(9)

Clinical problem

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g(t, v)

ﬂ

(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

g

(v)

= (

–

_≠

—

ln(v)) v

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

µ

V

p

(t)

“

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0

g

(t, V

0

)ﬂ(t, V

0

) = —(V

p

(t))

ﬂ

(0, v) = ﬂ

0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0 t œ]0, +Œ[, V œ]V

0

,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) = µV

p

(t)

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0

V

_œ]V

₀

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(

g

(t, V )

ﬂ

(t, V )) = 0 t œ]0, +Œ[, V œ]V

0

,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0

V

œ]V

0

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(g(v)ﬂ(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) = d(V

p

(t))

ﬂ

(0, v) = 0

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

8 >

<

>

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

g

(V

0

)ﬂ(t, V

0

) =

d

(V

p

(t))

Ä

+

s

_V+Œ₀

d

(v)ﬂ(t, v)dv

ä

ﬂ

(0, v) = ﬂ

0

g

(V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

+

⁄

_+Œ V0

—

(V )

ﬂ

(t, V ) dV

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

=

µ

V

p

(t)

g

(V

0

)ﬂ(t, V

0

) =

d

(V

p

(t))

Å

+

⁄

+Œ V0

d

(v)ﬂ(t, v)dv

ã

4

(10)

Lapse time from diagnosis 0

- t

diag Detectability threshold Dissemination: d Primary tumor growth g TTR lo g (T um o r siz e) Metastatic growth Pre-surgical history TTR

V

diag g_p

V

vis

Mathematical models

Growth rates of

primary

and

secondary

tumors

g

p

and

g

Dissemination rate

Size distribution of the metastases ρ(t,v)

[Iwata et al., 2000]

Metastatic burden (total number of metastatic cells)

÷

i

≥ N (0, Ê

2

)

◊

i

= ◊

pop

+ —

T

x

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

x

i

_{œ R}

p

N (t)

N

(t) =

⁄

_t 0

d

(V

p

(s))ds = E [N(t)]

ln(µ) = ln(µ

m

) + µ

‡

Á,

Á

≥ N (0, 1)

⁄

_+Œ x

ﬂ

(t, y)dy

M

(t) =

⁄

_+Œ V0

vﬂ

(t, v)dv =

⁄

_t 0

d

(V

p

(t ≠ s)) V (s)ds = E

S

U

ÿ

iØ1

V

i

(t)

T

V

T

n1, ... , nk≠1, j

:= T

n1, ... , nk≠1

+ ˜

_T

n1, ... , nk≠1, j

M

(t) =

⁄

_+Œ V0

vﬂ

(t, v)dv =

⁄

_t 0

d

(V

p

(t ≠ s)) V (s)ds

—

(V ) = mV

–

dV

p

dt

= (

–

p

≠

—

p

ln(V

p

)) V

p

dV

p

dt

=

g

p

(V

p

)

,

V

p

(t = 0) = V

i

dV

p

dt

=

g

p

(t, V

p

)

,

V

p

(t = 0) = V

i

dV

p

dt

= g

p

(t, V

p

)

3

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g(t, v)

ﬂ

(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) =

µ

V

_p

(t)

ﬂ

(0, v) = 0

g

(v)

= (

–

_≠

—

ln(v)) v

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) =

µ

V

p

(t)

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

((

–

≠

—

ln(v)) vﬂ(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) =

µ

V

p

(t)

“

ﬂ

(0, v) = 0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0

g

(t, V

₀

)ﬂ(t, V

₀

) = —(V

p

(t))

ﬂ

(0, v) = ﬂ

0

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(g(t, V )ﬂ(t, V )) = 0 t œ]0, +Œ[, V œ]V

0 ,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) = µV

p

(t)

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0 V

_œ]V

₀

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, V ) + ˆ

V

(

g

(t, V )

ﬂ

(t, V )) = 0 t œ]0, +Œ[, V œ]V

0 ,

+Œ[

g

(t, V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

t

œ]0, +Œ[

ﬂ

(0, v) = ﬂ

0 V

_œ]V

₀

,

+Œ[

8 <

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(g(v)ﬂ(t, v)) = 0

g

(V

₀

)ﬂ(t, V

₀

) = d(V

p

(t))

ﬂ

(0, v) = 0

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

8 >

<

>

:

ˆ

t

ﬂ

(t, v) + ˆ

v

(

g

(v)

ﬂ

(t, v)) = 0

g

(V

0 )ﬂ(t, V

0 ) =

d

(V

p

(t))

Ä

+

s

_V

+Œ

₀

d

(v)ﬂ(t, v)dv

ä

ﬂ

(0, v) = ﬂ

0 g

(V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

+

⁄

_+Œ

V

0

—

(V )

ﬂ

(t, V ) dV

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

=

µ

V

p

(t)

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

Å

+

⁄

+Œ

V

0

d

(v)ﬂ(t, v)dv

ã

4 ex: Gompertz

Number of metastases with size larger than the

visible size V

_vis

N

vis

(t) =

⁄

_+Œ Vvis

ﬂ

(t, v)dv =

⁄

t_≠·vis 0

d

(V

p

(t))dt

N

vis

(t) =

⁄

_+Œ Vvis

ﬂ

(t, v)dv

=

⁄

t≠·vis 0

d

(V

p

(t))dt

N

vis

(t) =

⁄

_+Œ Vvis

ﬂ

(t, v)dv

=

⁄

t≠·vis 0

µV

p

(t)dt

M

(t) = µ

⁄

_t 0

V

p

(t)V (t ≠ s)ds

µ

i

= µ

_pop

+ —

T

x

i

+ ÷

_i

,

÷

_i

_{≥ N (0, Ê}

2

)

ln

µ

i

= ln (µ

pop

) + —

µT

x

iµ

+ ÷

µi

,

÷

µi

≥ N (0, Ê

µ2

)

ln

–

i

= ln (–

pop

) + —

–T

x

–i

+ ÷

–i

,

÷

–i

≥ N (0, Ê

–2

)

–

i

= µ

pop

+ —

KiT 67

Ki67

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

µ

i

= µ

pop

+ —

T

Ki67

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

◊

i

= ◊

pop

+ —

T

x

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

x

i

ln

◊

i

= ln (◊

pop

) + —

T

x

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

◊

i

= ◊

pop

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

2 Time to relapse

(TTR) = time from diagnosis to first visible met

Volume (cells) 100 ₁₀5 ₁₀10 ₁₀15 Number of mets 0 5 10 15 20 25 30

Imaging detection limit

g(v)

÷

i

≥ N (0, Ê

2

)

◊

i

= ◊

pop

+ —

T

x

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2

)

x

i

œ R

p

N (t)

N

(t) =

⁄

t 0

d

(V

p

(s))ds = E [N (t)]

ln(µ) = ln(µ

m

) + µ

‡

Á,

Á

≥ N (0, 1)

⁄

_+Œ x

ﬂ

(t, y)dy

M

(t) =

⁄

_+Œ V0

vﬂ

(t, v)dv =

⁄

t 0

d

(V

p

(t ≠ s)) V (s)ds = E

"

ÿ

iØ1

V

i

(t)

#

T

n1, ... , nk≠1, j

:= T

n1, ... , nk≠1

+ ˜

_T

n1, ... , nk≠1, j

M

(t) =

⁄

_+Œ V0

vﬂ

(t, v)dv =

⁄

_t 0

d

(V

p

(t ≠ s)) V (s)ds

d

(V

p

) = µV

p“

—

(V ) = mV

–

dV

p

dt

= (

–

p

≠

—

p

ln(V

p

)) V

p

dV

p

dt

=

g

p

(V

p

),

V

p

(t = 0) = V

i

dV

p

dt

=

g

p

(t, V

p

),

V

p

(t = 0) = V

i

3

(11)

• Classical approach considers each

subject independently

Statistical procedure for model calibration: nonlinear mixed effects modeling

Subject 1 ≤ i ≤N, Time t

j

Likelihood maximization

Lavielle,CRC press, 2014

g

(V

₀

)ﬂ(t, V

₀

) =

—

(V

p

(t))

+

⁄

_+Œ

V

0

—

(V )

ﬂ

(t, V ) dV

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

=

µ

V

p

(t)

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

_p

(t))

Å

+

⁄

+Œ

V

0

d

(v)ﬂ(t, v)dv

ã

ˆ

t

ﬂ

(t, V ) + ˆ

V

(

g

(t, V )

ﬂ

(t, V )) = 0

ﬂ

(t, V ) ≥ e

⁄

0

t

(V )

⁄

_+Œ

V

0

ﬂ

(t, V )dV

⁄

_+Œ

V

0

V ﬂ

(t, V )dV

Y

_i

j

= M(t

j

_i

,

—

j

) + Á

j

_i

Y

_i

j

= M(t

j

_i

,

—

j

) + Á

j

_i

—

1 , . . . , —

N

_≥

_{LN (—}

µ

, —

Ê

)

,

—

µ

œ R

p

, —

Ê

œ R

p

◊p

y

_j

i

= M(t

i

_j

,

◊

i

) + ‡Á

i

_j

,

Á

i

_j

_{≥ N (0, 1)}

M

_j

i

= M(t

i

_j

,

◊

i

) + ‡(M(t

_j

i

,

◊

i

))Á

i

_j

,

Á

i

_j

_{≥ N (0, 1)}

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

y

j

_i

= M(t

j

_i

,

◊

j

) + Á

j

_i

,

◊

1 , . . . , ◊

N

_≥

_{LN (◊}

µ

, ◊

Ê

)

,

◊

µ

œ R

p

, ◊

Ê

œ R

p

◊p

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

◊

1 , . . . , ◊

N

_≥

_{LN (◊}

_µ

, ◊

_Ê

)

,

◊

_µ

_{œ R}

p

, ◊

_Ê

_{œ R}

p

◊p

ˆ◊

j

_{= min}

◊

j

ÿ

Ä

y

_i

j

_{≠ M(t}

_i

, ◊

j

)

ä

2 P(Mets) = P

Å

µ

⁄

_T

₁

0 V

(t)dt > 1

ã

5 g(V

0 )ﬂ(t, V

0 ) =

—

(V

p

(t))

+

⁄

_+Œ

V

0

—

(V )

ﬂ(t, V ) dV

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

p

(t))

=

µ

V

p(t)

g(V

0 )ﬂ(t, V

0 ) =

d

(V

p

(t))

Å

+

⁄

+Œ

V

0

d(v)ﬂ(t, v)dv

ã

ˆtﬂ(t, V ) + ˆV

(

g

(t, V )

ﬂ(t, V )) = 0

ﬂ

(t, V ) ≥ e

⁄

0

t

(V )

⁄

_+Œ

V

0

ﬂ(t, V )dV

⁄

_+Œ

V

0

V ﬂ(t, V )dV

Y

_i

j

= M(t

j

_i

,

—

j

) + Á

j

_i

Y

_i

j

= M(t

j

_i

,

—

j

) + Á

j

_i

—

1 , . . . , —

N

_≥

_{LN (—}

µ

, —

Ê

)

,

—µ

œ R

p

, —Ê

œ R

p

◊p

y

_j

i

= M(t

i

_j

,

◊

i

) + ‡Á

i

_j

,

Á

i

_j

_{≥ N (0, 1)}

M

_j

i

= M(t

i

_j

,

◊

i

) + ‡(M(t

_j

i

,

◊

i

))Á

i

_j

,

Á

i

_j

_{≥ N (0, 1)}

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

,

◊

1 , . . . , ◊

N

_≥

_{LN (◊}

µ

, ◊

Ê

)

,

◊

µ

œ R

p

, ◊

Ê

œ R

p

◊p

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

◊

1 , . . . , ◊

N

_≥

_{LN (◊}

µ

, ◊

Ê

)

,

◊µ

œ R

p

, ◊Ê

œ R

p

◊p

ˆ◊

i

_{= min}

◊

i

ÿ

y

_j

i

_{≠ M(ti}

, ◊

i

)

2 ˆ◊

j

= min

◊

j

ÿ

Ä

y

_i

j

_{≠ M(t}

i

, ◊

j

)

ä

₂

5 • Population

approach

Nvis(t) =

⁄

+Œ

V

vis

ﬂ(t, v)dv =

⁄

t

≠·

vis

0 d(Vp(t))dt

Nvis

(t) =

⁄

_+Œ

V

vis

ﬂ(t, v)dv

=

⁄

t

≠·

vis

0 d(Vp(t))dt

N

vis

(t) =

⁄

_+Œ

V

vis

ﬂ(t, v)dv

=

⁄

t

≠·

vis

0 µVp(t)dt

M

(t) = µ

⁄

_t

0 Vp

(t)V (t ≠ s)ds

µ

i

= µ

_pop

+ —

T

x

i

+ ÷

_i

,

÷

_i

_{≥ N (0, Ê}

2 )

ln

µ

i

= ln (µpop) + —

_µ

T

x

i

_µ

+ ÷

_µ

i

,

÷

_µ

i

_{≥ N (0, Ê}

_µ

2 )

ln

–

i

= ln (–pop) + —

_–

T

x

i

_–

+ ÷

_–

i

,

÷

_–

i

_{≥ N (0, Ê}

_–

2 )

–

i

= µ

pop

+ —

_Ki

T

67 Ki67

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2 )

µ

i

= µ

pop

+ —

T

Ki67

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2 )

◊

i

= ◊pop

+ —

T

x

i

+ ÷i

,

÷

i

≥ N (0, Ê

2 )

x

i

ln

◊

i

= ln (◊pop) + —

T

x

i

+ ÷i

,

÷i

_{≥ N (0, Ê}

2 )

ln ◊

i

_{= ln (◊}

pop

) + ÷

i

,

÷

i

≥ N (0, Ê

2 )

2

+

• Parameters to be estimated = 𝜃

_!"!

(𝑝), 𝜔

!(!$%)

'

and 𝜎

g

(V

0 )ﬂ(t, V

0 ) =

—

(V

p

(t))

+

⁄

_+Œ

V

0

—

(V )

ﬂ

(t, V ) dV

g

(V

0 )ﬂ(t, V

0 ) =

d

(V

p

(t))

=

µ

V

p

(t)

g

(V

₀

)ﬂ(t, V

₀

) =

d

(V

_p

(t))

Å

+

⁄

+Œ

V

0

d

(v)ﬂ(t, v)dv

ã

ˆ

t

ﬂ

(t, V ) + ˆ

V

(

g

(t, V )

ﬂ

(t, V )) = 0

ﬂ

(t, V ) ≥ e

⁄

0

t

(V )

⁄

_+Œ

V

0

ﬂ

(t, V )dV

⁄

_+Œ

V

0

V ﬂ

(t, V )dV

Y

_i

j

= M(t

j

_i

,

—

j

) + Á

j

_i

Y

_i

j

= M(t

j

_i

,

—

j

) + Á

j

_i

—

1 , . . . , —

N

_≥

_{LN (—}

_µ

, —

_Ê

)

,

—

_µ

_{œ R}

p

, —

_Ê

_{œ R}

p

◊p

y

_j

i

= M(t

i

_j

,

◊

i

) + ‡Á

i

_j

,

Á

i

_j

_{≥ N (0, 1)}

M

_j

i

= M(t

i

_j

,

◊

i

) + ‡(M(t

_j

i

,

◊

i

))Á

i

_j

,

Á

i

_j

_{≥ N (0, 1)}

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

,

◊

1 , . . . , ◊

N

_≥

_{LN (◊}

_µ

, ◊

_Ê

)

,

◊

_µ

_{œ R}

p

, ◊

_Ê

_{œ R}

p

◊p

y

_i

j

= M(t

j

_i

,

◊

j

) + Á

j

_i

◊

1 , . . . , ◊

N

_≥

_{LN (◊}

µ

, ◊

Ê

)

,

◊

µ

œ R

p

, ◊

Ê

œ R

p

◊p

ˆ◊

j

= min

◊

j

ÿ

Ä

y

_i

j

_{≠ M(t}

_i

, ◊

j

)

ä

2 P(Mets) = P

Å

µ

⁄

_T

₁

0 V

(t)dt > 1

ã

5

(12)

Validation on animal data

Nonlinear

mixed-effects

statistical model for

inter-animal variability

Benzekry et al. (Ebos), Cancer Res, 2016

⟹

same growth

for PT and mets:

α

_p

=

α

,

β

_p

=

β

Ebos lab Roswell Park Cancer Institute

N

vis

(t) =

⁄

_+Œ

V

vis

ﬂ

(t, v)dv =

⁄

_t

_≠·

_vis

0 d

(V

p

(t))dt

N

_vis

(t) =

⁄

_+Œ

V

vis

ﬂ

(t, v)dv

=

⁄

t

≠·

vis

0 d

(V

p

(t))dt

N

vis

(t) =

⁄

_+Œ

V

vis

ﬂ

(t, v)dv

=

⁄

t

≠·

vis

0 µV

p

(t)dt

M

(t) = µ

⁄

_t

0 V

p

(t)V (t ≠ s)ds

µ

i

= µ

_pop

+ —

T

x

i

+ ÷

_i

,

÷

_i

_{≥ N (0, Ê}

2 )

ln

µ

i

= ln (µ

_pop

) + —

_µ

T

x

i

_µ

+ ÷

_µ

i

,

÷

_µ

i

_{≥ N (0, Ê}

2 _µ

)

ln

–

i

= ln (–

_pop

) + —

_–

T

x

_–

i

+ ÷

_–

i

,

÷

_–

i

_{≥ N (0, Ê}

_–

2 )

–

i

= µ

pop

+ —

_Ki

T

67 Ki67

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2 )

µ

i

= µ

pop

+ —

T

Ki67

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2 )

◊

i

= ◊

_pop

+ —

T

x

i

+ ÷

_i

,

÷

_i

_{≥ N (0, Ê}

2 )

x

i

ln

◊

i

= ln (◊

pop

) + —

T

x

i

+ ÷

i

,

÷

i

≥ N (0, Ê

2 )

ln ◊

i

_{= ln (◊}

pop

) + ÷

i

,

÷

i

≥ N (0, Ê

2 )

2

• Main difficulty

: PDE model

• Solution

: fast Fourier transform for

computation of metastatic burden

(13)

Clinical data of individual breast metastatic relapse

Institut Bergonié, Bordeaux, FR

K =

21 features

n

=

642 pa

tie

nt

s

w

/o

a

dj

uva

nt

outcome

(14)

Descriptive power: fit to the data

Nicolò et al., JCO: Clin Cancer Inform, 2020

improve these metrics, a balanced, downsampled version

of the data set was constructed. This signiﬁcantly improved

positive predictive value and F1 as well as model calibration

(Appendix

Figure A5E-F

). We also compared our results

with classic Cox regression survival analysis, which was

found to exhibit similar predictive power (

Table 2

; Appendix

Figure A6

; Appendix

Table A2

).

DISCUSSION

We propose a mechanistic model for predicting metastatic

relapse after surgical intervention in patients with

early-stage breast cancer which, for the ﬁrst time, can simulate

the pre- and postdiagnosis history of the disease from data

available at diagnosis only. Notably, with a c-index of 0.65

(95% CI, 0.60 to 0.71), the mechanistic model achieved

performances similar to those of the RSF algorithm (95%

CI, 0.66 to 0.69) and Cox regression (95% CI, 0.62 to 0.67).

These were also comparable with the actual standard

prognosis model of breast cancer–speciﬁc survival

Adju-vant! (c-index, 0.71),

7

_{used for risk classiﬁcation in the}

MINDACT (ClinicalTrials.gov identiﬁer:

NCT00433589

)

trial.

39

_{Others also reported a similar c-index of 0.67 for}

prediction of relapse in untreated patients by using Cox

analysis.

40

_{Advanced deep learning algorithms did not}

outperform this predictive power (c-index, 0.68) for

pre-diction of survival, even though they integrated genomic

data.

13

_{A recent study that considered recurrence (local,}

regional, or distant) reported superior predictive power

(area under the curve, 0.81 for prediction of relapse

at 5 years v 0.73 in our analysis), which might be

explained by the much larger data set (15,314 patients)

and inclusion of epidemiologic data not available in our

analysis.

Grounded in the biology of the metastatic process, our

model provides insights not achievable by statistical

analysis alone. First, one of the most important and

well-known predictors of metastatic relapse—tumor size

23

_—is

directly incorporated into the model as an input parameter

used for calculation of the tumor age. Second, our model

allows testing whether covariates are associated with

Metastasis–Free Survival

(probability)

1.0 0.8 0.6 0 5 10 15 20 Kaplan−Meier estimate Model fit

Time Since Diagnosis (years)

A

Observed Probabilities

1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5

Predicted Probabilities

Calibration for 2-Year Outcome

Observed Probabilities

1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5

Predicted Probabilities

Observed Probabilities

1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5

Predicted Probabilities

B

FIG 3. Fit of the mechanistic model. (A) Mean of the conditional metastasis-free survival functions from the mechanistic model and Kaplan-Meier estimates with 95% CIs (shaded areas). (B) Calibration plots of metastasis-free survival for the ﬁnal mechanistic model with covariates. Vertical bars are 95% CIs.

Mechanistic Modeling of Metastatic Relapse in Breast Cancer

JCO Clinical Cancer Informatics 265

Metastasis-Free Survival

1.0

0.8

0.6

Time Since Diagnosis (years) 15 10 5 0 V_diag = 10 mm Metastasis-Free Survival 1.0 0.8 0.6

Time Since Diagnosis (years) 15 10

5 0

V_diag = 25 mm

FIG A3. Distant metastasis-free survival predictions of the mechanistic model and Kaplan-Meier estimates with 95% CIs stratiﬁed by values of the primary tumor size (diameter) at diagnosis.

1.0

0.8

0.6

Time Since Diagnosis (years) 15 10 5 0 Vdiag = 20 mm Metastasis-Free Survival 1.0 0.8 0.6

5 0

Vdiag = 25 mm

1.0

0.8

0.6

5 0

V_diag = 25 mm

1.0

0.8

0.6

5 0

Vdiag = 25 mm

the model remained within the Kaplan-Meier 95% CIs in all

cases, we observed that model predictions tended to be less

accurate as the tumor size increased, with overestimation of

the relapse risk for large PT sizes.

Mechanistic Covariate Analysis and Predictive Power of

the Mathematical Model

We next tested the covariates selected by the RSF analysis

in the mechanistic model. We built the covariate model by

using a backward elimination procedure, starting with the

full model with all the preselected covariates on both

pa-rameters α and µ. The ﬁnal model included Ki67 and CD44

on α, and EGFR on the dissemination parameter µ as

signiﬁcant covariates (

Table 1

). The cross-validated c-index

for this model was 0.65 (95% CI, 0.60 to 0.71). Calibration

plots for 2-, 5-, and 10-year outcomes demonstrated good

predictive accuracy of the model (

Fig 3B

). Nevertheless, as

in the RSF model, risk of relapse was overestimated in

high-risk groups. For the classiﬁcation task of predicting 5- and

10-year relapse, comparable results were obtained

be-tween the mechanistic model, the RSF, classiﬁcation

machine learning algorithms, and Cox regression (

Table 2

).

Predictive Simulations of the Mechanistic Model

The previous results allow calibration of the mechanistic

model parameters from variables available at diagnosis by

using only the covariate part in equation (2) and neglecting the

remaining unexplained variance. We used this to simulate the

natural cancer history for a number of representative

pa-tients in our data set. For each patient, the population-level

parameters α

pop

, µ

pop

, β

Ki67,α

, β

CD44,α,

and β

EGFR,µ

were

calibrated from an independent training set that did not

contain this patient (coming from the cross-validation

procedure). Simulations were then performed by using

a discrete version of the metastatic model

22

_{and are}

re-ported in

Figure 4

and Appendix

Figure A4

. Covariate

values used for prediction and the resulting inferred

individual parameters ˆα

i

and ˆµ

i

are reported in Appendix

Table A1

.

For patients 224 and 358 (

Fig 4A-B

), the model predicted

a time from the ﬁrst cell to detection (presurgical history) of

17.3 years and 3.61 years, respectively. In addition, the

model allowed prediction of the metastatic size distribution

at diagnosis (

Fig 4C-D

). Patient 224 was predicted to have

21 invisible metastases at the time of diagnosis. The largest

metastasis contained 2.32

× 10

4

_{cells and the smallest}

metastasis contained only one cell. The largest metastasis

was predicted to have initiated 12.9 years after the ﬁrst cell

of the PT (ie, 4.4 years before surgery). The model predicts

relapse 6.18 years after diagnosis, whereas true relapse

occurred at 4.88 years. At diagnosis, patient 358 was

predicted to have a total of 35 invisible metastases. The

largest metastasis contained 1.16

× 10

4

_{cells and the}

smallest contained 1.36 cells. According to this simulation,

the ﬁrst metastasis in this patient was emitted 2.6 years after

the PT onset (ie, 1.01 years before surgery). The model

predicted relapse at 1.97 years after diagnosis, but true

re-lapse occurred at 3.06 years. The differences in PT and

metastatic dynamics for these two patients are a result of the

different values of the covariates (Appendix

Table A1

). Distinct

levels of Ki67 cause distinct growth kinetics. Moreover, unlike

the tumor of patient 224, the tumor of patient 358 expresses

EGFR, which is associated with a higher metastatic potential.

Thus, although the tumor from patient 358 is much younger,

the total number of (invisible) metastases at surgery is

pre-dicted to be larger (35 v 21 metastases in patient 224). Model

predictions are also informative in the case of individuals who

were censored at the last follow-up. For instance, patient 70

(Appendix

Figure A4

) was censored at 17.7 years after

di-agnosis. This is consistent with our model, which predicts that

this patient was disease-free after PT resection and would

never have relapsed (TTR = +∞).

Comparison With Machine Learning Classiﬁcation

Algorithms and Cox Regression

We tested the predictive power of machine learning

clas-siﬁcation algorithms. These cannot account for

right-censored data. Thus, for this part, we focused on

pre-diction of 5-year metastatic relapse (yes or no). Best

per-formances were achieved by the random forest and logistic

regression models (

Table 2

and Appendix

Figure A5A-D

).

However, owing to the low event rate (9.25%), positive

predictive value and F1 scores were low (

Table 2

). To

TABLE 1.

Parameter Estimates

Parameter Estimate Relative Standard Error (%) P Model without covariates

log αpop −6.34 12.6

log µpop −26.8 3.68

σ 0.542 28.4

ωα 3.37 36.4

ωµ 3.78 15.9

Model with covariates

log αpop −9.01 10.8 βKi67,α 0.093 29.6 .001 βCD44,α 0.017 57.7 .083 log µpop −25.9 4.4 βEGFR,µ 0.053 38.1 .009 σ 0.606 24 ωα 2.75 22.1 ωµ 3.03 20.5

NOTE. Mechanistic mixed-effects models for the time to metastatic relapse were fitted to the data using likelihood maximization. Parameters α (growth) and µ (dissemination) were assumed to follow log-normal distributions with fixed effects (typical values) αpopand µpop, standard deviation of the random effects ωαand ωµ and standard deviation of the error model σ (see Methods). A first version was fitted without covariates and a second version included dependence of the individual parameters on individual covariates. P value refers to a Wald test for statistical significance of the covariate on the parameter value.

Nicol `o et al

(15)

Feature selection: random survival

forests

c-index = 0.69 (0.67 – 0.71)

M. Dmitrievsky, https://www.mql5.com/en/articles/3856

Ishwaran et al., Ann Appl Stat, 2008

⇒

nonlinear

effect of covariates and

non proportional

hazard