A biologically-based mathematical model for prediction of metastatic relapse

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https://hal.inria.fr/hal-01968980

Submitted on 3 Jan 2019

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A biologically-based mathematical model for prediction of metastatic relapse

Michalis Mastri, Cynthia Perier, Gaetan Macgrogan, John Ebos, Sébastien Benzekry

To cite this version:

Michalis Mastri, Cynthia Perier, Gaetan Macgrogan, John Ebos, Sébastien Benzekry. A biologically- based mathematical model for prediction of metastatic relapse. 17th Biennial Congress of the Metas- tasis Research Society, Aug 2018, Princeton, United States. �hal-01968980�

BACKGROUND & OBJECTIVES

AN ELEMENTARY THEORY OF METASTATIC DYNAMICS:

GROWTH + DISSEMINATION

A biologically-based mathematical model for prediction of metastatic relapse

G. MacGrogan ³ J. ML Ebos ¹ S.Benzekry ² M. Mastri ¹ C. Périer ²

• A biologically-based mathematical model was able to describe preclinical and clinical data of metastatic development in breast, lung and kidney cancer

• Machine learning algorithms used here don’t account for censored data + limited to prediction of relapse event at a fixed horizon

• The biological model accounts for these and could give predictions of the metastatic state at diagnosis and future evolution in order to guide therapeutic intervention

• Predictive power is only modest so far but only the primary tumor size was considered here as a feature with only other source of inter-subject variability being the dissemination parameter μ

==> include more features and link parameter μ and others to clinical features and biomarkers in a biologically relevant way

• Metastasis is the cause of 90% of deaths from solid tumors Chaffer and Weinberg, Science 2011

• ~ 20-30% of breast cancer patients will relapse with distant metastases EBCTG, Lancet, 2005

• For breast cancer, the current factors influencing decision for adjuvant therapy are: tumor size, nodal involvement, molecular factors (hormonal receptors and HER2 status), histological type and grade.

Poisson process for the dissemination with rate d(V

p

) = μV

pγ

Growth rates of primary and secondary tumors g

p

and g

Size distribution of the metastases ρ(t,v) Iwata et al., J Theor Biol, 2000

Figure 2

A ^Surgery

Metastatic burden (MB) Primary Tumor (PT)

Orthotopic Implantation

Isograft Model:

Kidney (Renca^luc+)

Xenograft Model:

Breast (LM2-4^luc+)

Surgery' (T=34) PT'

(presurgical) MET'

(postsurgical) Breast'Model'

(LM2=4^luc+)

Kidney'Model' (Renca^luc+)

Recommend(this(change(in(all(figures

Data primary tumor

Median model primary tumor

10th and 90th percentiles model primary tumor Data metastatic burden

Median model metastatic burden

10th and 90th percentiles model metastatic burden

Xenograft Model Isograft Model

Breast (LM2-4^luc+) Kidney (Renca^luc+)

B C

Fit

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=34) Pre-surgical

PT

Post-surgical MB

Time (days)

0 20 40 60 80

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=23) Pre-surgical

PT

Post-surgical MB

D E

Predict

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=38) Pre-surgical

PT

Post-surgical MB

Time (days)

0 20 40 60 80

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=30) Pre-surgical

PT

Post-surgical MB

1 Population fits

(nonlinear mixed-effects)

CLINICAL DATA OF METASTATIC RELAPSE PROBABILITY

2648 breast cancer patients screened for 20 years after primary tumor resection (no adjuvant therapy)

Koscielny et al., Br J Cancer, 1984




Cancer inception time can be inferred from PT volume V1 at diagnosis time T1 (Gompertz growth).

Lognormal distribution of metastatic parameter μ for inter-individual variability (2 degrees of freedom)

Diameter of primary tumor at

diagnosis (cm)

Proportion of patients developing

metastasis (%) Model fit

1 ÆD Æ2.5 27.1 25.5

2.5 < D Æ3.5 42.0 42.4

3.5 < D Æ4.5 56.7 56.3

4.5 < D Æ5.5 66.5 65.9

5.5 < D Æ6.5 72.8 74.3

6.5 < D Æ7.5 83.8 80.8

7.5 < D Æ8.5 81.3 85.7

8>

<

>:

ˆ_tfl(t, V) +ˆ_V(fl(t, V)g(V)) = 0 g(V0)fl(t, V0) =—(Vp(t))

fl(0, V) =fl⁰(V)

Z _+Œ

V₀ fl(t, V)dV

1

Z

_+Œ

V₀

V fl(t, V )dV

P (Mets) = P

Ç

µ

Z

_T1

0

V

_p

(t)dt > 1

å

m “

^µ

“

^‡

a K

2.61 · 10

^≠8

0.448 0.078 4.71 · 10

^≠4

10

¹²

G(V, K ) =

Ç aV ln ^Ä

_K^V

^ä bV ≠ dV

^2/3

K

å

dI

dt = pV

_p

(t) + ^Z pV fl(t, V, K )dV dK ≠ kI (1)

G(V, K ; V

_p

, fl) =

Ç aV ln ^Ä

_K^V

^ä

bV ≠ dV

^2/3

K ≠ eI (V

_p

, fl)K

å

dV

dt = –e

^≠—t

V

8 >

> >

> >

<

> >

> >

> :

ˆ

_t

fl(t, V, K ) + div(fl(t, V, K )G(V, K, fl, V

_p

)) = 0 ]0, T [ ◊ ]V

₀

, + Œ [ ◊ ]0, + Œ [

≠ G · ‹fl(t, V, K ) = ”

_{(V,K)=(V0,K0)}

^¶ — (V

_p

(t)) + ^R

_V⁺₀^Œ

^R

₀^+Œ

— (V )fl(t, V, K )dV dK ^© where G · ‹ < 0

fl(0, V ) = fl

⁰

(V ) ]V

₀

, + Œ [ ◊ ]0, + Œ [

fl = fl(t, V, K )

Z

_+Œ

V₀

Z

_+Œ

0

— (V )fl(t, V, K )dV dK

2

— ( V ) = mV

^–

8 >

<

> :

ˆ

_t

fl ( t, V ) + ˆ

_V

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

₀

, + Π[ g ( t, V

₀

) fl ( t, V

₀

) = — ( V

_p

( t )) + ^R

_V⁺₀^Œ

— ( V ) fl ( t, V ) dV t œ ]0 , + Œ [

fl (0 , v ) = fl

⁰

V œ ] V

₀

, + Π[

8 >

<

> :

ˆ

_t

fl ( t, V ) + ˆ

_V

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

₀

, + Π[ g ( t, V

₀

) fl ( t, V

₀

) = — ( V

_p

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

⁰

V œ ] V

₀

, + Π[

g ( V

₀

) fl ( t, V

₀

) = — ( V

_p

( t )) + ^Z

^+Œ

V₀

— ( V ) fl ( t, V ) dV

ˆ

_t

fl ( t, V ) + ˆ

_V

( g ( t, V ) fl ( t, V )) = 0

fl ( t, V ) ≥ e

^⁄0t

( V )

Z

_+Œ

V₀

fl ( t, V ) dV

Z

_+Œ

V₀

V fl ( t, V ) dV Y

_i^j

= M ( t

^j_i

, —

^j

) + E

_i^j

Y

_i^j

= M ( t

^j_i

, —

^j

) + E

_i^j

, —

¹

, . . . , —

^N

≥ N ( µ, Ê ) , µ œ R

^p

, Ê œ R

^p◊p

P (Mets) = P

Ç

m

Z

_T1

0

V ( t )

^“

dt > 1

å

dV

_p

dt = g

_p

( V

_p

( t ))

1

PERSONALIZED SIMULATIONS FOR DIAGNOSIS AND PROGNOSIS OF BRAIN METASTASES IN LUNG CANCER

PREDICTIVE PERFORMANCES FOR 5-YEARS METASTATIC RELAPSE FROM BREAST

S. Benzekry, A. Tracz, M. Mastri, R. Corbelli, D. Barbolosi, J.M.L. Ebos , Modeling spontaneous metastasis following surgery: an in vivo-in silico approach. Cancer Research, 76(3), 4931-40, 2016

CONCLUSION AND FUTURE DIRECTIONS

— (V ) = mV

^–

8 >

<

> :

ˆ

_t

fl(t, V ) + ˆ

_V

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

₀

, + Π[ g(t, V

₀

)fl(t, V

₀

) = — (V

_p

(t)) + ^R

_V⁺₀^Œ

—(V )fl(t, V ) dV t œ ]0, + Œ [

fl(0, v) = fl

⁰

V œ ]V

₀

, + Π[

8 >

<

> :

ˆ

_t

fl ( t, V ) + ˆ

_V

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

₀

, + Π[ g(t, V

₀

)fl(t, V

₀

) = —(V

_p

(t)) t œ ]0, + Œ [ fl (0 , v ) = fl

⁰

V œ ] V

₀

, + Π[

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) = fl

⁰

g ( V

₀

) fl ( t, V

₀

) = — ( V

_p

( t )) + ^Z

^+Œ

V₀

— ( V ) fl ( t, V ) dV

ˆ

_t

fl(t, V ) + ˆ

_V

(g(t, V )fl(t, V )) = 0

fl ( t, V ) ≥ e

^⁄0t

( V )

Z

_+Œ

V₀

fl(t, V )dV

Z

_+Œ

V₀

V fl ( t, V ) dV Y

_i^j

= M (t

^j_i

, —

^j

) + E

_i^j

Y

_i^j

= M (t

^j_i

, —

^j

) + E

_i^j

, —

¹

, . . . , —

^N

≥ N (µ, Ê), µ œ R

^p

, Ê œ R

^p◊p

P (Mets) = P

Ç

m

Z

_T1

0

V ( t )

^“

dt > 1

å

dV

_p

dt = g

_p

( V

_p

( t ))

1 Figure 2

A ^Surgery

Metastatic burden (MB) Primary Tumor (PT)

Orthotopic Implantation

Isograft Model:

Kidney (Renca^luc+)

Xenograft Model:

Breast (LM2-4^luc+)

Surgery' (T=34) PT'

(presurgical) MET'

(postsurgical) Breast'Model'

(LM2=4^luc+)

Kidney'Model' (Renca^luc+)

Recommend(this(change(in(all(figures

Data primary tumor

Median model primary tumor

10th and 90th percentiles model primary tumor Data metastatic burden

Median model metastatic burden

10th and 90th percentiles model metastatic burden

Xenograft Model Isograft Model

Breast (LM2-4^luc+) Kidney (Renca^luc+)

B C

Fit

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=34) Pre-surgical

PT

Post-surgical MB

Time (days)

0 20 40 60 80

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=23) Pre-surgical

PT

Post-surgical MB

D E

Predict

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=38) Pre-surgical

PT

Post-surgical MB

Time (days)

0 20 40 60 80

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=30) Pre-surgical

PT

Post-surgical MB

1

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=34) Pre-surgical

PT

Post-surgical MB

Time (days)

0 20 40 60 80

Primary tumor size (cells)

10⁴ 10⁶ 10⁸ 10¹⁰

Metastatic burden (cells)

10⁴ 10⁶ 10⁸ 10¹⁰ Orthotopic

implantation

Surgery (t=23) Pre-surgical

PT

Post-surgical MB

Time (days)

0 10 20 30 40 50 60 70

Primary tumor size (cell)

10⁵ 10⁶ 10⁷ 10⁸ 10⁹

Metastatic burden (cell)

10⁵ 10⁶ 10⁷ 10⁸ 10⁹ Orthotopic

implantation

Surgery (t=34) Pre-surgical

PT

Post-surgical MB

Xenograft model Breast (LM2-4^luc+)

Isograft model Kidney (Renca^luc+)

PRECLINICAL DATA AND POPULATION APPROACH

t = 18.0 years

Time (years)

0 5 10 15 20 25 30

Volume (cells)

10⁰ 10² 10⁴ 10⁶ 10⁸ 10¹⁰

Primary tumor

Volume (cells)

10⁰ 10⁵ 10¹⁰

Number of mets

0 5 10 15 20 25

Imaging detection limit Metastases

Volume (cells)

10⁰ 10⁵ 10¹⁰

Number of mets

0 5 10 15 20 25

Pre-diagnosis mets Post-diagnosis mets

Diagnosis

1

Department of Cancer Genetics and Medicine, Roswell Park Cancer Institute, Buffalo, New York, USA

2

Inria team MONC and Institut de Mathématiques de Bordeaux, Talence, France

3

Pathology department, Institut Bergonié, Centre de Recherche et de Lutte Contre le Cancer, Bordeaux, France

Time (days)

0 20 40 60 80

Primary tumor size (cell)

10⁴ 10⁵ 10⁶ 10⁷ 10⁸

Metastatic burden (cell)

10⁴ 10⁵ 10⁶ 10⁷ 10⁸

Orthotopic implantation

Surgery (t=26) Pre-surgical

PT

Post-surgical MB

Objective: establish a biologically- based mathematical model for

individualised prediction of metastasis

Total metastatic mass

Surgery Injection (or first cell)

Metastases Primary

Tumor (PT)

Dissemination law: d(Vp)= μ(Vp) ^γ PT growth law: gp(Vp)

Metastases growth law: g(V)

Pre-surgical Post-surgical

–

_P

, –

_ther

, –

_reb

µ, –

Z

M (t) = µ

Z _t

0

V

_p

(t)V (t ≠ s)ds

µ

ⁱ

= µ

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_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) =

^{Z t}

0

d(V

p

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

ln( µ ) = ln( µ

_m

) + µ

_‡

Á, Á ≥ N (0 , 1)

Z _+Œ

x

fl ( t, y ) dy

M (t) =

^{Z +Œ}

V₀

vfl(t, v )dv =

^{Z 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

(t , V

_p

)

1

Number of visible metastases

–

_P

, –

_ther

, –

_reb

µ, –

Z

N ( t ) = ^Z

^t≠·vis

0

d ( V

_p

( t )) dt M ( t ) = µ

Z

_t

0

V

_p

( t ) V ( t ≠ s ) ds

µ

ⁱ

= µ

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0 , Ê

²

)

◊

ⁱ

= ◊

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_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 ) = ^Z

^t

0

d ( V

_p

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

ln( µ ) = ln( µ

_m

) + µ

_‡

Á, Á ≥ N (0 , 1)

Z

_+Œ

x

fl ( t, y ) dy

M ( t ) = ^Z

^+Œ

V₀

v fl ( t, v ) dv = ^Z

^t

0

d ( V

_p

( t ≠ s )) V ( s ) ds

— ( V ) = mV

^–

dV

_p

dt = ( –

_p

≠ —

_p

ln( V

_p

)) V

_p

1

Individual parameter θ

ⁱ

–

_P

, –

_ther

, –

_reb

µ, –

Z

N ( t ) = ^Z

^t≠·vis

0

d ( V

_p

( t )) dt M ( t) = µ

Z

_t

0

V

_p

(t)V (t ≠ s)ds

µ

ⁱ

= µ

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0, Ê

²

)

◊

ⁱ

= ◊

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0, Ê

²

)

÷

_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 ) = ^Z

^t

0

d ( V

_p

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

ln( µ ) = ln( µ

_m

) + µ

_‡

Á, Á ≥ N (0 , 1)

Z

_+Œ

x

fl(t, y)dy

M (t) = ^Z

^+Œ

V₀

vfl(t, v)dv = ^Z

^t

0

d (V

_p

(t ≠ s)) V (s)ds

— (V ) = mV

^–

1

M

_jⁱ

= M ( t

ⁱ_j

, ◊

ⁱ

) + ‡ ( M ( t

ⁱ_j

, ◊

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

X Ä

y

_i^j

≠ M (t

_i

, ◊

^j

)

^ä2

P (Mets) = P

Ç

µ

Z _T1

0

V ( t ) dt > 1

å

dV

_p

dt = g

_p

(V

_p

(t))

Patient µ # metastases Patient µ # metastases

n

^¶

1 1.7 ◊ 10

^≠8

0 n

^¶

6 7.0 ◊ 10

^≠8

0

n

^¶

2 1.9 ◊ 10

^≠8

0 n

^¶

7 1.3 ◊ 10

^≠7

1

n

^¶

3 2.7 ◊ 10

^≠8

0 n

^¶

8 2.7 ◊ 10

^≠7

2

n

^¶

4 5.0 ◊ 10

^≠8

0 n

^¶

9 4.0 ◊ 10

^≠7

3

n

^¶

5 6.1 ◊ 10

^≠8

0 n

^¶

10 6.1 ◊ 10

^≠7

4

µ Protocole de Viens Optimized protocol

6 cycles 9 cycles 12 cycles 9 cycles 13 cycles 18 cycles 126 days 189 days 252 days 126 days 182 days 252 days

1.3 ◊ 10

^≠7

1 0 ı 0 ı ı

2.7 ◊ 10

^≠7

2 1 0 2 0 ı

4.0 ◊ 10

^≠7

3 2 1 3 1 0

6.1 ◊ 10

^≠7

5 4 3 4 3 1

8>

<

>:

ˆ

_t

fl(t, v ) + ˆ

_v

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

₀

, + Π[ g ( t, V

₀

) fl ( t, V

₀

) = µV

_p

( t ) t œ ]0 , + Œ [

fl (0 , v ) = 0 v œ ] V

₀

, + Π[

3

Data

Structural

model Error

model

A

Months post-diagnosis

0 10 20 30 40 50

Primary tumor diameter (mm)

0 5 10 15 20 25 30 35 40 45

TKI CT1 mTKI CT2

B

Months post-diagnosis

0 10 20 30 40 50

Metastases diameter (mm)

0 5 10 15 20 25 30 35 40 45

C

Months post-diagnosis

0 10 20 30 40 50

Number of visible metastases

0 2 4 6 8 10 12 14 16 18 20

C D

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

1

Model fit

Months post-diag

-20 0 20 40 60

Nb of visible mets

0 5 10 15 20 25

Model

Data

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20 Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

time

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

t = 19 t = 23

t = 28 t = 40

t = 43 t = 48

Model fit

Months post-diag

-20 0 20 40 60

Nb of visible mets

0 5 10 15 20 25

Model

Data

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20 Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

time

Met diameter (mm)

0 10 20 30 40 50

Number

0 2 4 6 8 10 12 14 16 18 20

t = 19 t = 23

t = 28 t = 40

t = 43 t = 48

Breast cancer data base (1057 patients), I. Bergonié

Years post diagnosis

-4 -3 -2 -1 0 1 2 3

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Years post-diagnosis

-6 -4 -2 0 2 4

Tumor size (cells)

10⁰ 10² 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹²

Figure 6: Clinical application

Simulations of the natural history

p tumor size clinical 0.00165 tumor size histological 0.4

grade 0.358

histology 0.0041

TNM T 0.146

TNM N 0.599

nb invaded ganglions 0.619 menopausal status 0.000471

ER 0.543

PR 0.34

Ki67 3.09e-05

HER2 0.17

HER2 intensity 0.482

CK56 0.793

EGFR 0.016

VIM 0.226

ALDH1 0.674

CD24 0.894

CD44 0.397

E cadherin 0.207

TRIO 0.0646

BCL2 0.727

age at diagnosis 0.0599 diagnosis year 0.101

radio 0.276

1

Features selection based on Cox regression

AUROC Accuracy NPV PPV Random forest 0.727 87 91.7 22.5 Logistic regression 0.772 90.1 91 25 Cox regression 0.728 87.4 91.1 16.7

Bio-based 0.641 89.7 91.3 31.2

1

AUROC = Area Under the ROC curve, NPV = Negative Predictive Value PPV = Positive Predictive Value

Time to relapse

T T R = inf { t > 0; N

_vis

( t ) > 1 }

–

_P

, –

_ther

, –

_reb

µ, –

Z

N ( t ) = ^Z

^t≠·vis

0

d ( V

_p

( t )) dt M ( t ) = µ

Z

_t

0

V

_p

( t ) V ( t ≠ s ) ds

µ

ⁱ

= µ

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0 , Ê

²

)

◊

ⁱ

= ◊

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0 , Ê

²

)

÷

_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 ) = ^Z

^t

0

d ( V

_p

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

ln( µ ) = ln( µ

_m

) + µ

_‡

Á, Á ≥ N (0 , 1)

Z

_+Œ

x

fl ( t, y ) dy

M ( t ) = ^Z

^+Œ

V₀

vfl ( t, v ) dv = ^Z

^t

0

d ( V

_p

( t ≠ s )) V ( s ) ds

1

T T R = inf { t > 0; N

_vis

( t ) > 1 }

–

_P

, –

_ther

, –

_reb

µ, –

Z

N

_vis

( t ) =

^{Z t≠·vis}

0

d ( V

_p

( t )) 1

_V

p(t)ÆV_d

dt

M ( t ) = µ

Z _t

0

V

_p

( t ) V ( t ≠ s ) ds

µ

ⁱ

= µ

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0 , Ê

²

)

◊

ⁱ

= ◊

_pop

+ —

^T

x

ⁱ

+ ÷

_i

, ÷

_i

≥ N (0 , Ê

²

)

÷

_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 ) =

^{Z t}

0

d ( V

_p

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

ln( µ ) = ln( µ

_m

) + µ

_‡

Á, Á ≥ N (0 , 1)

Z _+Œ

x

fl ( t, y ) dy

M ( t ) =

^{Z +Œ}

V₀

v fl ( t, v ) dv =

^{Z t}

0

d ( V

_p

( t ≠ s )) V ( s ) ds

1