Tumor growth: From mathematical models to clinical applications - - PowerPoint PPT Presentation

Tumor growth: From mathematical models to clinical applications

&

Christophe Letellier

Clément Draghi

Biomedical Engineer

Fabrice Denis

• ncologist

Alain Tolédano

• ncologist

 Why genetics cannot explain everything  A simple model taking into account the microenvironment  Spatial growth versus local dynamics  Cancer risk depends on the tissue  A web-mediated follow-up  A model for the PSA level in patients with prostate cancer Outline in few words

Incidence of cancers in US in 2014

Cancer risk strongly depends

• n the organ

External factors affecting cancer risk during lifetime

State of the tissue

Hereditary syndrome (genetic mutation) Lack of physical activity Obesity Bad quality food Tobacco addiction Alcohol addiction

10% Tissular dynamics = Cell interaction

Cancer risk: ‘’bad luck’’ or not?

Lifetime risk of cancer of many different types is strongly correlated with the total number

• f divisions of the normal self-

renewing cells…

r = 0.81, p < 410-8 The majority of the variations in cancer risk is due to bad luck, that is, random mutations arising during DNA replication in normal stem cells.

Age at death Average number of deaths per year

rman rwoman Nman Nwoman

Hypothesis: the probability to present tumor cells depends on the number

• f cell divisions during life
• Cancer incidence versus age

Corollary: The longer the life, the larger the cancer risk In the United-Kingdom

(all types of cancer included) 40 years

Evolution of life expectancy

year Life expectancy (year)

• The probability does not increase linearly with age…

40 years

Hypothesis: The probability to observe a malign cell depends on the number

• f cell divisions during the lifetime

Peto’s paradox

Corollary: bowhead whale should present much more cancers than humans…

• They frequently survive up to 250 years without cancer!
• Their immune system is 4 times more efficient!

Hypothesis: the probability to present malign cells depends on the number

• f cell divisions during the lifetime but detecting a tumour depends more on

the barriers developed by the microenvironment

Role of the microenvironment

Corollary 3: the probability to initiate a cancer depends more on the state

• f the microenvironment than on the presence of malign cells

Modification of the microenvironment

Host cells Tumor cells Immune cells

• + x
• ±

z +

• y

Interactions between cell populations in a single tumor site

A cancer model

contramensalism

Realistic parameter values?

Colons Prostate Breast Skin Lung 3.4 391 5.4 1.2 3.4 219

x 100 x 65

4.4 114 4.1

x 26

Time (days) Prostate Specific Antigen PSA (ng/ml)

 Chaotic oscillations in the cancer model

Tumor cells Host cells Growth rate rh = 0.54

 Very agressive cancers can also be observed

Host cells Tumor cells Growth rate rh =1.45 14 days later….

A clinical example

Man, 67 years Smoker Pulmonary adenocarcinoma

 Role of immune cells

Inhibition rate rti of tumor cells by immune cells

Tumor cells The action of immune cells does not affect the dynamics Influence of the inhibition of tumor cells by immune cells

• Sometimes immunotherapy is of a limited interest…

Woman, 64 years Smoker Adenocarcinoma with bone metastases anti PD-1 (Nivolumab) immunotherapy 3 month later….

Tumor cells Immune cells Tumor cells Immune cells

Spatial simulations of tumor growth

• Depends on the inhibition of immune cells by tumor cells

𝛽IT= 3.0 T ≈ 2.9 mm 𝛽IT= 1.9 T ≈ 1.3 mm After 6000 a.u.t. Proliferation Quiescence Necrosis Local dynamics

(proliferation)

In the lung of a woman

64 years, smoker, adenocarcinoma with a bone metastasis

Tumor cells Immune cells

Spatial simulations of tumor growth

• The tumor growth depends on the inhibition of immune cells by tumor cells

𝛽IT= 0,3015 rI = 5,05 D ≈ 1.7 mm After 74 800 a.u.t. Dynamics in each site

• Chaotic dynamics in each site
• Very few sites have tumor cells

Proliferation Quiescence Necrosis

• Very slow tumor growth

Tumor cells Immune cells

Spatial simulations of tumor growth

• When the local dynamics is chaotic

t = 74 800 a.u.t.

Local dynamics

• Heterogeneous
• Proliferation zone is thick
• Slow tumor growth
• Woman, 80 years, smoker up to 60 years: weakly evolutive nodule

6 March 2014 6 July 2015 Adenocarcinoma

 Dynamics more important than cell interaction

• Equivalent dynamics but different parameter values
• With equivalent dynamics, the role of the microenvironment is relevant
• The growth rate of host cells is a very influent parameter …

ρH = 0.4862 ρI = 5.05 αIT = 0.3015 ρH = 0.5015 ρI = 5.007 αIT = 0.504 ρH = 0.5015 ρI = 5.119 αIT = 0.624

𝐸 ≈ 1.7 mm 𝐸 ≈ 1.1 mm 𝐸 ≈ 1.1 mm

How depends cancer risk on the tissue?

• Our hypotheses
• The nature of tissue is distinguished by the growth rate rH of host cells
• The quality of the tissue (How strong are the barriers againt tumor

growth) depends on other parameters (which can be affected by external factors)

• Varying parameter values = creation of a cohort of patients randomly

chosen

• The growth rate rT of tumor cells in a given tissue is twice the growth

rate of host cells (rT = 2rH)

• How can cancer risk depend on normal cells?
• The probability to detect a tumor depends on the growth rate of host

cells, that is, on how competitive are the normal cells

Host cell growth rate rH Probability P for a tumor

Do we measure the right variable? Do we observe this correctly?  Evolution of a cancer…

 There is a mathematical background for addressing this question…

• Observability (from control theory)

 The observability matrix

Original state space where

• f(x) : RmRm is the dynamical system
• h(x) : RmR is the measurement function

Between the original and the reconstructed spaces, the change of coordinates where

     

x f x x

f f

x h h

j j

  

1

L L

are Lie derivatives Observability matrix The system f(x) is said to be observable if the observability matrix is full rank.

• Full observability

Φ = 𝑌 = 𝑀𝑔

0ℎ 𝒚 = 𝑡 𝑢

𝑍 = 𝑀𝑔

1ℎ 𝒚 = 𝑡 𝑢

𝑎 = 𝑀𝑔

2ℎ 𝒚 = 𝑡 𝑢

𝒚 = 𝒈 𝒚 𝑡 𝑢 = ℎ 𝒚 𝑃𝑡 𝒚 = 𝜖𝑀𝒈

0 ℎ 𝒚

𝜖𝒚 ⋮ 𝜖𝑀𝒈

𝑛−1 ℎ 𝒚

𝜖𝒚

• every states xi and xj are distinguishable with respect to the

measured variable s(t), that is, h(xi)  h(xj) iff xi  xj

 Conditions for a full observability

x y z Original state space

(Rössler system) Coordinate transformation

X Y Z Reconstructed space Diffeomorphism if Det JF 0

1 1 1 1 1 Det Det

2

             

F

a a a J

y

The Rössler system is fully observable from variable y(t)

• Jacobian matrix of Φ𝑧 ≝ 𝐩𝐜𝐭𝐟𝐬𝐰𝐛𝐜𝐣𝐦𝐣𝐮𝐳 𝐧𝐛𝐮𝐬𝐣𝐲

𝑦 = −𝑧 − 𝑨 𝑧 = 𝑦 + 𝑏𝑧 𝑨 = 𝑐 + 𝑦 − 𝑑 𝑌 = 𝑍 𝑍 = 𝑎 𝑎 = 𝐺 𝑌, 𝑍, 𝑎 Φ𝑧 = 𝑌 = 𝑧 𝑍 = 𝑧 = 𝑦 + 𝑏𝑧 𝑎 = 𝑧 = 𝑏𝑦 + 𝑏2 − 1 𝑧 − 𝑨

𝒆𝒔 = 𝟒 𝒆 = 𝟒

 Singular observability manifold

• Definition of the neighborhood of the singular observability manifold

from the determinant of 𝐾𝛸

• Assessing the probability for

𝜃𝑧

𝑁 = 1.00

𝑦 ∉ 𝐵 ∩ 𝑉𝑁𝑡

𝑝𝑐𝑡

 Observability of the cancer model

Hot cells Tumor cells Immune cells

> >

It should be more efficient to follow the tumor environment than the tumor itself! The worse! The best!

 Application to lung cancer

Non smoking Smoking

 Web-mediated follow-up: weekly self-evaluated symptoms

Environment = global patient’s state weekly self-evaluated

• rationale: since host cells provides a better
• bservability of the dynamic, could we follow the

environment rather than the tumor itself?

• A cohort of 121 patients treated for a lung cancer
• 1. Weight
• 2. Apetite loss
• 3. Weakness
• 4. Pain
• 5. Cough
• 6. Breathlessness
• 1. Pulmonary carcinome (stade 3-4)
• 2. Web access

 Web-mediated follow-up: weekly self-evaluated symptoms

 Web-mediated follow-up: weekly self-evaluated symptoms

• Patient without relapse

Man, 63 years Smoker, 90 kg, no exercise Treated by radiotherapy Relapse probability = 75 % November 23 August 9 Before radiotherapy

• Patient with relapse

August 19 January 6

• 2 months before the

routine imaging

 Web-mediated follow-up: weekly self-evaluated symptoms

Man, 65 years Smoker, 86 kg, no exercise Treated by chemiotherapy Relapse probability = 75 %

Confusion matrix

With relapse Without relapse Moovcare positive 13 3 Moovcare negative 25

Sensibility 100% 85% Specificity 89% 96%

• Reliability equivalent to a classical follow-up!
• Detecting cancer relapse five weeks earlier than using routine

imagings !

Other diseases

Breathlessness

 Web-mediated follow-up: weekly self-evaluated symptoms

 Weekly self-reported symptoms

• Phase III (randomized) clinical trial: survival curve
• Cancer relapses are treated earlier (5 weeks), leading to a more efficient

treatment and a better quality of life (less stress, less pain, …)

• More than 20% at 18 month

MoovCare 20%

• Evolution of prostate cancer: Tumor, Node, Metastasis Classification

Clinically inapparent tumor

T1 T2

Tumor confined within prostate

T3

Tumor extends through the prostatic capsule

T4

Tumor fixed or invades adjacent structures

• ther than seminal

vesicles Liver metastasis Bone metastasis Lymph node metastasis

• Biology of cancer in 2 min…
• Weakly competitive host tissue
• Chemical castration

Proliferation depends on the androgen level

Patient global status Androgen level Hormone-dependent Hormone-hypersensitive Anti-androgen drug LHRH analogs

 Evolution of prostate cancer

 Hormone-dependent tumor cells

• LHRH analogs

 Hormone-hypersensitive tumor cells

• Anti-androgen drug

 Hormone-independent tumor cells

• Chemotherapy

Hormone evasion

 A model for prostate cancer

• Reduce the level of androgens
• Prevent the effects of androgens

Er = 15%

 An easy case

75 years old Gleason score = 6 No particular medical history

• Model parameter values

No hormone-hypersensitive tumor cells

• Strong environment
• Aggressive cancer strongly

dependent on androgens

Balanced by a LHRH analogs

Er = 11%

 Another easy case

75 years old Gleason score = 6 Alcohol disorders & arterial hypertension

• Model parameter values

No hormone-hypersensitive tumor cells

• Weak environment
• Weak cancer

LHRH analogs (Small rd and ra)

Er = 18%

 An evolutive case

77 years old Gleason score = 6 No particular medical history

• Model parameter values

Hormone evasion initiated

• Slightly weak environment
• ‘’Normal’’ cancer

LHRH analogs

Anti-androgen

 A complicated case

• Radiotherapy is not

described by the model

66 years old Gleason score = 7 Smoker

Bone Metastatsis LHRH analogs LHRH analogs

Conclusion

• for understanding data

 In tumor growth

• The micro-environment is relevant

 Can be used for clinical follow-up  Mathematical models

• for optimizing intermittent treatment
• All of this allows to develop

individualized treatment and follow-up