Dr. Natalie Romanova 1

Dr. Natalie Romanova

1

Mathematical models of cancer

Modelling the tumour growth

Mathematical Modelling of metastases

Mathematical models to find changes of the physics-mechanical properties of cells Dr. Natalie Romanova

2




Model of the general population



Model of reaction-diffusion



Fractional order models


3


Cancer progresses as a result of the collective dynamics that emerge from interactions between tumour cells and their microenvironment.

 Sottoriva, A. et al.

A Bing Bang model of human colorectal tumor growth. Nat. Genet. 47, 209-216 (2015)

 Liotta, L.A. & Kohn, E.C. The microenvironment of the tumour-host interface. Nature 411, 375-379 (2001)  Mueller, M.M. & Fusenig, N.E. Friends or foes – bipolar effects of the tumor stroma in cancer. Nat. Rev. Cancer 4, 839-849 (2004)  Paget, S.

The distribution of secondary growths in cancer of the breast. Cancer Metastasis Rev. 8, 98-101 (1989) Dr. Natalie Romanova

4

Application of the General Population Model  x(t) is the population size at time t  bt  the growth rate decay exponentially as a  t   e

The growth curve of a cell population with a time-dependent growth rate can be written as:

x  t   e

 bt

x t 

dx  t 

or

dt

and the solution is x  t   x0e



k 1 e bt

 e bt x  t 



a where the initial size of the population is x0 and k  b Dr. Natalie Romanova

5

Application of the General Population Model Survival Model Andersson, T.M.L. et al. Estimating and modelling cure in population – based cancer studies within the framework of flexible parametric survival models. BMC Medical Research Methodology 2011, 11:96 Patient survival, the time from diagnosis to death, is the most important single measure of cancer patient care (the diagnosis and treatment cancer). The method of choice for studying cancer patient survival in a population-based setting is relative survival, R(t).

R t  

S t 

S  t 

S t   S  t  R t 

where S(t) is the observed (all-cause) survival among the cancer patients and S*(t) is the expected survival in a hypothetical group in the general population that is comparable to the cancer patients with respect to age, sex, calendar year and possible other covariates. Dr. Natalie Romanova

6

Fz  t      S  t   S  t     1       1    

where FZ(t) is a distribution function, a Weibull distribution is often used. S*(t) is assumed known and are usually obtained from routine data sources (eg. national or regional life tables). All patients diagnosed with colon adenocarcinoma in Finland 1953-2003, with follow-up until 2004. Patients that emigrated were censored at the date of emigration, and everyone still alive was censored 10 years after diagnosis. Patients that were incidentally diagnosed at autopsy or were registered solely on death certificate information were excluded. The cohort consists of 34,664 patients. Dr. Natalie Romanova

7

Application of the Reaction-Diffusion 2D Model Rejniak, K.A. & Anderson, A.R. Hybrid models of tumor growth. Wiley Interdiscip. Rev. Biol. Med 3, 115-125 (2011)

u  x,t  t

K

 2u  x,t  x

2

 vT  x,t  c  x,t   ac  x,t   bc  x,t 

where K is diffusion constant and a is constant of decay, T(x,y) is group size of tumour cells at position x at time t and at a rate v, given component is produced by macromolecule in the extracellular matrix at rate b.

Modelling tumour and normal cell populations and their genetic changes, such models require the incorporation of biophysical and environmental properties (diffusion of growth factors, harmones, nutrientints and oxygen) that affect tumour proliferation and invasion patterns. Dr. Natalie Romanova

8

Normal Diffusion A diffusion is a process in physics. Some particles are dissolved in a glass of water. At first, the particles are all near one corner of the glass. If the particles randomly move around ("diffuse") in the water, they eventually become distributed randomly and uniformly from an area of high concentration to an area of low concentration, and organized (diffusion continues, but with no net flux).

Fick’s second law

u  2u K 2 t x where u is the concentration u = u(x, t) is a function that depends on location x and time t t is time K is the diffusion coefficient in dimensions x is the position Dr. Natalie Romanova

9

Anomalous Diffusion (Sub or Super diffusion) Anomalous diffusion is a diffusion process with a non-linear relationship to time, in contrast to a typical diffusion process, in which the mean squared displacement (MSD) of a particle is a linear function of time.

Normal diffusion

2  t

Sub-diffusion

Super-diffusion

2  t  , 0    1

2  t  , 1    2

𝝈𝟐 shows a time dependence, which is significantly slower than linear

𝝈𝟐 shows a time dependence, which is significantly faster than linear

10 Dr. Natalie Romanova

Fractional order derivatives Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems Riemann- Liouville x

Daα f  x  

1 d  1  α  dx

d    sin   sin x       dx 2  

f  t  dt

 x  t , α

a

b

1 d Dbα f  x     1  α  dx

f  t  dt

 x  t

α

d xp  dx 

, α0

 c D0  f  x  

1  1   

  p  1 x p    p    1

x

Gerasimov-Caputo x



y  t  dt

 x  t



 x  0, 0    1

d  x e     e x   dx

0

Grünwald-Letnikov d f  x  t  a    lim  a Dt f  x    N   N  d t  x 

 N 1

k    1     k  f j 0


 t  a   x  k   N  ,   0   11

Fractional models of tumour growth u  x,t     x,t  K t x  0   1

where

Sub-diffusion

Super-diffusion 1   2    x,t  is Riemann- Liouville fractional derivative  x

Palocaren, A, and Drapaca, CD. Biomechanical modeling of tumor growth: its relevance to glioma research. Int J Num. Anal Model B. 2012; 3: 94–108. Meerschaert, M.M., Metzler, R., and Klafter, J. Fractional calculus, anomalous diffusion, and probability, fractional dynamics. Singapore: World Scientific, 2012: 265-284 Preziosi, L. and Tosin, A. Multiphase and multiscale trends in cancer modelling. Mathematical Model Nat. Phenom 2009, 4: 1-70

Zhuravkov, M. and Romanova, N. Review of methods and approaches for mechanical problem solutions based on fractional calculus. Mathematical & Mechanics of Solids, 2016 Dr. Natalie Romanova

12

Mathematical Modelling of metastases


13

Fractal Theory and Chaos A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales.

The Fractal Geometry of Nature by Benoît Mandelbrot; W H Freeman & Co, 1982; ISBN 0-7167-1186-9

Fractals are created by repeating a simple process over and over in an ongoing feedback loop.


14

Fractal or Self-similar Dimension Nr

D

N is quantity of self-similar parts D is a dimension

1

D=

r is scaling (magnification) factor

1 3

D =1

1 1 r= = 2 9 3

D=2

r=

1 1 r= = 3 27 3

log N 1 log r

D=3


15

Fractal Dimension of the Menger Sponge

D=

log N log 20 = = 2.7268 1 log 3 log r

Researchers at Queen Mary University of London use fractals to study the movement of bodies in complicated systems. These concepts have applications to everything from the chaotic motion of molecules in fluids to the movement of foraging animals. http://fractal.org/Life-Science-Technology/Fractal-Theory-Breast-Cancer.htm Fractal theory provided the basis for a unique software platform program that has been developed for use in conjunction with MRI, and is showing great promise in the early diagnosis and treatment of breast cancer. Using this unique method, a study has shown that in over 30 percent of patients there were additional tumors in the same breast, and in almost 10 percent of the patients there were tumors in the opposite breast. These tumors were not previously found using mammography or ultrasound. Dr. Natalie Romanova

16

Stochastic processes SP describe how random variable (or set of random variables) changes over time and/or space. A stochastic process ascribes a probability to each event and allows for the prediction of the probability of a certain outcome.

Markov process

Non-Markov process

A memoryless SP in which the conditional probability distribution over all future events dependents only on the present state


Stochastic processes with memory

17

Mathematical models of stochastic processes   u  x,t    x,t   K t  x

where

   x,t  , 0    1 is Gerasimov-Caputo fractional derivative  t

Iomin, A. Fractional transport of cancer cells due to self-entrapment by fission. In: Mathemat. Modeling of Biological Systems, 2007, 193–203. Dahari, H, et al. Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy. J. Theor. Biol., 2007; 247: 371–381. Preziosi, L, and Tosin, A. Multiphase and multiscale trends in cancer modelling. Mathematical Model. Nat. Phenom., 2009; 4: 1–70.

Ahmed, E, and El-Saka, HA. On fractional order models for Hepatitis C. Nonlinear Biomed. Phys., 2010; 4: 1–3. Dr. Natalie Romanova

18

Mathematical models to find changes of the physics-mechanical properties of cells


19

Biomechanical properties of cells

Healthy erythrocyte. Value of elastic modulus is equal to normal parameter

Unhealthy erythrocyte (friable) Value of elastic modulus is less than normal parameter

Unhealthy erythrocyte (heavy) Value of elastic modulus is greater than normal parameter Dr. Natalie Romanova

Changing of Hemorheological Properties (blood viscosity and blood viscoelasticity)

20

Different Techniques to estimate of Biomechanical properties

Schematic illustrations of the biomechanical assays used to probe subcellular regions are given in (a)–(c). Biophysical assays commonly used to probe the deformation of single cells are illustrated in (d)–(g). Techniques used to infer cytoadherence, deformation and mobility characteristics of populations of cells are schematically sketched in (h) and (i). Dr. Natalie Romanova

21

Atomic Force Microscopy Atomic Force Microscope


22

23

ATOMIC FORCE MICROSCOPE

Biomaterial ALGORITHMS to FIND the CONTACT POINT

 Crick algorithm  Modified Crick algorithm  New algorithm INDICATORS OF THE CONDITION AND DYNAMICS OF CHANGE FOR DIAGNOSTICS AND THERAPIES

DETERMINATION OF ELASTIC MODULUS VALUE     

Model 1 Model 2 Model 3 Model 4 Model 5 24

Modification of classical viscoelastic models on the base of fractional calculus  t  

t



t

J  t       d 

1 d  0 Dt f  t    1    dt

(1)



  t     0 Dt   t  

Fractional Maxwell model 1 1  D     D  G0

 D   1   R D

 R 

 G0



d

 0

Fractional model of Standard Linear Solid

(3)

0

     s e  s ds

(2)



 t   



t

  t    G  t       d

f  

(4)

0

(5)

Fractional Kelvin-Voigt model   G   D 

 C 

 G

   C G  t   G 1      1    t 

Multi-link fractional elements

1   R D   GR 1   C D   

GR  G0Gd /  G0  Gd 

 R   /  G0  Gd 

 C   / Gd 25

uiy  0 Dt ui  x, t 

(6)

h y  t   0 Dt h  t 

(7)

 for Hertz's solution

0

Dt h  t   P

E

2 3 3

2

9 1 2  1   16  E  R

3P 1   2 

4 R  0 Dt h  t  

1,5



(8) (9)

 Spherical-base indenter 3 2E  a2 b a 2  b2   b a  2 2  P a D h t   arccos     a b    2  0 t 2 tg  a 3 R 2 tg  3R   1       

0

E

Dt h  t  

a R





a 2  b2  a 



a b arccos    0 tg a

P 1 2



 a2 b a3 a 2  b2    2 2 b 2  a 0 Dt h  t   arccos   a b      2 tg  a 3 R 2 tg  3 R   

(10) (11)

(12)

26

Numerical Modelling

 on the basis of the fractional model of Kelvin-Voigt

Rheological approximations for multiple fractional models of the first and second types on the basis of the fractional Kelvin-Voigt model

J N  J0 

sin 



N 1



ln r m 1  et /m m 0



(13)

sin   L2  L1  N 1 e xi  t / e xi J  , , N  t   J 0  1  e   i 1 N



0 , ,  N

0   ,  N   i  

N i N



(14) 1

  N  , r    i N

 on the basis of Maxwell's generalized fractional model G , , N  t   G 

 ln r sin  N 1  t / i e   i 0

(15)

i

L2  L1 N 1   xi  t / exi ) G , , N  t   G  K e N  1 i 1  sin  K 

Rheological Approximations for multiple fractional models of the third and fourth types on the basis of the generalized Maxwell model

(13)

RMM 1

(15)

RMM 2

(16)

27

Numerical Model  Models contain Prony’s series for ANSYS t N   i G  t   G0  a   ai e  i 1 

   

ai 

N

i  e 

Gi G0

 for RMM 1 ai 

2sin    2(i 1) N / e  N

2( i 1)  N  / N

 for RMM 2 2 N sin  ai  e   N  1

  2 i  N  N N 1

i  e

 2i  N 

N

N 1

The behavior of the erythrocyte was considered in several approximations:  as an elastic material without viscoelasticity  taking into account the viscoelasticity within the rheological fractional model of RMM1 with nine elements (N = 9)  taking into account viscoelasticity within the rheological fractional model of RMM2 with nine elements (N = 9) 28

1. Before treatment RMM1

ai 0.2 0.173 0.153 0.136 0.12 0.1 0.095 0.084 0.074

i 0.05 0.097 0.188 0.367 0.716 1.4 2.7 5.3 10

RMM2 ai 0.21 0.18 0.157 0.136 0.12 0.1 0.091 0.08

i 0.07 0.15 0.32 0.687 0.716 1.45 3.1 14

 Visualization of indenter invasion into the erythrocyte

The values of the indenter depth for three types of materials. Experimentally: h = 39,3 nm Numerical modelling  without viscoelasticity h = 26,0 нм  RMM1 h = 41,8 нм  RMM2 h = 38,5 нм

29

2. After treatment РДМ3

ai 0.2 0.173 0.153 0.136 0.12 0.1 0.095 0.084 0.074

i 0.05 0.097 0.188 0.367 0.716 1.4 2.7 5.3 10

РДМ4

ai 0.21 0.18 0.157 0.136 0.12 0.1 0.091 0.08

i 0.07 0.15 0.32 0.687 0.716 1.45 3.1 14

 Visualization of indenter invasion into the erythrocyte without viscoelasticity

 Visualization of indenter invasion into the erythrocyte taking into account the viscoelasticity 30

Values of elastic modulus (acute coronary syndrome)

РАТК МАТК РАТК МАТК Модель 1

РАТК МАТК РАТК МАТК РАТК МАТК

Модель 2 до лечения

Модель 3

Модель 4

Модель 5

после лечения

Е, МПА

Е, МПА

90 80 70 60 50 40 30 20 10 0

Values of elastic modulus (lung cancer)

45 40 35

30 25 20 15 10 5 0 РАТК МАТК РАТК МАТК Модель 1

РАТК МАТК РАТК МАТК РАТК МАТК

Модель 2 до лечения

Модель 3

Модель 4

Модель 5

после лечения

31

Thank you 

32