We develop a field theory-inspired stochastic model for description of tumour growth based on an analogy with an SI epidemic model, where the susceptible individuals (S) would represent the healthy cells and the infected ones (I), the cancer cells. From this model, we obtain a curve describing the tumour volume as a function of time, which can be compared to available experimental data.
In a recent letter [
In the present work, we employ the very same methodology established in [
The rest of this work is organized as follows. In Section 2, we introduce the basic aspects of our model which allow us to obtain differential equations to describe the time evolution of the mean number of individuals in the interacting populations we are dealing with, i.e., normal (healthy) and cancer cells. The analytical solutions for these differential equations, besides the specific curve which describes the time evolution of the tumour volume and its comparison to experimental data are presented in Section 3. Finally, in Section 4 we present our concluding remarks.
We will start by considering interacting populations, whose total sizes are allowed to change, composed of two types of individuals: the normal (healthy) cells and the tumour (cancer) cells. Let us introduce N ( t ) and C ( t ) as random variables which represent, respectively, the number of normal and cancer cells at a given time instant t.
We will then consider a bivariate process { N ( t ) ; C ( t ) } t = 0 ∞ with a joint probability function given by
p ( n , m ) ( t ) = Prob { N ( t ) = n ; C ( t ) = m } . (1)
Our aim is to compute time-dependent expectation values of the observables N ( t ) and C ( t ) , which may be defined in terms of the configuration probability according to
〈 N ( t ) 〉 = ∑ n , m n p ( n , m ) ( t ) , 〈 C ( t ) 〉 = ∑ n , m m p ( n , m ) ( t ) . (2)
Let us represent the probabilistic state of the system by the vector
| μ , ν 〉 = ∑ n , m p ( n , m ) ( t ) | n , m 〉 , (3)
with the normalization condition ∑ n , m p ( n , m ) ( t ) = 1 .
As an example, the vector 1 4 ( | 1 , 1 〉 + | 2 , 1 〉 + | 1 , 2 〉 + | 2 , 2 〉 ) represents the probability distribution where there are 1 or 2 healthy/cancer cells present, each one with probability 1/4, i.e. p ( 1 , 1 ) = p ( 2 , 1 ) = p ( 1 , 2 ) = p ( 2 , 2 ) = 1 4 .
Since the configurations are given entirely in terms of occupation numbers ( n , m ) , which calls for a representation in terms of second-quantized bosonic operators [
[ h , h † ] = [ c , c † ] = 1 ,
[ h , c ] = [ h , c † ] = [ c , h † ] = [ h † , c † ] = 0. (4)
As usual in the second quantization framework, we say that h † and c † “create”, respectively, normal and cancer cells when applied over the reference
(vacuum) state | 0,0 〉 . This allows us to build our space from basis vectors of the form | n , m 〉 = ( h † ) n ( c † ) m | 0 , 0 〉 .
This vacuum state has the following properties: h | 0 , 0 〉 = c | 0 , 0 〉 = 0 (from which “annihilation” operators) and 〈 0 , 0 | 0 , 0 〉 = 1 (inner product).
Following the above definitions, we also have
h † | n , m 〉 = | n + 1 , m 〉 ; c † | n , m 〉 = | n , m + 1 〉 , h | n , m 〉 = n | n − 1 , m 〉 ; c | n , m 〉 = m | n , m − 1 〉 . (5)
At this point it is worth to note that h † h | n , m 〉 = n | n , m 〉 and c † c | n , m 〉 = m | n , m 〉 . Thus, the operators n = h † h and m = c † c just count the number of cells in a definite state. This is the main reason why they are usually called number operators. The vector state of our system may be then rewritten in terms of creation and annihilation operators as
| μ , ν 〉 = ∑ n , m p ( n , m ) ( t ) ( h † ) n ( c † ) m | 0,0 〉 . (6)
The time evolution of our system will then be generated by a linear operator � (called hamiltonian) which may be constructed directly from the transition rates present in our model according to
Transition | Description | Contribution to |
---|---|---|
birth of normal cell (rate | ||
birth of cancer cell (rate | ||
death of normal cell (rate | ||
death of cancer cell (rate | ||
change normal ® cancer (rate |
H = + ( b h + d h + λ ) n + ( b c + d c ) m − ( b h h † n + b c c † m + d h h + d c c + λ c † h ) . (7)
The notational advantage of this field theoretical description is made clear at this point if we observe that, from the above definitions, the equation which represents the flux of probability between states at rates defined by our model (the so-called master equation or forward Kolmogorov differential equation [
d d t | μ , ν 〉 = − H | μ , ν 〉 . (8)
We can get, after some algebra, a more common representation for the master equation by substituting the expressions for the hamiltonian, Equation (7), and the vector state, Equation (3), into Equation (8)
d d t p ( n , m ) = − [ ( b h + d h + λ ) n + ( b c + d c ) m ] p ( n , m ) + b h ( n − 1 ) p ( n − 1 , m ) + b c ( m − 1 ) p ( n , m − 1 ) + d h ( n + 1 ) p ( n + 1 , m ) + d c ( m + 1 ) p ( n , m + 1 ) + λ ( n + 1 ) p ( n + 1 , m − 1 ) . (9)
In order to compute the time-dependent expectation values of the observables N ( t ) and C ( t ) through the above master equation, we will follow the well- established methodology presented in [
M ( ϕ , θ ; t ) = 〈 e ϕ N ( t ) e θ C ( t ) 〉 = ∑ n , m p ( n , m ) e n ϕ + m θ . (10)
Note that from the above equation we have
[ ∂ M ∂ ϕ ] ϕ , θ = 0 = ∑ n , m n p ( n , m ) = 〈 N ( t ) 〉 , [ ∂ M ∂ θ ] ϕ , θ = 0 = ∑ n , m m p ( n , m ) = 〈 C ( t ) 〉 (11)
and, in general
[ ∂ k M ∂ θ k ] ϕ , θ = 0 = 〈 C k ( t ) 〉 ; [ ∂ k M ∂ ϕ k ] ϕ , θ = 0 = 〈 N k ( t ) 〉 . (12)
After multiplying Equation (9) by e x p ( n ϕ + m θ ) and summing on ( n , m ) , we are led, after some algebra, to
∂ M ∂ t = ∑ n , m = 0 d p ( n , m ) d t e n ϕ + m θ = + [ b h ( e ϕ − 1 ) + d h ( e − ϕ − 1 ) + λ ( e − ϕ + θ − 1 ) ] ∂ M ∂ ϕ + [ b c ( e θ − 1 ) + d c ( e − θ − 1 ) ] ∂ M ∂ θ . (13)
Finally, by differentiating the above equation with respect to ϕ and evaluating the result at ϕ = θ = 0 we get the following differential equation for 〈 N ( t ) 〉
[ ∂ 2 M ∂ t ∂ ϕ ] ϕ , θ = 0 = d d t 〈 N ( t ) 〉 = ( b h − d h − λ ) 〈 N ( t ) 〉 . (14)
On the other hand, if we differentiate Equation (13) with respect to θ and evaluate at ϕ = θ = 0 we get the following differential equation for 〈 C ( t ) 〉
[ ∂ 2 M ∂ t ∂ θ ] ϕ , θ = 0 = d d t 〈 C ( t ) 〉 = λ 〈 N ( t ) 〉 + ( b c − d c ) 〈 C ( t ) 〉 . (15)
By defining 〈 N ( 0 ) 〉 ≡ N 0 , 〈 C ( 0 ) 〉 ≡ C 0 , β h ≡ b h − d h and β c ≡ b c − d c we obtain the following analytical solutions for Equations (14) and (15)
〈 N ( t ) 〉 = N 0 e ( β h − λ ) t (16)
and
〈 C ( t ) 〉 = λ N 0 β h − λ − β c [ e ( β h − λ ) t − e β c t ] + C 0 e β c t . (17)
Finally, if we consider that the number of normal cells is approximately constant ( β h ≈ 0 ) and that the volume of a cancer cell (v) is approximately the same of a normal cell, we may write the following expression for the time evolution of the tumour volume
V c ( t ) = λ ( V h ) 0 − λ − β c [ e − λ t − e β c t ] + ( V c ) 0 e β c t , (18)
where V c ( t ) ≡ v 〈 C ( t ) 〉 , ( V h ) 0 ≡ v N 0 is the initial volume of the normal tissue, and ( V c ) 0 ≡ v C 0 is the initial tumour volume.
The above expression is compared to experimental data for liver cancer (average tumour volume) in
1A comprehensive review of the most common mathematical models for description of tumour growth may be found in [
We would like to finish this work by making a few comments about Equation (18). As far as we know, this is the first time that an expression describing the time evolution of the volume of tumours1 is obtained from basic assumptions about cancer as a phase transition and, most certainly, this is the very first model for description of tumour growth built by using standard field theoretical language commonly found in models describing fundamental interactions of elementary particles. In a future work we are going to present a qualitative
analysis of the behaviour of solutions for the system of first order linear differential equations composed by (14) and (15) in the phase plane, which we believe will shed more light on how our model is indeed connected to the idea of cancer as a phase transition.
Last but not least, other possible extensions for the present model should consider the inclusion of other kinds of dependence in the birth/death rates of cells, as temperature and/or concentration gradients of toxic carcinogens, for example.
The author would like to thank Prof. Jack A. Tuszynski for the valuable discussions and the very stimulating scientific environment shared at University of Alberta’s Li Ka Shing Applied Virology Institute, where the main results of this work were obtained.
Mondaini, L. (2017) Towards a Field Theoretical Stochastic Model for Description of Tumour Growth. Journal of Applied Mathematics and Physics, 5, 1092-1098. https://doi.org/10.4236/jamp.2017.55095