Vanier College Differential Equations Section ESP Project Tumor Modeling

Transcription

1 Vanier College Differential Equations Section ESP Project Tumor Modeling By: Mai Abdel-Azim Teacher: Ivan T. Ivanov

2 Introduction: Cancer is a disease in which a genetic change causes cells to undergo uncontrolled division forming a tumor. The genetic change could be a mutation to one of the genes responsible for cell division like Oncogenes and tumor suppressor genes. Another reason is chemical carcinogens such as tobacco, X-rays, and compounds from car exhausts, which are responsible for DNA damage. These genetic changes allow the cells to become cancerous by escaping the body s controls which include the control of birth and death processes, maturation and differentiation processes, and migration processes. Also these changes give cells the ability to stimulate production of their own nutrient supply, and to overcome the resulting immune response. Tumors cells then start multiplying at times where normal cells stop to multiply. For example, normal cells grow only when growth factors are attached to a receptor on the cell surface. In contrary, tumor cells may produce their own growth factors making growth possible all the time. However, growth is diffusion limited, which means that after the tumor grows to a certain size, diffusion is not enough to deliver sufficient nutrients to the tumor. This case stays until the tumor stimulates the formation of a blood vessel to supply it with enough nutrients, entering by that the vascular stage. In the pre-vascular stage, as the tumor grows the amount of nutrienst diffusing to the centre decreases and as a result the central cells proliferative rate (production of new cells) decreases and may cease; thus the cells become quiescent. Quiescent cells are still viable and can recover when sufficient nutrients are available again. The continued absence of nutrients in the central cells will cause them to die, forming a region of dead cells known as the necrotic core.

3 Figure(1) r 0 is the radius of the tumor. r n is the radius of the necrotic core. r c is the proliferating cells. Flux of nutrients is the rate of flow of atoms expressed as number of atoms per unit area and per unit time. Since diffusion of nutrients in the pre-vascular stage happens at a time scale of fractions of seconds, diffusion is described as steady state diffusion. This means that the flux doesn t depend on time and the radial flux of nutrients to the tumor follows the steady state diffusion equation which is Fick s first law J = D dc/dr. In this project, in the first section, mathematical models for the nutrient concentration in the steady state will be produced. Then in the next section, tumor growth is analyzed by including the perception of time producing mathematical models for the velocity of expansions of the tumor. This will lead us to conclusions about the limit size of the tumor in the diffusionlimited stage. To start the project, the following simplifying assumptions are made: 1-The tumor is spherically symmetric and grows in all directions. 2-The main nutrient limiting the system is oxygen. This is because the tumor s growth is mainly affected by oxygen.

4 3- There is a constant rate of oxygen uptake by the tumor, let it be K. 4-The radius of the tumor is constant over the time scale measurement. This is because the average time duration of cell doubling is much longer than the time of oxygen diffusion across the tumor.

5 Mathematical Models 1-Oxygen concentration in a steady state To describe how the surrounding oxygen diffuses into the tumor, the diffusion equation is used dc dt = k + D 1 d dr (r2 dc dr ) (1) c(r) is the concentration of oxygen at a distance r from the center of the tumor at time t. Since the diffusion of oxygen happens in a time scale of fractions of seconds which is much shorter than the tumor growth time scale, the steady state diffusion equation is acceptable: 0= -k + D 1 d dr (r2 dc dr )= -k+ DΔ2 c (2) Equation (2) applies in the region of living cells in the tumor thus when r1<r<r2. However when r<r1, meaning in the necrotic core, k=0 because oxygen is only taken up by living cells. D 1 d dr (r2 dc dr )= DΔ2 c= 0 (3) To be able to calculate the concentration of oxygen at any point in the tumor, at any value of r, we need to derive the equation of c(r) from steady state diffusion equation. To do so we multiply it by, integrate, then divide by, then integrate again. D 1 d dr (r2 dc )=k (2) dr D d dr (r2 dc dr )=kr2 d dr (r2 dc dr ) = K D r2 dc dr = kr 3D + A

6 c(r) = 1 k 6 D r2 + A + B (4) r When there is no necrotic core r1=0 Oxygen in this case diffuses to the center of the tumor thus concentration there is constant and greater than zero. It should also be greater than c1, the concentration at or below which cells die, c(0)>=c1. The concentration in the whole tumor is constant as well. As a result, boundary conditions are dc dr (0) = 0 c (r2) =c2 (5) When applying them to (4), we get k c(r) = 1 6 D (r22 ) + c2 (6) When there is a necrotic core r1>0 Necrotic core is dead cells; thus the concentration of oxygen is equal to c1 and there is no flow of oxygen. As a result, boundary conditions are C(r1)=c1, c(r2)=c2 J(r1)=0 (7) Applying them to (4) gives c1= 1 k r 6 D A B(a) r1 c2=1 k r 6 D A B(b) dc dr = 1 3 k D r 1 + A r 1 2=0 (c) From (c) A= 1 3 k r D 1 3

7 If we subtract the first equation from the second then substitute with the value of A, we get equation (8) from which we can find r1 c2 - c1= 1 k r 6 D 2 2 (1 + 2r 1 ) (1 r1 r2 )2 = 1 6 k D (1 + 2r 1 ) ( r 1 ) 2 (8) We can rewrite the first equality in (8) as: (1 r1 r2 )2 = c2 c1 1k 6D 2 (1+ 2r 1 r2 ) (9) We can see that as,(1 r1 r2 )2 = 0 thus r 1 = 1 and from the second equality in (8) ( r 1 ) must be a constant because c2 - c1 is a constant. ( r 1 ) 2 = h 2 = 2D K (c2 c1) (10) This implies that in a large tumor there is a shell of proliferating cells whose thickness depends on the how much excess concentration there is above threshold (c1) and not on the size of the tumor.

8 2- Tumor Growth In the previous section, we studied the diffusion of oxygen into the tumor and necrotic core formation; we shall now investigate tumor growth in time including Kinetics. In a tumor two processes are going at the same time, proliferation of living cells r 1 (t) < r < (t), and degradation of the necrotic core 0 r < r 1 (t). Proliferation is the increase of number of cells as a result of cell division which we assume to increase the cell volume at a rate P. Degradation of the necrotic core is breaking it down into a waste product which can diffuse out of the tumor. We assume degradation to decrease cell volume at a rate L. Applying the principle of conservation of mass to the tumor, we use the differential form of the continuity equation ρ =. (ρv) which states that, in a steady state process, the rate at which mass t enters a system is equal to the rate at which the mass leaves the system. Thus in the necrotic core: ρ t = ρl. J = ρl. (ρv) (11) Where ρ(r, t) is the density of the tumor, v(r, t) is the velocity vector field in the tumor, and J = ρv is the mass flux. In living cells we get: ρ t = ρp. J = ρp. (ρv) (12) In the tumor, proliferating cells absorb fluid from the surrounding for growth and division causing a net fluid flux into the tumor and thus an outward cell flux. When a necrotic core forms and then is degraded, it releases fluid that leaves the tumor causing a cell flux toward the interior of the tumor. After the tumor is large enough, the inward cell flux due to degradation balances the outward cell flux due to proliferation as described in figure (2)

9 Figure(2) As a result, we make a reasonable assumption that the density in the tumor is constant. 0 = ρl. (ρv) (13) 0 = ρp. (ρv) (14). v = L. v = P is the del operator which when applied to the velocity vector field in a spherical coordination gives the divergence of the velocity field as follows :. v = 1 ( v r ) + 1 r rsinθ (v θ sinθ) θ + 1 v (15) rsinθ Since the tumor is a symmetrical sphere, θ and are equal to zeros and. v = 1 ( v) r = L (16). v = 1 ( v) r = P (17) In 0 r < r 1 (t) In r 1 (t) < r < (t) v(r) is the velocity of tumor movement. Its equation can be derived from (16) and (17) by integration: In 0 r < r 1 (t)

10 1 ( v) r = L ( v) r = L v = L dr v = L r3 3 v(r) = Lr 3 (18) In r 1 (t) < r < (t) ( v) r = P Integrate both sides r v r r1 = P r1 dr = P( r3 3 r ) v(r) r 1 2 v( r 1 ) = P( r3 r 3 1 ) 3 3 v(r) r 1 2 v ( L r 1 3 ) = P(r3 r 3 1 ) 3 3 v(r) = Pr 1 (P + L) (r 1 r2) (19) Using the fact that the outermost cells are moving at the same velocity of growth of the tumor, from (19) v( ) = d dt = 1 Pr 3 2 (1 (P+L) ( r 1 3 )). When v(r P r 3 2 2) is zero; this means that the tumor stopped radial growing. When d = 0 1 (P+L) dt P ( r ) = 0

11 P = r 3 1 P+L r 3 2 Using r2-r1=h P P + L = ( h)^3 r h + 3 h 2 h 3 3 = P P(1 + L P ) 3 h 2 = 0 h 3 = 0 Because h has a negligible size compared to r h 3 = P P(1+ L P ) = 1 L P + (L P )2 + 3h = L P = 3hP L (20) From the results above we conclude that if L P, 3hP L as t. Therefore, the tumor cannot grow beyond that certain size because biologically its nutrient supply is insufficient. Alternative approach: We can get the same equation in (19) and derive the same conclusion as before with an alternative approach. First we integrate equations (11) and (12) over the volume of the tumor V(t) which divides into the necrotic core V1 (t) and the living cells V2 (t). J dv = (ρv) dv V(t) V(t) = ρldv V1(t) + ρpdv V2(t) (21)

12 According to the divergence theorem which states that the volume integral of the divergence of a function over is equal to the surface integral of. Thus V(t) (ρv) dv = ρv. nds = ρv. n. Area = ρ dv S(t) dt (22) Where S(t) is the surface of V (t). From (21) and (22) dv dt = V1(t) ρldv + ρpdv V2(t) dv dt = P(V V1) LV1 Taking the tumor to be a sphere V = 4 3 πr3 d( 4 3 π 3 ) = P ( 4 dt 3 π πr 1 3 ) L 4 3 πr r dr2 3 2 dt = P( 3 r ) Lr d dt = 1 Pr 3 2 (1 (P+L) ( r 1 3 )) P r 3 2 It is the same equation again to conclude that the limit at which the cell can t grow beyond is 3hP L. Summary: -After modeling the nutrient concentration in the steady state, we reached the conclusion that in a large tumor, there is a constant shell of proliferating cells. The width of the shell is related to the excess nutrient concentration above the threshold below which cells die.

13 -In the second section, we could model the velocity of expansion of the tumor to conclude that the tumor in the pre-vascular stage grows to reach a limit size of 3hP which is related to both the rate of proliferation and degradation in the tumor. L

14 Bibliography Britton, N. F. "Tumour Modelling." Essential Mathematical Biology" London: Springer, Print. Cristini, Vittorio, and John Lowengrub. "Multiscale Modeling of Cancer." Google Books Web. 19 May Crosta, Peter. "What Is Cancer? What Causes Cancer?" Medical News Today. MediLexicon International. Web. 19 May "Continuity Equation." Wikipedia. Wikimedia Foundation. Web. 19 May < Grimes, D. R., A. G. Fletcher, and M. Partridge. "Oxygen Consumption Dynamics in Steady-state Tumour Models." Royal Society Open Science (2014): Web. 19 May 2015.