Stochastic Model for Cancer Cell Growth through Single Forward Mutation

Transcription

1 Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com Follow his and addiional works a: hp://digialcommons.wayne.edu/jmasm Par of he Applied Saisics Commons, Social and Behavioral Sciences Commons, and he Saisical Theory Commons Recommended Ciaion Jayabalan, J. (2017). Sochasic model for cancer cell growh hrough single forward muaion. Journal of Modern Applied Saisical Mehods, 16(1), doi: /jmasm/ This Emerging Scholar is brough o you for free and open access by he Open Access Journals a DigialCommons@WayneSae. I has been acceped for inclusion in Journal of Modern Applied Saisical Mehods by an auhorized edior of DigialCommons@WayneSae.

2 Journal of Modern Applied Saisical Mehods May 2017, Vol. 16, No. 1, doi: /jmasm/ Copyrigh 2017 JMASM, Inc. ISSN Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy Puducherry, India A sochasic model for cancer cell growh in any organ is presened, based on a single forward muaion. Cell growh is explained in a one-dimensional sochasic model, and saisical measures for he variable represening he number of malignan cells are derived. A numerical sudy is conduced o observe he behavior of he model. Keywords: Muaion, probabiliy generaing funcion, differenial-difference equaion Inroducion Cancer has complex and sochasic cell growh mechanisms. Malignan cancer cells arise from several muaions in he gene of a cell. I has been shown ha a normal cell requires more han one sage o become a malignan cell (Tan & Brown, 1987). Sochasic growh is observed in malignan cells, and deerminisic exponenial growh is observed in normal cells. Mos developed models are mixed, represening boh deerminisic and sochasic cell growh. The ype of growh in a normal cell populaion depends on wheher muaion has aken place. Le [X(), > 0] be a sochasic process denoing he number of normal cells in an organ a ime, and [Y(), > 0] be a anoher sochasic process denoing he number of malignan cells in an organ a ime. Le us define a bivariae cell growh process {(X(),Y()), > 0} represening he number of normal and malignan cells a ime. The growh of cells can be sudied using he birh-anddeah process. In lieraure, he process of malignan cell growh has been sudied wih homogeneous and non-homogeneous birh, deah and muaion processes, bu i seems mos applicable when he sudy is conduced under a ime-dependen Jayabharahiraj Jayabalan is a Research Scholar in he Deparmen of Saisics. a jayabharahi8@gmail.com. 578

3 JAYABHARATHIRAJ JAYABALAN environmen. The birh-and-deah process has been used o sudy he sochasic growh of a populaion, and he average and variance of he size of a populaion has been obained for a given ime period (Kendall, 1949). A similar approach can be applied o obain he average and variance of he number of malignan cells in an organ a ime. Assume here are x 0 number of normal cells and y 0 number of malignan cells in any organ a ime = 0 (iniially). The model represening he cell division process for normal and malignan cells can be explained using eiher one or wo variables. Consider a single forward muaion process for he ransformaion of normal cells ino malignan cells, which reflecs in he growh of he cell populaion. If a malignan cell is formed from a normal cell, and i remains in he same sae ill exincion, hen here is no backward process of muaion. Le us assume ha he normal cell has deerminisic exponenial growh; he expeced number of cells in an organ a ime is hen defined by (Serio, 1984), = x 0 expí ò ( ) ( ) ˆX ì î 0 + m ( ) ü d ý þ (1) If i is assumed he malignan cells also show deerminisic growh, i.e., he cell growh a he malignan sage is deerminisic and exponenial, hen he expeced number of malignan cells is as follows, = y 0 expíò b M ( ) ( ) Ŷ ì î 0 ü d ý þ (2) Assumpions The model is developed based on he following assumpions: 1. Le he growh rae of normal cells from normal cells be (), and he probabiliy of growh of normal cells from normal cells in d be ()d + o(d). Le he deah rae of normal cells be d N () and probabiliy of he deah of normal cells in d be d N ()d + o(d). 2. Le he growh rae of malignan cells from he malignan cells be b M (), and probabiliy of growh of malignan cells from malignan cells be b M ()d + o(d). Le he deah rae of malignan cells be d M () 579

4 STOCHASTIC MODEL FOR CANCER CELL GROWTH and probabiliy of he deah of malignan cells in d be d M ()d + o(d). 3. Le he growh rae of he normal cell populaion be represened by {( () d N ()) + μ ()}, and he growh rae of he malignan cell populaion by (b M () d M ()). 4. In a very small inerval ( + d), le he probabiliy of a muaion which ransforms a normal cell ino a malignan cell be xμ ()d + o(d), where X() = x 0 a = 0. When a muaion akes place in a normal cell populaion, he number of normal cells is decreased by one and he number of malignan cells is increased by one. Assume ha he growh rae for normal and malignan cell populaions are differen. For any organ, a cerain number of cells is required for normal, proper funcioning; normal funcioning of any organ depends upon he number of cells. The expeced populaion size a ime can be described as X() + Y(). Assuming a deerminisic growh for normal and malignan cells, hen T c = ˆX ( ) + Ŷ ( ) ì = x 0 expí ò î 0 ì expíò b M î 0 ( ( )) + m ( ) ( ( )) ü d ý þ ü d ý þ (3) where X() = x 0 and Y() = y 0 a = 0. Hence, he number of normal cells in he populaion of an organ a ime is as follows = x 0 exp í ò ( ) ( ) ˆX ì î 0 ì exp íò î 0 b M + m ( ) ( ( )) ü d ý þ - Ŷ ü d ý þ (4) Assuming he above relaion holds, here exiss a sochasic dependence beween X() and Y(). There is no need o observe he variables X() and Y() as a 580

5 JAYABHARATHIRAJ JAYABALAN wo-dimensional sochasic process; i is enough o consider one-dimensional sochasic process for he malignan cell populaion [Y(), > 0] wih he above relaion. The above discussions deal wih a non-homogeneous environmen, and look more complex in mahemaical derivaions. For simpliciy, le us assume a homogeneous environmen wih respec o birh, deah and muaion parameers. Sochasic Model Le f M (y,) = P{Y() = y} denoe he probabiliy densiy funcion of Y(). Assume ha f M (y,) exiss and is differeniable wih respec o boh y and ; from Assumpion 4, obain he following relaion (Armiage, 1952): P y < Y + d < y + dy = f ( y, + d)dy ( ) = ( 1- m x) 1- + b M d +m æ ( ) ö ç f M ç y - + ( b M ) yd, dy ç è +m ø +m ( x +1) f M ( y -1, )dyd + (( Od) )dy + o dy By passing he limi on boh sides in above equaion, we obain he differenial-difference equaion in he form as follows (5) f M ( y,) + ( b M ) + m y f M ( ) = ( x +1)m f M ( y -1, ) b M +m +m x ( y, ) f M ( y,) (6) By using he relaions given in equaion (1), he above equaion becomes, 581

6 STOCHASTIC MODEL FOR CANCER CELL GROWTH f M ( y,) + ( ) + b M y f M ( y, ) y +m x 0 exp{ ( )} ì b = m exp M M í î +m - y +1 - ( ) + b M ü ý þ f M + m + m x ( y -1, ) f M ( y, ) (7) f M ( y,) + + ( b M ) y f M ( y, ) y +m x 0 exp{ ( ) + m } = m exp{ ( b M )} f M - y +1 ( ) + ( b M ) - + m æ x 0 exp N + m ç è ç exp ( b M M ) The boundary condiion for he above equaions is ( y -1, ) { + m } - y ö ø f M ( y,) (8) f M ( y,) = 0, for y < 0, ³ 0 lim y i f y M ( y,) = 0, for all i ³ 0, ³

7 JAYABHARATHIRAJ JAYABALAN The ineres is o obain he saisical momens such as mean and variance of malignan cells for a given ime. The probabiliy generaing funcion is y s, s f y, dy : 0 s 1 (9) The parial derivaive ϕ(s,) wih respec o s exiss and from he boundary condiion, ò s f y, ys y dy = - f s, + slogs s f s,. (10) Muliply boh sides of equaion (1) by s y and inegrae, which yields he following differenial equaion for he generaing funcion as s y ò f y, dy + ò ( ) + ( b M ) + m ys f y, y s { + m } x 0 exp = m exp{ ( b M )} ò s y f ( y -1, ) -x +1 ì ( ) + ( b M ) + m ü - x 0 exp{ ( ) + m + m } ò í exp{ ( b M )} - m y ýs y f y, î þ dy (11) 583

8 STOCHASTIC MODEL FOR CANCER CELL GROWTH f ( s,) + ( b M ) + m + x 0 exp = m ò exp ( b M ) +1 - ò m ys y f y -1, ì - í f s, î { + m } + ( b M ) + m + slog s s y f ( y -1, ) s f s, ì ü - x 0 exp{ ( ) + m + m } ò í ýs y f y, exp{ ( b M )} î þ + ò m ys y f y, ü ý þ (12) f ( s,) f ( s,) - -m s 2 + m s + = m x 0 exp N - m s + + ( b M ) + m { + m } exp b M + ( b M ) + m + m = m x 0 exp N f ( s,) s slog s ( s -1 )f s, exp b M f ( s,) slog s - m s2 s {} ( s -1 )f s, (13) To obain he momens, use he cumulan generaing funcion of y(). Le K(u,) = logϕ(s,), where s = e u. On simplificaion (Bharucha-Reid, 1960), K ( s,) - m + + ( b M ) + m + m x 0 exp = m exp ( b M ) eu -1 u - m eu K s, K ( s,) s (14) 584

9 JAYABHARATHIRAJ JAYABALAN Saisical Momens The momens of he model can be obained by expanding he cumulan generaing funcion K(u,) on boh sides of he expression as = ue Y ( ) K u; u2 Var ( Y ( ) ) + L, comparing he coefficien of he power of u s and v s, and equaing coefficiens on boh sides of he equaion. In his way we arrive a he following linear differenial equaions of consan parameers d Var Y d d E Y d = = 2 + ( b M ) E Y ( ) + m x 0 exp +m N exp ( b M ) + ( b M ) -m E Y ( ) Var Y + m x 0 exp + m N exp{ ( b M )} (15) (16) Solving he differenial equaion in (14) and (15) gives he average number of malignan cells and variance of number of malignan cells a a given ime. On solving above equaion, exp E Y C b d b d 1 N N M M b x0 exp bn dn y0 exp bm dm bm dm N dn 585

10 STOCHASTIC MODEL FOR CANCER CELL GROWTH { + ( b M ) } { + m } - m V Y = C exp 2 b 2 N N m x 0 ( b M )exp - ( b M ) + ( ) - m b M - m y b 0 ( N ) + m exp ( b M ) ( b M ) + 2( )( ) { + ( b M ) } + ( b M ) + C exp 2 b 1 N N The inegraion consans C 1 and C 2 will be obained using he boundary condiions of he differenial equaions. Numerical Sudy For he fixed parameers and changing ime, he changes are observed in he average, and expeced and variance numbers of malignan cells in any organ are presened. The numerical sudy was conduced using Mahemaica 8.0 sofware for solving he differenial equaions as given above in equaions (14) & (15) numerically. The average and variance of number of malignan cells for fixed values of he parameers, = , d N = , b M = 0.04, d M = , x 0 = , y 0 = , and varying values of muaion rae and ime are presened. For he good mainenance of normal cell level, growh rae and deah rae of normal cells are assumed o be equal, and large birh rae values for malignan cells and muaion rae are presened in he Figure 1. From he Figure, i is observed ha here is a posiive relaionship beween ime and average number of malignan cells; a posiive relaionship beween ime and variance of number of malignan cells a lower values of muaion raes, and so on. Conclusion Birh, deah, and single muaion processes wih differen growh raes are considered, o he sudy he growh of malignan cells by assuming X() is dependen on Y(). The usual wo dimensional models are replaced by a one dimensional model represening normal and malignan cells wih ineres in 586

11 JAYABHARATHIRAJ JAYABALAN a) μ = b) μ = c) μ = 0.06 d) μ = 0.08 e) μ = 0.1 f) μ = 0.15 Figure 1. Variaion in he momens wih respec o ime. 587

12 STOCHASTIC MODEL FOR CANCER CELL GROWTH malignan cell populaion. The saisical measure shows ha volailiy of he malignan populaion decreases as he muaion rae increases, and average number of malignan cells increases drasically as he muaion rae increases. The resuls of his sudy may help o undersand he behavior of malignan cells over a period of ime wih various decision parameers. References Armiage, B. P. (1952). The saisical heory of bacerial populaions subjec o muaion. Journal of he Royal Saisical Sociey. Series B (Mehodological), 14(1), Bharucha-Reid, A. T. (1960). Elemens of he heory of markov processes and heir applicaions. New York: McGraw-Hill. Kendall, D. G. (1949). Sochasic processes and populaion growh. Journal of he Royal Saisical Sociey. Series B (Mehodological), 11(2), Serio, G. (1984). Two-sage sochasic model for carcinogenesis wih imedependen parameers. Saisics & Probabiliy Leers, 2(2), doi: / (84) Tan, W. Y. & Brown, C. C. (1987). A non-homogeneous wo-sage carcinogenesis model. Mahemaical Modelling, 9(8), doi: / (87)