-℄n`P`>M Q+E 28" Rick Durrett P=L " p N <3g($%O_ = R &! R (Æ 0* = H /LN I ~. { }fP ( ? {z y jl H 80 S oj\q)* +I H5 =Hp+I Gompertz (L d) + ILN [21]. ~*B y m 8 \ Æ}j. ` 80 SLNo~ 8 ! T p/ v1>: [13]. 4QD}|,TDwC" A m :?*BÆ > R s y us (Agent-based) LNo\ ` 80 S0*GM b h3+ j\ —— I;bhÆ+I q)qZm [1]. e(CRs[0*p: [4]. us )}=t " ,\B$N 0* Ut [24], l = " V LN Q= 8 (: [16], [6], [8]). E,?FNt R 8w 5y% LN " N 8 R s SLN% 9 =N_A4//_a=2 HD Nu1 P( ? 9|F 4 HKm 1 N*tR APC l%.2 N*Rl%H3 N* ! KRAS (K )4 ), 5y 4 N*4 P53 mt R ~. e H R 9: Luebeck y Moolgavkar R Æ 5 [17] F Steven Frank J [12]. 5' Fr " <yT s H 3 R x_, ' % kItD8 w <3_ ` APC =tpq) h { R l% &i*mmk / 8 - _h7 KRAS =tpf t R l% ~. G .H R * Q Æ 4 sy^_mH&PÆ~.xZ = 8 *U~. NsT9 t = v P53 =~g 53 r &! rKJ g=t R U1 w"U* ? hhS .Q H 3 R \Y 1;l359so <34"O}|,~ BRCA Æ i ( 1g30v },6 e R8QHp e h& Il5t R 6{I /55 Q ?oFN 8 R s SLNt Re ; <3'k4t R hÆp3 f4~.Z!obg a ,~./ u . 4{= E LN O_*GUt UoTI =H R “e ;<3” 34i s 8w 4 20% HKmAPC <~.x y APC 46| uFWm f β– z&! W y 6 w =f ~. R0Ht<36 D9Ph q) DZ 0* & Notices of the AMS, Vol. 60 (2013), No. 3, p. 304–309, Cancer Modeling: A Personal Perspective, Rick Durrett, Figure number 3. Copyright c 2013 the American Mathematical Society. Reprinted with permission. All rights reserved. ? q Qa ÆGGZYp ÆG> Durham rsa James B. Duke QaBGf7 `> rtd@math.duke. i edu. · 340 · rr ? 0 4 " N 8 R s SLNmi N* V/Æ b , +Y 6 ~.u (/Æ a , G + i + 1 N* (/Æ u . 4HÆHD(/=t R [/5 Markov (9 & o ;) {/{ t = 0 /tR*4 t Rw/5V,\H/= a t + o(t). ( Z (t) = i N*4 t /s O 0 N* , \QÆ}R&Æ5 ~" [B : Athreya y Ney E R Æ zv [2]. r λ = a − b =[;I/T; 0 N*4 t / s "kO= i i i+1 i i i r a i i EZ0 (t) = eλ0 t Z0 (0). 0 > b0 , T; 0 N*9Hg H /Æ λ /a ; g<Hg: 0 e −λ0 t 0 Z0 (t) → V0 = exponential (a0 /λ0 ). HÆ “→” 19O!~ H / 1 D V = exponential (r) & “V aO/ r”, H { v ≥ 0 / P (V > v) = e . * N ( ?wC 0 → 1 → 2 → · · ·, H Rs S Xii8w x_ 1 N *~./5 .<}938wZ 1 N* k ^: −rv t EV0 eλ0 s · u1 · eλ1 (t−s) ds. EZ1 (t) = 0 R U Y4/s s, 0 N* m L 2 R =~./ 3 R =4 t /s E ~.su O x= r 0 < λ0 < λ1 , H R k ^ EZ1 (t) rr Ma 1 N* O 8 uo \ 2EN 0 /5x4P u1 mH R\ 2= 6 H (10−5 W "), g_<4 p 1Admtnk ^t R G+t R6 Hr H r #<< 4 8 ! " N s S ( ?/x$m4: V0eλ s ≈ 1/u1, y 1 N*~O[ /G+/5?WÆC Vb_sS 51 N*~.=6nh Poisson (7X) s S Poisson s S 'LR y Laplace ( i U) . 8w,\o~QI 1 0 z lm α = λ /λ . H R H r 4 V ^: e−λ1 t Z1 (t) → V1 , 0 1 Ee−θV1 = (1 + cu1 θα )−1 , 0 E(e−θV1 | V0 ) = exp (−cu1 V0 θα ) / W & I (5y& k k > 1 N). 9: [11]. 8 ,=t R w, . Laplace . HD 8 :4t j\+ H /uSm oB Æg=,r p t= 8 “ _. (S − b )/a 0 y oW"4=1;” 4Hp$S, . Y1 (oX) =D( Qh xO9?i g f~. 6 e/= 6 H ( R0 E # 6 Iq) jR O. ~ so 4t; Barrett 2 h #aj\m [19], C M Rf jR&a q){' e 5&ÆÆo~"DlgP $ RM? gq)jRÆb=Q Y * tv y^jR ;I q)/ u jl* oWR W r 't 0 EqtDI * 9| Nq)*U5 ZM jR I~% n n n · 341 · 1 uS# O9StTn r* kS I>ndo6NO l u S/t ,J0 >mv,J02 uS5 [0, .05] 6 ug9; 2:F/t,J0z5 d 0tmvQ Q y q 1 O+ _R kS+ #Q Jasmine Foo G+jR t r 9: [20] (4 PubMedCentral =D 7 , ). q) " nR t R a0Simpson a0 R , o~ Æ R _^9 k N* '6|~. H / y^_PUÆ | s 70 " [ tD e . <3o~QI k N* t R6 H8w H r ER = 1 − α , mn a = λ /λ . z s Mt< X o~,\ R `i % &l 8 : 9: [10]. R Z6HDHrQÆoW (i) 9|~.P* 6 \ 2 Poisson s S19 —— H= "4 V Æ/e; : 1, tRLX" N8Rs Smq)jR 9 (ii) Poisson-Dirichlet ( q) 8 :'j\ O_1 e Y14 T p T n rmi+_m 4 LN_^9Z! H ^ Rj W" 9: Pitman J [23]. 1. P)LM " N 8 R s SLNWt R 9* % 5 ~v xl3R W Æ so t R 8w qr_'m+ Kaveh Danesh Æp qv_mJ D Gmv+ q t R r % X Evan Myers y Laura Havriletsky :T “Æ C 2 l2No " ℄ t h2NE” ÆIfH R r=t R" N 8 R s SLN 4H R LNm0 N* 2NI2 h m # ( q) 1 N* > 4 A um2 N* =TD}R C 4 A O * 2 N*)*j o{z (℄ p f g{grk 4/P&o Æ III F x 2 3ONOn$ ) + gO (45) B P+ g 4H R LNmi 0 N 1 N O ( 5) z|{+ gO (V5) nN Q) y i 1 N 2 N {.$%* F `y> [T , T ], mn T >$ ) r* ZX 0.5cm 1 S 06T > 10 S |{+ (* 1 r) sz%6 ~. 06 Kaveh Danesh 2N t R r=X " 4 k k k k k−1 k i 0 2 2 · 342 · 0 9 4 III IV F ?_Kl 5 [k/9 30%, % U6( I F q) F 5 [k/oo 90%. g r PW&wh4 I F % 9_W^X " + K (Ye6 . #a II F —— 952N / I2 h q) —— 59RHY 14 I F y III F U5 ) E _x= T —T , lmT = t h# ( q)YW= 0.5cm /5 (igz s R L*7Eo~i :), T = th 10 R 2 N* (6{tq) /5 HD = _ x [ 8 ~ U Ylg/5 H r }T q)+I y ~./ Y H5=U_= 2.9 [r= 88 :$m4 [2.5, 3] [U5 gÆ"C=[Uo thE 9 : [7]. 2 U Yt < * 4 O0 ;I 2. ,FÆ 7A= W (ÆWO_ 0*8 ! 8LN 47kXs+emj\)+L N y Ising (xH) LN 4_ Ising LNm= R x#℄QÆE QÆo~= +1 (=) −1 (=,), y℄ + {[/f$N . =℄x# O { 4t R J3 xwm #M v5mQR0=℄l℄U5 6 5 6 ( +46$ #5 % P h H D8Ising LN=W`R.e6. Rj 't0EP+_1zH W (ÆTD!u 6e*GLN Cristini y Lowengrub J [6] 6 A FN3}h#.LNHp.=OR3 H: "D F6us =L4 ^ k -*.y*5Tp6 oG+9 , 24 R BO?LX4h 1mm } h mlLNq W℄ ' xa %6 h : 3 LX ^ LN ? 0 WMQ 3 J 8 !xl 4 13R 4 2012 [ 8 + 7—10 > p - f O a SIAM(Xs%O__) +Km_ (Life Sciences meeting) =y Paul Macklin lH 3 ~ i $ NY + x >A R LN LX,\= { ~a}vfC [18]. (&U% (TC) + r=4 1 X j Camo: 3} h mV* >y<_+ >1 I w o(*k4 ! g *W ({pW ) + > W (W ) + > Z a,&x[1 !r H=t R so r hAJVhA t> 9 Æ 50% =R s 1 q)x[M ^R*W 2 Æabl5 (5~ d i nK } & hEMlH= Æ 1 h59= * & ). 5 WaGn 2:J |[ T NY 06 (j -): 0, r ZM W Æ9* LN W" I~ 7, 14, 21 z 28 u b [18], 5 B,O_v_+_ (Mathematical Medicine and 6,a1e -Y-|2 2 0 0 2 9 · 343 · 2012 [ 1 dlm 2009 [urÆ “P,\LNMU1}” SIAM 0 Njn (SIAM three-part minisymposium) 8# H =L 0* 9i? | *Wx=Hp 0*~ soO4P r℄4H 0 Ed) 8LN y * GRLN68W 8LNo}B*GRLN Y C% *GRLN:oiY8 LN ^ 8# 5yWi P BlBO 3. 5JTG <Hn t_A<tD!If r?=9N g?qY R r L! 8 ! y }, 8 *M [22], ?t R 3* 8 Æ 6 R fN 0* }R& ( ?o V b/ 0. —— f!ZHsz (estrogen receptor/ER), /{Hs (progesterone receptor/PR) y HER2 q) 8 }R G lC r x=v w k4 Foq= I0 = 16% q)09<HD0.5 o Æ [27]. 2012 [ .+ y t6; Marc Ryser 4: qPj\ j\% Kimberly Blackwell /I't R r q)jRq)a n'oW9 . ~. g :oI3jR v5H ^ ~-MQ W 83 Wxa7E q_=T:o+D9W 8 nq)jR}f 3 R (false Biology) negatives). FU5yF&e3jR ~" <}=T [25]. %Phr Kq) DZj\x= ^ =LN U eE &4 f4h X " <}= 9T ~ ~ UT iSÆh7 ÆtJO_1 0* =j\8 v5LNi % I g D jRL8= { wLNBO ~Hp 0 8 Yo~ T pP N H r 4. I(3DC$O 2012 [ 8 + 23 (Nature) 0d\: (p. 462—463 “I g ” >o), 146 Qb1Iq) K * (CSC) k4CSC W ` >iz K *G+OH W TnG+q)* CSC k4tYk4M z[j\1I{ * &x#KsV/ ~ " oPg YHn H5 q)* ta =f4 % U |j aY?*il+I Vmz[oW G .lQÆ g9: [29]. 3 _I5lm_ ( 14 ([5], [9]), 't_ ( 14m_ (Science) [26], tpP j)~ y^*M41 q)m~*i % +DHp . : Hp3lA*M=~wGr" O T,y Ig=~U % 6Lo N |\ K * _3%= : 3G+3lm g D QÆO_1?o so :oI : * ~B, [28] F il 1, m (4 CSC y R Æ LN, q)Pj ^ x = &sm ~v k4., OH K *tD8w<3q= T g℄ 10 R HKÆm=t R =x 0 a ,== R Æm 4—20 R OH K * b `m K *G+*lmtD4 { /s =T$ xT$ 7 · 344 · /oW= 10% FqoW= 90%. ~It/5 ÆÆK*=9 P8 "G+tRK*y tR* % N j\1I /* 8 " H r = RK */= R * [15]. ( 1 = F 8 !6e{= Bramson y Griffeath 30 [tE t_ e ^ aLN A [3], e? E H r HKÆOb t)}m 5. 1'#9H<:N-& +_9= _ },_ _ 6 5 y / 5}RqY ? 0 O_ r< T5 O_1qo~ ~ hJ j\HDr % ~4+j_ y^_14=Pr 25 [ X Rli+m 80% r=oIf “ r=1;”, 61;O_ X T= { Bt R ` 8w LNMQI! j\5yWTItD 39 " 6. 40KR!,D.*>IU6*/ vZ =ms 2012 [ 8 + 7—10 4-fOEh SIAM +Km_4 n5~"PLN :* 2014 [W"| V 2014—2015 [O_+ m_)?qZtRMP=L&!8wx T/ZW `Bk2m=8 54 ritDAF4 http://michorlab.dfci.harvard.edu/ index.php/publications z s| Franziska Michor , Wk/ )!Y o A=(/5 " N4Yb MathSciNet (> p O_g5 (Mathematical Reviews) 7 —— x) /U Y 2000 [~ ( 1 2423 _ e P 5 y 212 _ e 3 5e 3 5 " O=}, 8 ! ` 80 S ToW4 100 " _ e b h +Q 5mB W" arXiv (O_2 z —— x) m +_= 8 P=L5xqLÆbD$ E =h 5 PubMedCentral (> pp +j\) NIH q Z t; = k|+v_+Km_m j2 —— x), NIH ℄CUFUo4HE ( :e 5 H/=t Rv ?$q =t R ?$ 2010 [~40rm u “3” 5 3800 _ ^ Zo~i0d y J#mrL$x=RtÆbMO_+_&! v0* =Bt+_1v_j\ %n5 n= ℄ 5A+ v5C BYZW D1;we j\59x= T^ <} ~ "= SB (4) %;? [ 1 ] T. 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