Modelling Intermittant Androgen Deprivation Therapy. Alex Browning Dr Matthew Simpson Queensland University of Technology

Transcription

1 Modelling Intermittnt Androgen Deprivtion Therpy Alex Browning Dr Mtthew Simpson Queenslnd University of Technology

2 Introduction Previous wor by Klotz [], hypothesised tht prostte cncer cells my be clssified s either sensitive or prtilly insensitive to the level of ndrogen present in the ptient. Motivted by the schemtic presented by Klotz, this project ims to cpture some of the ey behviour in corresponding mthemticl model using ordinry di erentil equtions. Anlysis is performed on the ssocited system nd ey behviours re identified nd quntified. Klotz s model presents three cell types: stem cells, prostte cncer cells tht re prtilly insensitive to the ndrogen concentrtion s well s those tht re sensitive. The deth rte of these cncer cells is therefore driven by the presence of ndrogen. If the ndrogen concentrtion is bove certin threshold (di erent for sensitive nd insensitive cells), the cncer cells do not die. The ndrogen concentrtion cn be regulted through continuous nd intermittent ndrogen deprivtion therpy, the ppliction of which is investigted in this report. Figure : Schemtic demonstrting the interprettion of Klotz 23 model The mthemticl model presented in this report utilises ordinry di erentil equtions, ignoring the sptil structure of the cncer. Only temporl is investigted, nd the popultion of the two cncer cell types nd the concentrtion of ndrogen re considered. Suppose tht S(t) represents the mss of sensitive cncer cells t time t, I(t) represents the mss of insensitive cncer cells nd A(t) represents the concentrtion of the ndrogen. If we ssume tht the popultion of stem cells is lrge, nd the density pproximtely constnt, then we cn pproximtely 2

3 describe the dynmics of the system s follows: ds(t) = D S S (A, t)s(t), di(t) = D I I (A, t)i(t), da(t) = P (t) A(t), () where D S > is the rte t which the stem cells di erentite into sensitive tumour cells, D I > isthe rte t which the stem cells di erentite into insensitive tumour cells, S > is the rte t which the sensitive cells die, I > is the rte t which the insensitive cells die, P (t) > is the production rte of ndrogen nd > is the decy rte of ndrogen. We ssume D S,D I nd re constnt. The ey feture of Klotz schemtic is tht the deth rtes of the two tumour cell types depend on the concentrtion of the ndrogen, A(t), nd this dependence cts over di erent thresholds. For exmple where R> is constnt, setting S = if A(t) >AS, R if A(t) <A S, (2) I = if A(t) >AI, R if A(t) <A I, nd setting A S >A I reflects the ey elements of Klotz s model, nmely If A(t) <A I then I = S = R nd the rtes t which both types of tumour cells die re equl, If A I <A(t) <A S then S = R nd I = nd so the rte t which both types of tumour cells die re di erent, with the sensitive popultion dying due to ndrogen suppression wheres the prtilly insensitive popultion is un ected, nd If A(t) > A S then I = S = reflecting the hypothesis tht the ndrogen concentrtion is su ciently high tht both cell popultions do not die. Androgen deprivtion therpy cn be incorported into the model by ssuming on ndrogen deprivtion therpy, P (t) = p o ndrogen deprivtion therpy. 2 Anlysis 2. Stedy Stte The stedy stte is defined s the popultion or concentrtion wherein system of ordinry di erentil equtions suggest no chnge with time. Mthemticlly, the time-derivtives in the set of ODEs re equl to zero. A system of ODEs such s those developed here my or my not hve stedy stte. In the cse where the ODEs re not utonomous (there is dependence on time, such s the ppliction of the therpy), it my be tht no stedy sttes exist. If the ppliction of the tretment occurs periodiclly, it is possible (s will be explored lter on) tht the long term behviour is periodic. The stedy stte of the system is (S, I) =(S,I DS )=, D I () S I (3) (4) (6) 3

4 provided S 6=, I 6=. If S = or I =, the relevnt popultion increses without bound. It is immeditely obvious tht the existence of stedy stte is dependent on, which is in turn dependent on the concentrtion of ndrogen. If the ndrogen concentrtion is bove the corresponding threshold, then = nd the stedy stte does not exist. In this cse ech popultion increses without bound. Initilly, di erent cses re considered, dependent on whether or not the level of ndrogen exceeds ech threshold. This will determine whether the model will result in stedy stte. The stedy stte for ndrogen cn be shown to be: ( p A if P (t) p, = (7) if P (t). Thus, if P (t) (the tretment is pplied constntly), the ndrogen concentrtion tends to. This mens tht the cncer cells deth, nd zero stedy stte will be reched. if P (t) p (no tretment is pplied), the ndrogen concentrtion tends towrd non-zero stedy stte. The vlue of this stedy stte will determine whether or not stedy sttes exist in either cncer cell popultion. It is lso noted tht A pple p. In other words, the ndrogen concentrtion cn never exceed p, unless they initilly strt bove tht vlue. It is therefore cler tht IADT will never cuse the ndrogen concentrtion to permnently become zero. Cesing the tretment will cuse the ndrogen level to rise, tending towrd the stedy stte given bove. It is cler tht the stedy stte of ech of the cncer cell types is not directly ected by the tretment ppliction. Rther it is the ct of pplying the tretment to eep the ndrogen concentrtion below the threshold tht ensures stedy stte is reched, interpreted s the cncer not growing. It my therefore be cost e ective to pply the tretment in such wy tht the ndrogen concentrtion is just below the threshold, s there is currently no mthemticl indiction tht mintining the levels ny lower hs ny benefit, bsed on the nlysis developed here. 2.2 Tretment Regimes It is relistic to ssume tht the ndrogen concentrtion in ptient with no tretment will remin pproximtely constnt. At the strt of the tretment, it is therefore resonble to ssume tht A() = A = p. Note tht the choice of units for the time, t, is rbitrry. Thus the prmeters D S,D I, nd R re given in [units]/[unit of time]. The prmeters chosen ccount for the following ssumptions:. The ndrogen concentrtion strts t the stedy stte (i.e. with no tretment, they will remin constnt). 2. The stem cells di erentite into sensitive cells fster thn into prtilly insensitive cells. 3. The initil popultions of the sensitive nd prtilly insensitive cells re equl (it is observed tht the initil popultions hve no e ect on the long term behviour). 4. The exponentil decy rte, =.. Globl prmeters used in ll cses, nd throughout this report, unless otherwise specified re: 4

5 Initil popultion of sensitive cells S() Initil popultion of prtilly insensitive cells I() Androgen decy rte. Sensitive cell threshold A S Prtilly insensitive cell threshold A I 2 Sensitive cell di erentition rte D S 2 Insensitive cell di erentition rte D I Constnt tumour deth rte R.2 Hence, the only prmeters tht will chnge between cses re the initil ndrogen popultion (lwys equl to A ) nd the norml ndrogen production rte, p. The periodicity of the tretment pplictions cn then be modified to explore the e ect of the tretment Cse : Androgen remins below both thresholds Here nd we ssume A <A I <A S, p =8,A() = 6. In this cse, the ndrogen level lwys remins below both thresholds, so tht both cells lwys experience growth nd decy. This cse represents the sitution where ptient hs cncer tht is no longer growing. This leds to the popultion of both cncer cell types tending towrds stedy sttes. No ndrogen deprivtion 6 4 S I A Production Figure 2: No ndrogen deprivtion therpy, where A <A I <A S When no tretment is pplied, the ndrogen level lso remins constnt t the stedy stte. As lredy stted, the ndrogen level is below both thresholds, cusing both cell popultions to tend towrd non-zero stedy sttes.

6 Continuous ndrogen deprivtion 6 4 S I A Production Figure 3: Continuous ndrogen deprivtion therpy, where A <A I <A S When continuous ndrogen deprivtion tretment is pplied, it is ssumed tht no ndrogen is produced, cusing the ndrogen concentrtion to decy. Agin, s the ndrogen concentrtion cnnot exceed the thresholds, both cell popultions tend towrds non-zero stedy sttes. Intermittent ndrogen deprivtion Figure 4: Intermittent ndrogen deprivtion therpy, where A <A I <A S 6

7 Intermittent tretment cuses the ndrogen concentrtion to vry, eventully becoming periodic (this is exmined lter). The mximum level however does not exceed the thresholds, cusing both cncer cell popultions to gin tend towrd stedy stte, regrdless of the periodicity of the tretment ppliction Cse 2: Androgen my only exceed prtilly insensitive threshold Here nd we ssume No ndrogen deprivtion A I <A <A S, p =, A() = S I A Production Figure : No ndrogen deprivtion therpy, where A I <A <A S With no tretment, the ndrogen concentrtion will remin constnt. This leds to the sensitive cells reching their stedy stte popultion rpidly, s they re ble to undergo both prolifertion nd deth. Prtilly insensitive cells do not die t this level of ndrogen nd hence grow without bound. The increse is liner, t constnt rte given by the di erentition rte from the stem cells. 7

8 Continuous ndrogen deprivtion 3 2 S I A Production Figure 6: Continuous ndrogen deprivtion therpy, where A I <A <A S With continuous tretment, the ndrogen concentrtion depletes, llowing both cncer cell popultions to experience deth. This, in turn, gin cuses both cell popultions to rech their corresponding stedy sttes. Intermittent ndrogen deprivtion Figure 7: Intermittent ndrogen deprivtion therpy, where A I <A <A S 8

9 When intermittent ndrogen deprivtion therpy is pplied, the insensitive cells re ble to experience periods of growth only, s well s periods of net deth. This cuses the popultion of insensitive cells to vry in periodic mnner. The ndrogen level lwys remins below the threshold for the sensitive cell popultion, cusing these cells to rech their stedy stte Cse 3: Androgen my exceed both thresholds Here nd we ssume A I <A S <A, p = 3, A() = 6. This cse presents more cliniclly relistic scenrio where the ndrogen level is llowed to exceed both thresholds. Here the bsence of tretment cuses both cncer cell popultions to continue to grow without bound. No ndrogen deprivtion S I A Production Figure 8: No ndrogen deprivtion therpy, where A I <A S <A As seen previously, when there is no tretment the ndrogen level remins constnt. In this cse, the ndrogen level remins bove both thresholds, cusing both cncer cell popultions to permnently undergo experience no deth. The only chnge in popultion is the increse s result of the di erentition rte of ech cell. Both cell popultions increse linerly, the rte determined by the di erentition rte. 9

10 Continuous ndrogen deprivtion 6 S I A Production Figure 9: Continuous ndrogen deprivtion therpy, where A I <A S <A When continuous tretment is pplied, the ndrogen level decys to zero. The popultions of ech cncer cell tend towrd their respective stedy sttes. Intermittent ndrogen deprivtion Figure : Intermittent ndrogen deprivtion therpy, where A I <A S <A When intermittent tretment is pplied, the ndrogen level fluctutes in periodic mnner. As the

11 ndrogen concentrtion is llowed to go bove nd below ech threshold, ech cell type experiences periods of growth nd deth, chrcterised by the prmeters governing the cell behviour for the current level of ndrogen. 2.3 Long term behviour Consider the governing di erentil eqution for ndrogen: where P (t) = da(t) = P (t) A, p if t pple t pple t +, if t + <t<t + + b, P (t +( + b)) = P (t) (8) Here is the time o tretment, nd b the time on. It ws observed in numericl simultions tht periodic behviour my occur in the vribles A, S, nd I. Periodic behviour will lso lwys hve period ssocited with the tretment P (t). Thus, ll periodic behviour must hve period T = + b. It follows from the uniqueness theorem, tht, provided A(t) =A(t +( + b)) s t!, ech function S(t),I(t) will lso tend to periodic behviour s t!. While it my be di cult to find n exct solution for ny vrible, it is simple to solve ech vrible for given smll intervl, during which the DE cn be regrded s utonomous. Forcing the function to be continuous, nd forcing S(t) =S(t +( + b)) (nd similr for I), it is possible to gther vitl informtion bout the solutions. For exmple, the minimum, mximum nd mplitude of ech vrible in its periodic stte is found, in ddition to the prmeter vlues tht llow periodic behviour. Firstly, it my be useful to determine the durtion of time,, tht A(t) is bove certin vlue, nmely, : = log e (+b) e b p. (9) By considering the durtion of time bove A S (the mount of time S(t) experiences no decy nd only liner growth), the mplitude S nd the mximum vlue of S(t) in its periodic stedy stte cn be found: nd S = D S, () mx = D S R + D S. () e R(+b ) A similr result ws found for I(t). It cn be shown tht there is discontinuity in the mximum when: + b = log e (+b) e b p. (2) A S This corresponds to the minimum time o the tretment,, tht ndrogen concentrtion will exceed A S for some time.

12 2.3. Sensitive cell nlysis The e ect of chnging the tretment period is now exmined numericlly nd we present the results in the form of plots. All plots eep the prmeter vlues unchnged from previous nlysis, s in Cse 3 where A(t) is llowed to exceed both A S nd A I. Cse Here the time on-tretment is equl to the time o tretment so tht = b. Mx Min Figure : Minimum nd mximum vlue of S(t) in its periodic stedy stte, for = b In this cse, there is no discontinuity nd the mximum nd minimum vlues for I(t) re smooth s chnges. However, it is noticed tht t the point the mplitude becomes negtive (when the minimum is greter thn the mximum), the threshold is not reched, cusing stedy stte to occur. Also, there is no minimum mximum in this cse: the mximum is lwys decresing s the period gets smller. Cse 2 Here the time on-tretment is twice the time o -tretment so tht b =2. Mx Min Figure 2: Minimum nd mximum vlue of S(t) in its periodic stedy stte, for 2 = b 2

13 Cse 3 Here the time on-tretment is hlf the time o -tretment so tht b = 2. Mx Min Figure 3: Minimum nd mximum vlue of S(t) in its periodic stedy stte, for =2b Prtilly insensitive cell nlysis As the threshold A I is less thn tht for the sensitive cells, A S, di erent behviour is observed. This is becuse with the chosen prmeter vlues, it is more liely tht the ndrogen concentrtion will exceed the threshold for some time. Cse Here the time on-tretment is equl to the time o tretment so tht = b. Mx Min Figure 4: Minimum nd mximum vlue of I(t) in its periodic stedy stte, for = b The mximum hs discontinuity when 2.2. For vlues of less thn this, the mximum vlue is negtive. This does not correspond to negtive cell popultions, but rther indictes tht periodic 3

14 solution does not exist. In this cse, the vlue of I(t) tends to infinity: the prtilly insensitive cells grow without bound. It is lso cler tht minimum mximum vlue exists (pproximtely t =6.4 for the chosen prmeter vlues). The closer to the symptote the vlue of is, it is cler tht the ndrogen concentrtion is more controlled - the mplitude is smller. Cse 2 Here the time on-tretment is twice the time o -tretment so tht b =2. Mx Min Figure : Minimum nd mximum vlue of I(t) in its periodic stedy stte, for 2 = b When the time on the tretment is twice the time o the tretment, the minimum mximum vlue is less thn observed in Cse, occurring t vlue of closer to. It is hypothesised tht s the time on the tretment becomes lrger, the minimum mximum will become smller, nd the intervl on the tretment to chieve this will become smller. This is to be expected, s the longer the tretment is pplied, the lower the ndrogen level will remin. However, incresing the time on the tretment my not be desirble for the ptient. Cse 3 Here the time on-tretment is hlf the time o -tretment so tht b = 2. 4

15 Mx Min Figure 6: Minimum nd mximum vlue of I(t) in its periodic stedy stte, for =2b When the time on the tretment is hlf of the time o the tretment, the opposite from Cse 2 is observed. In this cse, the vlue of producing both the minimum mximum nd the discontinuity becomes lrger. Finding the optiml period The gol of intermittent ndrogen deprivtion therpy is to control the cncer cell levels (to be below some threshold), while mximising the time o the tretment. It is cler from previous nlysis tht minimum mximum cell level exists for I(t). For given n such tht b = n, it is optiml for the tretment to be pplied such tht this minimum mximum is chieved (chnging in this cse does not chnge the percentge of time o the tretment, while the mximum cell level cn be reduced without ny negtive e ects experienced by the ptient). While the resultnt eqution is too complicted to be solved using nlyticl techniques, the loction of the minimum mximum cn be computed using numericl methods.

16 n Figure 7: Minimum possible mximum vlue of I(t) s function of n As n is representtive of the proportion of time the ptient spends o the tretment, it is desirble for n to be s low s possible. This, we conclude with this nlysis tht it is possible to choose vlue of n tht produces minimum mximum vlue tht is less thn the desired threshold for the cncer cell popultion, while eeping n s low s possible. 2.4 Exct Solution 2.4. Exct Solution for ndrogen level t ny time In the cse where = b, da(t) cn be expressed in such wy tht n exct solution cn be found using Lplce trnsforms. Here, the ODE for the concentrtion of ndrogen, A(t) is given by where E(t) = The exct solution is given by: da(t) = pe( t ) A, if pple t pple, if <t<2, A(t) =A()e t + p e t + p E(t +2) =E(t). (3) X ( ) n e (t n) U(t n), (4) where U(t) is the Heviside step function. This function cn be thought of s two components. The first: A (t) =A()e t + p e t 6

17 is the nturl decy/growth of the ndrogen, un ected by the tretment. It is the solution to da(t) = p A, nd will lwys hed towrd the only fixed point A = p/. The second component cn be thought of s the e ect of the tretment - the e ect of turning the growth rte o, compred to hving it on. A 2 (t) = p X ( ) n e (t n) U(t n) It is possible to demonstrte tht A 2 (t), nd hence A(t) tends to periodic behviour s t gets lrge. This is shown in ppendix Conclusion The mthemticl interprettion of the model presented by Klotz (23) llows nlysis of the immedite nd long term behviour of the cncer, in reltion to the ppliction of continuous nd intermittent ndrogen deprivtion therpy. Possible future extensions my include the cncer developing resistnce to the tretment, s sensitive cells evolve into insensitive cells. Further, irregulr, intermittent pplictions of the tretment my lso be explored. In conclusion, the mthemticl model llows the optiml tretment ppliction to be chosen to mximise ptient welfre (defined s the mount of time o -tretment), while minimising the mximum size of the cncer. 7

18 4 Appendices 4. Exct solution for ndrogen concentrtion Lplce trnsforms were used to find the exct solution for the ndrogen concentrtion t ny time, A(t). da(t) = pe( t ) A, where E(t) is defined by (3). If we denote the Lplce trnsform of f(t) byl{f(t)}, then We now tht so L{E( t )} = e 2s = = = Z 2 e st E( t ) ( e s )( + e s ) ( e s )( + e s ) ( e s )( + e s ) = ( e s )( + e s ) = s( + e s ) Z e st pple s e st pple s s pple e s s e s +x = X ( ) n x n provided x <, n= +e s = X ( ) n e ns. n= We now pply the Lplce trnsform technique to the eqution L{ da(t) } = L{pE( t ) A}. Then sl{a} A = pl{e( t } L{A}, p so tht (s + )L{A} = s( + e s ) + A, where A = A(). p Then L{A} = s(s + )( + e s ) + A (s + ) = p X ( ) n e ns + A s s + (s + ) n= = A (s + ) + p X ( ) n e ns p X ( ) n e ns s s + n= n= = A (s + ) + p + p X ( ) n e ns s s + s p X ( ) n e ns s + 8

19 We cn find the inverse of the trnsform to obtin: A(t) =A e t + p e t + p = A e t + p e t + p X ( ) n U(t n) p X ( ) n e (t n) U(t n) X ( ) n e (t n) U(t n). 4.2 Technique for obtining the mximum Consider A(t) in its periodic stte, such tht: A(t) = A (t) pple t pple, A 2 (t) <t<+ b, A(t +( + b)) = A(t). () Then for pple t pple This cn be solved to give where C is constnt. For <t<+ b so tht da A (t) = p da 2 = p A. C e t, = A, A 2 (t) =C 2 e t, where C 2 is constnt. It is noted, for A(t) to be continuous nd periodic, we require Solving these simultneously gives, A () = A 2 ( + b), A () =A 2 (). C = pe (e b ) e (+b), nd C 2 = p e C. Firstly, we determine the time,, tht A(t) is bove vlue To do so, we solve A (t )=A 2 (t + )=, 9

20 for some t. A (t )= A 2 (t + )= = p C e t = C 2 e t e () ) e t = C2 e (2) (2)! () = p C C 2 e ) = log e (+b) e b p Assuming tht S(t) isperiodicwithperiodt = + b, the mplitude nd mximum vlue of S(t) inits periodic stte cn now be found. Consider S(t) = S (t) pple t pple, S 2 (t) <t<+ b, S(t +( + b)) = S(t). (6) Note tht S 2 hs been trnslted in time so s to strt t t = for simplicity. Then for pple t pple where C 3 is constnt. For <t<+ b ds = D S, which cn be solved to give S (t) =D S t + C 3, ds 2 so tht S 2 (t) = D S R = D S RS, C 4 e Rt, where C 4 is constnt. Note tht for S(t) to be continuous, nd for periodic behviour, S () = S 2 ( + b S ( ) = S 2 () = mx, ) = min, where is the durtion of time bove the threshold A S. Substituting the vlues in, the following equtions re found: mx = min +D S min = mx D S R e (+b ) + D S R It is strightforwrd to see tht the mplitude, s = mx min is given by: s = D S 2

21 When solved simultneously for min nd mx, the following solution cn be derived: mx = D S R + D S e R(+b ) Replcing D S with D I nd considering the mount of time bove the threshold A I, it is possible to derive similr formul for the mximum nd mplitude of I(t). 4.3 Proof A(t) tends to periodic behviour By considering the long term behviour of ech component of A(t) =A (t)+a 2 (t) where nd A 2 (t) = p A (t) =A()e t + p e t, X ( ) n e (t n) U(t n), it is possible to demonstrte tht the long term behviour of A(t) is periodic. Consider A (t) s the long term behviour of A(t), such tht Firstly, consider A (t) = lim t! A(t) = lim t! A (t)+ lim t! A 2 (t). lim A (t) =A() lim e t + p t! t! = p, p lim t! e t provided >, which is lredy stisfied. When considering A 2 (t), it is observed tht As such, A 2 (t) cn be simplified to finite sum: U(t n) =,t n >, j t ) n< t. A 2 (t) = p X ( ) n e (t n). It is noted tht t is the number of full hlf periods preceding t. Let A 2(t) be the long term behviour of A 2 (t), such tht A 2(t) = lim A 2 (t) t! 2j t = p lim 6X 4 ( ) n t! j t X ( ) n e (t n) 3 7. It is observed tht lim t! e (t n) =, provided t n > ) n< t. 2

22 Also but It follows t t pple t, < t. 2j t A 2(t) = p lim 6X 4 ( ) n t! j t X 2j t = p lim 6X t 4 ( ) n ( )j t! ( ) n e (t n) ( )j t e (t j t ) 3 7 e (t j t ) 3 7 Since By letting t lim t! j t X ( ) n e (t n) =. t = t,wheret is the distnce to the nerest preceeding hlf period, it cn be seen: 2j t 3 A 2(t) = p lim 6X t 4 ( ) n ( )j 7 e t t! Consider A 2(t +2) A 2(t) = p lim 6 4 t! 2 j t+2 X ( ) n ( ) j t+2 e t j t X t ( ) n +( )j e t 3 7 Since the distnce to the nerest hlf period, t remins the sme for t +2. It follows: A 2(t +2) 2 A 2(t) = p lim 6 X 4 t! j t +2 = p lim X 6 4 ( ) n + t! 2 j t j t ( ) n ( ) +2 e t j t +2 X j t n= + j t X t ( ) n +( )j ( ) n ( )j t = p pple t t lim ±+ ( )j e t +( )j e t t! = p lim t! =. ( ) 2 e t e t j t 3 7 X t ( ) n +( )j e t

23 So Consider A (t) = p + A 2(t). A (t +2) A (t) = p + p A 2(t +2) = A 2(t +2) A 2(t) = A 2(t) Thus, A (t) is periodic, indicting tht A(t) tends to periodic behviour s t gets lrge. 4.4 Anlyticl nd numericl solution comprison The following figure compres the nlyticl solution (4) with tht computed with the RK4 lgorithm implemented in MATLAB. 6 Anlyticl Solution Numericl Solution Figure 8: Comprison between nlyticl nd numericl solution References. Klotz L. 23. Intermittent versus continuous ndrogen deprivtion therpy in dvnced prostte cncer. Current Urology Reports, 4: Croo JM, O Cllghn CJ, Duncn G, Dernley DP, Higno CS, Horwitz EM, Frymire E, Mlone S, Chin J, Nbid A, Wrde P, Corbett T, Angylfi S, Goldenberg SL, Gospodrowicz MK, Sd F, Logue JP, Hll E, Schellhmmer PF, Ding K, Klotz L. 22. Intermittent ndrogen suppression for rising PSA level fter rdiotherpy. The New Englnd Journl of Medicine, 367: