Mathematical modeling of follicular development in bovine estrous cycles

Transcription

1 Konra-Zuse-Zentrum für Informationstechnik Berlin akustraße 7 D-495 Berlin-Dahlem Germany MANON BONDOUY AND SUSANNA RÖBLIZ Mathematical moeling of follicular evelopment in bovine estrous cycles ZIB-Report 2-26 (August 22)

2 Herausgegeben vom Konra-Zuse-Zentrum für Informationstechnik Berlin akustraße 7 D-495 Berlin-Dahlem elefon: elefax: bibliothek@zib.e URL: ZIB-Report (Print) ISSN ZIB-Report (Internet) ISSN

3 Mathematical moeling of follicular evelopment in bovine estrous cycles Manon Bonouy Susanna Röblitz August 6, 22 Abstract Normal follicular evelopment is a prerequisite for successful fertilization in airy cows. During the last ecaes, however, prevalence of anestrus has been increasing for ifferent reasons. In this stuy we use a mathematical moel to investigate mechanisms that lea to anestrus. his moel is erive by coupling two previously publishe moels: a small moel for the evelopment of multiple follicles (Smith et al., 24), an a large estrous cycle moel (Stötzel et al., 22). We first investigate the influence of synchronization protocols on the time-shift of ovulation. In a secon scenario we simulate an extene perio of anestrus as it typically occurs after calving. AMS MSC 2: 9C3, 92C3, 92C42, 93A3 Keywors: orinary ifferential equations, systems biology, reprouction, airy cows Zuse Institute Berlin, akustraße 7, 495 Berlin, Germany Corresponing author. susanna.roeblitz@zib.e

4 Contents Introuction 3 Biological backgroun 4. he phases of an estrous cycle Reprouctive hormones Description of previous moels 6 2. A large moel: he moel by Stötzel et al A small moel: he moel by Smith et al A tiny moel: he moel by Soboleva et al A submoel with multiple follicles 9 3. Moel equations Results he couple moel 4 4. Moel equations Results Moeling of external influences 2 5. Synchronization protocols Postpartum anestrus Conclusion 29 A Equations of the couple moel B List of Hill functions C List of parameters an initial values i iv v 2

5 Introuction Concurrent with selection for increase milk yiel, a ecrease in airy cow fertility has been observe uring the last ecaes [4]. his ecline in fertility is shown by, e.g., alterations in hormone patterns uring the estrous cycle, reuce expression of estrus behavior an lower conception rates [9]. However, the antagonistic relationship between milk yiel an reprouctive performance is a multifactorial issue [5]. New strategic irections for genetic selection incluing fertility-relate traits will take time to be effective. In the mi-term, mathematical moeling of the involve mechanisms is expecte to improve insight into the biological processes unerlying the bovine estrous cycle, an coul thereby help to evelop strategies against ecline fertility in airy cows. Although the enocrine an physiologic regulation of the bovine estrous cycle is stuie extensively, mathematical moels of cycle regulation are scarce an of limite scope [, 6]. A number of moels have been evelope for other ruminant species, especially ewes [6, 9], but these moels o not contain all the key components that are require to simulate follicle evelopment an the accompanying hormone levels throughout consecutive cycles. In a cooperation between scientists from ZIB an animal scientists from Wageningen, a new moel was evelope that integrates the major tissues an hormones involve in the ynamics of follicular evelopment [4]. his moel serve as a basis for the examination of follicular wave patterns [3], as well as for the stuy of synchronization protocols [8]. It inclues the processes of follicle an corpus luteum (CL) evelopment an the key hormones that interact to control these processes: gonaotropin-releasing hormone (GnRH), luteinizing hormone (LH), follicle stimulating hormone (FSH), estraiol (E2), progesterone (P4), oxytocin (O), an prostaglanin F2α (PGF2α). his moel generates successive estrous cycles with a varying number of follicular waves per cycle. Follicular evelopment is moele with a single equation that represents the state of follicular maturation an the ability of follicles to prouce steroi hormones. wo other moels have been publishe by Soboleva et al. [6] an Smith et al. [5]. hese two moels are very similar in that they both escribe the growth of many follicles by using one equation for each follicle. Large follicles suppress the growth of small ones, thus giving rise to usually one ominant follicles. he moel in [6] escribes ovarian follicular evelopment following commitment throughout one follicular wave. he moel in [5] aitionally escribes the interaction among follicles, LH, FSH, an E2. his moel has especially been constructe to simulate postpartum anestrus. Hence, simulations stop with ovulation an o not epict sequences of ovulatory cycles. he aim of this stuy is to couple these moels in orer to simulate the evelopment of multiple follicles throughout consecutive cycles. he paper is organize as follows. We start with an explanation of the biological backgroun in Section. hereafter, Section 2 briefly escribes the moels in [8, 5, 6], before we erive a new submoel in Section 3. In Section 4, we present the coupling between this new sub-moel an the large estrous cycle moel from [8]. Simulation results for synchronization protocols an postpartum anestrus are iscusse in Section 5. 3

6 Biological backgroun he usual length of an estrous cycle is about 2. he cycle can be ivie into four perios, which are separate accoring to hormone levels in the bloo: estrus, met-estrus, i-estrus, an pro-estrus. Ovulation is regulate by the interplay of various reprouctive hormones.. he phases of an estrous cycle Estrus refers to the phase when the female is sexually receptive. Since the corpus luteum unerwent lysis earlier in time, the progesterone level in the bloo is very low. E2 levels are ecreasing from the high levels that were reache just prior to estrus. Finally, LH spikes to a high level uring estrus, which initiates ovulation. During met-estrus, the corpus luteum starts to form. It is still small an oes not yet have the capacity to prouce large amounts of progesterone at this stage of evelopment. herefore, progesterone levels are still low, but increase slightly uring met-estrus. Di-estrus is characterize by the activity of the corpus luteum an the prouction of large amounts of progesterone. Small follicles emerge from a cohort of antral follicles, but o not procee further. Pro-estrus is characterize by regression of the CL an eviation of an ovulatory follicle. he follicle matures uner the influence of FSH an pulsatile LH, secrete from the pituitary. FSH plays an important role in the beginning of follicular growth, whereas LH is important for maturation up to ovulation. Pro-estrus an estrus form the follicular phase, whereas met-estrus an i-estrus make up the luteal phase. Figure : Progesterone level an ovulation [].2 Reprouctive hormones he hypothalamus controls the release of pituitary hormones by releasing small peptie hormones that travel own along short bloo vessels from the hypothalamus to the pituitary. In particular, the hypothalamic hormone GnRH causes 4

7 the pituitary to release LH an FSH. Other hypothalamic hormones either inhibit or inuce the release of aitional pituitary hormones (growth hormone, ACH, SH, MSH an prolactin). hese hormones have some affect on reprouction, but we will focus on FSH an LH because they more irectly regulate the estrous cycle. Estraiol an progesterone have a negative feeback on the hypothalamus. Corpus Luteum an progesterone he LH surge inuces ovulation an ifferentiation of cells of the follicular wall into luteal cells [2]. he primary purpose of the CL is to prouce P4. In general, as the CL increases in size at the beginning of the luteal phase, progesterone prouction also increases. As long as P4 levels are high, the ovaries continuously show follicular growth an ecay in a ynamic wave pattern up to the eviation phase, but ovulation is inhibite. When the progesterone level is low, ovulation can occur (Figure ). Follicular waves an follicle stimulating hormone he follicular growth is not linear, but occurs in waves (about 3 waves per cycle). Uner the influence of FSH, a cohort of 8-4 growing follicles emerge []. From this group of follicles, one follicle is allowe to grow to a much larger size than the others. his large follicle is calle the ominant follicle, because it has the ability to regulate an restrict the growth of the smaller follicles. A few after reaching maximum size, the ominant follicle begins to egenerate an ies. As the ominant follicle egenerates, its ability to restrict the other follicles is reuce. herefore, if FSH levels are high enough, a new follicular wave is initiate. As a consequence of this ynamic process, follicles of all sizes exist on each ay of the estrous cycle. he large follicle that eventually ovulates is ientifiable on the ovary only 48 hours before estrus. Estraiol As follicles grow they prouce increasing amounts of estraiol. Circulating concentrations of estraiol increase an ecrease as follicular waves grow an regress. Luteinizing hormone he main function of LH is to trigger ovulation, thereby releasing an egg from the ominant follicle, an to initiate the conversion of the resiual follicle into a corpus luteum. 5

8 2 Description of previous moels In the following we briefly escribe the large estrous cycle moel with a single follicle equation [8] an the two smaller moels for multiple follicles [5, 6], which form the basis for the couple moel later on. 2. A large moel: he moel by Stötzel et al. GnRH Hypothalamus GnRH Pituitary Estraiol Progesterone Inhibin LH Pituitary FSH Pituitary Follicles LH Bloo FSH Bloo Oxytocin proctn Oxytocin effect Enzymes PGF2 α Corpus Luteum Figure 2: Mechanisms of the large bovine estrous cycle moel publishe in [8]. Boxes represent the 5 key components of the system. Differential equations are erive for these 5 components. Arrows enote functional epenencies. Stimulating an inhibiting effects are inicate by an, respectively. represents a threshol. A large moel for the bovine estrous cycle has been publishe in [8]. Its mechanisms are illustrate in Figure 2. he moel comprises 5 orinary ifferential equations an generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It escribes the growth an ecay of follicles an corpus luteum, as well as the change of the key reprouctive hormones an enzymes over time. he moel has been valiate with ata from synchronization stuies. Although this moel takes into account several interactions between hormones, it has the isavantage of only simulating the accumulative behavior of the follicles. herefore we consiere more etaile but smaller moels that escribe the ynamics of many follicles at the same time [6, 5]. 6

9 2.2 A small moel: he moel by Smith et al. LH Pulse LH Bloo FSH Bloo Estraiol y Figure 3: Schematic outline of the hormonal interactions in the moel by Smith et al. he moel in [5] escribes the ynamics of many follicles at the same time an their interactions with LH an FSH. Follicular size is moele inirectly via E2 prouction per follicle. he moel oes not account for the time after ovulation an the influence of progesterone. Figure 3 epicts the interactions in this moel, which are escribe by the following equations: FSH ( FSH (t) = β FSH β ) t t max(x i) E2 prouction (x i is the estraiol prouce by follicle i, i =, 2,..., N) t x i(t) = k x i ( x i ) [y κ LH (µ x i ) Sensitivity of the hypothalamus to E 2 () N x j ] (2) j= LH t y(t) = a (max(x i) mean(x i )) b t { t LH (t) = l ν p c LH, ν p νp, ν p =, ν p < (3) (4) Pulse generator ( ) ( α b t ) sin(p) (5) t ν p(t) = A. max(x i ) t p(t) = ν p (6) Parameters Parameters β an β characterize the rate of FSH increase an ecrease. he function y κ LH supports follicular evelopment. A follicle ovulates if this function approaches µ 2, whereby µ escribes the rate of E2 prouction by each follicle. κ characterizes the sensitivities of the ovaries to LH. he parameter b etermines the ecrease in y, until y becomes low enough for ovulation. Parameters l, A an α escribe the amplitue of LH pulses, an c is the egraation rate of LH. 7

10 Functionality FSH oes not have a irect influence on the other equations, but it triggers an event: Whenever the level of FSH reaches some threshol, a new commitment begins. y an FSH are re-set to their initial values. Moreover, the number of follicles an the initial values of x i are selecte ranomly an inepenently for each new wave. he corresponing istributions are not given in [5]. A follicle is consiere to reach ovulation after many waves when x i =. Results Unfortunately, the publication [5] oes not contain information on the initial values of the species. he value of the FSH-threshol is missing as well. When we trie to re-implement the moel an to ientify initial values from publishe simulation results, we figure out that the simulation results are very sensitive with respect to the initial values. Hence our results were not conclusive. In particular, a high LH frequency challenges any aaptive stepsize control, an we obtaine ifferent results with ifferent numerical integrators (MALAB s oe45 an oe23s). herefore we ecie to go one step back an to implement a simpler moel that was publishe five years earlier [6]. 2.3 A tiny moel: he moel by Soboleva et al. A ynamic moel to escribe ovarian follicular evelopment following commitment has been evelope in [6]. Follicular evelopment is escribe by a series of i equations (i =, 2,..., N), t x i(t) = k λ2 E x i (E x i ) [λ 2 (λ x i ) where x i is the amount of E2 prouce by the ith follicle. N x j ] (7) Parameters he rate of estraiol prouction by each follicle is escribe by λ. he term λ 2 (λ x i ) N j= x j escribes the interaction among the follicles. he parameter k is a measure of time. he factor k x i (E x i ) escribes the logistic law of growth, with E as maximum level of estraiol. Functionality As in the moel by Smith et al., follicular growth is ientifie with estraiol prouction, an in a group of follicles the ominant follicle suppresses the growth of the smaller ones. Note that with this moel only one wave can be simulate, which usually ens with ovulation. Different initial conitions can lea to single ovulation, multiple ovulations, or even an anovulatory wave. In aition, the moel in [6] was moifie slightly to escribe the effect of exogenous FSH on the evelopment of follicles. Results We were able to reprouce the results publishe in [6]. Since we are still intereste in simulating multiple follicular waves, we will in the next step combine the two moels [5] an [6] an exten them towars the interaction with enogenous FSH an P4. j= 8

11 3 A submoel with multiple follicles Progesterone FSH Bloo y Estraiol Figure 4: Schematic outline of the hormonal interactions in our moel. Stimulating an inhibiting effects are inicate by an, respectively. Figure 4 epicts the interactions that we want to inclue in our moel. FSH an P4 are moele as input curves (Figure 5), which represent the typical change of these hormones in the bloo over time..8 FSH P Moel equations Figure 5: FSH an P4 input curves. FSH an P4 are ae into the equations for follicular growth via hill functions as in [4]. A new follicular wave is initialize if one of the two following events occurs: either all follicles unergo atresia or at least one follicle becomes large enough. In this case the integration is re-starte with new initial values of every follicle, an y is re-set to its initial value. FSH an P4 input functions F SH(t) = exp( (mo(t, 2) 3) 2 /5) exp( (mo(t, 2) ) 2 /5) exp( (mo(t, 2) 7) 2 /5) exp( (mo(t, 2) 24) 2 /5) (8) P 4(t) = exp( (mo(t, 2) ) 2 /34) (9) Follicular maturation t x i(t) = k λ2 E FSH (t) 5 5 FSH (t) 5 x i (E x i ) [y (λ x i ) N x j ] () j= 9

12 Sensitivity of the hypothalamus to E 2 t y(t) = a [max(x i) mean(x i )] b () P 4(t) 2 able contains parameters an initial val- Parameters an initial values ues use for the simulations. Parameter Value k µ a 7.2 b.7357 E Component x x 2 x 3 x 4 x 5 x 6 x 7 x 8 y Initial value able : Parameters an Initial Values Events We consier the follicles as being atretic if their level x i is smaller than.5 for all follicles, an we reckon ovulation if max(x i ) >.99. Whenever such an event occurs, we re-set the x i to their initial values in able, multiplie by a uniform ranom variable from the interval (, ). o obtain reproucible results with Matlab, we fix the internal state of the pseuoranom number generator with the comman ran( state,rancounter). he integer variable rancounter is increase by one every time a new follicular wave begins. he variable y is always re-set to its original initial value. 3.2 Results Regular cycles First, we present the results for follicles, FSH, P4, an y from a simulation with 5 follicles (Figures 6(a) an 6(b)). he moel generates successive estrous cycles with a length of about 22. As long as P4 is high, ifferentiation of a ominant follicle is inhibite, but ovulation takes place if P4 levels are low. Note that here an in the following, all components are imensionless.

13 x x5 fsh p (a) Follicles, FSH, an P (b) y(t) Figure 6: Simulation results for the new submoel: Regular estrous cycles.

14 Influence of the number of follicles As Figure 7 an Figure 8 show, the regular cycle pattern is maintaine if we change the number of follicles x fsh p Figure 7: Simulation with 4 follicles x x5 x6 x7 x8 fsh p4 Figure 8: Simulation with 8 follicles. 2

15 Influence of initial values If we change the initial values, cycles with varying numbers of follicular waves can occur. his can also happen in reality x x5 x6 x7 x8 fsh p4 Figure 9: Simulation results for the new submoel: Estrous cycles with ifferent numbers of follicular waves, resulting from varying initial values of the follicles. Influence of FSH perioicity If we change the frequency of FSH, we o not obtain estrous cycles anymore, because P4 an FSH are not correctly correlate in time to promote ovulation. his example illustrates that timing is very important to obtain regular cycles x x5 x6 x7 x8 fsh p4 Figure : Higher FSH frequency results in irregular cycles. o conclue, our sub-moel is robust to changes in the initial value an in the number of follicles, but not to timing of FSH. 3

16 4 he couple moel We now integrate the small moel with multiple follicles, which was introuce in Section 3, into the large estrous cycle moel that was presente in Section 2.. In particular, FSH an P4, which playe an important role in the submoel, are now the result of interactions between the hormones an no longer provie as input curves. he mechanisms of the large couple moel are epicte in Figure. GnRH Hypothalamus GnRH Pituitary Estraiol Progesterone Inhibin LH Pituitary FSH Pituitary LH Bloo FSH Bloo Follicle Follicle 2 Follicle N Oxytocin proctn Oxytocin effect Enzymes PGF2 α Corpus Luteum Figure : Schematic outline of the hormonal interactions in the fully couple moel. 4. Moel equations We briefly introuce the equations that have been aapte from the estrous cycle moel in [8], together with the new equations from the submoel. he fully couple moel comprises 6 ifferential equations plus one ifferential equation per follicle. he equations for follicles i =, 2,..., N are taken from the submoel in 4

17 Section 3: t x i(t) = k λ2 E H 2 (FSH ) x i (E x i ) [y (λ x i ) N x j ] j= (2a) t y(t) = a [max(x i) mean(x i )] H3 (P 4) (2b) hese equations are couple to the prouction of E2 an Inh in the following way: t E 2(t) = c E2 x i Inh(t) = cinh x t i N x i (t) c E2 E 2 (t) (3) i= N x i (t) c Inh Inh(t) (4) i= hus, estraiol an inhibin prouction are irectly correlate with the sum of all follicles. he remaining equations coincie with the equations in the estrous cycle moel in [8], except the following: LH-synthesis is boune by a maximum capacity LH max. LH-release is aitionally inhibite by P4. Rel LH (t) = b LH H (GnRH Pit) H (P 4) LH Pit (t) he growth of CL is not irectly initiate by LH anymore, but triggere by an event. he etaile list of equations an parameters can be foun in the appenix. Events In the same way as in the non-couple moel, a new follicular wave begins every time the ominant follicle ecays (max(x i ) reaches.5 from above), or reaches a size large enough for ovulation (max(x i ) reaches.9 from below). In both cases, the E2-sensitivity y(t) is re-set to its initial value, an the follicles x i are reassigne with new ranom initial values as in the submoel (Section 3). If the ominant follicle becomes larger than.9 an the LH level is higher than a certain threshol (.2), the event is ientifie as ovulation. In this case, also the CL is re-set to its initial value (.), an a new corpus luteum starts to evelop. 4.2 Results For most of the parameters, we kept the values from the submoel an the estrous cycle moel [8]. However, some parameters an initial values ha to be aapte to obtain regular estrous cycles. he final list of parameter values an initial values can be foun in the appenix. In the following, we present an iscuss some of the simulation results. We will first consier simulations with four follicles, before we analyze the behavior of the moel with respect to changes in the initial values an ifferent numbers of follicles 5

18 4.2. Estrous cycle with 4 follicles.5 x Figure 2: Follicular evelopment in the couple moel. Follicular waves As one can see in Figure 2, the moel prouces a solution with regular cycles. he cycle length varies between 8 an 24, an ovulation occurs in every thir wave. 2.5 E2 Inh Figure 3: E2 an Inh prouction. E2 an Inh prouction As expecte from the moel equations, Inh an E2 increase proportionally to the follicles (Figure 3). 6

19 FSH P Figure 4: P4 an FSH prouction. FSH an P4 he FSH an P4 levels result from interactions between the hormones an are not efine as input curves anymore. In Figure 4, it can be observe that the solutions are less smooth than in the sub-moel. he influence of P4 is still significant, but now FSH levels in the first follicular wave are always very low. o maintain, nevertheless, the influence of FSH on follicular maturation, we ae an artificial term.3 into the Hill function H 2 (FSH ) = h (.3FSH (t); xi FSH, 5) (Appenix B). Since this term is larger than the threshol xi FSH, the Hill function will always be close to one, so that it can be omitte as factor in Equation (2a). his is certainly a eficiency in the current moel an nees to be improve in future work..2 LH bloo GnRH pit Figure 5: GnRH an LH peaks. LH an GnRH he increase in GnRH inuces the LH peak shortly before ovulation (Figure 5). 7

20 .5 x CL Figure 6: Growth of the corpus luteum after ovulation. Corpus luteum he corpus luteum starts to grow right after ovulation an ies about ten later, which might be too early compare to reality Change of initial values It is possible to change the ranom initial values assigne to the follicles after each event by changing the see of the ranom number generator. By simulating ifferent cycles, we can thus evaluate the robustness of the couple moel. In fact, the moel often resists changes in the initial values an shows ifferent regular cycle patterns (Figure 7)..5 x CL (a).5 x CL (b) Figure 7: Follicular evelopment in two simulations with ifferent ranom initial values for the follicles. 8

21 4.2.3 Different numbers of follicles By increasing the number of follicles, the estrous cycle becomes less regular, as one can observe in Figures 8 an 9. In case of 3 follicles, we obtain regular cycles with two follicular waves per cycle, whereas for 9 follicles sometimes 4 waves occur, sometimes ovulation is elaye by many. It seems that the follicles become smaller if there are more of them. One might argue that the larger the number of follicles, the higher the competition between them. However, the moel is certainly eficient in this point an nees improvement. x CL Figure 8: Simulation of the evelopment of 3 follicles x x5 x6 x7 x8 x9 CL Figure 9: Simulation of the evelopment of 9 follicles. 9

22 5 Moeling of external influences o emonstrate the usefulness of this moel an to support future expansions concerning meical requirements, it is necessary to valiate the moel by simulating real-worl experiments. 5. Synchronization protocols Estrous synchronization protocols are commonly use in cattle. hey imply the successive aministration of ifferent hormones or their analogues following a precise orer. heir aim is to synchronize the cycles of iniviual females in orer to facilitate the timing of artificial insemination without consiering the estrous cycle phase at the protocol beginning. However, this kin of protocol is still not entirely reliable in clinical practice. Scientific reviews on this subject report that the protocols o not always result in the expecte synchronization in all animals, though they are still able to increase the fertility rate [7]. Moreover, in many cases the factors leaing to failure are not completely unerstoo, an quantitative measures are missing. Mathematical moels might help in improving an preicting the outcome of synchronization protocols. 5.. Prostaglanin aministration In veterinary meicine, PGF2 α an its analogues, PGF syn, are aministere mainly to make use of their luteolytic action. hus they play an important role in synchronization protocols. he treatment effect epens on the estrous stage which etermines the CL s receptivity for PGF syn. Within a few after ovulation, PGF syn aministration oes not lea to luteolysis. hereafter, a suen rise of PGF syn results in an immeiate ecay of the CL an a ecrease of P4 plasma levels. We will emonstrate in the following that our moel nicely captures this behavior, thus reproucing the results from [8]. o moel the aministration of PGF syn, we use the same formalism as in [8]: t PGF syn(t) = D β 2 t mo (t) exp( β t mo (t)) c PGFsyn PGF syn (t) (5) he parameter D represents the amount of rug, scale to obtain the esignate height of the relative level of PGF syn. he parameter c PGFsyn enotes the clearance rate constant of PGF syn. he moifie time function t mo is given as: t mo (t) = max(, t t D ) he rise of PGF syn is large right after osing time an approaches zero quickly thereafter, leaing to a rapi ecay of the function (Figure 2). 2

23 3 PGF amin Figure 2: Aministration of PGF syn at time t D = 2 (D = 3.7, β =, c PGFsyn = 5.5)..5 x LH bloo PGF CL Figure 2: Visualization of the cycle without treatment. We will now consier the influence of PGF syn on the cycle. For this purpose, the action of PGF2 α in the moel equations is replace by PGF2 α PGF syn. Figure 2 shows the estrous cycle without artificial aministration of PGF syn. PGF syn aministration right after ovulation slightly elays the first wave, but oes not have an influence on the next ovulation (Figure 22). Aministration 4 after ovulation (Figure 23) results in a next ovulation within the same wave six later. If aministration takes place when the follicles are ecaying, ovulation oes not occur until the next wave (Figure 24). 2

24 .5 x LH bloo PGF CL Figure 22: Aministration of PGF syn 3 after ovulation slightly elays the cycle, but has no effect on ovulation..5 x LH bloo PGF CL Figure 23: Aministration of PGF syn 4 after ovulation results in ovulation in the first wave..5 x LH bloo PGF CL Figure 24: Aministration of PGF syn 6 after ovulation leas to ovulation in the secon wave. 22

25 5..2 Combine therapies We have teste the influence of continuous P4 aministration uring a couple of, followe by a single aministration of PGF syn as escribe in [7]. For the protocol being effective in our simulations, we reuce the P4 aministration from seven [7] to five. his requirement might be ue to the short life-span of the corpus luteum. It is in fact a combination of GnRH, PGF syn an P4, that is use in [7], since GnRH aministration at the beginning of the protocol is reporte to increase the ovulation rate. In our moel, however, this aministration oes not have any influence. he aministration of GnRH is therefore omitte in the following PGF2α P4 PGF syn P4 amin Figure 25: Visualization of the protocol for 26 to x LH bloo PGF CL Figure 26: Effect of the protocol on 26 to 3. As shown in Figure 25, the simulate protocol consists of a continuous aministration of P4 followe by a single injection of PGF syn. his protocol was applie at ifferent stages of the cycle. In a first experiment, P4 aministration starts shortly before ovulation (Figure 26). his aministration prevents ovulation. Finally, PGF syn aministration at the en of the protocol causes ovulation not until ay 47 because follicular evelopment is very slow. However, as state in [7], the protocol oes not always lea to ovulation on the 23

26 esire ay x LH bloo PGF CL (a) Stanar protocol on 29 to x LH bloo PGF CL (b) Aministration of PGF syn on ay 34. Figure 27: Comparison between the stanar protocol (5 P4 plus a single ose PGF syn ) an a single ose PGF syn without preceing P4 treatment. We now compare the influence of P4 aministration between 29 an 34 before PGF syn aministration, with the single aministration of PGF syn on ay 34 without preceing P4 treatment. Figure 27(a) shows the simulation with P4 aministration. At P4 removal, a new wave begins which leas to ovulation on ay 4. Without this P4 aministration (Figure 27(b)), ovulation occurs one ay later. Compare to the stanar protocol, one can observe aroun ay 33 that the follicles are larger if P4 is not aministere. If aministere uring the last follicular wave of a cycle (Figures 28 an 29), P4 inhibits the growth of the follicles an thus ovulation. herefore the cycle counts three waves before the ovulating wave. Depening on the timing of the protocol, it might also happen that ovulation is elaye by some weeks (Figure 29). o conclue, P4 aministration inhibits the maturation of follicles an prevents ovulation uring the treatment perio. PGF syn causes the ecay of the corpus luteum, an the following follicular wave will lea to ovulation. 24

27 x LH bloo PGF CL Figure 28: Effect of the stanar protocol on 4 to x LH bloo PGF CL Figure 29: Effect of the stanar protocol on 45 to Postpartum anestrus Dairy cows are pregnant most of the lactation perio. Pregnancy an lactation are accompanie by marke cariovascular en enocrine aaptations. A negative energy balance uring early lactation in airy cows can lea to metabolic ysfunctions such as infertility. If estrus has not been observe in a airy cow by 6 post partum the conition is efine as Post Partum Anestrus (PPA). Recovery of cyclicity after calving is influence by boy conition score at calving an nutritional status uring early lactation [3]. Insulin is an important hormone in this process. It controls the pulsatile secretion of LH an increases the sensitivity of the ovary to this gonaotropin [3]. Low insulin levels promote a long anestrus perio. o observe this effect in our moel, we have to change parameter values an initial values in such a way that LH frequency is low at the beginning of the simulation, an slowly increases until the first ovulation. LH frequency can be reuce by lowering the amount of GnRH in the pituitary, which can again be achieve by reucing GnRH frequency. 25

28 5.2. Change of moel parameters In orer to moel the post partum anestrus, we ecrease the basal GnRH frequency from /ay to b GnRH :=.3/ay. Since this reuces the release from the hypothalamus, we increase the maximum capacity from 6 [GnRH Hypo ] to GnRH max Hypo := 32 [GnRH Hypo ]. Finally, in orer to stimulate the LH frequency even if the GnRH concentration is smaller we ecrease the threshol GnRH LH from.69 [GnRH ] to LH GnRH =. [GnRH ]. Keeping all other parameter values an initial values unchange, we can observe anestrus for a perio of about 4 at the beginning of the simulation (Figure 3). Even though a ominant follicle evelops in every wave, the level of LH at the beginning of the simulation is too low to cause ovulation (Figure 3). his is ue to lowere levels of GnRH in the hypothalamus (Figure 32) an the pituitary. However, the solution equilibrates within about 4 an returns to the quasi-perioic normal cycle for the remaining time x LH bloo PGF CL Figure 3: Follicles uring anovulatory an ovulatory cycles. 26

29 .8 LH bloo Figure 3: LH bloo level uring anovulatory an ovulatory cycles GnRH hypo Figure 32: GnRH level in the hypothalamus uring anovulatory an ovulatory cycles Change of initial values he change of parameters in the previous section is still biologically reasonable since the solution equilibrates towars regular cycles. o obtain such regular cycles alreay at the beginning of the simulation, we nee to change the initial values. Except for the follicles an the variable y(t), we select values from the previous simulation at some point in time when the cycle alreay recovere to a regular pattern. hese values are liste in able 2. hey coul be interprete as the state of a healthy cow, whereas the previous values correspon to a cow after gestation. 27

30 Component Initial value GnRH Hypo GnRH Pit 2 4 FSH Pit FSH Bloo LH Pit LH Bloo E PGF2 α CL P Inh.654 Enz O.385 IO.55 2 able 2: Initial values leaing to regular estrous cycles. As expecte, these changes lea to regular cycles right from the beginning (Figure 33). However, these changes have not been valiate ue to the lack of empirical ata x LH bloo PGF CL Figure 33: Estrous cycles without post partum anestrus. 28

31 Conclusion his work continues a series of previous moeling activities on the bovine estrous cycle. hanks to the robustness of the large estrous cycle moel by Stötzel et al. [8], it can easily be couple with other moels. In the stuy presente here we have extene the moel by incluing multiple follicles. he new moel can generate regular estrous cycles with multiple follicular waves per cycle, as well as anovulatory waves by changing certain parameters an initial values. In aition, the moel correctly preicts the effect of prostaglanin aministration on ovulation. However, there are still some eficiencies. For example, the influence of the gonaotropins on follicular evelopment is still not moele satisfactory. LH etermines at least the timepoint of ovulation, whereas FSH has nearly no influence on follicular maturation. his nees to be improve in future work. Moreover, several stuies inicate that consecutive follicular waves are not separate, but that they overlay each other [2, 7]. his feature is not yet capture by the moel. Besies, the moel coul be refine by incluing the influence of other hormones, e.g. insulin, which has a large impact on LH frequency. Nevertheless, we hope that the moel presente here will serve as a starting point for other researchers in this area. In particular, we inten to valiate the moel with experimental ata. Acknowlegment he authors woul like to thank Clauia Stötzel for proviing her latest results on the bovine estrous cycle moel an for her helpful comments on the manuscript. 29

32 A Equations of the couple moel In this part we list all equations that have been use an aapte from the estrous cycle moel in [8], together with our new equations. he complete moel comprises 6 ifferential equations plus one ifferential equation per follicle. Equation of GnRH he amount of GnRH in the hypothalamus is a result of synthesis in the hypothalamus an release into the pituitary, t GnRH Hypo(t) = Syn GnRH (t) Rel GnRH (t). (A.) GnRH synthesis epens on its current level in the hypothalamus. If this level approaches a specifie threshol, synthesis ecreases until zero. his effect is moele as logistic growth, ( Syn GnRH (t) = c GnRH,. GnRH ) Hypo(t) GnRH max. (A.2a) Hypo As long as GnRH is far below its maximum, the factor GnRH Hypo(t) GnRH max Hypo has only a small impact. he release of GnRH from the hypothalamus to the pituitary epens on its current level in the hypothalamus an the frequency. E2 inhibits GnRH frequency uring the luteal phase, i.e. if P4 an E2 are high at the same time uring the luteal phase. his is escribe by H (P 4&E 2 ). Aitionally, the release of GnRH is inhibite by P4 only. Rel GnRH (t) = F req GnRH GnRH Hypo (t) (A.2b) F req GnRH = b GnRH (H (P 4&E 2 ) H 2 (P 4)) (A.2c) Changes in GnRH levels in the pituitary are epenent on the release amount from the hypothalamus, but also on the presence of E2. E2 increases the number of GnRH receptors in the pituitary. his effect is inclue in the equation as a positive Hill function. GnRH clearance from pituitary portal bloo is proportional to the GnRH level in the pituitary, i.e. GnRH clearance is represente by c GnRH,2 GnRH P it (t), in which c GnRH,2 is a constant, t GnRH Pit(t) = Rel GnRH (t) H 3 (E 2) c GnRH,2 GnRH Pit (t). (A.2) Equation of FSH FSH is synthesize in the pituitary an release into the bloo, t FSH Pit(t) = Syn FSH (t) Rel FSH (t). (A.3) FSH synthesis rate in the pituitary is only epenent on inhibin, as in [8]. FSH is synthesize when the Inh level is low, i.e. high Inh levels inhibit FSH synthesis, which is inclue as a negative Hill function, Syn FSH = H 4 (Inh). (A.4a) FSH release from the pituitary to the bloo is stimulate by P4 an GnRH, an inhibite by E2, Rel FSH = (b FSH H 5 (P 4) H 6 (E 2) H 7 (GnRH Pit)) FSH Pit (t). (A.4b) i

33 Concluing, FSH serum level is a result of the ifference between the release amount from the pituitary an clearance from the bloo, t FSH Bloo(t) = Rel FSH (t) c FSH FSH Bloo (t), where c FSH is the FSH clearance rate constant. (A.4) Equation of LH Like FSH, the LH serum level epens on synthesis in the pituitary, release into the bloo an clearance thereof, t LH Pit(t) = Syn LH (t) Rel LH (t). LH synthesis in the pituitary is stimulate by E2 an inhibite by P4, ( ) Syn LH (t) = (H 8 (E 2) H9 (P LH (t) 4)). LH max (A.5) (A.6a) Synthesis stops if a maximum capacity LH max is reache. LH release epens on the frequency, which is stimulate by GnRH an inhibite by P4. Rel LH (t) = F req LH LH Pit (t) F req LH = b LH H (GnRH Pit) H (P 4) Summarizing, LH in the bloo is obtaine as t LH Bloo(t) = Rel LH (t) c LH LH Bloo (t), where c LH is the LH clearance rate constant. (A.6b) (A.6c) (A.6) Maturation of the i-th follicle is mo- Equations of follicular evelopment ele as t x i(t) = k λ2 E H 2 (FSH ) x i (E x i ) [y (λ x i ) t y(t) = a [max(x i) mean(x i )] H 3 (P 4). N x j, ] j= (A.7) (A.8) Equation of CL he CL evelopment epens on three mechanisms : an initiating impulse from an ovulating follicle, a self-growth, an the ecay ue to the inter-ovarian factor (IOF). CL changes as follows in the moel : At the time of ovulation, CL is re-set to an initial size CL CL sufficient for self-growth. hen it grows until the rise of PGF2α an thus in IOF inuces its ecay. t IOF (t) = H 8 (PGF2 α) H 9 (CL) c IOF IOF (t) t CL(t) = H 2 (CL) H 2 (IOF ) CL(t) (A.) (A.9) ii

34 Equations of P4, E2, an Inh he rise of P4 epens quaratically on CL to obtain lower P4 levels at the beginning of CL growth compare to later luteal stages [8]. he prouction of E2 an Inh is assume to be proportional to follicular function. t P 4(t) = c P4 CL CL(t)2 c P4 P 4 (t) N t E 2(t) = c E2 x i x i (t) c E2 E 2 (t) Inh(t) = cinh x t i i= N x i (t) c Inh Inh(t) i= (A.) (A.2) (A.3) he parameters c P4, c E2 an c Inh enote the respective clearance rate constants. Equation of PGF2α PGF2α initiates the functional regression of the CL, an thereby the ecrease in P4 levels. P4 has a stimulating effect on the enzyme prouction, E2 activates oxytocin (O) synthesis in the granulosa cells, an O initiates PGF2α peaks. t Enz(t) = H 4 (P 4) c Enz Enz(t) t O (t) = H 5 (E 2) CL(t) 2 c O O (t) t PGF2 α(t) = H 6 (Enz) H 7 (O ) c PGF2 α PGF2 α(t) (A.4) (A.5) (A.6) Detaile notations for the Hill functions an parameters are given in Appenices B an C, respectively. iii

35 B List of Hill functions ( H (P4 &E2 ) := m P4 &E2 h (P4 (t); H 2 H 3 H 4 H 5 H 6 GnRH, P4 h GnRH, (P4 (t);,2 (P4 ) := mgnrh P4 h GnRH,2 (P4 (t), P4, 2),2 (E2 ) := mgnrh E2 h GnRH,2 (E2 (t), E2, 5) (Inh) := mfsh Inh (P4 ) := mfsh P4 (E2 ) := mfsh E2 h (Inh(t), FSH Inh, 5) h (P4 (t); FSH P4, 2) h (E2 (t); FSH E2, 2) H 7 (GnRH Pit) := m FSH GnRH h (GnRH Pit (t); FSH GnRH, ) H 8 H 9 (E2 ) := mlh E2 h (E2 (t); LH E2, 2) (P4 ) := mlh P4 h (P4 (t); LH P4, 2) H (GnRH Pit) := m LH GnRH h (GnRH Pit (t); LH GnRH, 5) H (P4 ) := mfreq P4 h (P4 (t); P4 LH (t), 2), H 2 (FSH ) := h (.3 FSH (t); xi FSH, 5) H 3 (P4 ) := h (P4 (t); xi P4, 2) H 4 (P4 ) := menz P4 h (P4 (t); P4 Enz, 5) H 5 (E2 ) := mo E2 h (Enz(t); E2 O, 2) H 6 α (Enz) := mpgf2 Enz h (Enz(t), Enz PGF2 α, 5) H 7 α (O ) := mpgf2 O h (O (t); O PGF2 α, 2) H 8 (PGF2 α) := miof PGF2 α h (PGF2 α(t); PGF2 IOF α, 5) H 9 (CL) := miof CL h (CL(t); CL IOF, ) H 2 (CL) := mcl h (CL(t); CL CL, 5) H 2 (IOF ) := mcl IOF h (IOF (t); IOF CL, 5), 2) h (E2 (t), GnRH, E2, 2) ) P4, 2) h GnRH, (E2 (t), E2, 2) iv

36 C List of parameters an initial values Component Initial value GnRH Hypo 5.87 GnRH Pit FSH Pit FSH Bloo.72 LH Pit LH Bloo 4.65 E PGF2 α CL P Inh 8.84 Enz 3.58 O.83 2 IO 2.53 y x x x x x x x x x 9 able 3: Initial Values v

37 able 4: List of parameters. [ ] enotes the physical unit. In the present moel, the time unit [t] is equal to. Parameter Value Unit GnRH max Hypo.6e [GnRH Hypo ] c GnRH,.925 [GnRH Hypo ]/[t] b GnRH /[t] m P4&E GnRH, E e-2 [E2 ] GnRH, P 4 3.5e- [P4 ] GnRH,2 mp GnRH,2 P e- [P4 ] GnRH,2 me 2 9.9e- [GnRH Pit ]/[GnRH Hypo ] GnRH,2 E e- [E2 ] c GnRH, /[t] m FSH FSH Inh [FSH ]/[t] Inh.8e- [Inh] b FSH 6.636e- /[t] m FSH P 4 2.5e- /[t] P F SH 4.52e- [P4 ] m FSH E e- /[t] E FSH 2 3.2e- [E2 ] m FSH GnRH.2 /[t] GnRH FSH 7.8e-2 [GnRH ] c FSH.9 /[t] m LH LH m LH LH E 2 [LH ]/[t] E e- [E2 ] P [LH ] P e-2 [P4 ] LH max 5e [LH ] m LH GnRH 2.22 GnRH LH 6.9e- [GnRH ] m freq P b LH 2.464e- /[t] c LH 8.4 /[t] k 3.36e /(t [x i ] 4 ) λ 5.476e- [x i ] E [x i ] xi F SH 2.5e- [FSH ] a [x i ]/[t] xi P e-2 [P4 ] PGF2 α m PGF2 α PGF2 α O 3 [O ] Enz 3.774e [PGF2 α]/[t] Enz.43 [Enz] c PGF2 α 8.6e- [t] Continue on next page... vi

38 able 4 continue from previous page Parameter Value Unit m Enz Enz P [Enz]/[t] P 4 7.7e- [P4 ] c Enz 2.86 /[t] m O E 2.3 [O ]/([CL] 2 [t]) E O 2.43e- [E2 ] c O 4.58e- /[t] m CL CL m CL CL m IOF IOF IOF CL.7e- [CL]/[t] CL e- [CL] IOF 2.573e [CL]/[t] IOF 4e- [IOF ] P GF 2α 2.778e [t] P GF 2α.22 [PGF2 α] CL 6e- [CL] c IOF 2.86e- /[t] c P CL e- [P 4]/([CL] [t]) c P4 8.97e- /[t] c E2 x i 4.45 [E2 ]/([x i ] [t]) c E /[t] c Inh x i.323 [Inh]/([x i ] [t]) c Inh.568 /[t] vii

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