Using a Model of Magnesium Dynamics in Cows to Predict the Risk of Tetany in Dairy Herds

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1 Usng a Model of Magnesum Dynamcs n Cows to Predct the Rsk of Tetany n Dary Herds A.E. McKnnon a, S.T. Bell b and A.R. Sykes b a Appled Computng, Mathematcs and Statstcs Group, Lncoln Unversty, New Zealand Emal: mcknnon@lncoln.ac.nz b Anmal and Food Scences Dvson, Lncoln Unversty, New Zealand Abstract: The onset of tetany, when dary cattle have nsuffcent magnesum, has a huge mpact both economcally and on anmal welfare. To ad our understandng of the factors that determne magnesum status n dary cattle, we have adapted and mproved a model of magnesum dynamcs n sheep. As t stands, ths model s of lttle practcal use to a dary farmer who s prmarly concerned wth the status of the herd and the rsk that anmals wll contract tetany. In ths paper we descrbe how we have attempted to use the model to gve the dary farmer more useful nformaton. Our approach s to calbrate the model usng easlyobtaned measurements such as urne and mlk magnesum fluxes and then use Monte Carlo smulatons n whch model parameters are vared randomly accordng to ther statstcal dstrbutons. The result of each smulaton s classfed as the lkelhood of tetany occurrng based on the value of the smulated magnesum concentraton n cerebrospnal flud. Wth suffcent smulatons we can estmate the rsk of tetany for the herd as a whole. A senstvty measure was used to determne the extent to whch each parameter contrbutes to tetany and hence the mportance of knowng ts dstrbuton more accurately. Intal results show that our approach has potental. However, the process whereby we use measurements of urnary magnesum flux to calbrate some of the model parameters needs to be mproved and a comparson of the calculated rsk wth actual feld data needs to be carred out. Keywords: Magnesum model; Tetany rsk n dary herds; Monte Carlo smulaton; 1. INTRODUCTION The onset of tetany, when dary cattle have nsuffcent magnesum, has a major mpact both economcally and on anmal welfare (Feyter et al. 1986). Because the tme between the onset of the symptoms of tetany and death s only a few hours, a farmer needs to be able to assess whether anmals n a herd are at rsk of succumbng to the dsease before any symptoms are apparent. Although a great deal of expermental work has been done to elucdate the factors whch cause tetany, the farmer stll does not have the tools to provde a useful assessment of the rsk that anmals n the herd wll contract the dsease. We have modfed a model of magnesum dynamcs n sheep (Robson et al., 1997) so that t represents magnesum dynamcs n lactatng dary cows. In ths paper we show how we are attemptng to use ths modfed model to provde useful rsk nformaton for the farmer. 2. THE APPROACH There s no sngle experment that records all the nformaton necessary to construct a model such as the one we have developed. Rather, the process of buldng a typcal bologcal model nvolves gatherng together nformaton from a wde range of expermental studes and, after careful nvestgaton and analyss, usng t to construct the model equatons and determne parameter values. Valdaton s carred out both for ndvdual parts of the model and hopefully for the model as a whole. The expermental studes used to buld the model are lkely to come from varous breeds of anmal and may even come from dfferent speces. A model of magnesum dynamcs n dary cattle developed usng ths process may well exhbt behavour consstent wth the speces as a whole but s unlkely to be able to represent exactly the behavour of an ndvdual anmal. The modeller hopes that the equatons n the model are correct and that reasonable varaton n model parameters s suffcent to explan the dfferences between anmals.

2 Rumen Hndgut Det Slow Degradng Salva Lqud Sold Q Hg Lqud Sold Faeces Plasma Mlk Cerebral spnal flud Kdney Urne Fgure 1. Schematc dagram of the model of magnesum dynamcs n dary cows. The arrows ndcate magnesum fluxes. Full detals of the model equatons are avalable from the authors In a partcular herd, parameters wll vary between anmals, so that some wll be more lkely to contract tetany than others. If we have a dsease crteron expressed n terms of one of the model's state varables, we can carry out Monte Carlo smulatons to assess the rsk that anmals n a herd wll contract the dsease. Parameter values are selected randomly from ther dstrbutons; for each smulaton we can determne whether that "anmal" wll contract the dsease by comparng ts state varable value wth the dsease crteron. If the number of dseased anmals s N d and the total number of smulatons s N, then provded N s suffcently large, the rsk of dsease s calculated as: N R = d d N (1) There are some ssues that need to be addressed when takng ths approach: 1. There are a large number of parameters n the model some of whch may have only a small effect on whether the anmal contracts the dsease. It would be sensble to carry out a senstvty analyss and dentfy the parameters that most affect rsk so that further effort can be put nto mprovng knowledge of ther dstrbutons rather than those of parameters that have lttle effect. 2. The value of R d from (1) would be for the speces as a whole as determned by the overall model and would not gve a farmer any useful nformaton about the rsk for a partcular herd. The model needs to be calbrated to the partcular herd under study. The sectons that follow descrbe the model of magnesum dynamcs n cattle and how we have dealt wth the ssues above as we have attempted to use the model to estmate the rsk of tetany n dary herds. 3. THE MODEL A schematc dagram of the model s shown n Fgure 1. The modfcatons of the prevous model pertanng to sheep (Robson et al. 1997) are manly n the areas of lactaton, hndgut secreton and absorpton, urnary magnesum loss, salva flow nto the rumen and the transfer of magnesum to the cerebral spnal flud that s ts ste of acton as far as tetany s concerned. The model s expressed as a set of ordnary dfferental equatons. Dstrbutons for model parameters were taken from publshed data where ths was avalable or else they were set followng dscusson wth relevant scentsts. Very lttle expermental work s avalable n relaton to the hndgut absorpton and secreton of magnesum so there s sgnfcant uncertanty n relaton to those

3 parameters. Full detals of the model equatons and parameters are avalable from the authors. Each smulaton run on whch ths work s based was for a perod of 10 days. Feedng perods were programmed to occur followng normally observed patterns. Values used n the rsk calculaton were taken just before the commencement of the fnal feedng perod, when they would be at ther most crtcal. 4. RISK CRITERION Authors such as Allsop & Paul (1975) and Meyer & Scholz (1972) have shown that tetany occurs when the concentraton of magnesum n the cerebral spnal flud (M CSF ) falls below about 0.6 mmol/l. Accordngly, a smulated anmal was consder to have contracted tetany when M CSF <= 0.6 mmol/l. (2) Calculaton of rsk usng equatons (1) and (2) gves an assessment of the rsk f the herd contnues to be subject to the same envronmental condtons. But a farmer would also want to know what the rsk would be f the stuaton was perturbed, perhaps due to bad weather causng a feedng perod to be mssed or f an ncrease n mlk producton was mposed. 5. SENSITIVITY ANALYSIS To dentfy the parameters that predomnantly nfluence the rsk of dsease, Monte Carlo smulatons were carred out allowng all parameters to vary accordng to ther dstrbutons. The result of each smulaton was classfed as havng contracted tetany (T) or not as determned by equaton (2). For the th parameter, X T ε T the mean and standard devaton of values for whch tetany occurred were calculated and compared to the mean X and standard ε devaton for all values for that parameter used n the Monte Carlo smulatons. The comparson was done usng a t-test. Parameters not showng a sgnfcant dfference between the two categores after 20,000 smulatons were not vared n subsequent analyses. The senstve parameters are lsted n Table MODEL CALIBRATION To make the rsk assessment as specfc as possble to the herd under study, we would lke to be able to obtan nformaton about the dstrbuton of every model parameter n relaton to that herd. That s clearly mpractcal. There are a few herd-specfc measures avalable and we must use ths nformaton as best we can. Table 1. Parameters n the model of magnesum n dary cows that predomnantly nfluence the rsk of tetany. Parameter Symbol C D S HG C K 6.1. Mlk Producton Descrpton Concentraton of magnesum n the det Hndgut absorpton constant Concentraton of potassum n the rumen Daly mlk producton for the herd as a whole s routnely measured as part of normal herd montorng. Ths provdes an estmate of the mean of the mlk producton parameter. The mlk producton of ndvdual anmals s measured as part of a herd test 2 or 3 tmes per season, provdng an estmate of the standard devaton of the daly mlk producton for the herd. The mean of the dstrbuton (assumed to be Normal) s adjusted accordng to the herd's daly producton data. The magnesum concentraton n the mlk of each anmal s relatvely stable (Thelen, 2000) but there s sgnfcant varaton between anmals (McCoy, et al., 2001). Ths nformaton s also potentally avalable from the herd test Urnary Magnesum Kts are avalable for routne estmaton of the daly flux of urnary magnesum (M u ) of ndvdual anmals. Ths provdes an estmate of the mean and standard devaton of M u for the herd. However, M u s a model output, not a parameter. Therefore, to use these measurements to mprove our knowledge of the dstrbutons of model parameters, we need to carry out some sort of fttng process. As shown n Fgure 2, dstrbutons for the parameters {P, 1..N} wll produce a dstrbuton for M u as a result of the Monte Carlo smulatons. The challenge then s to adjust the dstrbuton of each parameter P untl the smulated and measured dstrbutons for M u match. In ths ntal study, we have taken the followng smple manual approach to ths fttng problem: 1. Rank the three parameters that predomnantly determnne tetany (Table 1) n the order n whch they are lkely to be most strongly related to M u. The rankng was done by nspecton of the model equatons. Ths resulted n a ranked set of parameters {P, 1..3}

4 Dstrbutons for parameters P Fgure 2. The relatonshp between the dstrbutons of model parameters P and the dstrbuton of urnary magnesum flux M ths fttng problem:ank the three parameters u that 2. Wth the mean of each P ntally at ts standard value and the standard devatons of P set to zero, teratvely adjust the mean of each parameter n turn to gve the best ft between the means of the measured and smulated M u values. The teratve adjustments were constraned to prevent parameter means gong outsde predetermned physologcally reasonable ranges. 3. Usng the mean parameter values resultng from step 2, teratvely adjust the standard devaton for each parameter n turn untl the best ft between the measured and smulated standard devatons for M u s obtaned. The teratve adjustments were constraned to prevent the standard devatons from beng greater than 30% of ther respectve means values. 4. Refne the mean values of the parameter dstrbutons prevously determned as the means and standard devatons are not ndependent n ther effect on the calculated M u. Wth ths smple approach to refnng the parameter dstrbutons, no real meanng can be assgned to the values that result. The rankng of the parameters as set out n step 1 of the process descrbed above prevents that. Also, because the parameters were adjusted one at a tme and n sequence, any covaraton that mght exst between model parameters s prevented from havng an effect. However, our objectve s to calbrate the model to the measured M u data only so that we can calculate the rsk of tetany n the herd. To assess whether our approach to fndng the parameter dstrbutons mpacts the rsk calculaton, we changed the rank of the three parameters n step one and repeated the process. 7. RESULTS Monte Carlo smulatons M u dstrbuton As an ntal assessment of ths approach to determnng the rsk of tetany n a dary herd, we appled t to a fcttous case as specfed n Table 2. Table 2. Specfcatons for the fcttous herd used to assess ths approach to estmatng rsk. These dstrbutons are consstent wth a normal healthy herd. Dstrbuton of mlk producton Dstrbuton of urne magnesum flux (M u ) Mean mlk magnesum concentraton 7.1. Model Calbraton Normal wth mean 18.5 L/day and standard devaton 2 L/day Normal wth mean 1.0 gm/day and standard devaton 0.5 gm/day 85 mg/l As explaned prevously, the dstrbuton of daly mlk producton could be drectly assgned to the dstrbuton of the relevant model parameter. The results of fttng the dstrbutons of the three model parameters to the dstrbuton of urne magnesum flux are gven n Table 3. Table 3. Means and standard devatons of the (Normal) dstrbutons of model parameters obtaned by fttng to the dstrbuton of urne magnesum flux. The results for two dfferent rankngs (A and B) of the parameters are shown. Parameter C D (gm/kg Dry Matter) Rank A 1 S HG 2 C K (mmol/l) Rsk Calculaton Ftted Dstrb µ σ Rank B Ftted Dstrb µ σ Wth the parameter dstrbutons for C D, S HG and C K determned as descrbed, and wth all other model parameters allowed to vary accordng to ther dstrbutons, t s now possble to calculate the rsk of tetany accordng to equaton (1) usng the crteron (2). As prevously dscussed, the farmer may well be nterested n how the rsk mght vary when the envronmental condtons are

5 perturbed n some way. To demonstrate ths, n Fgure 3 we show the rsk of tetany as a functon of the perturbaton of the mlk magnesum concentraton from the value determned durng the calbraton of the model. Rsk (%) Multpler Appled to Mlk Mg Conc Fgure 3. The rsk of tetany as a functon of a multpler appled to the mlk magnesum concentraton. The values were obtaned usng the parameters from Rank A (Table 3) and the values were obtaned usng the parameters from Rank B. 8. DISCUSSION AND CONCLUSIONS The ntal motvaton for developng a model such as one descrbng magnesum dynamcs n dary cattle s to explore quanttatvely the varous mechansms that are thought to be nvolved, and to provde a bass for further expermental nvestgatons. When such a model has matured to the pont that t explans the mportant observatons and mechansms, t s temptng to see f the model can be used n a predctve way n a practcal envronment - n our case to estmate the rsk of tetany n a dary herd. The crtcal factor n carryng ths out, s calbratng the model so that, as much as possble, ts parameter dstrbutons represent the actual herd under study. In the case of the present model, the only nformaton avalable to do ths s daly mlk producton for the herd as a whole and some sample measurements of urnary magnesum flux. The mlk producton nformaton allowed the correspondng model parameter to be estmated drectly. However, urne magnesum flux s a model output so a fttng procedure had to be used. We have ntally taken a very smple manual approach to ths problem where we ftted each parameter one at a tme ndependent of the others. Wth ths approach, no meanng can be attrbuted to the parameter dstrbutons that result, but t was our expectaton that the effect on the subsequent rsk calculatons would be small. Table 3 shows that the means of the ftted dstrbutons are not sgnfcantly affected by the order n whch the parameters were ftted and Fgure 3, shows lttle effect on the resultng rsk calculaton. In both cases the rsk seems reasonable. Nevertheless, we recognse the smplcty of our fttng process. We envsage some stuatons where the ftted parameters would be sgnfcantly affected by the order of fttng. Therefore, we plan to use a more robust approach smlar to those used n dscrete event modelng. Despte the smplcty of the model calbraton, we have demonstrated that our approach to usng the model to estmate rsk n a practcal farmng stuaton has potental. However, a number of successful feld trals on a range of herds where there s some ndependent assessment of rsk would need to be undertaken to confrm ths. 9. REFERENCES Allsop, T.F. & J.V. Paul. Responses to the lowerng of magnesum and calcum concentratons n cerebrospnal flud of unanaesthetsed sheep, Australan Journal of Bologcal Scences, 28, , Feyter, C., M.B. O'Connor, P.W. Young, and C.B. Dyson, Magnesum status of dary herds n Matamata county, New Zealand. 2. Assocatons of mlkfat producton and magnesum status, New Zealand Journal of Expermental Agrculture, 14, , McCoy, M.A.; T. Hutchnson, G. Davson, D.A. Ftzpatrck, D.A. Rce, D.G. Kennedy, Postmortem bochemcal markers of expermentally nduced hypomagnesaemc tetany n cattle. The Veternary Record, 148, , Meyer, H. and H. Scholz, Pathogeness of hypomagnasaemc tetany I. Relatonshp between Mg content n blood and cerebrospnal flud of sheep. Deutsche Terärztlche Wochenschrft, 79, 55-61, 1972 Robson, A.B., A.C. Feld, A.R. Sykes, and A.E. McKnnon, A model of magnesum metabolsm n young sheep. Magnesum absorpton and excreton, Brtsh Journal of Nutrton, 78, , Thelen, M., Investgaton of Urnary Magnesum Excreton as an Indcator of Magnesum Status n Dary Cows, Ph.D Thess, Lncoln Unversty, New Zealand, 2000.