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A Model for Reproductive Efficiency of Dairy Bulls1
A Model for Reproductive Efficiency of Dairy Bulls 1 W. J. KOOPS2 Department of Animal Husbandry Agricultural University PO Box 338 6700 AH Wageningen, The Netherlands M. GROSSMAN3 Department of Animal Sciences University of Illinois Urbana 61801
J.H.G. DEN DAAS Holland Genetics PO Box 5073 6802 EB Arnhem, The Netherlands
probability of completing gestation after insemination (calving rate) through the relationship of nonreturn rate to the concentration of spermatozoa at insemination and the time after insemination. The model is illustrated with three bulls, using nonreturn rates by 28, 56, and 84 d after insemination. (Key words: calving rate, conception rate, mathematical model, nonreturn rate)
ABSTRACT
Reproductive efficiency of bulls is usually measured by nonreturn rate, which is commonly defined as the proportion of cows that were inseminated and did not return for another service within a specified number of days. The AI organizations use nonreturn rate to evaluate fertility of a bull or performance of a technician. Measures derived from nonreturn rate, such as conception rate and calving rate, might be more reliable for evaluation than nonreturn rate itself. Estimated conception rate is a better early measure of efficiency than nonreturn rate, because conception rate depends on the population of spermatozoa at insemination and not on developmental potential of the conceptus after insemination. A mathematical function is presented to model reproductive efficiency of bulls by estimation of the probability of conception at time of insemination (conception rate) and the
INTRODUCTION
Received June 24, 1994. Accepted December 5, 1994. lSupponed in part by the Illinois Agricultural Experime?t Station, .Hatch Project 35-367, the Department of Animal Breedmg, Wageningen Agricultural University, and Holland Genetics. 2Reprint requests. 3Research conducted while at the Department of Animal Breeding, Wageningen Institute of Animal Science Wageningen Agricultural University. ' 1995 J Dairy Sci 78:921-928
Reproductive efficiency of dairy cattle depends on ability of the bull to fertilize the cow thus initiating the process of pregnancy, and o~ ability of the cow and the conceptus to sustain pregnancy to complete gestation. Reproductive efficiency of bulls is measured by the response of cows to which bulls are bred (4), usually by nonreturn rate. Nonreturn rate is commonly defined as proportion of cows that were inseminated during a given period and did not return for another service within a specified number of days, such as 28 or 56 d (6), .or commonly 60 to 90 d (4). The AI organizations use nonreturn rate to evaluate the fertility of a bull or the performance of a technician (6, 9). Nonreturn is the result of two events: conception at or near time of insemination and gestation after conception. The joint event of conception and gestation will be termed gestation after insemination. Conception at insemination depends on, among other factors, the availability of ova and their quality and on the
921
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KOOPS ET AL
population of spennatozoa inseminated and its characteristics (1). Factors that influence conception include number or concentration of spennatozoa inseminated. technician. parity, and herd management. Gestation after insemination depends on conception, maternal environment, and maternal and paternal contributions to developmental potential of the conceptus (1). Factors that influence gestation after insemination include failure of conception. failure of implantation. embryonic or early fetal death. and abortion (4). Nonretum rates can be used to derive useful measures of the reproductive efficiency of a bull. such as conception rate and calving rate. These measures might be more reliable than nonreturn rate itself to evaluate reproductive efficiency of a bull or perfonnance of an AI technician. Conception rate, defined as the probability of conception at time of insemination. is difficult to observe directly, but calving rate, the probability of completing gestation after insemination. can be observed directly. but only at calving. The objective of this paper is to present a mathematical function to model reproductive efficiency of bulls by estimation of conception rate and calving rate through the relationship of nonreturn rate to concentration of spermatozoa at insemination and time after insemination. Using data from the Royal Netherlands Cattle Syndicate on nonreturn rates by 28, 56. and 84 d after insemination. the model is illustrated with three bulls. MATERIALS AND METHODS
after insemination and at concentration s of spennatozoa. The development of the model considers first the relationship of nonreturn with time, ignoring concentration, using probability of nonreturn by time t after insemination CPt,-); second, the relation of nonreturn with concentration, by a given time after insemination, using probability of nonreturn at concentration s of spennatozoa (P-,s); and finally. the joint relationship of nonreturn by time t after insemination and at concentration s of spermatozoa, using joint probability
PO.-
of conception at time of insemination, or conception rate, and 1 - PO,- = probability of nonconception at time of insemination, or nonconception rate; and time at calving (t Pc,-
1 - Pc,-
Theory
A mathematical function is presented to model the reproductive efficiency of bulls, using nonreturn rates and their relationship to concentration of spennatozoa at insemination and to time after insemination. Estrus detection is assumed to be perfect, and early embryonic death is assumed to be negligible. To understand better the development of the model, the following notation is used for the joint relation of nonreturn with concentration of spermatozoa and time after insemination: PI,S = probability of nonreturn by time t after insemination and at concentration s of spennatozoa, and 1 - PI,S = probability of return by time t Journal of Dairy Science Vol. 78, No.4. 1995
= probability
= c):
= probability
=
of completing gestation after insemination, or calving rate, and probability of not completing gestation after insemination, or noncalving rate.
From Figure 1, total probability of unity can be partitioned as 1
= Pc,-
+
CPt.- -
Pc.-) + (1 - PI,-)
where CPt.- - Pc.-) is the joint probability of nonreturn and failure to complete gestation after insemination. This partition is best understood by summarizing these probabilities in Table 1. The only accurate measure of calving rate of a bull is to observe the completed gestation. Gestation need not be completed, however, before calving rate of a bull is measured. In-
923
A MODEL FOR REPRODUCTIVE EFFICIENCY
oL.-----L---------ca Iving
The relationship of nonretum rate and time after insemination can be described by a nonlinear function that relates the decline in nonreturn rate to the increase in time after insemination (2. 3, 7, 8). The greatest decline In nonreturn rate is during the first 90 dafter insemination, followed by a slight decline from 90 to 180 d (3, 7). Therefore, Oddst,_ is assumed to have an exponential relation to time, so that In(Oddst,_). or logit, is assumed to have a linear relationship to time:
concept ion
Time Figure I. Partition of total probability of unity into probability of conception at time of insemination
stead, calving rate (Pc,-), before completed gestation, is estimated using the probability of not completing gestation after insemination (1 - Pc,-). From Figure 1 and Table 1, (1 - Pc,-) can be written in terms of nonreturn rate by each time t CPt,-) as 1 - Pc,-
= CPt,- -
Pc,-)
+ 0 -
Pt,-)·
[l]
IJPt ,- - Pc,-] = lJPo,- - Pc,-] _ bt 1 - Pt,1 - Po,-
'1
'1
where In[(Pt,_ - Pe,-)l(l - Pt,-)] is In(Oddst ,_) by time t after insemination. In[(po,- - Pc,-YO Po,-)] is In(Oddso,_) at insemination (t = 0) and, therefore, is the intercept of the line. and b is the slope of the line. Taking antilog of both sides yields:
where e is the base of natural logarithms. Finally, addition of 1 to both sides to solve for nonreturn rate by time t after insemination CPt,-) yields:
Equation [1] can be normalized to yield conditional probabilities: 1
=P
l ,- -
Pc ,- +
1 - pc.-
- Pt ,- Pc,-
or simply
where 8t,- = CPt,- - Pc,-)lO - Pc,-) is the probability of nonreturn before completed gestation and (1 - 8t _) = (1 - Pt -)1(1 - Pc _) is the probability of return before completed gestation. The ratio of 8t,- to (1 - 8t,_) compares the probability of nonreturn with the probability of return, before completed gestation, and is termed the odds ratio (Oddst,_): Oddst,_
= 8t ,.J(1 =
- 8t,_)
Pc.-)1(1 - Pt.-)·
[2] which is a logistic function, where b is rate of decline in probability of nonreturn. Such a function constrains values for probability of nonreturn CPt,-) to be between 0 and 1 and to be symmetric between an upper asymptote of unity and a lower asymptote of Pc,- (Figure 1). If observations on nonreturns are available at several times after insemination, then Equation [2] can be used to model reproductive efficiency of a bull. On the right-hand side of Equation [2], conception rate CPo,-). calving rate (Pc,-), and rate of decline (b) are the three parameters to be estimated. With only three nonreturn rates for a bull, the three parameters can be estimated. but without a measure of error; therefore. parameters cannot be tested for statistical significance. At least four nonJournal of Dairy Science Vol. 78, No.4, 1995
924
KOOPS ET AL.
TABLE I. Summary of probabilities for nonretum
<1\,-) and for completing gestation
(Pc.-)'
Completed gestation
Nonretum
Yes No Totals
=
S
[3]
where P-,oo is upper asymptotic value of nonreturn rate by a given time, and d is rate at which that asymptote is approached. Nonretum Versus Time and Concentration. For the relationship of nonreturn with time and concentration, Pt.s is defined as the joint probability for nonreturn by time t after insemination and at concentration s of spermatozoa inseminated. Although nonreturn rate by 28 d can be considered to be a reliable early measure of reproductive efficiency of a bull (2), we consider conception rate to be a better early measure of efficiency because conception rate depends on the population of spermatozoa at insemination, not on developmental potential of the conceptus after insemination. Although conception rate is difficult to observe, it can be estimated as Po,- in Equation [2], ignoring the Journal of Dairy Science Vol. 78, No.4, 1995
Totals
Pc.-
Pt.- - Pc.1 - Pt.1 - Pc.-
Pt,-
Pc,-
d
P-.s
No
o
return rates for a bull are necessary to test the significance of parameters in Equation [2]. Nonretum Versus Concentration. For the relationship of nonreturn with concentration, by a given time after insemination, P-.s is defined as probability of nonreturn at concentration s of spermatozoa. Nonreturn rate at a given time has a curvilinear relationship with concentration of viable spermatozoa inseminated (8). This relationship has been modeled based on kinetic theory (12). based on concentration of viable spermatozoa having a Poisson distribution (10) and on an assumed upper asymptote for the curve (5). Although other nonlinear functions may also describe the data, the exponential function was suitable to fit nonreturn rates of 56 d (1) and 90 d (5):
p_,ooe
Yes
I - Pt,1
concentration of spermatozoa. If spermatozoa concentration is known, however, then P-,oo can be estimated from Equation [3] as the asymptotic value at a given time. With information on time after insemination and concentration of spermatozoa, an asymptotic conception rate, PO.co, can be estimated by substitution of Equation [3] at time 0 for PO,- in Equation [2]:
If observations on nonreturns are available at several times after insemination and at several concentrations of spermatozoa, then Equation [4] can be used to model the reproductive efficiency of a bull. Data
Data file 1 included three bulls (A, B, and C) that were selected to illustrate the model in
Equation [2]. Bulls were used for first or second inseminations between September and December 1993, and nonreturn rate was computed in April 1994. For each cow that returned to service, the interval between first and second or between second and third inseminations was recorded in days. From these intervals, nonreturn rate was computed for three multiples of 28 d: 28, 56, and 84 dafter insemination (Table 2). The number of inseminations ranged from about 12,500 to 37,400; inseminations were assumed to be randomized over various effects, such as cows and herds. Data file 2 included the same three bulls as in data file 1 selected to illustrate the model in Equation [4]. Bulls were used for first and
925
A MODEL FOR REPRODUCTIVE EFFICIENCY TABLE 2. Data file 1: number of inseminations (n) and observed nonreturn rate <1\-), by bull and time t after insemination (t = 28, 56, or 84 d). Bull
n
P28,-
P56,-
PM,-
A B C
15,555 12,478 37,392
.783 .764 .813
.672 .656 .718
.624 .606 .670
second inseminations between May and November 1993, and nonretum rate was computed in January 1994. The 28-d nonretum rate was computed for cows inseminated between June through November, 56-d nonretum rate for cows inseminated between May through October, and 84-d nonretum rate for cows inseminated between May through September. Data were from a study to measure effect of concentration of spermatozoa at insemination on nonreturn rate. Five or six different concentrations of spermatozoa per dose were used: about 5.0, 7.5, 10, 12.5, and 15 (xlO6 ) for bulls A and B, and, in addition to those five concentrations, about 2.5 (xl()6) for bull C (fable 3). Number of inseminations per concentration ranged from about 700 to about 2000; inseminations were assumed to be randomized over various effects, such as cows and herds.
Estimation of Parameters
For data file 1, the nonreturn rate by 28, 56, and 84 d after insemination (Table 2), Equation [2] was fitted for each bull. For data file 2, the nonreturn rate by 28, 56, and 84 d after insemination and the concentration of spermatozoa at insemination (fable 3), Equation [4] was also fitted for each bull. Parameters were estimated by nonlinear regression, using an adaptive nonlinear least squares algorithm [NLREG; (11)]. A default value of 1 x lO-10 was used for the tolerance factor, which specifies the convergence criterion for the iterative estimation procedure. All estimations converged within 50 iterations. Because of the relatively large numbers of inseminations used for each bull, data were not weighted during analysis. Goodness of fit was measured by residual standard error when possible. RESULTS
For data file 1 (Table 2), estimates for conception rate (AJ _), calving rate (Pc _), and rate of decline in no~etum (b) by bull ~e in Table 4. Residual standard error could not be estimated because only three data were available to estimate the three parameters. Conception
TABLE 3. Data file 2: number of inseminations (Ilt) and nonreturn rate after insemination (t = 28, 56, or 84 d), for bulls A, B, and C. s (x 106)
n28
P28,s
CPt s) by concentration (s) at insemination and time .
n56
P56,s
n84
P84,s
1304 1309 2068 1142 1046
.638 .652 .688 .675 .670
1282 1279 1955 1117 1029
.581 .59B .636 .611 .626
879 987 1014 1023 923
.654 .655 .653 .659 .663
838 945 959 993 895
.598 .594 .595 .609 .599
1147 1171 1167 1180 1139 1182
.664 .699 .711 .731 .721 .724
1119 1156 1150 1169 1124 1158
.620 .648 .674 .680 .666 .670
Bull A 5.0 7.5 10.0 12.5 15.0
880 906 1624 814 841
.747 .772 .BOI .800 .779
5.0 7.5 10.0 12.5 15.0
779 863 859 881 715
.759 .767 .782 .779 .783
25 5.0 7.5 10.0 12.5 15.0
889 885 B05 776 725 808
.772 .805 .810 .821 .828 .811
Bull B
Bull C
Journal of Dairy Science Vol. 7B, No.4, 1995
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KOOPS ET AL.
TABLE 4. Estimates of conception rate CPo.-), calving rate (fIc,-). and rate of decline in nonreturn rate (b) by bull from data file 1. Bull A B C
1.0 , .9 ,,~.
~O,-
~c,-
b
.908 .889 .914
.605 .583 .646
.050 .046 .045
"",,/'... -,-
,,\
.8
",,"' . .....
.7
"'>::::
. . . . . ;:::::_==---------------------------------
.6 .5
ratec,-) averaged about .61, which means that about 61% of cows were expected to complete gestation. Nonreturn rates and predicted probability of nonreturn by time after insemination for bulls A, B, and C are in Figure 2. Bulls ranked the same for conception rate and calving rate. Bulls A and C had similar conception rates, which were higher than that for bull B, and each had a calving rate higher than B. For similar conception rates, however, A had a lower calving rate than C: i.e., fewer cows conceived to A were expected to complete gestation compared with cows conceived to C. Rate of decline in nonreturn (b) averaged .047. Bull A had a faster rate of decline than B or C, which had similar rates (Figure 2). For bull A, for example, nonreturn rate at 84 d was .624 (Table 2), and estimated calving rate was .605 (Table 4): i.e., about 97% of decline in nonreturn rate to completed gestation (.605/ .624) was achieved by 84 d; results for B and C were 96%. For data file 2 (Table 3), estimates for conception rate c,-), rate of decline in nonreturn rate (b), rate of approach to asymptotic nonreturn rate (d), and residual standard error are in Table 5. Conception rate
_._._._._._._._.__._._._._._._._-
i _-.-1..'_-'--_......J.
_
0285684
Time (d) Figure 2. Nonretum rates for bulls A (+), B (A), and C (e) and predicted probability of nonretum for bulls A (-), B (-), and C (- - -) by time after insemination.
trations, about 92% of cows conceived at time of insemination. Calving rate ([>c.-) averaged about .58: i.e., about 58% of cows were expected to complete gestation. Bulls A and C had similar conception rates, but A had a lower calving rate than C. Rate of decline in nonreturn rate (b) averaged .041. Rate of decline was slower for bull C than for bulls A or B, which were similar to one another. Rate of approach to the asymptote (d) averaged .22 and was slowest for A: i.e., A reached the asymptote at a higher concentration of spermatozoa than did B or C. For bull C, which had six levels of concentration, nonreturn rates, and predicted probability of nonreturn by concentration are in Figure 3. The model proposed in Equation [4] fitted best data from B; residual standard error was .005, about one-half to one-third that for A or C. To compare results from data on only time after insemination (Table 4) with results from data on time and concentration of spermatozoa (Table 5), bulls ranked differently for concepA
TABLE 5. Estimates and standard err~rs of conception rate CPo.-), calving rate (fIc.-), rate of decline in nonretum rate (b). rate of approach to the asymptote (d). and residual standard error (RSE) by bull from data file 2.
~o.-
Bull
A B C
a
b
~c.-
RSE
Est.
SE
Est.
SE
Est.
SE
Est.
SE
.930 .913 .921
.020 .006 .013
.574 .572 .605
.021 .006 .022
.042 .046 .035
.007 .002 .005
.351 .132 .189
.110 .030 .037
Journal of Dairy Science Vol. 78, No.4. 1995
.015 .005 .012
A MODEL FOR REPRODUCTIVE EFFICIENCY 1.0
i;..---- j----~----~---~----+ .... _ _ -''"'''1- _.-.- T _._._._ .... _.- - , - - - - 1.
~ .._ .._..r'·_··_··.!.··_··_··!··_··_··.... ·_··_··J
• .5 l - - .
o
2.5
_ 5.0
7.5
10.0 12.5 15.0
Concentration (x 10 6 ) Figure 3. Nonreturn rates by 28 d (+), S6 d (.A), and 84 d (e) and predicted probability of nonreturn by 0 d (-), 28 d (---), S6 d (---), and 84 d (----) by concentration of spermatozoa for bull C.
tion rate but similarly for calving rate. Differences in ranking can be explained by use of different data files and by use of additional information on concentration of spermatozoa. For example, Po,- in Table 4 presents an estimate of mean conception rate at a mean concentration; Po,co in Table 5 estimates asymptotic conception rate at a high concentration. DISCUSSION
The nonreturn rate of a bull is a measure of his reproductive efficiency and is used to evaluate performance for bulls and technicians in AI organizations. Assuming perfect detection of estrus and negligible embryonic death, nonreturn is the result of conception and gestation after conception. Conception is influenced by, among other factors, characteristics of the population of spermatozoa inseminated, and gestation is influenced by maternal environment and by maternal and paternal contributions to developmental potential of the conceptus. An estimate of conception rate, therefore, would be expected to be a more reliable measure to evaluate bulls on their reproductive efficiency or AI technicians on their performance than nonretum rate by a given time, say 28 d, after insemination. The dairy producer is interested in calving rate of bulls used, because cows are kept to produce calves and milk. A high calving rate means that the producer has increased opportu-
927
nity not merely to replace cows in the herd but also to select replacements. The producer, however, must wait until gestation is completed to observe calving rate. Therefore, an early estimate of calving rate means that producers can select bulls earlier on their calving rate. An AI organization is interested in reproductive efficiency of bulls used and in the effect that concentration of spermatozoa has on efficiency. Models presented herein can be used to determine concentration of spermatozoa necessary at time of insemination for a bull to achieve, say, 95% of asymptotic conception rate
928
KOOPS ET AL
known, which will enable modeling of estrus detection and early embryonic death. CONCLUSIONS
Reproductive efficiency of bulls is usually measured by nonretum rate. The AI organizations also use nonretum rate to evaluate fertility of a bull or performance of a technician. The mathematical function presented herein to model reproductive efficiency of bulls estimates conception rate and calving rate through the relationship of nonretum rate to concentration of spermatozoa at insemination and time after insemination. These estimates might be more reliable for evaluation than nonretum rate itself. ACKNOWLEDGMENTS
The authors thank Lucia Lansbergen and Janneke van Wagtendonk, Holland Genetics, for their cooperation and helpful discussions and J.B.M. Wilmink, the Royal Netherlands Cattle Syndicate, for his cooperation in providing data used in this research. REFERENCES
I den Daas, N. 1992. Laboratory assessment of semen characteristics. Anim. Reprod. Sci. 28:87. 2 Erb, R. E., and F. H. Flerchinger. 1954. Influence of fertility level and treatment of semen on nonreturn
Journal of Dairy Science Vol. 78, No.4, 1995
decline from 29 to 180 days following artificial service. 1. Dairy Sci. 37:938. 3 Foote, R. H.. and R. W. Bratton. 1952. The influence of antibiotics on delayed returns in artificial breeding. J. Dairy Sci. 35:261. 4 Olds, D. 1978. Conception rate and factors affecting its magnitude. Chapter 19 in Physiology of Reproduction and Artificial Insemination of Cattle. 2nd ed. G. W. Salisbury, N. L VanDemark and 1. R. Lodge, ed. W. H. Freeman and Co., San Francisco, CA. 5 Pace, M. M., J. 1. Sullivan, F. I. Elliot, E. F. Graham. and G. H. Coulter. 1981. Effects of thawing temperature, number of spermatozoa and spermatozoal quality on fertility of bovine spermatozoa packaged in .5-ml French straws. 1. Anim. Sci. 53:693. 6 Reurink, A., J.H.G. den Daas, and J.B.M. Wilmink. 1990. Effects of AI sires and technicians on nonreturn rates in the Netherlands. Livest. Prod. Sci. 26: 107. 7 Salisbury, G. W., R. W. Bratton, and R. H. Foote. 1952. The effect of time and other factors on the nonreturn to service estimate of fertility level in artificial insemination of cattle. J. Dairy Sci. 35:256. 8 Salisbury, G. W., and N. L VanDemark. 1961. Pages 434 and 478 in Physiology of Reproduction and Artificial Insemination of Cattle. W. H. Freeman and Co., San Francisco, CA. 9 Schaeffer, L R. 1993. Evaluation of bulls for nonreturn rates within artificial insemination organizations. 1. Dairy Sci. 76:837. 10 Schwartz, D., P.DM. Macdonald, and V. Heuchel. 1981. On the relationship between the number of spermatozoa and the probability of conception. Reprod. Nutr. Dev. 21:979. 11 Sherrod, P. H. 1992. Nonlinear Regression Analysis Program (NLREG), Version 1.8. 4410 Gerald Place. Nashville. TN. 12 van Duijn. c., Jr. 1965. Sperm number and fertility: a kinetic approach. Neth. J. Agric. Sci. 13:378.
J.H.G. DEN DAAS Holland Genetics PO Box 5073 6802 EB Arnhem, The Netherlands
probability of completing gestation after insemination (calving rate) through the relationship of nonreturn rate to the concentration of spermatozoa at insemination and the time after insemination. The model is illustrated with three bulls, using nonreturn rates by 28, 56, and 84 d after insemination. (Key words: calving rate, conception rate, mathematical model, nonreturn rate)
ABSTRACT
Reproductive efficiency of bulls is usually measured by nonreturn rate, which is commonly defined as the proportion of cows that were inseminated and did not return for another service within a specified number of days. The AI organizations use nonreturn rate to evaluate fertility of a bull or performance of a technician. Measures derived from nonreturn rate, such as conception rate and calving rate, might be more reliable for evaluation than nonreturn rate itself. Estimated conception rate is a better early measure of efficiency than nonreturn rate, because conception rate depends on the population of spermatozoa at insemination and not on developmental potential of the conceptus after insemination. A mathematical function is presented to model reproductive efficiency of bulls by estimation of the probability of conception at time of insemination (conception rate) and the
INTRODUCTION
Received June 24, 1994. Accepted December 5, 1994. lSupponed in part by the Illinois Agricultural Experime?t Station, .Hatch Project 35-367, the Department of Animal Breedmg, Wageningen Agricultural University, and Holland Genetics. 2Reprint requests. 3Research conducted while at the Department of Animal Breeding, Wageningen Institute of Animal Science Wageningen Agricultural University. ' 1995 J Dairy Sci 78:921-928
Reproductive efficiency of dairy cattle depends on ability of the bull to fertilize the cow thus initiating the process of pregnancy, and o~ ability of the cow and the conceptus to sustain pregnancy to complete gestation. Reproductive efficiency of bulls is measured by the response of cows to which bulls are bred (4), usually by nonreturn rate. Nonreturn rate is commonly defined as proportion of cows that were inseminated during a given period and did not return for another service within a specified number of days, such as 28 or 56 d (6), .or commonly 60 to 90 d (4). The AI organizations use nonreturn rate to evaluate the fertility of a bull or the performance of a technician (6, 9). Nonreturn is the result of two events: conception at or near time of insemination and gestation after conception. The joint event of conception and gestation will be termed gestation after insemination. Conception at insemination depends on, among other factors, the availability of ova and their quality and on the
921
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KOOPS ET AL
population of spennatozoa inseminated and its characteristics (1). Factors that influence conception include number or concentration of spennatozoa inseminated. technician. parity, and herd management. Gestation after insemination depends on conception, maternal environment, and maternal and paternal contributions to developmental potential of the conceptus (1). Factors that influence gestation after insemination include failure of conception. failure of implantation. embryonic or early fetal death. and abortion (4). Nonretum rates can be used to derive useful measures of the reproductive efficiency of a bull. such as conception rate and calving rate. These measures might be more reliable than nonreturn rate itself to evaluate reproductive efficiency of a bull or perfonnance of an AI technician. Conception rate, defined as the probability of conception at time of insemination. is difficult to observe directly, but calving rate, the probability of completing gestation after insemination. can be observed directly. but only at calving. The objective of this paper is to present a mathematical function to model reproductive efficiency of bulls by estimation of conception rate and calving rate through the relationship of nonreturn rate to concentration of spermatozoa at insemination and time after insemination. Using data from the Royal Netherlands Cattle Syndicate on nonreturn rates by 28, 56. and 84 d after insemination. the model is illustrated with three bulls. MATERIALS AND METHODS
after insemination and at concentration s of spennatozoa. The development of the model considers first the relationship of nonreturn with time, ignoring concentration, using probability of nonreturn by time t after insemination CPt,-); second, the relation of nonreturn with concentration, by a given time after insemination, using probability of nonreturn at concentration s of spennatozoa (P-,s); and finally. the joint relationship of nonreturn by time t after insemination and at concentration s of spermatozoa, using joint probability
PO.-
of conception at time of insemination, or conception rate, and 1 - PO,- = probability of nonconception at time of insemination, or nonconception rate; and time at calving (t Pc,-
1 - Pc,-
Theory
A mathematical function is presented to model the reproductive efficiency of bulls, using nonreturn rates and their relationship to concentration of spennatozoa at insemination and to time after insemination. Estrus detection is assumed to be perfect, and early embryonic death is assumed to be negligible. To understand better the development of the model, the following notation is used for the joint relation of nonreturn with concentration of spermatozoa and time after insemination: PI,S = probability of nonreturn by time t after insemination and at concentration s of spennatozoa, and 1 - PI,S = probability of return by time t Journal of Dairy Science Vol. 78, No.4. 1995
= probability
= c):
= probability
=
of completing gestation after insemination, or calving rate, and probability of not completing gestation after insemination, or noncalving rate.
From Figure 1, total probability of unity can be partitioned as 1
= Pc,-
+
CPt.- -
Pc.-) + (1 - PI,-)
where CPt.- - Pc.-) is the joint probability of nonreturn and failure to complete gestation after insemination. This partition is best understood by summarizing these probabilities in Table 1. The only accurate measure of calving rate of a bull is to observe the completed gestation. Gestation need not be completed, however, before calving rate of a bull is measured. In-
923
A MODEL FOR REPRODUCTIVE EFFICIENCY
oL.-----L---------ca Iving
The relationship of nonretum rate and time after insemination can be described by a nonlinear function that relates the decline in nonreturn rate to the increase in time after insemination (2. 3, 7, 8). The greatest decline In nonreturn rate is during the first 90 dafter insemination, followed by a slight decline from 90 to 180 d (3, 7). Therefore, Oddst,_ is assumed to have an exponential relation to time, so that In(Oddst,_). or logit, is assumed to have a linear relationship to time:
concept ion
Time Figure I. Partition of total probability of unity into probability of conception at time of insemination
stead, calving rate (Pc,-), before completed gestation, is estimated using the probability of not completing gestation after insemination (1 - Pc,-). From Figure 1 and Table 1, (1 - Pc,-) can be written in terms of nonreturn rate by each time t CPt,-) as 1 - Pc,-
= CPt,- -
Pc,-)
+ 0 -
Pt,-)·
[l]
IJPt ,- - Pc,-] = lJPo,- - Pc,-] _ bt 1 - Pt,1 - Po,-
'1
'1
where In[(Pt,_ - Pe,-)l(l - Pt,-)] is In(Oddst ,_) by time t after insemination. In[(po,- - Pc,-YO Po,-)] is In(Oddso,_) at insemination (t = 0) and, therefore, is the intercept of the line. and b is the slope of the line. Taking antilog of both sides yields:
where e is the base of natural logarithms. Finally, addition of 1 to both sides to solve for nonreturn rate by time t after insemination CPt,-) yields:
Equation [1] can be normalized to yield conditional probabilities: 1
=P
l ,- -
Pc ,- +
1 - pc.-
- Pt ,- Pc,-
or simply
where 8t,- = CPt,- - Pc,-)lO - Pc,-) is the probability of nonreturn before completed gestation and (1 - 8t _) = (1 - Pt -)1(1 - Pc _) is the probability of return before completed gestation. The ratio of 8t,- to (1 - 8t,_) compares the probability of nonreturn with the probability of return, before completed gestation, and is termed the odds ratio (Oddst,_): Oddst,_
= 8t ,.J(1 =
- 8t,_)
Pc.-)1(1 - Pt.-)·
[2] which is a logistic function, where b is rate of decline in probability of nonreturn. Such a function constrains values for probability of nonreturn CPt,-) to be between 0 and 1 and to be symmetric between an upper asymptote of unity and a lower asymptote of Pc,- (Figure 1). If observations on nonreturns are available at several times after insemination, then Equation [2] can be used to model reproductive efficiency of a bull. On the right-hand side of Equation [2], conception rate CPo,-). calving rate (Pc,-), and rate of decline (b) are the three parameters to be estimated. With only three nonreturn rates for a bull, the three parameters can be estimated. but without a measure of error; therefore. parameters cannot be tested for statistical significance. At least four nonJournal of Dairy Science Vol. 78, No.4, 1995
924
KOOPS ET AL.
TABLE I. Summary of probabilities for nonretum
<1\,-) and for completing gestation
(Pc.-)'
Completed gestation
Nonretum
Yes No Totals
=
S
[3]
where P-,oo is upper asymptotic value of nonreturn rate by a given time, and d is rate at which that asymptote is approached. Nonretum Versus Time and Concentration. For the relationship of nonreturn with time and concentration, Pt.s is defined as the joint probability for nonreturn by time t after insemination and at concentration s of spermatozoa inseminated. Although nonreturn rate by 28 d can be considered to be a reliable early measure of reproductive efficiency of a bull (2), we consider conception rate to be a better early measure of efficiency because conception rate depends on the population of spermatozoa at insemination, not on developmental potential of the conceptus after insemination. Although conception rate is difficult to observe, it can be estimated as Po,- in Equation [2], ignoring the Journal of Dairy Science Vol. 78, No.4, 1995
Totals
Pc.-
Pt.- - Pc.1 - Pt.1 - Pc.-
Pt,-
Pc,-
d
P-.s
No
o
return rates for a bull are necessary to test the significance of parameters in Equation [2]. Nonretum Versus Concentration. For the relationship of nonreturn with concentration, by a given time after insemination, P-.s is defined as probability of nonreturn at concentration s of spermatozoa. Nonreturn rate at a given time has a curvilinear relationship with concentration of viable spermatozoa inseminated (8). This relationship has been modeled based on kinetic theory (12). based on concentration of viable spermatozoa having a Poisson distribution (10) and on an assumed upper asymptote for the curve (5). Although other nonlinear functions may also describe the data, the exponential function was suitable to fit nonreturn rates of 56 d (1) and 90 d (5):
p_,ooe
Yes
I - Pt,1
concentration of spermatozoa. If spermatozoa concentration is known, however, then P-,oo can be estimated from Equation [3] as the asymptotic value at a given time. With information on time after insemination and concentration of spermatozoa, an asymptotic conception rate, PO.co, can be estimated by substitution of Equation [3] at time 0 for PO,- in Equation [2]:
If observations on nonreturns are available at several times after insemination and at several concentrations of spermatozoa, then Equation [4] can be used to model the reproductive efficiency of a bull. Data
Data file 1 included three bulls (A, B, and C) that were selected to illustrate the model in
Equation [2]. Bulls were used for first or second inseminations between September and December 1993, and nonreturn rate was computed in April 1994. For each cow that returned to service, the interval between first and second or between second and third inseminations was recorded in days. From these intervals, nonreturn rate was computed for three multiples of 28 d: 28, 56, and 84 dafter insemination (Table 2). The number of inseminations ranged from about 12,500 to 37,400; inseminations were assumed to be randomized over various effects, such as cows and herds. Data file 2 included the same three bulls as in data file 1 selected to illustrate the model in Equation [4]. Bulls were used for first and
925
A MODEL FOR REPRODUCTIVE EFFICIENCY TABLE 2. Data file 1: number of inseminations (n) and observed nonreturn rate <1\-), by bull and time t after insemination (t = 28, 56, or 84 d). Bull
n
P28,-
P56,-
PM,-
A B C
15,555 12,478 37,392
.783 .764 .813
.672 .656 .718
.624 .606 .670
second inseminations between May and November 1993, and nonretum rate was computed in January 1994. The 28-d nonretum rate was computed for cows inseminated between June through November, 56-d nonretum rate for cows inseminated between May through October, and 84-d nonretum rate for cows inseminated between May through September. Data were from a study to measure effect of concentration of spermatozoa at insemination on nonreturn rate. Five or six different concentrations of spermatozoa per dose were used: about 5.0, 7.5, 10, 12.5, and 15 (xlO6 ) for bulls A and B, and, in addition to those five concentrations, about 2.5 (xl()6) for bull C (fable 3). Number of inseminations per concentration ranged from about 700 to about 2000; inseminations were assumed to be randomized over various effects, such as cows and herds.
Estimation of Parameters
For data file 1, the nonreturn rate by 28, 56, and 84 d after insemination (Table 2), Equation [2] was fitted for each bull. For data file 2, the nonreturn rate by 28, 56, and 84 d after insemination and the concentration of spermatozoa at insemination (fable 3), Equation [4] was also fitted for each bull. Parameters were estimated by nonlinear regression, using an adaptive nonlinear least squares algorithm [NLREG; (11)]. A default value of 1 x lO-10 was used for the tolerance factor, which specifies the convergence criterion for the iterative estimation procedure. All estimations converged within 50 iterations. Because of the relatively large numbers of inseminations used for each bull, data were not weighted during analysis. Goodness of fit was measured by residual standard error when possible. RESULTS
For data file 1 (Table 2), estimates for conception rate (AJ _), calving rate (Pc _), and rate of decline in no~etum (b) by bull ~e in Table 4. Residual standard error could not be estimated because only three data were available to estimate the three parameters. Conception
TABLE 3. Data file 2: number of inseminations (Ilt) and nonreturn rate after insemination (t = 28, 56, or 84 d), for bulls A, B, and C. s (x 106)
n28
P28,s
CPt s) by concentration (s) at insemination and time .
n56
P56,s
n84
P84,s
1304 1309 2068 1142 1046
.638 .652 .688 .675 .670
1282 1279 1955 1117 1029
.581 .59B .636 .611 .626
879 987 1014 1023 923
.654 .655 .653 .659 .663
838 945 959 993 895
.598 .594 .595 .609 .599
1147 1171 1167 1180 1139 1182
.664 .699 .711 .731 .721 .724
1119 1156 1150 1169 1124 1158
.620 .648 .674 .680 .666 .670
Bull A 5.0 7.5 10.0 12.5 15.0
880 906 1624 814 841
.747 .772 .BOI .800 .779
5.0 7.5 10.0 12.5 15.0
779 863 859 881 715
.759 .767 .782 .779 .783
25 5.0 7.5 10.0 12.5 15.0
889 885 B05 776 725 808
.772 .805 .810 .821 .828 .811
Bull B
Bull C
Journal of Dairy Science Vol. 7B, No.4, 1995
926
KOOPS ET AL.
TABLE 4. Estimates of conception rate CPo.-), calving rate (fIc,-). and rate of decline in nonreturn rate (b) by bull from data file 1. Bull A B C
1.0 , .9 ,,~.
~O,-
~c,-
b
.908 .889 .914
.605 .583 .646
.050 .046 .045
"",,/'... -,-
,,\
.8
",,"' . .....
.7
"'>::::
. . . . . ;:::::_==---------------------------------
.6 .5
rate
_._._._._._._._.__._._._._._._._-
i _-.-1..'_-'--_......J.
_
0285684
Time (d) Figure 2. Nonretum rates for bulls A (+), B (A), and C (e) and predicted probability of nonretum for bulls A (-), B (-), and C (- - -) by time after insemination.
trations, about 92% of cows conceived at time of insemination. Calving rate ([>c.-) averaged about .58: i.e., about 58% of cows were expected to complete gestation. Bulls A and C had similar conception rates, but A had a lower calving rate than C. Rate of decline in nonreturn rate (b) averaged .041. Rate of decline was slower for bull C than for bulls A or B, which were similar to one another. Rate of approach to the asymptote (d) averaged .22 and was slowest for A: i.e., A reached the asymptote at a higher concentration of spermatozoa than did B or C. For bull C, which had six levels of concentration, nonreturn rates, and predicted probability of nonreturn by concentration are in Figure 3. The model proposed in Equation [4] fitted best data from B; residual standard error was .005, about one-half to one-third that for A or C. To compare results from data on only time after insemination (Table 4) with results from data on time and concentration of spermatozoa (Table 5), bulls ranked differently for concepA
TABLE 5. Estimates and standard err~rs of conception rate CPo.-), calving rate (fIc.-), rate of decline in nonretum rate (b). rate of approach to the asymptote (d). and residual standard error (RSE) by bull from data file 2.
~o.-
Bull
A B C
a
b
~c.-
RSE
Est.
SE
Est.
SE
Est.
SE
Est.
SE
.930 .913 .921
.020 .006 .013
.574 .572 .605
.021 .006 .022
.042 .046 .035
.007 .002 .005
.351 .132 .189
.110 .030 .037
Journal of Dairy Science Vol. 78, No.4. 1995
.015 .005 .012
A MODEL FOR REPRODUCTIVE EFFICIENCY 1.0
i;..---- j----~----~---~----+ .... _ _ -''"'''1- _.-.- T _._._._ .... _.- - , - - - - 1.
~ .._ .._..r'·_··_··.!.··_··_··!··_··_··.... ·_··_··J
• .5 l - - .
o
2.5
_ 5.0
7.5
10.0 12.5 15.0
Concentration (x 10 6 ) Figure 3. Nonreturn rates by 28 d (+), S6 d (.A), and 84 d (e) and predicted probability of nonreturn by 0 d (-), 28 d (---), S6 d (---), and 84 d (----) by concentration of spermatozoa for bull C.
tion rate but similarly for calving rate. Differences in ranking can be explained by use of different data files and by use of additional information on concentration of spermatozoa. For example, Po,- in Table 4 presents an estimate of mean conception rate at a mean concentration; Po,co in Table 5 estimates asymptotic conception rate at a high concentration. DISCUSSION
The nonreturn rate of a bull is a measure of his reproductive efficiency and is used to evaluate performance for bulls and technicians in AI organizations. Assuming perfect detection of estrus and negligible embryonic death, nonreturn is the result of conception and gestation after conception. Conception is influenced by, among other factors, characteristics of the population of spermatozoa inseminated, and gestation is influenced by maternal environment and by maternal and paternal contributions to developmental potential of the conceptus. An estimate of conception rate, therefore, would be expected to be a more reliable measure to evaluate bulls on their reproductive efficiency or AI technicians on their performance than nonretum rate by a given time, say 28 d, after insemination. The dairy producer is interested in calving rate of bulls used, because cows are kept to produce calves and milk. A high calving rate means that the producer has increased opportu-
927
nity not merely to replace cows in the herd but also to select replacements. The producer, however, must wait until gestation is completed to observe calving rate. Therefore, an early estimate of calving rate means that producers can select bulls earlier on their calving rate. An AI organization is interested in reproductive efficiency of bulls used and in the effect that concentration of spermatozoa has on efficiency. Models presented herein can be used to determine concentration of spermatozoa necessary at time of insemination for a bull to achieve, say, 95% of asymptotic conception rate
928
KOOPS ET AL
known, which will enable modeling of estrus detection and early embryonic death. CONCLUSIONS
Reproductive efficiency of bulls is usually measured by nonretum rate. The AI organizations also use nonretum rate to evaluate fertility of a bull or performance of a technician. The mathematical function presented herein to model reproductive efficiency of bulls estimates conception rate and calving rate through the relationship of nonretum rate to concentration of spermatozoa at insemination and time after insemination. These estimates might be more reliable for evaluation than nonretum rate itself. ACKNOWLEDGMENTS
The authors thank Lucia Lansbergen and Janneke van Wagtendonk, Holland Genetics, for their cooperation and helpful discussions and J.B.M. Wilmink, the Royal Netherlands Cattle Syndicate, for his cooperation in providing data used in this research. REFERENCES
I den Daas, N. 1992. Laboratory assessment of semen characteristics. Anim. Reprod. Sci. 28:87. 2 Erb, R. E., and F. H. Flerchinger. 1954. Influence of fertility level and treatment of semen on nonreturn
Journal of Dairy Science Vol. 78, No.4, 1995
decline from 29 to 180 days following artificial service. 1. Dairy Sci. 37:938. 3 Foote, R. H.. and R. W. Bratton. 1952. The influence of antibiotics on delayed returns in artificial breeding. J. Dairy Sci. 35:261. 4 Olds, D. 1978. Conception rate and factors affecting its magnitude. Chapter 19 in Physiology of Reproduction and Artificial Insemination of Cattle. 2nd ed. G. W. Salisbury, N. L VanDemark and 1. R. Lodge, ed. W. H. Freeman and Co., San Francisco, CA. 5 Pace, M. M., J. 1. Sullivan, F. I. Elliot, E. F. Graham. and G. H. Coulter. 1981. Effects of thawing temperature, number of spermatozoa and spermatozoal quality on fertility of bovine spermatozoa packaged in .5-ml French straws. 1. Anim. Sci. 53:693. 6 Reurink, A., J.H.G. den Daas, and J.B.M. Wilmink. 1990. Effects of AI sires and technicians on nonreturn rates in the Netherlands. Livest. Prod. Sci. 26: 107. 7 Salisbury, G. W., R. W. Bratton, and R. H. Foote. 1952. The effect of time and other factors on the nonreturn to service estimate of fertility level in artificial insemination of cattle. J. Dairy Sci. 35:256. 8 Salisbury, G. W., and N. L VanDemark. 1961. Pages 434 and 478 in Physiology of Reproduction and Artificial Insemination of Cattle. W. H. Freeman and Co., San Francisco, CA. 9 Schaeffer, L R. 1993. Evaluation of bulls for nonreturn rates within artificial insemination organizations. 1. Dairy Sci. 76:837. 10 Schwartz, D., P.DM. Macdonald, and V. Heuchel. 1981. On the relationship between the number of spermatozoa and the probability of conception. Reprod. Nutr. Dev. 21:979. 11 Sherrod, P. H. 1992. Nonlinear Regression Analysis Program (NLREG), Version 1.8. 4410 Gerald Place. Nashville. TN. 12 van Duijn. c., Jr. 1965. Sperm number and fertility: a kinetic approach. Neth. J. Agric. Sci. 13:378.