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Some De nitions in Reproductive Biology Discussion of a bit of reproductive biology may be helpful in the understanding of study designs to evaluate reproductive and developmental toxicity Tests to assess general reproductive performance and fertility are generally conducted using rats In the rat, multiple eggs are ovulated from mature follicles in the ovary The follicle that an egg leaves behind develops into glandular tissue known as a corpora lutea The corpora lutea secretes progesterone, a hormone needed to maintain pregnancy (unlike humans in which progesterone is secreted by the placenta) Corpora lutea are visible as blisterlike protuberances on the ovary A count of the corpora lutea in the ovary allows one to determine the maximum number of potential offspring for that pregnancy Fertilized eggs develop into zygotes that may attach to the wall of the uterus (implantation).

Because of the anticipated large number of sources, the actual bandwidth process should be closely approximated by its mean, by the law of large numbers (LLN) As in Duf eld and Whitt [14], the conditional mean can be thought of as a deterministic uid approximation; for example, see Chen and Mandelbaum [8] Since the withinlevel variation process has mean zero, the within-level variation process has no effect on this conditional mean Hence, the conditional mean of the aggregate bandwidth process is just the sum of the conditional means of the component level processes Unlike the more elementary M =G=I model considered in Duf eld and Whitt [14], however, the conditional mean here is not available in closed form In order to rapidly compute the time-dependent conditional mean aggregate demand, we exploit numerical inversion of Laplace transforms.

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The discrete areas of implantation may be observed and counted upon examination of the uterus at C-section Calculation of pre- and postimplantation loss are important end points in a reproductive toxicity study Pre-implantation loss is the death of a fertilized ova prior to implantation in the uterine wall Postimplantation loss (ie, resorption and/or fetal death) is the death of the conceptus after implantation in the uterine wall and prior to parturition Postimplantation loss can be broken down into early and late resorptions and fetal death A late resorption has discernable features such as limbs, eyes, and nose, whereas an early resorption has none of these features Single- and Multiple-Generation Tests Fertility and general reproductive performance can be evaluated in single and multiple generation tests These tests are usually conducted using rats.

ACKNOWLEDGMENT The author would like to thank Dr. Catherine Laughlin for her critical reading of the manuscript. Thanks also to Linda Cooney and Darryl Metzler for their assistance in information management.

It follows quite directly from the classical theory of semi-Markov process that explicit expressions can be given for the Laplace transform of the conditional mean More recently, it has been shown that numerical inversion can be an effective algorithm; see Abate et al [1] For related discussions of transient analysis, design and control, see s 13, 16, and 18 in this volume 173 OUTLINE OF THE CHAPTER.

Fertility is de ned as the ability to produce a pregnancy, while the ability to produce live offspring is known as fecundity An abbreviated protocol for a single-generation test is shown in Figure 212 In typical tests 25 males per dose group are treated for 70 days prior to mating and 25 females per dose group are treated for 14 days pre-mating The number of animals is chosen to yield at least 20 pregnant females per dose group including controls The treatment durations are selected to coincide with critical times during which spermatogenesis and ovulation occur It takes approximately 70 days in the rat for spermatogonial cells to become mature sperm capable of fertilization.

The rest of this chapter is organized as follows. In Section 17.4, we show that the Laplace transform of the mean of the transient conditional aggregate demand can be expressed concisely. This is the main enabling result for the remainder of the chapter. The conditional mean itself can be very ef ciently computed by numerically inverting its Laplace transform. To carry out the inversion, we use the Fourier-

1. Morens, D. M.; Folkers, G. K.; Fauci, A. S. The challenge of emerging and re-emerging infectious diseases. Nature 2004, 430, 242 249. 2. Feldmann, H.; Czub, M.; Jones, S.; Dick, D.; Garbutt, M.; Grolla, A.; Artsob, H. Emerging and re-emerging infectious diseases. Med. Microbiol. Immunol. 2002, 191, 63 74. 3. Fauci, A. S. Infectious diseases: considerations for the 21st century. Clin. Infect. Dis. 2001, 32, 675 685. 4. Gomez, L.; Clavel, A.; Castillo, J.; Seral, C.; Rubio, C. Emerging and reemerging pathogens. Int. J. Antimicrob. Agents 2000, 16, 335 339.

In the female rat the estrus cycle length is 4 to 5 days and a 14-day dosing period is considered suf cient time to detect potential effects on hormonal or other systems that may effect ovulation In some study designs both males and females are treated for 70 days pre-mating..

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