A MATHEMATICAL MODEL FOR THE SECRETION OF VASOPRESSIN USING FUZZY TRUNCATED NORMAL DISTRIBUTION

The present article is devoted to investigate the increase of the hormone vasopressin due to the administration of adenosine tri-phosphate and phenylephrine in the explants. A mathematical model using fuzzy truncated normal distribution was developed and used this model to calculate the mean values of vasopressin release in the truncated time intervals. The result shows that a synergistic effect of adenosine tri-phosphate and phenylephrine on vasopressin. AMS Subject Classification: 60A86, 62A86, 62H10


Introduction
Truncated distributions occur in many practical situations, particularly in several industrial settings.The truncated normal distribution is an important distribution in the world of probability and statistics.It appears quite naturally when the normal distribution itself arises.Most of the available literatures tend to focus on the mean and variance of the truncated normal distribution.This is partially provoked from the statistical community.They were paying interest in obtaining unbiased mean and variance estimators for data that is screened or threshold.Truncated distributions can also be used to model intensity statistics in the study of atomic heterogeneity.A mathematically defensible way to protect the main features of the normal distribution while avoiding extreme values involves the truncated normal distribution, in which the range of definition is made finite at one or both ends of the interval.It is the purpose of this paper to describe the truncation process, to consider how certain basic statistical properties of the new distribution can be determined.
Informally, the truncated normal probability density function is defined in two steps.We choose a general normal probability density function by specifying parameters µ and σ 2 , and then a truncation range (a; b).The probability density function associated with the general normal distribution is modified by setting values outside the range to zero, and uniformly scaling the values in the range so that the total integral is 1.
The truncation range can be classified into four classes.Truncated sample of normal distribution arise, in practice, with various types of experimental data in which recorded measurements available over only a partial range of the variable.The Maximum likelihood estimation of singly truncated and doubly truncated normal distribution was considered by Cohen [2], [11].Numerical solution to the estimators of the mean and variance for singly truncated samples were computed with an auxiliary function which is tabulated in Cohen [3].The properties and applications of the fuzzy probabilities have been studied by many authors, including Zadeh [4], [7], Yager, R.R. et al [5], [8], J.J.
Buckley, E. Eslami [14], [15].Diverse effect of norepinephrine on vasopressin release was reported by C. Sladek, C. Yagil [10] and the effect of actinomycin and cycloheximide on stimulation of vasopressin was studied by Z. Song, H.E. Sidorowicz, C.D. Sladek [12].In this paper, we investigate the increase of the hormone vasopressin (VP) due to the administration of adenosine tri-phosphate(ATP) and phenylephrine (PE) in the explants using fuzzy truncated normal distribution.
Here ϕ (.) and φ(.) are the probability density function and cumulative density function of the standard normal distribution respectively The mean value of truncated normal distribution is

Fuzzy Truncated Normal Distribution
Let Ā be a fuzzy event in Here ϕ (.) and φ(.) are the probability density function and cumulative density function of the standard normal distribution respectively The mean value of truncated normal distribution is

Application
Vasopressin is a hormone that is secreted by the posterior pituitary gland.In the body, vasopressin act on the kidneys and blood vessels.Vasopressin helps prevent the loss of water from the body by reducing urine output and serving the kidneys reabsorb water in the body.Vasopressin also raises blood pressure by constricting (narrowing) blood vessels.It acts as a neurotransmitter in the brain to control circadian rhythm, thermoregulation, and adrenocorticotrophic hormone (ACTH) release.The therapeutic use of vasopressin has become important in the critical care environment in the management of cranial diabetes insipidus, bleeding abnormalities, oesophageal variceal hemorrhage, asystolic cardiac arrest, and septic shock.The A1 catecholamine neurons of the caudal ventrolateral medulla transmit hemodynamic information to the vasopressin neurons in the hypothalamus.Simultaneous exposure of explants to 100 µM adenosine tri-phosphate (ATP) and 100 µM phenylephrine(PE) resulted in synergistic responses that were delayed, larger and sustained relative to the responses observed with ATP or PE individually.We truncate the time intervals into two cases such that from 120 min.to 180 min.is taken as the first case and 180 min.to 240 min.is taken as the second case and the mean value of truncated normal distribution are calculated accordingly.The mean values of truncated normal distribution for the interval 120 min.to 180 min.are calculated by using    where

Conclusion
Here we proposed two cases to study, the mean value of vasopressin releasing level due to the administration of ATP and PE by using fuzzy truncated normal distribution.The membership function of the mean values showed that, after the administration of ATP and PE the releasing level of vasopressin is increased in both the two cases.These results suggest that the release of cotransmitters may be responsible for maintaining a sustained increase in plasma vasopressin in response to blood pressure.

µ
Mean value of normal distribution; σ Standard deviation of normal distribution; ϕ pdf of normal distribution; φ cdf of normal distribution; a lower truncation interval; b upper truncation interval µ[α] alpha cut of mean; σ[α] alpha cut of standard deviation; E[α] mean value of fuzzy truncation normal distribution.

Table 5
.1 shows the % basal of vasopressin release due to the administration of ATP and PE.

Table 5 .
1: Vasopressin release due to the administration of ATP and PE