Population doubling equation

The population doubling time has been given in Equation 7.6.8c as... 

The term X, describes the age distribution of the population, it is a term which varies slowly with population doubling time with a minimiun value of 0.693 in steady state tissues and a maximum value of 1.38 if the S phase cxrcupied the complete cell cycle. In tumours a value of 0.8-1 is common. The volmne doubling time represents the time interval in which a tumour doubles its volmne and will be subject to cell loss and the growth fraction, as well as the influence of any host cells contained within the tumour mass, and is related to Tp by the equation ... 

Tab. 9.6. Full set of population balance equations for the PVAc problem in the case of a maximum of one terminal double bond per chain.

Tab. 9.11. The general (N, M)th double moment formulation of the population balance equation set of Table 9.10, obtained by multiplying by the TDB number and branching number indices, / and and subsequent summation over these indices.

FIGURE 12.6 Measurement of full agonist affinity by the method of Furchgott. (a) Dose-response curve to oxotremorine obtained before (filled circles) and after (open circles) partial alkylation of the receptor population with controlled alkylation with phenoxybenzamine (10 jiM for 12 minutes followed by 60 minutes of wash). Real data for the curve after alkylation was compared to calculated concentrations from the fit control curve (see arrows), (b) Double reciprocal of equiactive concentrations of oxotremorine before (ordinates) and after (abscissae) alkylation according to Equation 5.12. The slope is linear with a slope of 609 and an intercept of 7.4 x 107 M-1. 

One of the interesting features of difference equations such as these is that they are much more unstable than the differential equations we have used throughout this text. If the reproduction rate exceeds a certain value, the system is predicted to exhibit population oscillations with high and low populations in alternate years. At higher birth rates, the system exhibits period doubling, and at even higher birth rates, the system exhibits chaos, in which the population in any year cannot be predicted from the population in a previous year. Similar phenomena are observed in real ecological systems, and even these simple models can capture this behavior. 

Equation (7.5) shows that the population of each eigenstate oscillates with its transition frequency as a function of r. For B transition of the iodine molecule that we will discuss later, the pump laser wavelength is 600 nm, which corresponds to the oscillation period of 2fs. If we require Ittx 1/10 stability for the relative phase between the two interfering WPs, attosecond stability is necessary for the delay t. The details of the experimental setup to prepare the phase-stabilized double pulses will be described in the following section [38, 39,47,48]. 

This behavior is exploited in SEP experiments [51] where the lowering of the population of level 2 for double-resonance conditions is probed by laser-induced fluorescence (LIF) or ion detection (ion dip experiments) by ionizing the molecules in level 2 with a third laser pulse. It is obvious from the rate equations that no dip depth larger than 50% of the maximum off-resonant signal can be obtained as long as no fast decays of the final levels must be considered. (However, for fast-decaying final levels deeper dips can be expected and the dip depth has been used for an estimate of the decay rate [53].)... 

Let us now come back to equation (1). So far we have used it to calculate 35ci NQR frequencies from the computed 3pz orbital populations of two simplified models. Then we compared the calculated figures with the observed resonances of the more complicated pyranosyl chlorides. Conversely, we may start from these observed resonances, and use formula (1) to derive experimental values of the 3pz orbital populations of pyranosyl chlorides, so as to gain direct insight into their electronic structure. If no double bonding is involved, and if a constant value of 2 is adopted bor b9 we can accept as an experimental observation that on all pyranosyl chlorides examined, the 3pz orbital population on axial chlorine is higher than the 3pz orbital population on equatorial chlorine by an amount of about 5% (up to 7% in the case of mannose).Some people may consider that a statement in terms of ionicity i is more suggestive. Equation (1) may be written ... 

We summarize recent work showing that condensation can be derived as a natural consequence of the Poisson-Boltzmann equation applied to an infinitely long cylindrical polyelectrolyte in the following sense Nearly all of the condensed population of counter-ions is trapped within a finite distance of the polyelectrolyte even when the system is infinitely diluted. Such behavior is familiar in the case of charged plane surfaces where the trapped ions form the Gouy double layer. The difference between the plane and the cylinder is that with the former all of the charge of the double layer is trapped, while with the latter only the condensed population is trapped. 

The three lowest states in the CH3F system, (p2,Ps) were recognized as forming a subsystem that could be isolated from the remaining vibrational manifold and treated independently. A solution of the kinetic rate equations for this three-level system will yield expressions for the population evolution that are double exponentials however, experimentally signal quality and apparatus constraints precluded a full double exponential analysis of fluorescence signals. 

This equation should be compared to Eq. (4.20) which describes the time dependence of a superposition state for the double-well tunneling problem. The two equations are identical except in Eq. (8.12), (t) is multiplied by (0) = Cjil/i + C2i 2 d includes damping from unimolecular decay, that is, the exp (-F t/2ft) terms. The probability of populating the superposition state versus time, that is, P(t) = is given by... 

A double exponential model presented with Equation (32.3) is proposed for this response. The model implies the existence of a heterogeneous population of cells, which can be roughly divided into two subpopulations, differing in the rate of cell reaction to the toxin exposure. It is interesting to note that the visual appearance of fast responding cells ( star-like cells) and slow responding cells ( sheet-like cells) can be easily discriminated under microscope and with the aid of the shape-recognition software. 

In tumours, the term potential doubling time (TpoJ is used to describe the shortest time a cell population can double its number taking into account growth fiaction but in the absence of cell loss using the equation ... 

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