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Parameters of a random unit graph are normally the number of nodes N and the common transmission radius R However, R may be a misleading parameter when used in simulations, since all generated graphs may be either dense or sparse, which has signi cant impact on the performance (for example, of routing protocols) It is therefore preferable to use another parameter that can be easily interpreted and matched directly with the graph density That parameter is the average number of neighbors d per each node There exists a relation between R and d, since an approximate radius R can be obtained from the formula d (N 1)pR2 =A, where A is the area of the region where nodes are placed In case of a square of edge length a, A a2.

Figure 14.2. Rotating wall vessel bioreactors (RWV). Left: Slow Turning Lateral Vessel (STLV); right: High Aspect Ratio Vessel (HARV). Photos courtesy of Synthecon Inc.

O2 diffusion to the cell, uptake by the cell, and convection along the length of the ow region. Both analytical and numerical models were shown to correlate closely with measured oxygen outputs. A detailed description of the mathematical modeling for this system has been reported previously [Allen and Bhatia, 2003]. Given an inlet O2 concentration representative of physiologic periportal levels, 76 mmHg (10% O2 ), a ow rate of 0.5 ml/min was determined to result in the exposure of cells in the nal 50% of the chamber length to perivenous O2 levels (<35 mmHg). Subsequently, this O2 distribution resulted in zonated hepatocyte phenotypes within the bioreactor.

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No closed-form solution exists for estimating logistic regression coef cients Thus, we must turn to maximum likelihood estimation, which nds estimates of the parameters for which the likelihood of observing the observed data is maximized A saturated model is a model that which contains as many parameters as data points and so predicts the response variable perfectly with no prediction error We may then look upon the observed values of the response variable to be the values predicted by the saturated model To compare the values predicted by our tted model (with fewer parameters than data points) to the values predicted by the saturated model, we use likelihood of the tted model deviance = 2 ln likelihood of the saturated model The resulting hypothesis test is called a likelihood ratio test The deviance represents the error left over in the model, after the predictors have been accounted for.

This formula is obtained by nding the expected number of nodes inside a circle with radius R (the formula multiplies number of other nodes with probability of being generated inside the region) In the literature, random unit graphs are normally generated by selecting each of N nodes at random locations inside a square or circle In the case of a square region, this means generating x and y coordinates at random in an interval [0, a) Then all N (N 1)=2 potential edges in the network among the N nodes are sorted by their length in ascending order The radius R that corresponds to a chosen value of d is equal to the length of the N d/2-th edge in the sorted order Any edge no longer than R will remain in the graph Other edges are eliminated from the graph.

As such, it is analogous to the sum of squares error in linear regression To determine whether a particular predictor is signi cant, nd the deviance of the model without the predictor, and subtract the deviance of the model with the predictor, thus: G = deviance(model without predictor) deviance(model with predictor) likelihood without predictor = 2 ln likelihood with predictor The test statistic G follows a chi-square distribution with 1 degree of freedom (ie, 2 =1 ), assuming that the null hypothesis is true that 1 = 0 Odds may be de ned as the probability that an event occurs divided by the probability that the event does not occur The odds ratio (OR) is de ned as the odds that the response variable occurred for records with x = 1 divided by the odds that the response variable occurred for records with x = 0.

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