Flexibility index calculation

A bond flexibility index calculated as the mean of the atom flexibility indices ... 

A flexibility index, calculated from molecular shape descriptors [378, 379]. 

The problem of calculating the flexibility index can be mathematically formulated as... 

When the critical point must be a vertex of the hyperrectangle 0(F), the simplest approach to calculating the flexibility index F is to maximize s in each vertex direction 61 (Fig. 5) by the following (N)LP (Swaney and Grossmann, 1985a) ... 

Example 4 (from Grossmann and Floudas, 1987). The flexibility index of the HEN structure in Fig. 4 is to be calculated with respect to an expected uncertainty range of + 10 K in all stream supply temperatures. 

The RI can be scaled in terms of maximum expected uncertainties like the flexibility index. Then the RI would be an upper bound on the flexibility index, since the RI only considers uncertainties in each stream individually, while the flexibility index considers uncertainties in all streams simultaneously. However, for HENs it seems reasonable to assume that the uncertainty in each stream s supply temperature is roughly inversely proportional to that stream s heat capacity flow rate (i.e., that each stream is subject to roughly the same load uncertainty ). Thus the RI is calculated in terms of load uncertainty 81 or 8t, where... 

The flexibility index can be calculated by determining the largest scaled uncertainty sk which the network can tolerate in the direction of each of the k corner points of the uncertainty range. Then the flexibility index is... 

Different algorithms are required if the HEN resilience problem is nonlinear. Special algorithms were presented for testing the resilience of minimum unit HENs with piecewise constant heat capacities, stream splits, or simultaneous flow rate and temperature uncertainties. A more general algorithm, the active constraint strategy, was also presented which can test the resilience or calculate the flexibility index of a HEN with minimum or more units, stream splits and/or bypasses, and temperature and/or flow rate uncertainties, but with constant heat capacities. 

Novartis uses the In Silico Profiling web tool. Available properties include the octanol water partition coefficient log P, molar refractivity, flexibility index, hydrogen bonding characteristics and molecular polar surface area. Various drug properties, such as intestinal absorption, BBB permeability or Plasma Protein Binding (PPB) are calculated based on in-house models. 

The calculation methods for the flexibility index discussed below can be applied to step 3 of the worst-case design optimization procedure to maximize constraint violation instead of minimizing 8. 

Under the vertex solution assumption, the flexibility index can be calculated by evaluating the maximum 8 along each vertex direction and taking the minimum of the results. This approach becomes computationally impractical for more than about 15-20 parameters, so two procedures are presented which give upper bounds on F more efficiently (a vertex search method and a branch-and-bound method). Of the two methods, the vertex search method is found to be more efficient on the examples considered and therefore seems the best candidate for use to get an approximate solution for the worst-case vertex where appropriate. The vertex search procedure is given below. 

The Diverse Property-Derived method (DPD method) is based on the partitioning of six noncorrelated molecular descriptors and physico-chemical properties [Ashton, Jaye et al, 1996]. These are a lipophilicity descriptor (CLOGP), an electrotopological index calculated as normalized sum of the squares of the atomic electrotopological state indices, the number of hydrogen-bond acceptors (HBA), the number of hydrogen-bond donors (HBD), a flexibility index defined as the ratio of the —> Kier shape descriptors over k, and the aromatic density defined as the number of aromatic rings over the molecular volume (Table Cl). 

Using the shape indices Kj and K2, the Kier algorithm calculates the flexibility index, O. 

The Wiener index [86] can be expressed in terms of the distance matrix [87] and equals the half-sum of all distance matrix entries. Randi(5 [88] and Kier and Hall indices of order 0-3 [89] are calculated from coordination numbers of atoms or from values of atomic connectivity. The Kier shape index (order 1-3) [90] depends on the number of skeletal atoms, molecular branching, and the ratio of the atomic radius and the radius of the carbon atom in the sp hybridization state. The Kier flexibility index [90] is derived from the Kier shape index. The Balaban index depends on the row sums of the entries of the distance matrix and the cyclomatic number [92,93]. The information content index and its derivatives (order 0-2) are based on the Shannon information theory [95]. Modifications of the information content index are structural information content, complementary information content, and bond information content [96],... 

This is an index encoding information about the flexibility of a bond and calculated as the mean of the atom flexibility indices d> of the two atoms forming the bond [von der Lieth, Stumpf-Nothof et al., 1996] ... 

Mid-Arm Anthropometry. The arm muscle circumference (AMC), also called the mid upper arm muscle circumference (MUAMC), reflects both caloric adequacy and muscle mass and can serve as a general index of marasmic-type malnutrition. The arm circumference is measured at the midpoint of the left upper arm by a fiberglass flexible-type tape. The arm muscle circumference can be calculated from a formula that subtracts a factor related to the skinfold thickness (SFT) from the arm circumference ... 

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