Homa B Formulation
As with the HOMA method for insulin resistance (see Chapter 3), the calculation of the HOMA (cell function index (HOMAB) requires only fasting glucose and insulin measurements. A modelbased version of the method (HOMA2, http://www.dtu.ox.ac.uk/homa/ (web search keywords: homa calculator,site:dtu.ox.ac.uk) is recommended, although a formula is also available: HOMAB = constant X fasting insulin/(fasting glucose  3.5) (Matthews et al. 1985). The advantages of the HOMA2 computer index have not been demonstrated (Mari 2006).
HOMAB is an empirical index that does not represent a specific (cell characteristic. Its performance in comparison to the hyperglycaemic clamp is mediocre (Hanson et al. 2000). HOMAB, being proportional to fasting insulin, is expected to be inversely related to insulin sensitivity in normotolerant subjects, as is AIR (see below). Thus adjustment for insulin sensitivity is necessary, but if only fasting measurements are available, the correction cannot be made (Mari et al. 2005a).
Modelling methods
Modeling methods deserve specific attention because they are often used not only with tests in which the glucose levels cannot be controlled, such as the OGTT, but also in standardised tests such as the IVGTT or the hyperglycaemic clamp. Furthermore, Pcell models have had an important historical role in the understanding of Pcell function (Bergman & Urquhart 1971; Grodsky 1972; Licko 1973; Cerasi et al. 1974). See Mari (2002) for a more comprehensive review. This section discusses only models used to determine Pcell function parameters; models for simple insulin secretion are discussed in previous reviews (Pacini 1994; Mari 2002).
The models currently used for Pcell function assessment are simplifications tailored for the clinical tests of historical Pcell models. Figure 2.7A shows the common paradigm of these models, which is based on the Cpeptide methodology discussed previously. The leftmost block represents the Pcell model, which is a mathematical representation of the dynamic relationship between glucose concentration (the input to the model block) and insulin secretion (the output). The Pcell model contains parameters representing specific Pcell characteristics, such as the doseresponse (see below). The Pcell model is coupled to a model of Cpeptide kinetics (the rightmost block), which describes the dynamic relationship between insulin secretion and Cpeptide concentration. The Cpeptide model is assumed to be known, the typical model being that used for simple deconvolution (Van Cauter et al. 1992). Thus, the two blocks represent the dynamic relationship between glucose and Cpeptide concentration, which are both measured variables. Cpeptide concentration depends on glucose concentration (the input function) and on the parameters of the Pcell model. Once glucose and Cpeptide concentration are obtained from a test, the Pcell model parameters are estimated by fitting the model to the data.
The most frequently used Pcell model variants have the structure depicted in Figures 2.7B and C. A common feature is the Pcell doseresponse, which is the model estimate of the doseresponse that can be obtained with the graded glucose infusion test (with the proviso that intravenous and oral tests do not yield the same doseresponse). Some models include additional secretion components. In particular, some models provide parameters describing the anticipation of insulin release related to the first phase mechanisms (Breda et al. 2001; Cretti et al. 2001; Mari et al. 2002a, 2002b) (Figures 2.7B and C). These models, however, describe differently other aspects of the dynamic relationship between glucose concentration and insulin secretion, i.e. they include a delay between glucose concentration and secretion (Breda et al. 2001; Cretti et al. 2001) (Figure 2.7B) or a factor accounting for potentiation phenomena (Mari et al. 2002a, 2002b) (Figure 2.7C). See Mari (2002) and Ferrannini & Mari (2004) for a more comprehensive discussion.
The modeling methods are a valuable tool for interpreting tests that do not have standardised glucose levels, such as the oral tests, as they are based on a physiological representation of the P cell, though simplified. At least in principle, modeling approaches are preferable to empirical ones, in which normalisation to the glucose levels is not based on physiological principles. In addition, some models provide multiple parameters, with which a more complete characterisation of Pcell function can be achieved (Breda et al. 2001; Mari
Figure 2.7 Illustration of modeling methods: A) the combination of a pcell model (containing Pcell function parameters to be estimated) and a model of Cpeptide kinetics (assumed to be known) provides a mathematical relationship between observed glucose and Cpeptide concentrations. This mathematical relationship is used to estimate the pcell function parameters from glucose and Cpeptide data; B) pcell model incorporating a doseresponse, a description of the anticipation of insulin release related to the first phase mechanisms and a delay between glucose concentration and insulin secretion (Breda et al. 2001; Crettiet al. 2001). The model (Hovorka et al. 1998) has a similar structure but includes only the doseresponse; C) pcell model incorporating a doseresponse, a description of the anticipation of insulin release related to the first phase mechanisms and a description of potentiation phenomena that modulate the doseresponse (Mariet al. 2002a, 2002b).
Figure 2.7 Illustration of modeling methods: A) the combination of a pcell model (containing Pcell function parameters to be estimated) and a model of Cpeptide kinetics (assumed to be known) provides a mathematical relationship between observed glucose and Cpeptide concentrations. This mathematical relationship is used to estimate the pcell function parameters from glucose and Cpeptide data; B) pcell model incorporating a doseresponse, a description of the anticipation of insulin release related to the first phase mechanisms and a delay between glucose concentration and insulin secretion (Breda et al. 2001; Crettiet al. 2001). The model (Hovorka et al. 1998) has a similar structure but includes only the doseresponse; C) pcell model incorporating a doseresponse, a description of the anticipation of insulin release related to the first phase mechanisms and a description of potentiation phenomena that modulate the doseresponse (Mariet al. 2002a, 2002b).
et al. 2002a, 2002b). But the practical application of these methods requires specialised software and expertise.
pcell function and insulin sensitivity
It is an old observation that Pcell function adapts to insulin resistance in order to maintain glucose tolerance normal (see Kahn 2003; Ahren & Pacini 2004; Mari et al. 2005a for reviews). This observation led to the concept that Pcell function cannot be correctly assessed unless insulin sensitivity is also measured and the Pcell function parameters are adjusted for the degree of insulin resistance. Failure to account for insulin resistance may produce false results, as well illustrated in Bergman et al. (2002).
The most widely used approach to account for insulin resistance is the socalled disposition index, derived from the use of the IVGTT and the minimal model for insulin sensitivity. According to this approach, the index of Pcell function corrected for insulin sensitivity (the disposition index) is the product of AIR and the index of insulin sensitivity S1.
The disposition index approach rests on the assumption that the Pcell adaptation to insulin resistance follows precisely a hyperbolic law. If P and a denote the Pcell function and insulin sensitivity indices respectively, it is assumed that in a normal subject P = k/a, where k is a constant for that individual. Thus, while P differs in different states of insulin resistance (because of Pcell adaptation), Pa, i.e. the disposition index, is a Pcell function index that does not change if the subject's insulin sensitivity changes. Quantitative support for this tenet has come from the analysis of the relationship between the acute insulin response in an IVGTT and the minimal model insulin sensitivity index SI (Kahn et al. 1993) (Figure 2.2F).
This correct principle has however been subject to abuse, particularly because it has been applied indiscriminately to all Pcell function and insulin sensitivity indices without previous testing of the appropriateness of the assumptions. To ensure that this principle is correctly applied (see Mari et al. 2005a for more details) it is necessary that: 1) the insulin sensitivity and Pcell function tests yield indices that are substantially independent, i.e. are not intrinsically correlated. The HOMA indices are intrinsically dependent, but so also may be the indices resulting from the use of the hyperglycaemic clamp for assessment of second phase secretion and insulin sensitivity; 2) it must be verified that there is a relationship between the specific Pcell function and insulin sensitivity indices in a population of control subjects, because otherwise correction for insulin resistance is unjustified; 3) it must be verified that the relationship is precisely the hyperbola of equation P = k/a; otherwise the disposition index calculated as the simple product is not valid. If the relationship is not a hyperbola, alternative formulations can be used (see Mari et al. 2005a for discussion); 4) caution should be used in the interpretation of the relationships between the disposition index and other physiological variables, as these may reflect relationships with insulin sensitivity, a component of the disposition index, rather than with Pcell function.
Comparative evaluation of methods
In contrast to the assessment of insulin sensitivity  for which the euglycemic hyperinsulinemic glucose clamp is commonly reputed the gold standard  for the evaluation of Pcell function no true gold standard exists. This is due to the complexity of Pcell response, which cannot be disclosed by a single test. In addition, the most informative tests are considerably complex and thus have limited applicability. Table 2.1 summarises the characteristics of the tests illustrated in this chapter, while Table 2.2 gives some general suggestions for their practical realisation.
Intravenous standardised tests have the advantage that they do not need normalisation of the secretory response to a variable stimulus, as do the oral tests, which may be difficult or imprecise. Among the intravenous tests, the hyperglycaemic clamp is possibly the most convenient compromise between experimental complexity and accuracy of outcome. The graded glucose infusion test gives a better assessment of the Pcell doseresponse, but is
test 
(icell function characteristics tested 
specific equipment 
insulin sensitivity1 
Cpeptide2 
complexity3 
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