Indirect link models

In the direct-link model, concentration-effect relationships are established without accounting for the intrinsic pharmacodynamic temporal behavior, and the relationships are valid only under the assumption of effect site, prereceptor equilibrium H3. In contrast, indirect-link models are required if there is a temporal dissociation between the time courses of concentration and effect, and the observed delay in the concentration-effect relationship is most likely caused by a functional delay between the concentrations in the plasma and at the effect site. 

When a lag time of E (t) is observed with respect to the c (t) time course, the use of a combined pharmacokinetic-dynamic model, the indirect-link model, is needed to relate the drug concentration c (t) to the receptor site drug concentration y (t) (which cannot be measured directly) and the y (t) to the pharmacological response E (t).1... 

The time symbols t, t denote the temporal dissociation between the time courses of concentration and effect, respectively. For various types of drug administration, the function c(t) is known and therefore analytic solutions for y(t) have been obtained using the integral defined above. Substituting (10.12) into (10.7), we obtain the fundamental equation for the Emax indirect-link model ... 

The basic feature of the indirect-link model is the counterclockwise hysteresis loop that is obtained from plotting the observed values of the effect vs. the observed plasma drug concentration values, Figure 10.2. In other words, the effect is delayed compared to the plasma drug concentration and this is reflected in the effect-concentration state space. 

Figure 10.2 Normalized effect-plasma drug concentration state space for the indirect link model. As time flows (indicated by arrows) a counterclockwise hysteresis loop is formed. The rate constant for drug removal from the effect compartment ky characterizes the temporal delay, that is, the degree of hysteresis.

An example of the indirect-link model after bolus intravenous injection can be seen in Figure 10.3. The arrow indicates the time flow. Each point represents... 

Figure 10.3 Indirect link model with bolus intravenous injection. (A) The classical time profiles of the two variables c(t) (solid line) and E (t) (dashed line) for dose qo = 0.5. (B) A two-dimensional phase space for the concentration c(t) vs. effect E (t) plot using three doses 0.5, 0.75, and 1 (solid, dashed, and dotted lines, respectively).

The standard effect-compartment model, usually characterized as an atypical indirect-link model, also constitutes an example of what we will call a direct-response model in contrast to the indirect-response models. Globally, the standard direct-response models are models in which c(t) affects all dynamic processes only linearly. 

The second perspective might be that of the leader of some large project where chemical analyses are just a side issue, where sample numbers are large and chemical niceties might be completely swamped by, say, biological variability here a statistician will be necessary to make sense of the results in the context of a very complex model. Chemistry is a bit harder to relate to than many other industries in that the measured quantities are often abstract, invisible, and only indirectly linked to what one wants to control. 

Two-compartment PK model the indirect PD response model is a more appropriate approach for modeling the PK/PD of insulin than the effect-compartment link model... 

Lin, S. and Ghien, Y.W., Pharmacokinetic-pharmacodynamic modeling of insulin comparison of indirect pharmacodjmamic response with effect-compartment link models, J. Pharm. Pharmacol, 54, 791-800, 2002. 

More complex integrated PK/PD models are necessary to link and account for a possible temporal dissociation between the plasma concentration and the observed effect. Four basic attributes may be used to characterize PK/PD models First, the link between measured concentration and the pharmacological response mechanism that mediates the observed effect (direct versus indirect link) second, the response mechanism that mediates the observed effect (direct versus indirect response) third, the information used to establish the link between measured concentration and observed effect (hard versus soft link) and, fourth, the time dependency of the involved PD parameters (time variant versus time invariant) (Danhof et al., 1993 Steimer et al., 1993 Aarons, 1999 Lees et al., 2004). The expanded and early use of PK/PD modeling in drug discovery and development is highly beneficial for increasing the success rate of drug discovery and development and will most likely improve the current state of applied therapeutics. 

It is rather obvious that an indirect response mechanism, whatever the detailed processes involved, results in a counterclockwise hysteresis loop for the effect-concentration relationship, Figure 10.2. Here, however, the elaboration of the observed response is usually secondary to a previous time-consuming synthesis or degradation of an endogenous substance(s) or mediator(s). Since both the indirect-link and indirect response models have counterclockwise hysteresis effect-concentration plots, an approach based on the time of the maximum effect has been applied to furosemide data [440] for indirect (link or response) model selection. 

When one looks into the basic functions of the link and indirect response models, it is clear that one of the differences resides in the input functions to the effect and the receptor protein site, respectively. For the link model a linear input operates in contrast to the indirect model, where a nonlinear function operates. For the link model the time is not directly present and the pharmacological time course is exclusively dictated by the pharmacokinetic time, whereas the indirect model has its own time expressed by the differential equation describing the dynamics of the integrated response. 

PK and PD can be linked directly through a measured concentration that is directly linked to an effect site. The direct link model does not work well when there is a temporal relationship between a measured concentration and effect, as when hysteresis is present. When this is the case, an indirect link between the measured concentration and effect must be accounted for in the model. This has been done in... 

Models of varying complexities can be used to describe the placebo effect. Models for drug effect can be semimechanistic (i.e., link model) (20) or mechanistic (i.e., indirect response model) (21). A semimechanistic drug effect model can be expressed as... 

The model ensures a set of simple non-dominated networks (Pareto optimal) while minimizing the numbers of indirect links in the networks. The total number of indirect links is minimized to keep the network sparse (i.e., to keep the network simple). The model also maximizes the percentage difference between clustering coefficients to maintain one of the small world network (SWN) characteristic and minimizes the sum of squared errors to fit the experimental data. One of the constraints is included to ensure the minimum connectivity in all non dominated networks. The other constraint is included to make the L(avgshortestpath) nearly equal to log -if so that SWN characteristic can be ensured. Mathematically, the model (Ganta, 2007) is ... 

ExpressionValueDimension is an abstraction of all classes that can be directly or indirectly linked to ExpressionValues. Subclasses of ExpressionValueDimension are ExpressionValueType, Element, CompositeElement, Sample, Composite-Sample, ImageAnalysis. Such abstraction was introduced to permit easier modeling of Transformations (described below). 

Direct response models are characterized by a direct correlation between the effect site concentration of the drug and the observed effect without time lag. Direct response models can comprise either a directly linked direct response model or an indirectly linked indirect response model. Indirect response models are models for drugs whose mechanism of action consists of either inhibition or stimulation of a physiological process involved in the elucidation of the clinical expression of the observed effect, as discussed in detail in the previous section. 

Mathematical models are the link between what is observed experimentally and what is thought to occur at the molecular level. In physical sciences, such as chemistry, there is a direct correspondence between the experimental observation and the molecular world (i.e., a nuclear magnetic resonance spectrum directly reflects the interaction of hydrogen atoms on a molecule). In pharmacology the observations are much more indirect, leaving a much wider gap between the physical chemistry involved in drug-receptor interaction and what the cell does in response to those interactions (through the cellular veil ). Hence, models become uniquely important. 

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