Understanding the Hysteresis Loop Conundrum in Pharmacokinetic / Pharmacodynamic Relationships

Hysteresis loops are phenomena that sometimes are encountered in the analysis of pharmacokinetic and pharmacodynamic relationships spanning from pre-clinical to clinical studies. When hysteresis occurs it provides insight into the complexity of drug action and disposition that can be encountered. Hysteresis loops suggest that the relationship between drug concentration and the effect being measured is not a simple direct relationship, but may have an inherent time delay and disequilibrium, which may be the result of metabolites, the consequence of changes in pharmacodynamics or the use of a non-specific assay or may involve an indirect relationship. Counter-clockwise hysteresis has been generally defined as the process in which effect can increase with time for a given drug concentration, while in the case of clockwise hysteresis the measured effect decreases with time for a given drug concentration. Hysteresis loops can occur as a consequence of a number of different pharmacokinetic and pharmacodynamic mechanisms including tolerance, distributional delay, feedback regulation, input and output rate changes, agonistic or antagonistic active metabolites, uptake into active site, slow receptor kinetics, delayed or modified activity, time-dependent protein binding and the use of racemic drugs among other factors. In this review, each of these various causes of hysteresis loops are discussed, with incorporation of relevant examples of drugs demonstrating these relationships for illustrative purposes. Furthermore, the effect that pharmaceutical formulation has on the occurrence and potential change in direction of the hysteresis loop, and the major pharmacokinetic / pharmacodynamic modeling approaches utilized to collapse and model hysteresis are detailed. This article is open to POST-PUBLICATION REVIEW. Registered readers (see “For Readers”) may comment by clicking on ABSTRACT on the issue’s contents page. __________________________________________________________________________________________


INTRODUCTION
A central tenet of clinical pharmacotherapeutics is that there often exists a relationship between drug concentration and pharmacological and toxicological effects for drugs. The most common pharmacokinetic-pharmacodynamic (PK-PD) models assume that plasma concentration is in equilibrium and proportional with the effect site (biophase) concentration. In its simplest form a plasma drug concentration versus effect graph demonstrates a direct linear relationship between the two variables where effect is directly proportional to drug concentrations at the active site and this relationship is independent of time [1] (Figure 1a).Where equation 1 is: E is the effect, C is the drug concentration and S is the slope parameter. This linear function model does not predict a maximum pharmacological effect (E max ). The relationship between drug concentration and pharmacological effect more often follows a sigmoidal E max model (Hill equation) (Figure 1b). This simple mathematical relationship is based on receptor theory that defines the drug concentration effect relationship with two parameters E max and EC 50 (the concentration producing 50% of the maximum effect). It allows for differences in the shape of this relationship, where n is the number of molecules combining with each receptor molecule that affects the shape of the curve. The relationship between drug concentration at the receptor and the response is defined using equation 2. _________________________________________ n n n C EC E is the observed effect, E max is the theoretical maximal effect that can be attained, C is the concentration, EC 50 is the C value that produces an effect equivalent to 50% of the theoretical maximal effect and n is a slope factor parameter that determines the steepness of the curve. The time courses of drug effect and concentrations may not be completely superimposable. Time-dependent concentration-effect relationships exist with a time lag present between measurable effect and measurable concentration. In these cases, when pharmacodynamics and drug concentration data are connected in time series at a later point compared with a previous time point there is a discordance in the plasma concentration versus effect relationship with respect to time. Hence, the magnitude of pharmacological effect either increases or decreases at any given plasma drug concentration. The term "hysteresis" has been utilized to describe this time lag. The term "hysteresis" is derived from the Greek husterēsis or husteros meaning 'shortcoming, to come late or to come behind'. Hysteresis is the dependence of a system on both its current and past environments. Figure 1c and d present the graphical evidence of a temporal relationship of dependence between the pharmacological effect and the drug plasma concentration. As the data modeling field in pharmaceutical science examining the concentration versus effect relationships and simulations has grown, there has been some debate regarding the terminology used to describe these phenomena when encountered. It has been suggested that instead of using the term clockwise hysteresis, the moniker "proteresis" should be employed. "Proteresis" is a term also derived from the Greek language with proteros meaning 'former, before or to mark an earlier event'. Similarly, instead of stating that a "counter-clockwise" or "anticlockwise" hysteresis is present it was proposed to simply state the vernacular of 'hysteresis' to avoid redundancy [2]. However, the term 'proteresis' has not become the conventional lexicon and most studies in the literature still utilize the appellatives 'clockwise' or 'counter-clockwise' hysteresis. For consistency and clarity in this review clockwise hysteresis will be used instead of proteresis, and counter-clockwise hysteresis instead of simply hysteresis or anti-clockwise hysteresis.
In the counter-clockwise scenario (Figure 1 c) there is often non-instantaneous distribution of a drug to the effect site (biophase), as the drug appearance is delayed into the pharmacodynamic (PD) effect site at a slower rate than that in which it appears in plasma, this temporal delay in delivery results in a mismatch between declining concentrations and the response [3,4]. When the biophase is not in the central compartment, it exhibits a counter-clockwise hysteresis loop when followed over time (Figure 1c). In this instance, there is a small effect at a given drug concentration; however, after some time has passed the same drug concentration gives rise to a greater measured effect than expected. Thus, the same drug concentration produces two different magnitudes of pharmacological effects depending on the temporal sequence in which the effect is measured. Counterclockwise hysteresis has been generally defined as the process in which effect increases with time for a given drug concentration [5]. These phenomena can be caused by uptake into an active site, cascade activity, active metabolites or sensitization (Table  1) [5].
In the opposite scenario a hysteresis loop may also occur where a clockwise hysteresis loop is evident, for example if tolerance is developed to a drug such as an opioid (Figure 1d). Here it can be seen that the same plasma concentration has a greater effect early on in temporal sequence and that after some time the effect diminishes at the same plasma concentration. In the case of clockwise hysteresis the measured effect decreases with time. These phenomena can be mechanistically induced by active antagonistic metabolites, tolerance, learning effects, or feed-back regulation etc. (Table  1) [5].
An analysis of the pharmaceutical literature using PubMed, EMBASE and Google Scholar searches indicates that there are many plausible mechanistic proposed reasons for explaining the findings of hysteresis loops in drug concentration versus effect plots (Tables 1-3). Situations that may lead to a clockwise hysteresis are the development of tolerance to a drug, antagonistic metabolites that are formed as the drug is metabolized, downregulation of receptors and feedback regulation [6]. Some potential causes for counter-clockwisehysteresis include distribution delay between the plasma and effect site, response delay, sensitization of receptors, the formation and subsequent accumulation of active metabolites through drug metabolism as well as up regulation of receptors after ongoing exposure [6,7]. Once this type of hysteresis loop relationship has been discovered further mathematical modeling, such as the effect compartmental modeling, or one of its many modifications such as indirect modeling, can be applied to the data to take into account lag times, formation of active metabolites or multiple receptor sites in order to mechanistically define and simplify the concentration-effect relationship [1].
As hysteresis loops become readily evident during attempts to correlate the pharmacokinetic (PK) measurement [concentration (C)] of a drug with its measured PD measurement [effect (E)], an accurate determination of the PD measurement is critical. [8] In general, most of the clinical pharmacology study designs include in vivo PK and in vivo PD endpoint measurements; however, in the case of antibiotics or immunosuppressants it is common to have in vivo non-steady state dosing of drug, and the sampled matrix is used to determine in vitro PD effect, which can make the interpretation of hysteresis more straightforward. However, in early stages of drug discovery/development analytical assays are sometimes incapable of differentiating parent drug from its metabolite(s) and therefore would not be able to account for the presence/degree of in vivo pharmacological active metabolite(s) [8]. From the pharmacological and mechanistic point of view, counter-clockwise and clockwise hysteresis loops are a phenomenon that occurs under specific conditions, and the amount, activity and potency of parent and metabolite ratio is a key concept in the development and direction of a determined hysteresis.
Counter-clockwise hysteresis loops in PK-PD relationships could be explained and defined [319] as a consequence of a number of factors illustrated below in Figure 2.
Counter-clockwise Hysteresis 1. Disequilibrium caused by a temporal displacement can occur because of rate limiting steps: Step A: Instantaneous equilibrium between C and effect site concentrations (C e ) is not attained, which results in a temporal displacement (between C and C e ), where C (t) in plasma and C e in the effect site are not identical and C (t) > C e . [8] Step B: The rate of change of C e is much greater than the rates of pharmacological receptor activation/deactivation (R*). For instance the number of activated receptors (R*) is not reflective of C e at time (t), resulting in a temporal displacement (between C e > and R*). [8] Step C: The overall rate of conversion of signal transduction induced by R* to measured E is much less than the rate of change of R* so that E (t) is not equal to R* (t) , resulting in temporal displacement between R* >and E. The concentration (C) of drug binding with a receptor (R) forms a transient (CR) drug-receptor complex and this altered complex is affected by the receptor association and dissociation on and off the receptor. K d is the receptor dissociation rate constant and is equal to the ratio of K off /K on .
2. Transformation of a drug or prodrug into a metabolite(s) with agonist actions (MPCA / MPPCA or MPNCA) are formed from the parent drug (D or E) and are included in the contribution of the combined measurement of E.
Step D: Common Receptor-Common Transduction Mechanism MPCA or MPPCA is a competitive agonist or competitive partial agonist with lower intrinsic activity, for which it occupies the same receptor (R*) as the parent drug concentration [EmaxMPCA or Emax MPPCa is less than Emax(C)] Step E: Separate Receptor-Common Transduction Mechanism. Metabolite with noncompetitive agonist actions (MNCA) occupies a different receptor but the same effect is achieved through a similar mechanism.
Step J: Separate Receptor Separate Transduction Mechanism MNCA is a noncompetitive agonist, for which a different pharmacological receptor (R2) is occupied but the same E achieved through a different mechanism. 3.
Step K: Changes in PD over time. PD has a distinct temporal component. For instance, up regulation of receptor response (R*) (Step K) (sensitization) when the EC50 might reduce over time.
4. Indirect physiological mechanism of drug action with changes in PD over time. A biosensor process leads to a positive modulation in production Kin or a negative change in degradation Kout of the biosignal may occur and be transduced (Step L) into an E. 5. The measurement of C (t) is not specific and analysis should seek to measure the active component.
A. Total (unbound and bound) drug analyzed is being measured over time rather than unbound concentration of drug being measured over time and time-dependent protein binding due to protein binding changes over time are occurring such that unbound pharmacologically active concentration reduces with time. [9] B. Total achiral (all enantiomers) drug concentration is being measured over time rather than enantiospecific concentration of a racemic drug through the use of non-stereospecific assays [10].
Stereospecific pharmacokinetics and pharmacodynamics are occurring leading to 1. Stereospecific disequilibirum 2. Stereospecific metabolism and/or 3. Stereospecific changes in PD occurring. a

) Common Receptor-Common Transduction Mechanism
Stereospecific Competitive Agonist/Partial Agonism. b) Separate Receptors-Common Transduction Mechanisms. One enantiomer with non-competitive agonist actions occupies a different receptor but the same effect is achieved through a similar mechanism. c) Separate Receptor Separate Transduction Mechanism. One enantiomer is a non-competitive agonist, for which a different pharmacological receptor (R2) is occupied but the same E achieved through a different mechanism.
C. Total (parent drug + metabolites) concentration together are being measured by a non-specific analytical assay (i.e. radioimmunoassay, radioactive assay, or high performance liquid chromatographic assay, etc.), which is non-specific for the parent drug such that an assumed E vs. C plot is in actuality an E vs. C+MPCA or E vs. C+MPPCA plot. 5). A specific analytical assay has been developed to measure C, however, an unknown or unidentified agonist metabolite is adding to the E.

Clockwise hysteresis
Clockwise hysteresis loops in PK-PD relationships could be explained and defined as a consequence of a number of factors illustrated below in Figure 3. 1. Disequilibrium caused by a temporal displacement that occurs because of a major rate limiting step: Step A: Disequilibrium occurs when C e temporally precedes C (t) . For instance, the rate of equilibration between arterial plasma concentrations (compartment delivering drug to effect site) and venous plasma concentrations (sampling compartment for drug concentration analysis) is significantly less between arterial concentrations and C e such that C (t) is not equal to Ce (t) , resulting in temporal displacement (between C (t) and C e .) 2. Metabolite(s) with antagonistic actions (MpCAn) are formed.
Step D: Common Receptor-Common Mechanism. MpCAn is a competitive antagonist with no intrinsic activity, for which the same receptor (R*) is occupied as the administered drug concentration but it lacks pharmacological activity [E max(MANTCA)~0 <<<< E max(C) ] so that receptor blockade occurs. [8] Step G: Common Receptor-Common Transduction Mechanism. MPPA is a partial competitive agonist with low intrinsic activity, for which it occupies the same receptor (R*) as the parent drug concentration [E max(MPOa) is less than E max(C) ] Step J: Separate Receptor-Separate Mechanism. MpCA is a non-competitive antagonist (reverse agonist), for which it interacts with a different receptor (R1) than the drug concentration administered and has opposing effects over the measured E (t) .
Step N. Changes in PD over time. PD has a measurable temporal component. For instance, with tolerance down regulation of receptors leads to EC 50 to increase and/or the E max would decrease overtime.
3. Indirect mechanism of drug action with changes in PD over time. A biosensor process leads to a negative modulation of production K in or a positive increase in degradation in K out of the biosignal may occur and be transduced (Step O) into E.
5. The measurement of C (t) is not-specific. A. Total (unbound and bound) drug measured is being measured over time rather than unbound free concentration of drug being measured over time and time-dependent protein binding such as the protein binding changes over time are occurring such that unbound pharmacologically active concentration decreases with time [9].  Factors Influencing Counter-clockwise Hysteresis C p(t) = Plasma parent drug concentration, C e(t) = "Effect site" concentration, R* (t) = Receptor site, E (t) = Effect, MPCA(t) = Metabolite(s) in plasma which are competitive agonists, MeCA(t) = Metabolite(s) in "Effect site" which are competitive agonists, MPNCA(t) = Metabolite(s) in plasma which are competitive agonists /(MPPCA) partial agonists which have noncompetitive agonist action acting on a different receptor BUT same Effect, MeNCA(t) = Metabolite(s) in "Effect site" which are competitive agonists which have non-competitive agonist action acting on a different receptor BUT same Effect, R2 = Alternate receptor site (with same Effect).
B. Total achiral (all enantiomers) drug concentration is being measured over time rather than enantiospecific concentration of a racemic drug through the use of non-stereospecific assays [10].
Stereospecific pharmacokinetics and pharmacodynamics are occurring leading to 1. Stereospecific Disequilibrium, 2. Stereospecific Metabolism and/or 3. Stereospecific Changes in PD occurring.
A. Common Receptor-Common Mechanism Competitive Antagonism, where one enantiomer is active and the other has affinity but no activity. B. Total (parent drug + metabolites) concentration together are being measured by a non-specific analytical assay is being utilized (i.e. radioimmunoassay, radioactive assay, or high performance liquid chromatographic assay, etc.) that is non-specific for the parent drug such that an assumed E vs. C plot is in actuality an E vs. C+MpCAN or E vs. C+MpCA plot. A specific analytical assay has been developed to measure parent compound concentration, however, an unknown or unidentified antagonist metabolite is adding to the E.  The following overview of hysteresis loops aims to provide a comprehensive rather than exhaustive appraisal of the available pharmaceutical and biomedical literature in which hysteresis in either direction has been observed or studied. The goal of this article is to provide the reader with a more comprehensive understanding of the mechanistic reasons underlying why these phenomena can occur, provide examples of which drugs and group of drugs have been reported to exhibit these characteristics (Table 2 and 3), the effect that pharmaceutical formulation may have on the occurrence and change in direction of a hysteresis loop, and the main pharmacokinetic-pharmacodynamic modeling approaches utilized to further understand hysteresis relationships.

COUNTER-CLOCKWISE HYSTERESIS
A counter-clockwise hysteresis loop may signify an increasing pharmacological effect compared with earlier temporal pharmacological effects obtained with the same plasma concentration of drug. There are a variety of examples in the literature that suggest this type of pharmacokinetic / pharmacodynamic relationship as demonstrated in Table 2 [ 11-79, 298-312, 318, and 327-329]. A counter-clockwise hysteresis may mechanistically manifest due to a variety of underlying mechanisms as discussed below.

Distribution Delay into Site of Effect
Counter-clockwise hysteresis loops can occur because of a distribution delay between the systemic drug concentration and the time to reach the effect site. This is the most commonly encountered underlying mechanism responsible for the finding of a counter-clockwise hysteresis loop. Effect-concentration is time-dependent and an indirect link is made between the two variables.
For example, a delay for a drug to be transported from the systemic circulation to its site of action to elicit a response has been reported for ∆-9-tetrahydrocannabinol (THC) after intravenous, oral or smoking administration, and after various intravenous doses [107]. It was observed that counter-clockwise hysteresis was evident after intravenous and smoking administration because THC takes a finite time in order to equilibrate with one of its sites of action (brain). However, it was also observed that after oral administration no hysteresis loop was evident because more time was allowed for penetration into the brain [107]. In the case of intravenous and smoking administration, the time necessary to reach equilibrium was approximately 15 minutes ( Figure 4a). Furthermore, it can be observed that the counter-clockwise hysteresis loop was evident after all the intravenous doses (Figure 4b), indicating that this phenomenon is both dose and route-independent [107]. It can be observed that the location and the protective barriers surrounding the active site, in this case the brain, plays a critical role in the occurrence of hysteresis. As the brain possesses multiple protective barriers such as the blood-brain barrier (BBB) it could be expected that a delay in reaching the site of action would occur. This type of hysteresis was also observed for morphine after subcutaneous administration (14 µmol/kg) to rats with renal failure in which counter-clockwise hysteresis was developed between anti-nociceptive activity and morphine plasma concentrations, which correlated with an equilibrium delay as the consequence of the ability of morphine to cross the BBB [58].
The organic nitrate isosorbide dinitrate (ISDN) exhibited counter-clockwise hysteresis after intravenous infusion (0.133 mg/min for 15 min) or sublingual (5 mg) administration ( Figure 5) [44]. It was observed that the changes in standing systolic pressure were greater during the declining phase than in the ascending phase after IV and sublingual administration [44], this correlated with previous studies after oral administration [108]. The proposed mechanism for this hysteresis was a delay in distribution to the active site in tissue, a delay in saturation at the receptor level because it is a non-instantaneous event, or contribution of vasoactive metabolites [109,110] which are less active than parent drug [44]. However, the main factor responsible appears to be the delay in equilibrium between the plasma and the site of action [44].

Slow Receptor Kinetics
Drug receptor theory states that as drug concentration increases the occupancy of the receptor will increase rapidly at first but then it will progressively decrease as the receptors become occupied, and that the drug concentrations necessary to achieve maximal effect (E max ) can be many fold higher than that necessary to produce a 50% response.
[5]. However, not all drug receptor interactions can be described by an E max model since there are limitations in the type of binding, regulation, type of receptors, and the use of surrogate sample-feasible biological matrices (i.e. blood) instead of the actual receptor binding site [5]. However, besides the limited access of drugs to the site of action the presence of slow receptor kinetics are recognized as one of the main causes of counter-clockwise hysteresis [111]. It has been reported that in the case of anti-psychotic drugs they need to traverse not only the BBB but also the transporters that reside in this barrier [12]. The rate at which drugs bind to the receptor (k on ) and the rate at which it dissociates from a receptor (k off ) determine the kinetics of a drug such as in the case of anti-psychotic drugs and their relationship with the dopamine D 2 receptor [12]. For these types of drugs the k on values show low variability for various drugs, but the k off can vary within a 1000-fold range [112]. This switching movement has been evaluated by positron emission tomography (PET) studies in dopamine receptor occupancy after single oral administration of aripiprazole (2, 5, 10 or 30 mg) to healthy subjects, which reported that a high receptor occupancy was present after administration (lower arm of hysteresis), but low receptor occupancy was observed at later time points post-drug administration (upper arm of hysteresis) [12].
In the case of telmisartan, an angiotensin receptor antagonist, counter-clockwise hysteresis was observed between plasma concentration and angiotensin II response after oral administration (20, 40 or 80 mg) following an angiotensin II challenge [74]. It was determined that the hysteresis could be explained because of the tight binding and subsequent slow dissociation of telmisartan from the receptor (AT 1 ) on the vascular smooth muscle cells [74], which was based on the 3 H-telmisartan binding to rat lung AT 1 receptors and slow dissociation (t 1/2 = 5.9 h) from the binding sites [113]. Furthermore, the slow dissociation from the AT 1 receptor can also contribute to the antagonism of telmisartan [74,114]. Similarly, candesartan cilexetil and losartan after oral dosing exhibited counter-clockwise hysteresis after angiotensin II challenge in healthy subjects, and it was reported that candesartan exhibited a slower off-rate from the AT1 receptor than losartan [22]. However, the extent of insurmountable antagonistic activity [115][116][117] or differences in distributional phenomena could also occur. The slow onset of the inhibitory effect on blood pressure for candesartan [23,[118][119] could result in a longer than expected PD effect based on the plasma concentrations [22]. On the contrary, in the case of irbesartan the pharmacological effects in the renin-angiotensin system (RAS) are related to the blockade of AT 1 receptors by increasing the plasma angiotensin II and plasma renin activity for which an actual clockwise hysteresis was reported [92]. It was reported that the duration of the antagonism of AT1 receptor for irbesartan would be longer than predicted using plasma concentrations [92] as demonstrated after single 150 mg PO for which the antagonism lasted for 2 days, which was much longer than valsartan and losartan [120].

Delayed or Modified Activity
The pharmacological response of a drug is not only bound by the rate of binding to a specific receptor, but can also be related to a progressive series of stochastic events that could cause a modification or delay in pharmacological activity. [63]. For both regular and neutral protamine Hagedorn (NPH) insulin after a single subcutaneous injection of 10 U or 25 U, the time to reach maximum infusion rate of glucose infusion (R max ), namely TR max , occurs at a later time than t max indicating a delay between maximum serum concentrations and the maximum PD effect. This delay was more obvious for regular insulin, and when the serum concentrations were correlated with glucose infusion rate (GIR) values, a counter-clockwise hysteresis loop was observed for both types of insulin. As the difference in delay between regular and NHP insulin is appreciable, the hysteresis loop was greater for regular insulin than NPH insulin [63].
In the case of molsidomine [a prodrug for the formation of nitric oxide (NO)] it first requires biotransformation (rapid hydrolysis) to its active metabolite SIN-1, which downstream will release NO [56]. It is because of this metabolic delay in the formation of NO from SIN-1 that counter-clockwise hysteresis has been reported in the change of diastolic diameter after a single oral dose (4 mg) of molsidomine to coronary artery disease (CAD) patients [121]. These findings correlated with a separate study in which finger pulse curve as a PD effect exhibited counter-clockwise hysteresis after administration of molsidomine (8 mg) to healthy subjects [122].

Active Agonist Metabolite
As the existence of first pass metabolism occurs predominantly in the liver, the route of administration may play a critical role in the appearance of a hysteresis loop. Hysteresis is possible because a drug can be converted to an active metabolite that has a C max and a combined peak activity at a later time point compared to the parent drug [5]. For instance, midazolam exhibited a slower reaction time when administered orally compared to intravenous administration, and when the concentrations were combined (both oral and intravenous routes) a counter-clockwise hysteresis loop was evident. However, when the active metabolite α-hydroxy midazolam was analyzed, their combined concentrations gave similar reaction times regardless of the route of administration [123].
Itraconazole (ITZ) is a chiral oral active triazole anti-fungal agent that has non-linear PK in rats and humans and dose-dependent first pass metabolism [124][125][126][127], and it is also metabolized by CYP3A to the major chiral metabolite hydroxyitraconazole (OH-ITZ) that has similar anti-fungal activity compared to ITZ [125,128]. Counter-clockwise hysteresis was observed between the ITZ and OH-ITZ concentrations entering the liver (expressed as an average of portal venous and aortic concentrations, assuming that the liver receives 25% of total blood flow via the hepatic artery and 75% via the portal vein) after duodenal infusion of ITZ (5 or 40 mg/kg) to chronically catheterized rats [46]. Once the change in hepatic availability (F H ) versus ITZ concentrations were plotted over time, a counter-clockwise hysteresis loop was observed indicating an equilibration delay between ITZ and effect (F H ) or another factor that would control F H such as the production of an active metabolite. The importance of metabolism was evident because of the lack of hysteresis and only a direct hyperbolic relationship between F H and OH-ITZ. This lack of hysteresis indicates that this metabolite or some other factor with a similar time course would be the key regulator of CYP3A inhibition and the hepatic availability (F H ) of ITZ. Similar relationships were obtained at the 40 mg/kg dose [46]. However, although analytical assays were capable of measuring the parent compound and its metabolite, no stereospecific analysis was undertaken to delineate the importance of chirality of this racemic drug and the influence of stereoselective metabolism, which should be considered in the interpretation of the mechanism underlying the hysteresis loop.
The cholinesterase inhibitor eptastigmine was administered to healthy volunteers as a single oral dose (10, 20 or 30 mg), and counter-clockwise hysteresis was observed between plasma eptastigmine concentrations and both red blood cell acetyl-cholinesterase inhibition and plasma butyrylcholinesterase inhibition [32]. It was evident that eptastigmine is more effective and provides a longer duration in inhibition of cholinesterase in RBC (acetyl-cholinesterase) than in plasma (butyrylcholinesterase), which is similar to previous reports in young [129,130] and elderly subjects [131]. However, these previous findings do not correlate with in vitro studies in which it has been reported that eptastigmine is more active on butyrylcholinesterase compared to acetyl-cholinesterase [132], which could be attributed to the formation of active metabolites such as 3'-and 5'-carboxylic acid derivatives and 3'-carboxylic acid-1-demethyl derivative [133]. Thus, the therapeutic drug monitoring should not be performed using the unchanged eptastigmine concentrations [32]. Furthermore, the observed counter-clockwise hysteresis in RBCs indicates that eptastigmine does not develop acute tolerance, which could be explained by the formation of active metabolites but also because eptastigmine slowly dissociates from acetyl-cholinesterase in RBCs [134]. The observed invertible character of the hysteresis loop in plasma butyryl-cholinesterase inhibition s suggests that eptastigmine reaches immediate equilibrium with the enzyme [32]. Furthermore, as presented by Gupta et al.
[8] the potency of parent compound and the agonistic metabolite (MA) was estimated using generated plasma concentration-time and plasma concentration-effect curves. The degree of counterclockwise hysteresis increases as the agonistic metabolite accumulates [as elimination rate of the metabolite (k mo ) decreases and is not formation rate limited]. Furthermore, the degree of hysteresis is also reflective of the potency of the parent compound and agonist metabolite M A since as the ratio of potency parent compound/agonist metabolite decreases in magnitude (potency of agonist metabolite increases), the degree of hysteresis will increase.

Indirect Physiological Response
Often drugs act via an indirect mechanism of action and the pharmacologic effect takes considerable time to become evident as concentrations of drug are decreasing. Response is governed by the stimulation or inhibition of factors which can modulate the response [320]. There are two indirect situations following drug administration where the response measured when related to drug concentrations will produce a counter-clockwise hysteresis. Counter-clockwise hysteresis occurs when input is stimulated (i.e. stimulating the release of an endogenous physiological factor) or the output is inhibited (inhibiting or degrading the release of an endogenous physiological factor). For example stimulation of insulin or prolactin leads to a counter-clockwise hysteresis and the inhibition of anticholinesterase or diuresis leads to a counterclockwise hysteresis [321][322][323][324][325][326][327][328][329][330] Terbutaline is a bronchodilator that increases cyclic AMP this in turn leads to bronchial smooth muscle dilation. Pyridostigmine and other agents inhibit cholinesterase preserving acetylcholine leading to an increase in muscular activity leading to a gain in muscular response. An indirect response can result in counter-clockwise hysteresis [321][322][323].

CLOCKWISE HYSTERESIS
A clockwise hysteresis loop may signify a decreasing pharmacological effect compared with earlier temporal pharmacological effects obtained with the same drug concentration. There are a variety of examples in the literature that suggest this type of pharmacokinetic/ pharmacodynamic relationship as reported in Table 3 [ 80-102, 313-317, and 330]. A clockwise hysteresis may mechanistically manifest due to a variety of underlying mechanisms as discussed below.

Venous vs Arterial Blood Concentrations.
Drug concentration is often measured in venous blood sampling sites and the site of effect equilibrates with arterial concentrations at different rates. When the effect site (i.e. the brain or heart, etc.) concentration at the receptor site (leading to drug effect) equilibrates faster with arterial concentration than forearm venous blood concentration, clockwise hysteresis may occur. In the case of the opioid remifentanil after IV infusion (3 µg/kg/min for 10 min), it was observed that as opioid concentration increases the spectral edge decreases in the form of counter-clockwise hysteresis as a result of an equilibrium delay between arterial remifentanil concentration and the site of action (brain) (Figure 6a) [69] . However, a significant difference in arteriovenous concentrations in healthy subjects was reported and the direction of the hysteresis loop was reversed in venous drug concentrations ( Figure 6b) and it was determined that the venous concentration lag behind the drug effect (clockwise hysteresis) [69]. It was suggested that the arterial drug concentration and effect site reach equilibration faster than the equilibration between arterial and venous concentration [69,[103][104][105][106].

Tolerance (Down Regulation of Receptors)
Tolerance is a time-dependent loss of intrinsic activity that can occur within the time course of a single dose, and is called acute tolerance or tachyphylaxis. In the case of pharmacodynamic tolerance intrinsic responsivity of the receptor diminishes over time. Many drugs present clockwise hysteresis due to tolerance because they present a reduced pharmacological effect at the same concentration than earlier leading to an increased effect [140][141][142][143][144][145]. After oral administration of the benzodiazepine diazepam (0.28 mg/kg) clockwise hysteresis was observed between tracking or digit-symbol substitution impairment and unbound diazepam concentrations relative to receptor occupancy [85]. Acute tolerance to the psychomotor effects of other benzodiazepines like alprazolam [146,147], midazolam [148], and triazolam [149] has been reported. However, it needs to be acknowledged that the actual mechanism of tolerance development to benzodiazepines is poorly understood [81]. There is no consensus delineating the actual mechanism but there are reports that consider that a decrease in binding potential and/or decrease in receptor functionality could explain the appearance of tolerance [150]. However, other mechanisms such as desensitization associated with receptor phosphorylation, uncoupling, and protein internalization or degradation have been proposed [151].
In the case of temazepam (30 mg oral dose), clockwise hysteresis was observed between plasma concentration and sedation score with or without the co-administration of aluminum hydroxide gel (30 mL) in end-stage renal patients (Figure 7) [101]. The hysteresis loops were very similar with or without the co-administration of aluminum hydroxide gel, but the main difference was the presence of the lag time of 1 h when temazepam was administered alone, but it was concluded that aluminum hydroxide gel had no effect on the absorption of temazepam [101]. The clockwise hysteresis was attributed to tolerance [152], which could be a consequence because of a discrepancy between its effective t 1/2 and receptor binding affinity [153]; however, psychological adaptation [154] and functional disturbance [155] can also play a role since sedation has some subjectivity in a b scoring, as a patient can force themselves to stay awake after an entire morning of sleep or the opposite may occur [101]. Interestingly, when temazepam is administered as a racemate a clockwise hysteresis is evident in its sedation. It is likely that stereospecific pharmacokinetics and pharmacodynamics contribute to this relationship [9-10]. Use of a nonstereospecific assay for a chiral drug could lead to an apparent clockwise hysteresis loop where this relationship might not be apparent if the stereospecific concentrations were measured.
In the case of morphine after intragastric administration as a single dose (15 mg/kg) to rats, it was observed that clockwise hysteresis was evident between unconjugated and conjugated morphine concentrations and anti-thermal pain effect. It was proposed that the fast and short-lasting anti-thermal pain effect of unconjugated and conjugated morphine was due to tolerance [59]. Morphine administered as an IV bolus to rats caused less tolerance than IV infusions at two different rates, with tolerance stronger for the higher infusion rate [156]. Similarly when morphine was administered SC (14 µmol/kg) to rats tolerance (clockwise hysteresis) developed between antinociceptive activity and morphine brain concentrations in renal failure-induced rats, which has been suggested to most likely involve a post-opioid receptor transduction mechanism [58,157].
In the case of diltiazem after a single oral dose (120 mg) to healthy subjects, clockwise hysteresis was observed in 4 out of 6 subjects ( Figure 8) [86]. Significant effects on arteriovascular (AV) conduction were observed as expressed by the prolongation of PR interval, and because of the inverse relationship between PR interval and heart rate [158] diltiazem would decrease the heart rate in order to increase the PR interval [86]. Furthermore, acute tolerance has been reported for diltiazem after a single oral dose (180 mg) of sustained-release formulations to healthy subjects based on the observed clockwise hysteresis for PQ and PR interval prolongation [87]. However, previous reports indicated counter-clockwise hysteresis after a single IV administration of diltiazem [159][160][161][162], which could occur because a delay before equilibrium is reached between systemic and site of action concentrations, or the contribution of active metabolites [163]. But also because tolerance [164], arteriovenous differences during sampling [69,165,166], or the presence of inhibitory metabolites with increasing metabolite-to-parent concentration ratio [164]. The arteriovenous equilibrium differences would result in time-dependence over short time intervals, but hysteresis occurred over an extended interval of many hours [86]. The metabolites Ndesmethyldiltiazem and desacetyldiltiazem [167][168][169] are produced and have been reported to have equal or lower hemodynamic effects than parent drug [170,171]; however, it is unknown how active the unbound metabolites compete/react with the receptor active sites in contrast with diltiazem [86]. The proposed tolerance of diltiazem is not caused by compensating physiological mechanisms because a decrease in blood pressure would indirectly increase the heart rate; however, this does not appear to be evident [86].

Feedback Regulation
Mammalian physiology has multiple feedback mechanisms to control various pathophysiological processes such as biochemical, nerve and enzymatic functions [5]. In the case of clockwise hysteresis these negative feedback mechanisms decrease the activity for the same concentration. For instance, when almitrine bismesylate (a respiratory stimulant) was infused (0.5 mg/kg) over 30 minutes to phase II chronic obstructive lung disease (COLD) patients, it was observed that almitrine concentrations increased to a plateau around 500 ng/mL at 30 minutes but rapidly decreased after the infusion was over. Furthermore, oxygen levels (PaO 2 ) reached a maximum around 15 minutes and rapidly declined exhibiting a clockwise hysteresis loop [172]. It has also been reported that in the case of almitrine, the production of a feedback mechanism would be present at higher concentrations than 500 ng/mL [173]. Furthermore, the formation of almitrine inhibitory metabolites has also been proposed as a mechanism, but these have been isolated and synthesized and reported to have poor uptake by the carotid body and have little activity in rats [174]. However, studies in cats have reported that almitrine stimulates the carotid body under the feedback mechanism of the automatic sympathetic ganglioglomerular nerve (GGN) and on a lesser intensity by the contralateral carotid nerve (CCN) [175].

Active Antagonistic Metabolite
The occurrence of antagonistic metabolites is rare in the pharmacokinetic-pharmacodynamic literature. Typically metabolite concentrations are lower than parent drug in humans, and generally metabolites are more polar and less active than parent and may go unnoticed. However, if an antagonistic metabolite is present and with sufficient potency, the parent drug would appear to be less effective and could have a shorter activity [5]. This has been reported for oxyphenylbutazone affecting the elimination of phenylbutazone [176], 5-hydroxypentobarbital and pentobarbital [177], and hydroxydiphenylhydantoin and hydantoin [178]. Clearly, the greater the potency of the antagonistic metabolite relative to the parent compound, and the slower the elimination rate constant relative to the parent drug, the larger the hysteresis loop [8].

Indirect Physiological Response
Often drugs act in an indirect mechanism of action and the pharmacologic effect takes considerable time to become evident and response is governed by the stimulation or inhibition of factors that modulate the response [320]. There are two situations following drug administration where the response measured when related to drug concentrations will produce a clockwise hysteresis. Clockwise hysteresis occurs when input is inhibited or the output is stimulated. For example, acid secretion is inhibited by H 2 -receptor antagonists, and the formation of angiotensin II is inhibited by angiotensin converting enzyme inhibitors and certain anticoagulants such as warfarin that inhibit prothrombin complex activity, methylprednisone and other corticoids that inhibit cortisol. Similarly, diuretics such as furosemide may stimulate secretion of electrolytes into urine, and warfarin inhibits coagulation through prothrombin complex activity. In these situations we would readily expect the appearance of clockwise hysteresis if an effect versus concentration relationship is plotted [321][322][323].

RATE: PHARMACEUTICAL FORMULATION EFFECTS AND THE DIRECTION OF HYSTERESIS LOOPS
As various formulation efforts are designed to provide a desired therapeutic profile, variations in pharmacokinetics and pharmacodynamics are common. Thus, by altering the formulation in which a drug is prepared and thereby altering its input rate may also affect the direction of the hysteresis loop at various steps in the process (Figures 2 and 3). For instance, the loop diuretic bumetanide (1 mg) was administered orally to healthy subjects as tablets and as retarded capsules (containing sustained release granules) [20]. In the case of the tablet formulation a counter-clockwise hysteresis was present and it was determined that this was caused due to the time lag between plasma concentration and diuretic effect because bumetanide acts directly in the renal tubule or because of variations in absorption rate from the GI tract [20]. However, in the case of the bumetanide retarded capsules a clockwise hysteresis was evident in the relationship between urinary excretion rate and urine flow rate. In this case the urine flow rate maximum was achieved before the plasma C max or maximum of urinary excretion in the case of bumetanide [20]. Similar results were obtained with furosemide (another loop diuretic) for which minimal counterclockwise hysteresis was observed for plain tablets (Furix®) while controlled release formulations (Furix Retard® and Lasix Retard®) exhibited clear clockwise hysteresis when the diuretic effect versus furosemide excretion rate were correlated [38]. The occurrence of the counter-clockwise hysteresis indicated that after the administration of the plain tablet there was a minimal delay of the effect related to furosemide excretion rate. However, in the case of the clockwise hysteresis of the two controlled release formulations, tolerance and upregulation of the biosignal of the Na-Cl-K transporter protein may be the main mechanism of action responsible for the effect [38]. These results were parallel to the ones observed in a similar study where a regular tablet and a retarded furosemide capsules were administered to healthy subjects [39]. Therefore, the higher diuretic effect (related to the amount of excreted furosemide) could have been the result of the slower output of drug from the controlled release formulations compared to the plain tablet, indicating that the diuretic response is independent of intrinsic activity and maximum response [135].
Another relevant example includes lisdexamphetamine mesylate (prodrug that gets metabolized to D-amphetamine and L-lysine) and immediate-release (IR) D-amphetamine when it was administered intraperitoneally (IP) to rats at equivalent doses (1.5 mg/kg D-amphetamine base) [28]. Counter-clockwise hysteresis between Damphetamine (from lisdexamfetamine and IR Damphetamine) plasma concentrations and striatal dopamine efflux was evident [28]. The counterclockwise hysteresis was evident because the magnitude of the increase in extracellular dopamine was greater when the concentrations of Damphetamine were decreasing instead of increasing. When the D-amphetamine plasma concentrations were related with the locomotor activity, it was observed that lisdexamfetamine (1.5 or 5.0 mg/kg IP) presented counter-clockwise hysteresis, whereas there was no hysteresis for IR D-amphetamine, demonstrating the important effect of formulation in the PK-PD relationship. Hysteresis was also analyzed between the striatal extraneuronal dopamine concentration and locomotor activity [28]. In this case counter-clockwise hysteresis was evident for lisdexamphetamine; however, in the case of IR D-amphetamine clockwise hysteresis was observed because there was a greater locomotor response during the ascending portion of the dopamine concentration profile [28].
The observed differences in pharmacokinetics and hysteresis between lisdexamphetamine and IR D-amphetamine could be explained because the prodrug lisdexamphetamine is hydrolyzed by red blood cells and by a rate-limited enzymatic reaction to D-amphetamine [136]. This would cause a more sustained gradual release of D-amphetamine compared to the IR formulation causing a more prolonged and sustained efficacy [137][138][139]. The counter-clockwise hysteresis observed was linked with the requirement of D-amphetamine to cross the blood-brain barrier in order to enter the striatal nerve terminals before releasing dopamine to produce locomotor activity (functional outcome) [28]. Therefore, lisdexamphetamine would be a lesser stimulant than an equivalent dose of IR Damphetamine. However, lisdexamphetamine offers a later, more gradual and more sustained increase of striatal dopamine compared to a rapid achieving but rapidly declining effect for IR D-amphetamine [28].
The calcium sensitizer levosimendan when administered as a single dose via different routes: IV infusion (2 mg for 5 min), slow-release tablet (SR, 2 mg), conventional tablet (CT, 2 mg) or conventional capsule (CC, 2 mg) to healthy humans exhibited counter-clockwise hysteresis in all the formulations for the electromechanical systole corrected for heart rate (QS2i) (Figure 9) [49]. It was observed that the SR formulation resulted in lower concentrations and generally weaker effects compared to the other formulations. The observed hysteresis was proposed to occur because of an equilibration delay that reflects the time that the drug is required to distribute from the plasma to its site of action (heart) and the difference between formulations may be due to the physiological barriers and physicochemical properties of the actual formulations that would render different absorption and distribution profiles [49]. Furthermore, counter-clockwise hysteresis in QS2i has also been reported in severe congestive heart failure patients after IV infusion (0.2 μg/kg/min for 6 hours) or oral dose (2 mg), for which also the fact that levosimendan has inotropic and vasodilatory effects could contribute to development of hysteresis [48]. It is evident that in all cases the later effect is higher at 0 ng/mL than before the study commenced. These is exactly what would be predicted if you have a specific assay for the parent drug and have an active metabolite that is not detected or accounted for. More recent literature has demonstrated that levosimendan has two active metabolites OR-1896 and OR-1855 that have mean elimination half-lives of 72.6 and 81.3 hours, respectively, compared to the elimination half-life of parent drug that ranges between 1.1 to 1.4 hours [324].

Theoretical and Practical Considerations: Clockwise and Counter-clockwise Hysteresis and the Importance of Specific Measurement of Concentration
Total (drug concentration + metabolites) together can be measured by a non-specific analytical assay method. This would most often occur with the use of a radioimmunoassay, or by measuring radiolabelled drug in early pharmaceutical development. Using non-specific methods of analysis, drug concentration and concentration of metabolite would be measured simultaneously and could be plotted together versus effect. As presented in simulations by Gupta et al. [8] the potency of the parent compound and the generated agonistic metabolite (MA) were estimated using generated plasma concentration-time and plasma concentration-effect curves. Different derived equations were used to describe parent and MA PK, and plasma concentration-effect profiles using PD models in which the effect was considered a linear function of parent and MA (equation where, P PAR is the pharmacological potency of the parent compound and P MA is the pharmacological potency of the metabolite. A competitive agonist E max model was developed using equation 4  (4) E vs. C plot is in actuality an E vs. C+CMA in these instances ( Figure 10). When the parent compound and metabolite are equipotent no hysteresis was observed (PPAR/PMA ratio = 1 or EC 50 /EC 50 MA) ( Figure 10a) Clockwise hysteresis was present when C > MA in potency (Figure 10 b) and counter-clockwise hysteresis was reflective of MA > C in potency ( Figure 10c). Similar findings were demonstrated and applicable using Equation 3. Therefore, the potency of the metabolite relative to parent compound is the key to the hysteresis and its direction, and it needs to be considered that nonspecific analytical assays such as RIA or achiral analytical methods in PK-PD studies could cause interpretational problems but also warrants the need to identify all of the active metabolites or stereoisomeric forms. What is apparent is that if using a non-specific method of analysis ( Figure 11b), a direct linear relationship could be interpreted between concentration and effect ( Figure 1a, Figure 11b), however employing a specific method of analysis ( Figure 11a) demonstrates the existence of a counter-clockwise hysteresis loop. Likewise, a clockwise hysteresis loop could be incorrectly assigned to a situation where a counter-clockwise hysteresis is occurring (Figure 10a and Figure 11a). Finally, a larger counter-clockwise hysteresis may be evident (Figure 10c) than if a specific method of concentration analysis utilized ( Figure 11a).

Time-Dependent Protein Binding
In a simulation study the time-dependent protein binding can occur as a consequence of a time dependent decrease in protein concentration in serum, by displacement of a metabolite. When pharmacological effect was plotted versus total drug concentration in serum counter-clockwise hysteresis was evident; however, when concentration of free drug in serum, which was correlated with pharmacological effect, was plotted no hysteresis was evident [9]. Time-dependent protein binding that can occur as a consequence of an increase in protein concentration in serum can lead to a decrease in free fraction of drug. When pharmacological effect was plotted versus total drug concentration in serum (i.e. free and bound drug) clockwise hysteresis was evident; however, when concentration of free drug in serum was correlated with pharmacological effect no hysteresis was evident [9]. Despite these theoretical simulations and modeling no examples of studies in the literature demonstrating this phenomenon are apparent to date.

Racemic Drug and Chirality
The utility of using non-stereoselective assay methodology for measuring concentration of a racemic drug can lead to interpretation errors in the concentration versus pharmacological effects correlation and the assignment of a hysteresis loop [10,179]. Indeed as stated by Ariens 30 years ago [297] an analytical assay that does not measure the enantiomers of a racemic drug and attempts to relate total concentrations to effect without stereochemical knowledge is "highly sophisticated scientific nonsense". However, many examples from studies in both Tables 2 and 3 continue to produce this achiral scientific gibberish. After oral administration when enantiomers differ substantially in total body clearance and when an active enantiomer has lower clearance, counterclockwise hysteresis was evident between plasma concentration of total drug and its pharmacological effect. The active enantiomer would constitute a greater proportion of the total concentration as time progresses. In addition, when the V d of an active enantiomer is larger than the inactive enantiomer and a different half-life of the enantiomers ensues, the proportion of the active enantiomer in the total concentration would be higher over time. In the case of zero-order absorption, which could be possible when enantiomers are orally absorbed and transported via carriers, and when the K a of the active enantiomer is less than that of the inactive enantiomer hysteresis was evident. The implications of chirality on pharmacodynamics modeling were also simulated when enantiomers acted as competitive agonists, partial agonists, competitive agonists, enantiomers may also have affinity and activity and intrinsic activity at separate receptors, separate transduction mechanisms or affinity and intrinsic activity at separate receptors but with a common transduction mechanism. When the more active enantiomer had higher clearance or a smaller volume of distribution plots of pharmacological effect versus non-stereospecific plasma concentration produced anti-clockwise hysteresis loops [179].
When a racemic drug's active enantiomer has a higher total body clearance a clockwise hysteresis describes the relationship between total concentration and pharmacological effect as the active enantiomer would be a lower proportion of the total concentration over time. In addition, when the volume of distribution of active enantiomer is smaller than the inactive enantiomer and a different half-life of the enantiomers ensues, the proportion of the active enantiomer in the total concentration would be lower over time. In the case of zero-order absorption which could be possible when enantiomers are orally absorbed and transported via carriers and when the K a of the active enantiomer is greater than that of the inactive enantiomer hysteresis was evident.
Many studies identifying hysteresis using racemic drugs (i.e. Tables 2 and 3) and that have utilized non-stereospecific assays may therefore require further evaluation of their underlying mechanisms. The implications of chirality on pharmacodynamics modeling extended the importance of pharmacodynamics to the hysteresis relationship [179]. There are a variety of possible pharmacological interactions between enantiomers that were evaluated through the use of simulation of the pharmacological effect-time profile and ultimately clockwise hysteresis was also evident. Enantiomers may act as competitive agonists, partial agonists or competitive antagonists. Enantiomers may also have affinity and activity and intrinsic activity at separate receptors, separate transduction mechanisms or affinity and intrinsic activity at separate receptors but with a common transduction mechanism. In cases where the more active enantiomer had higher clearance and a smaller volume of distribution, plots of pharmacological effect versus non-stereospecific plasma concentration produced clockwise hysteresis loops. The plots outlined in Figures 10 and 11 are also applicable to achiral analysis of total enantiomers of a racemate [8,10,179]. Depending on both the pharmacokinetic and pharmacodynamic behaviours of the enantiomers, the less active enantiomer may significantly affect the observed effect and therefore the reliability of any hysteresis loop obtained with the use of suspected achiral concentration data. All of the eight different clockwise hysteresis examples that use racemic drugs in Table 3 may be similar to Figure 10b but could in fact produce counter-clockwise hysteresis if stereospecificity was considered in the analysis.

PHARMACOKINETIC-PHARMACODYNAMIC (PK-PD) MODELING
The general assumption is that drug in the surrogate biological matrix, such as plasma, and the drug at the biophase are at equilibrium [5]. However, this assumption may not be correct because the drug concentrations change as a result of the innate pharmacokinetics of the drug, and the pharmacodynamics could also change independently or in an opposite direction to the drug concentration. Various approaches for simultaneous PK-PD modeling have been explored [180], including compartmental models [181], system dynamics models [182], distributed log analysis [183], or numerical deconvolution [184]. All of these approaches have advantages and disadvantages due to the complexity of the inherited mathematical equations utilized. [185,[325][326].

Effect Compartment Model
The most commonly used PK-PD model is the effect compartment model (Figure 12), which assumes that the active site compartment receives a negligible amount of drug and has a negligible volume [4,164,[186][187][188][189].

Figure 12. Effect Compartment Model
This approach has now been implemented in various modeling software with the so called nonparametric or parametric link model [190].The use of an effect compartment model has been widely used to collapse the hysteresis loop, which is generally performed by linking it to the PK model as it was originally proposed by Segre [191] and by Galeassi et al. [192], and later elaborated and described by Holford and Sheiner [164] and by Sheiner et al. [3]. The effect compartment model has been described by equation 5.
where, C e is the effect compartment concentration, KA is the absorption rate constant, K is the elimination rate constant, K 21 is the transference rate constant from the peripheral to the central compartment, V d /F is the volume of distribution corrected by the bioavailability of the oral dose D, α and β are the hybrid rate constants corresponding to the initial and terminal slope factors, respectively, and K e0 is the constant of the disappearance of the effect [29,164].
The main assumption necessary to make the hysteresis loop collapse is that the effect depends on the drug concentration in an effect compartment rather than in the systemic compartment.
Furthermore, the effect is correlated to C e by the sigmoidal E max PD model using equation 6 [29].
where, E is the observed effect, E max is the theoretical maximal effect that can be attained, C e is the effect-compartment concentration, EC 50 is the C e value that produces an effect equivalent to 50% of the theoretical maximal effect and h is a parameter that determines the steepness of the curve.
The fitting procedures then can be performed using a PK/PD modeling software. The effect compartment model has been applied to the observed counter-clockwise hysteresis between diclofenac blood concentrations and functional index (FI) recovery after oral diclofenac administration to male Wistar rats (Figure 13a) [29]. This hysteresis loop has been previously reported to be due to the formation of active metabolites; however, diclofenac metabolites do not exhibit anti-nociceptive activity [193,194] and local administration of diclofenac causes an antiinflammatory effect [195]. Another proposed mechanism of action was a cascade of physiological events [196] because the anti-nociceptive effect of diclofenac is an indirect response from the inhibition of prostaglandin synthesis [197]; however, it has been reported that diclofenac has a rapid pharmacodynamic effect when administered locally [195] indicating that once it reaches the site of action it has a rapid pharmacological response without delay. Therefore, the lag in anti-nociceptive effect onset occurs because there is a slow equilibrium kinetics between blood concentration in the central and effect compartment [29]. The use of the effect compartment model results in the collapse of the hysteresis loop (Figure 13b), because the derived effect data exhibited good fit as a function of the estimated C e [29].
Tolerance Model: Incorporation of the Hypothetical Non-Competitive Antagonist A tolerance model was developed by Prochet et al. [199] in which a hypothetical non-competitive antagonist is included to represent the factor driving tolerance (Figure 14), and this has been applied to diltiazem  Relationship between the measured blood concentration of diclofenac and the observed anti-nociceptive effect expressed as FI recovery after oral administration of a 10 mg/kg sodium diclofenac dose to rats that were injected with uric acid in the right hind knee. (b) Relationship between the observed anti-nociceptive effect, measured as FI recovery, and calculated effect-compartment Diclofenac concentrations corresponding to PO administration of 0.56, 1, 1.8, 3.2, 5.6 and 10 mg/kg of sodium diclofenac. Reprinted by permission from American Society for Pharmacology and Experimental Therapeutics: The Journal of pharmacology and experimental therapeutics, [29], copyright 1997. where, E is the measured PR interval, E 0 is the baseline PR interval, C is the drug plasma concentration, S is the slope of the linear relationship between effect and concentration in the absence of antagonist, C ant is the concentration of the hypothetical antagonist, and C ant50 is the concentration of hypothetical antagonist resulting in 50% inhibition of effect. Hypothetical antagonist concentration units are those of steady-state drug concentrations [86].
In the case diltiazem after a single oral dose (120 mg) to healthy subjects, the same model was not only applied to parent drug but also to the metabolites N-desmethyldiltiazem and desacetyldiltiazem. The best fit of the various options tested was obtained with the incorporation of the hypothetical non-competitive antagonist rather than the use of any of the metabolite concentration ( Figure 15) [86]. Panel B is the best fit. The 400-800 min times demonstrate a lack of weighting.

Indirect Physiological Response Turnover Model.
As the pharmaceutical industry has diversified from small molecules into administration of proteins and peptides we have seen effects that are more discordant in time and production or degradation of a mediator that is often responsible for drug action. [321][322][323][324][325][326][327][328][329][330] As the mechanism of action of many drugs involves protein synthesis, a drug may affect the net response measured by altering either then K in or K out that will evoke the response measured ( Figure 16). Indirect models allow for a later T max with larger doses of drug. Of course partial inhibition or synergism can be adapted, circadian variation accounted and cascade models could be developed and transduction effects incorporated into these models. As many xenobiotics act indirectly through physiological and biochemical mediators and enzymes there is broad applicability of this approach. Jusko has pioneered the work in turnover model systems by pointing out that four main mechanisms are involved in stimulating or inhibiting production of the biosignal that is measured as the effect, or inhibiting or stimulating its removal [321][322][323].These models can be further extended by adding more transit compartments which are similar to steps in the transduction of the progression of the measured effect.

DISCUSSION
The appearance of hysteresis loops in PK-PD analysis indicates that the relationship between drug concentration and the effect being measured is not direct but has an inherent time delay and disequilibrium. As hysteresis depends on both pharmacokinetics and pharmacodynamics including all the innate factors affecting either of them this has a critical role in the appearance, direction, magnitude and collapse of a hysteresis loop. Some of these factors include the equilibrium / disequilibrium between sampled PK concentration and effect site concentration, rate of pharmacological receptor activation/deactivation, rate of signal transduction at the receptor level, presence of agonist or antagonist active metabolites, upregulation/downregulation of pharmacological response, rate of equilibration between arterial plasma concentrations (compartment delivering drug to effect site) and venous plasma concentrations (sampling compartment for concentration analysis), among others.  In addition, the study design can play a major role since the availability of a specific analytical method plays a critical role in the ability to detect the pharmacologically relevant analyte (parent vs. metabolite, or racemate vs. enantiomer). Also, it is critical to understand the nature of the activity of a metabolite (namely agonist or antagonist) because generally an agonist metabolite would aid in the development of a counter-clockwise hysteresis, while an antagonist metabolite would do the same for clockwise hysteresis loops.
It can be observed that hysteresis loops are present for a wide range of drugs and the mechanism of action (MOA) sometimes overlap between each other. In the case of clockwise hysteresis the most common MOA is tolerance, which is a constant concern in the therapeutic use of benzodiazepines, opioids and CNS drugs. However, tolerance has also been reported for loop diuretics and nitrates. Feedback mechanisms can also play a critical role in hysteresis because they control various physiological processes and it has been reported that they can also decrease the pharmacological effect for the same drug concentrations, which could cause inhibition and/or depletion at the terminal/receptor level [93,201,202]. Another factor to consider is drug and effect location and the protective barriers surrounding the active site such as the brain. As the brain possesses a protective blood-brain barrier (BBB) it would be expected that a delay in reaching the site of action would occur. Another factor that may be neglected in PK-PD interpretation is the potential differences in arteriovenous concentrations of a drug, because arterial blood delivers the drug to the effect site and venous blood is typically the sampled matrix. This for instance has been observed for thiopental in which concentrations were higher in the arterial samples during infusion but became comparable to venous samples after the infusion, and at the time of adding the pharmacodynamics component (EEG frequency reduction by spectral edge analysis), it was observed that the hysteresis loop was evident for arterial but not for venous blood [165].
The current ability to measure receptor binding using positron emission tomography (PET) or an equivalent technology can help us understand better the rate at which drugs bind to the receptor (k on ) and the rate at which it dissociates from a receptor (k off ) to determine the kinetics of drugs such as antipsychotics, in which the k on values show low variability, but the k off can vary within a 1000-fold range [112]. This interplay is critical because with the help of PET the dopamine receptor occupancy after single oral administration of aripiprazole was evaluated. It was observed that high receptor occupancy was present after the administration (lower arm of hysteresis) but low receptor occupancy was observed at later time points post drug administration (upper arm of hysteresis) [12].
The relevance of using different pharmaceutical formulations and routes of administrations has been presented to illustrate the need to be considered in order to achieve the desired therapeutic profile. For instance, bumetanide as a tablet exhibited counterclockwise hysteresis because a time lag between plasma concentration and diuretic effect is evident since the drug acts directly in the renal tubule or because of variations in absorption rate from the GI tract. However, in the case of the retarded capsules a clockwise hysteresis was present because the maximum urine flow rate was achieved before the plasma C max or before the maximum of urinary excretion [20]. Thus, it can be clearly observed that the pharmaceutical formulation may change the pharmacodynamics of a drug. However, the change from one formulation to another does not follow a constant pattern in the direction of occurrence of a hysteresis loop as this is dependent on the drug itself and the actual effect site. Other drug delivery formulation approaches have centered on the modification of the lipophilicity of a drug and having a closer delivery to the site of action in order to try to circumvent biological barriers. For instance, morphine and fentanyl were formulated into a pressurized olfactory delivery (POD) device. Clockwise hysteresis was observed after POD administration of both morphine and fentanyl, but counter-clockwise hysteresis was observed after nasal drops and IP administration of morphine, while no clear hysteresis after nasal drops and IP administration of fentanyl [57]. These observed differences could be attributed to significant differences in hydrophobicity and ability to penetrate the BBB, which not only affected the systemic plasma concentrations but also the delivery to the nasal olfactory epithelium.
With the relevance that hysteresis loops have, various modeling approached have been proposed to collapse hysteresis and allow for adequate PK and PD estimates, and the most commonly used model remains the effect compartment model, which assumes that the active site compartment receives a negligible amount of drug and has negligible volume [4,164,[186][187][188][189]. However, variation of this model such as the tolerance model has also been implemented [102] where a tolerance (use of a linear PD model) and pseudo-tolerance (use of an effect compartment model) PK-PD model were evaluated for different drugs [104,105,203]. Also as proposed by Gupta et al. [8], the potency of the metabolite relative to parent compound is the key to the hysteresis and its direction, and it needs to be considered that non-specific analytical assays such as RIA or achiral analytical methods in PK-PD studies could cause interpretational problems but also warrants the need to identify all of the active metabolites and enantiomers.
Ultimately it can be seen that the presence of a hysteresis loop provides guidance on how to model pharmacokinetic-pharmacodynamic relationship of a particular drug, it allows the pharmaceutical scientist to design studies more appropriately when arteriovenous drug versus venous concentration differences are large and to provide a more rational basis for dosage individualization. A very clear example is the case of piritramide, for which it is recommended that it should be initially administered as an intravenous bolus of at least 5 mg to circumvent its pronounced hysteresis [96].

CONCLUSIONS
The linking of pharmacokinetics and pharmacodynamics is taking on greater relevance because of the necessity to understand the concentration-time profiles of drugs and the need for the ability to determine dosing regimens that will achieve the necessary concentrations for optimal efficacy. These complex relationships have allowed us to be able to detect hysteresis loops and to begin to understand the various mechanisms of action, metabolic and rate limiting steps that cause them. It can be observed that there are various modeling alternatives to collapsing hysteresis loops when determining PK and PD estimates. Special attention needs to be placed on the study design with the various caveats that could arise from the selection of PD estimates as well as the selection of formulation and route of administration. Interdisciplinary approaches are warranted to aid in the further understanding of hysteresis loops to help us develop drugs with a clearer understanding of their complicated pharmacokinetic-pharmacodynamic interactions and behaviours.

ACKNOWLEDGMENTS
The authors would like to acknowledge the assistance of Nicole R. Messinbird BSc(Pharm) for initial collation of some references utilized for this paper and Dr. Matthew R. Wright for his seminal discussions and insights on this topic in the early 1990's.   (1994) (1987) (1988) Table 2 and 3