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In this tutorial, we introduce basic concepts in dynamical systems analysis, such as phase-planes, stability, and bifurcation theory, useful for dissecting the behavior of complex and nonlinear models. A precursor-pool model with positive feedback is used to demonstrate the power of mathematical analysis. This model is nonlinear and exhibits multiple steady states, the stability of which is analyzed. The analysis offers insight into model behavior and suggests useful parameter regions, which simulations alone could not., (© 2016 The Authors CPT: Pharmacometrics & Systems Pharmacology published by Wiley Periodicals, Inc. on behalf of American Society for Clinical Pharmacology and Therapeutics.)

Quantitative and systems pharmacology (QSP) is increasingly being applied in pharmaceutical research and development. One factor critical to the ultimate success of QSP is the establishment of commonly accepted language, technical criteria, and workflows. We propose an integrated workflow that bridges conceptual objectives with underlying technical detail to support the execution, communication, and evaluation of QSP projects., (© 2016 The Authors CPT: Pharmacometrics & Systems Pharmacology published by Wiley Periodicals, Inc. on behalf of American Society for Clinical Pharmacology and Therapeutics.)

Target-mediated drug disposition (TMDD) is the phenomenon in which a drug binds with high affinity to its pharmacological target site (such as a receptor) to such an extent that this affects its pharmacokinetic characteristics.1 The aim of this Tutorial is to provide an introductory guide to the mathematical aspects of TMDD models for pharmaceutical researchers. Examples of Berkeley Madonna2 code for some models discussed in this Tutorial are provided in the Supplementary Materials.

The level of the endocannabinoid anandamide is controlled by fatty acid amide hydrolase (FAAH). In 2011, PF-04457845, an irreversible inhibitor of FAAH, was progressed to phase II clinical trials for osteoarthritic pain. This article discusses a prospective, integrated systems pharmacology model evaluation of FAAH as a target for pain in humans, using physiologically based pharmacokinetic and systems biology approaches. The model integrated physiological compartments; endocannabinoid production, degradation, and disposition data; PF-04457845 pharmacokinetics and pharmacodynamics, and cannabinoid receptor CB1-binding kinetics. The modeling identified clear gaps in our understanding and highlighted key risks going forward, in particular relating to whether methods are in place to demonstrate target engagement and pharmacological effect. The value of this modeling exercise will be discussed in detail and in the context of the clinical phase II data, together with recommendations to enable optimal future evaluation of FAAH inhibitors.CPT: Pharmacometrics Systems Pharmacology (2014) 3, e91; doi:10.1038/psp.2013.72; published online 15 January 2014.

In addition to methodological Tutorials,(1) CPT:PSP has recently started to publish software Tutorials.(2,3) Our readership and authors may be wondering what kind of format or product is expected, and the review of submissions we have already received prompted several discussions within the PSP Editorial Team. This editorial reflects on these discussions and summarizes their salient points. It aims at providing some details about the current vision of CPT:PSP for software tutorial articles. In addition, it brings some clarity on the topic of what role commercial software tutorials can have in CPT:PSP and how CPT:PSP tutorials differ from publications which describe the software itself, as those which can be found in other computer science journals. Finally, the discussion includes reproducibility considerations and the general use of commercial and noncommercial software in CPT:PSP publications. We hope our thoughts, and especially a stated requirement to publish user input to the software to aid in reproducibility, will help in guiding our authors and will stimulate healthy debate among our readers about the evolving nature of our science, how it can be facilitated using software and associated databases as a conduit, and what role this journal can play in fostering both the best modeling and simulation practices and the best scientific approaches to computational modeling, to bring the advantages of modeling and simulation to all regular practitioners, and not to just a (self) selected few.

Zileuton, a 5-lipoxygenase (5LO) inhibitor, displays complex pharmaokinetic (PK)-pharmacodynamic (PD) behavior. Available clinical data indicate a lack of dose-bronchodilatory response during initial treatment, with a dose response developing after ~1-2 weeks. We developed a quantitative systems pharmacology (QSP) model to understand the mechanism behind this phenomenon. The model described the release, maturation, and trafficking of eosinophils into the airways, leukotriene synthesis by the 5LO enzyme, leukotriene signaling and bronchodilation, and the PK of zileuton. The model provided a plausible explanation for the two-phase bronchodilatory effect of zileuton-the short-term bronchodilation was due to leukotriene inhibition and the long-term bronchodilation was due to inflammatory cell infiltration blockade. The model also indicated that the theoretical maximum bronchodilation of both 5LO inhibition and leukotriene receptor blockade is likely similar. QSP modeling provided interesting insights into the effects of leukotriene modulation.CPT: Pharmacometrics & Systems Pharmacology (2013) 2, e74; doi:10.1038/psp.2013.49; advance online publication 11 September 2013.

Endometriosis is a gynecological condition resulting from proliferation of endometrial-like tissue outside the endometrial cavity. Estrogen suppression therapies, mediated through gonadotropin-releasing hormone (GnRH) modulation, decrease endometriotic implants and diminish associated pain albeit at the expense of bone mineral density (BMD) loss. Our goal was to provide model-based guidance for GnRH-modulating clinical programs intended for endometriosis management. This included developing an estrogen suppression target expected to provide symptomatic relief with minimal BMD loss and to evaluate end points and study durations supportive of efficient development decisions. An existing multiscale model of calcium and bone was adapted to include systematic estrogen pharmacologic effects to describe estrogen concentration-related effects on BMD. A logistic regression fit to patient-level data from three clinical GnRH agonist (nafarelin) studies described the relationship of estrogen with endometrial-related pain. Targeting estradiol between 20 and 40 pg/ml was predicted to provide efficacious endometrial pain response while minimizing BMD effects.CPT: Pharmacometrics & Systems Pharmacology (2012) 1, e11; doi:10.1038/psp.2012.10; advance online publication 17 October 2012.

Agonists for the 5-hydroxytryptamine (HT)(1A) receptor induce a hypothermic response that is believed to occur by lowering of the body's set-point temperature. We have developed a physiological model that can be used to predict the complex time course of the hypothermic response after administration of 5-HT(1A) agonists to rats. In the model, 5-HT(1A) agonists exert their effect by changing heat loss through a control mechanism with a thermostat signal that is proportional to the difference between measured and set-point temperature. Agonists exert their effect in a direct concentration-dependent manner, with saturation occurring at higher concentrations. On the basis of simulations, it is shown that, depending on the concentration and the intrinsic efficacy of a 5-HT(1A) agonist, the model shows oscillatory behavior. The model was successfully applied to characterize the complex hypothermic response profiles after administration of the reference 5-HT(1A) agonists R-8-hydroxy-2-(di-n-propylamino)tetralin (R-8-OH-DPAT) and S-8-OH-DPAT. This analysis revealed that the observed difference in effect vs. time profile for these two reference agonists could be explained by a difference in in vivo intrinsic efficacy.

The aim of this study was to evaluate whether the operational model of agonism can yield independent estimates of agonist affinity (pK(A)) and efficacy (log tau) when Furchgott's method of irreversible receptor inactivation is employed. For this purpose, the interaction between noradrenaline and phenoxybenzamine was studied in rat small mesenteric artery using a paired-curve design. Phenoxybenzamine pretreatment produced a significant rightward shift and depression of the upper asymptote of the noradrenaline concentration-effect (E/[A]) curve. Although the operational model of agonism appeared to provide an adequate fit of the individual E/[A] curves, a highly significant correlation was found between the estimates of pK(A )and log tau (r = -0.80, p < 0.0001), inconsistent with the assumption that affinity and efficacy are independent parameters (best line fit: pK(A) = -0.96 x log tau + 6.75). The pK(A) and log tau estimates were not correlated with either the pEC50s of the control curves or upper asymptotes of the phenoxybenzamine-treated curves. Simulations showed that the correlation between affinity and efficacy can be explained by the effect on the outcome of the analysis of random errors in the response measurements. Therefore, although in theory the operational model of agonism should provide independent estimates of agonist affinity and efficacy, this is unlikely to be the case with experimental data.

We have developed a fitting procedure, based on nonlinear mixed effect modelling and original work by Richards (1959, J Exp Botany 10, 290-300), to describe the degree of asymmetry of concentration-effect E/[A] curves and analysed the shape of E/[A] curves obtained with alpha1-adrenoceptor agonists in rat aorta. The four-parameter Richards model provided a significantly better fit of the data than the standard logistic/Hill model for all ligands investigated, which implies that E/[A] curves were asymmetrical. With the exception of ST 587, the asymmetry parameter (delta) tended toward zero and the Richards model could be replaced without significant loss of goodness-of-fit by the three-parameter, asymmetrical Gompertz model. The alpha1-adrenoceptor antagonist, prazosin (10 nM), had no effect on the asymmetry of the noradrenaline E/[A] curve but significantly increased the slope at the point of inflection. In contrast, pretreatment with the irreversible antagonist, phenoxybenzamine (60 nM), produced a shift of the delta estimate for noradrenaline from zero to unity, indicating a change from an asymmetrical to a symmetrical curve. Therefore, detailed statistical analysis of E/[A] curve asymmetry demonstrates that alpha1-adrenoceptors in rat aorta do not operate as a homogenous one-receptor-one-transducer system. This conclusion could not have been reached by either an analysis with the standard logistic/Hill model or visual inspection of experimental data. Overall, the curve-fitting analysis developed in this study provides a quantitative and sensitive measure of asymmetry and a novel method for the objective discrimination of agonist action on the basis of curve shape. The method is generally applicable to other pharmacological assays and provides a new tool in receptor classification studies.

Recently, Calderone and Martinotti (J Pharmacol Toxicol Meth 40:57-62, 1998) presented a simple equation for the estimation of agonist dissociation equilibrium constants from data obtained from receptor inactivation experiments using an irreversible, competitive antagonist. In the present paper, however, it is demonstrated that application of this equation will result in significant over- and underestimation of agonist affinity in the case of flat and steep concentration-effect curves, respectively. Therefore, accurate estimation of agonist affinity using irreversible receptor inactivation requires detailed curve-shape information from both control and antagonist-treated tissues.

1. The exact nature of the receptor subtype(s) involved in the action of arg-vasopressin (AVP) on the rat aorta and small mesenteric artery (SMA) is controversial. Therefore, we have studied the effects of the selective V1A receptor antagonists, OPC 21268 and SR 49059, and the oxytocin (OT) receptor antagonist, atosiban, on the AVP- and OT-induced contractions of the two vessels. 2. AVP and OT displayed similar intrinsic activities in the rat aorta and SMA, but AVP was approximately 130 fold and approximately 500 fold more potent than OT, respectively. In the rat aorta, Hill slopes (nH) were similar for OT and AVP. However, in rat SMA, the OT concentration-effect (E/[A]) curve was significantly steeper than the AVP E/[A] curve (nH, = 3.3+/-0.20, 2.3+/-0.15; P<0.001). 3. In the aorta OPC 21268, SR 49059 and atosiban competitively antagonized the AVP and OT E/[A] curves. Except for atosiban and SR 49059 against AVP, competitive antagonism was also observed in the SMA. Atosiban caused concentration-dependent steepening of the AVP E/[A] curve, whereas SR 49059 decreased the upper asymptote. 4. Schild analysis yielded affinities indicative of V1A receptor involvement in both vessels: pKB/ pA2=9.20 9.48, 7.56 7.71 and 6.19 6.48 for SR 49059, OPC 21268 and atosiban, respectively. 5. Neither AVP nor OT relaxed U46619 pre-contracted aorta or SMA in the presence of SR 49059, suggesting no interference of a vasodilatory component. 6. Despite predominant involvement of V1A receptors in both vessels, the different Hill slopes of AVP and OT E/[A] curves as well as the steepening of the AVP E/[A] curves by atosiban are indicative of receptor heterogeneity in the rat SMA.

Recently, Zernig and colleagues (1996) (J Pharmacol Toxicol Meth 35: 223-237) suggested that for the estimation of agonist affinity and efficacy, the method of simultaneously fitting of concentration-effect curves from control and irreversible antagonist-treated tissues to the operational model of agonism is superior to other analytical approaches. In the present study, we have evaluated the limitations of this simultaneous curve fitting method. Simulation studies showed that this method can be only employed with confidence when the upper asymptotes of the control curves display minimal variation between tissues, which makes its practical utility rather limited. The unreliability of the simultaneous fitting procedure was further underscored with the analysis of experimental data obtained from the interaction between noradrenaline and phenoxybenzamine in rat isolated aorta. The lack of robustness of the parameter estimates showed that under standard experimental conditions the outcomes of simultaneous model fitting are highly dependent on between-tissue variations of the upper asymptotes of the control curves and, therefore, may be unreliable. Therefore, whenever possible, a multiple curve design should be adopted, in which control and treated curves are obtained in one tissue and provide enough information for an independent estimation of affinity and efficacy that is free of intertissue differences.

1. In this study, we examined the interaction between noradrenaline (NA) and phenylephrine (PE) with seven antagonists (prazosin, tamsulosin, phentolamine, WB-4101, 5-methylurapidil, spiperone and HV 723) in an attempt to characterize the alpha 1-adrenoceptor population of the rat isolated small mesenteric artery (SMA) preparation. 2. Six of the seven antagonists investigated produced concentration-dependent, parallel, rightward shift of the NA concentration-effect (E/[A]) curves. The exception was tamsulosin, which produced significant decrease of the upper asymptote. In the case of 5-methylurapidil and HV723, the Schild plot slope parameters were not significantly different from unity over the range of concentrations used. However, the Schild plot slopes obtained for the other antagonists were all significantly greater than unity, inconsistent with expectations for simple competitive antagonism. 3. HV723, prazosin and tamsulosin were also tested using PE as an agonist. All three antagonists produced concentration-dependent, parallel, rightward shifts of the PE curves and Schild analysis yielded slope parameters not significantly different from unity. The pKB estimates obtained for tamsulosin and prazosin were not significantly different from the pA2 values obtained when NA was used as agonist. In the case of HV723, the 95% confidence intervals for the pKB values yielded with NA and PE did not overlap (pKB = 8.80-9.13 and 8.15-8.77 for NA and PE, respectively). 4. In the absence of evidence to indicate that the steep Schild plots were due to failure to satisfy the basic criteria for quantitative analysis in a one-receptor system, we considered the possibility that the complexity was caused by an action of NA at inhibitory D1 receptors. The selective D1 receptor antagonists, SCH-23390 (10 nM), had no significant effect on the NA E/[A] control curve, but the apparent potency of 100 nM prazosin was reduced by approximately 3.5 fold. 5. This study indicates that the steep Schild plots obtained from the interaction between NA and alpha 1-adrenoceptor antagonists were due to the simultaneous activation of inhibitory D1 receptors by NA. Notwithstanding this complexity, our explanatory model of the system (see Appendix) suggests that the antagonist affinity values estimated in the absence of D1 receptor block were not significantly affected by this other action of NA. The low affinity estimate obtained for prazosin suggests that the pharmacologically-defined alpha IL-subtype operates in the SMA.

1. In this study, the effect of seven alpha 1-adrenoceptor antagonists (tamsulosin, phentolamine, prazosin, WB-4101, 5-methylurapidil, spiperone and HV723) have been examined on the contractile response to noradrenaline (NA) and phenylephrine (PE) in rat isolated aorta. 2. NA and PE, when administered using a cumulative dosing schedule, both produced concentration-dependent contraction of aortic rings. It was possible to fit the individual concentration-effect (E/[A]) curve data to the Hill equation to provide estimates of the curve midpoint location (p[A]50 = 7.74 +/- 0.10 and 7.14 +/- 0.18), midpoint slope (nH = 0.82 +/- 0.03 and 0.99 +/- 0.10) and upper asymptote (alpha = 3.2 +/- 0.3 and 3.1 +/- 0.2 g) parameters for NA and PE, respectively. However, the Hill equation provided a better fit to the E/[A] curve data obtained with another contractile agent, 5-hydroxytryptamine (5-HT) (p[A50] = 6.09 +/- 0.08, nH = 1.49 +/- 0.09, alpha = 2.6 +/- 0.3 g), as judged by calculation of the mean sum of squares of the differences between the observed and predicted values. 3. All of the antagonists investigated produced concentration-dependent inhibition of the contractile responses of the aorta to NA and PE. Although no significant effects on the upper asymptotes of the E/[A] curves of any of the antagonists tested were detected, only tamsulosin and 5-methylurapidil did not have a significant effect on the slope (nH) of the NA and PE E/[A] curves. The other antagonists produced significant steepening of the curves obtained with NA and/or PE. 4. Notwithstanding the fact that one of the basic criteria for simple competitive antagonism at a single receptor class was not always satisfied, the individual log [A]50 values estimated in the absence and presence of antagonist within each experiment were fitted to the competitive model. The Schild plot slope parameters for the antagonism of NA and PE by phentolamine and HV723 were found to be significantly less than unity. The Schild plot slope parameters for the other antagonists were not significantly different from unity. 5. In the absence of evidence to suggest that the deviations from simple competitive antagonism were due to failure to satisfy basic experimental conditions for quantitative analysis, an attempt was made to see whether the data could be accounted for by an existing two-receptor model (Furchgott, 1981). The goodness-of-fit obtained with the two-receptor model was significantly better than that obtained with the one-receptor model. Furthermore, with the exception of the data obtained with phentolamine, the pKB estimates for the two receptors were independent of whether NA or PE was used as agonist. 6. To determine which alpha 1-adrenoceptor subtypes may be associated with those defined by the two receptor model, the mean pKB estimates obtained from the two-receptor model fit were compared with affinities measured by Laz et al. (1994) for rat cloned alpha 1-adrenoceptor subtypes expressed in COS-7 cells. The sum of squared differences of the data points from the line of identity was smallest for both pKB1 and pKB2 in the case of the alpha 1a/d-adrenoceptor (now referred to as alpha 1d-adrenoceptor; Hieble et al., 1995). Therefore, the complexity exposed in this study may be due to the expression of closely-related forms of the alpha 1d-adrenoceptor. However, relatively good matches were also found between pKB1 and alpha 1c and between pKB2 and alpha 1b. Therefore, on the basis of these data, it is not possible to rule out the involvement of all three alpha 1-adrenoceptors. The conflicting reports concerning the characteristics of the alpha 1-adrenoceptor population mediating contraction of the rat aorta may, at least in part, be due to the lack of highly selective ligands and to between-assay variation in the expression of multiple alpha 1-adrenoceptors.

1. Previously, we reported that noradrenaline (NA), in addition to its alpha 1-adrenoceptor-mediated contractile effect, may relax the rat small mesenteric artery (SMA) in order to account for steep Schild plots obtained with compounds classified as alpha 1-adrenoceptor antagonists. In this study, a relaxant action of NA has been exposed in the rat isolated, endothelium-denuded SMA precontracted by the thromboxane A2-mimetic, U46619. 2. NA, but not the selective alpha 2-adrenoceptor agonist, UK14304, produced concentration-dependent contraction of the SMA (pEC50 = 5.7 +/- 0.1). After precontraction with 0.1 microM U46619, 10 nM-30 microM NA produced a further contraction (pEC50 = 6.1 +/- 0.2), while higher concentrations of NA produced small, but significant, relaxant responses. 3. In the presence of 1 microM prazosin, 0.1-30 microM NA produced concentration dependent relaxation (pIC50 = 5.9 +/- 0.1) after precontraction with 0.1 microM U46619. The NA relaxation concentration-effect curve was completely inhibited by 1 microM of the beta 1/beta 2-adrenoceptor antagonist, timolol. However, when the concentration of prazosin was increased by 10 fold (10 microM), NA once again produced concentration-dependent relaxation (pIC50 = 4.5 +/- 0.2). This relaxation concentration-effect curve was not blocked by a 10 fold higher concentration of timolol (10 microM), nor by the presence of idazoxan (10 microM), cyanopindolol (10 microM), NG-nitro-L-arginine methyl ester (L-NAME, 100 microM), indomethacin (10 microM) or sulpiride (1 microM). However, haloperidol (10 microM) and (+/-)-SCH-23390 (10 nM) produced significant inhibition of the relaxation, suggesting the involvement of dopamine D1 receptors. 4. Dopamine also produced concentration-dependent relaxation following U46619 precontraction (pIC50 = 5.4 +/- 0.1) which was significantly inhibited by haloperidol and (+)-SCH-23390. Pretreatment with 10 microM phenoxybenzamine for 60 min produced a significant inhibition of the dopamine and NA relaxation curves and application of the operational model of agonism yielded estimates of the affinity (pKA = 5.3 +/- 0.2 and 4.4 +/- 0.2) and efficacy (log gamma = 0.06 +/- 0.11 and 0.01 +/- 0.10) for dopamine and NA, respectively, at D1 receptors. 5. HV723 (0.1 and 1 microM), a ligand that yielded a Schild plot slope parameter of unity as an antagonist of NA in the contractile assay, produced concentration-dependent inhibition of the NA-mediated relaxation (pA2 approximately 8). 6. The results of this study indicate that NA can activate D1 receptors mediating relaxation in the rat SMA at concentrations which were encountered in our previous receptor classification experiments using competitive alpha 1-adrenoceptor antagonists.