Mechanistic Model for Blood Pressure and ... - Wiley Online Library

Basic & Clinical Pharmacology & Toxicology, 2018, 122, 139–148

Doi: 10.1111/bcpt.12856

Mechanistic Model for Blood Pressure and Heart Rate Changes Produced by Telmisartan in Human Beings Dongwoo Chae1,2, Mijeong Son1,2, Yukyung Kim1,2, Hankil Son1,2 and Kyungsoo Park1,2 1

Department of Pharmacology, Yonsei University College of Medicine, Seoul, Korea and 2Brain Korea 21 Plus Project for Medical Science, Yonsei University, Seoul, Korea (Received 20 October 2016; Accepted 17 July 2017) Abstract: Telmisartan, an angiotensin receptor blocker (ARB), is indicated for the treatment of essential hypertension. This study aimed to develop a mechanistic model of telmisartan drug effect in human beings using non-invasive markers. Data were acquired from a previous study where telmisartan 80 mg was given once daily for 6 days. Systolic (SBP) and diastolic blood pressure (DBP) and heart rate (HR) were measured before dosing for days 1–5 and serially after the last dose. Mean arterial pressure (MAP) and pulse pressure (PP) were calculated from SBP and DBP. Relationships between MAP, PP, HR and total peripheral resistance (TPR) were developed. Circadian variation was incorporated into PP and HR, and TPR was assumed to adjust itself in response to changes in PP and HR based on baroreflex mechanism. Drug effects were then described as lowering the set point of MAP through TPR with a physiological feedback effect stimulating HR and PP. Drug concentrations were described by a two-compartment disposition model with first-order absorption and lag time, and first-order elimination. Circadian variation was described by cosine functions, having periods of 12 and 24 hr. A log-linear model was used to describe drug effect, with estimated drug effect parameter of 0.051/hr. Estimated fractional turnover rate of PP, HR and TPR was 11.2 hr. The model successfully described the time courses of these cardiovascular variables. This work demonstrated the feasibility of using non-invasive cardiovascular measurements to derive a mechanistic model for telmisartan in human beings. The model may be suitable for other ARBs.

Telmisartan is an antagonist of the angiotensin II type 1 (AT1) receptor that is used for the treatment of hypertension. It selectively inhibits stimulation of the AT1 receptor by angiotensin II. It has a high volume of distribution and is eliminated mainly by the kidneys which results in a long terminal elimination half-life. The compound is not metabolized by cytochrome P450 isoenzymes and has a low risk for P450based drug interactions [1,2]. The action of telmisartan is known to lead to inhibition of aldosterone secretion and reduced vasoconstriction. The time course of its effect involves both rapid and slow responses. The rapid response is mainly due to antagonizing angiotensin II vasoconstriction. The slow response is due to antagonizing angiotensin-mediated aldosterone secretion which leads to a natriuretic effect, followed by decreased plasma volume and consequent decrease in cardiac stroke volume. Both the rapid and slow effects contribute to lowering of blood pressure [1]. A population pharmacokinetic (PK) model for telmisartan was first published in 2003 [3] and reported that a two-compartment disposition model with first-order absorption and elimination described the data. A PKPD model of telmisartan has been published [4] in spontaneously hypertensive rats. To the authors’ knowledge, no PKPD model of telmisartan has been published in human beings. Previous PKPD models used either an empirical modelling approach whereby telmisartan concentration was linked to blood pressure or heart rate (HR) without

considering the physiological mechanism [4,5] or mechanistic models that required measurements of cardiac output and total peripheral resistance (TPR) [6,7]. Because of the complexity of interactions among different cardiovascular components, empirical PKPD models may lack predictive utility under different disease and dosing conditions. At the same time, invasive monitoring of cardiovascular indices in human beings is often not feasible so the development of model using cardiovascular measurements which are simple to obtain would be desirable. In this study, a physiological PKPD model with consideration of interactions between blood pressure, pulse pressure (PP), HR and TPR is proposed. In particular, the proposed model uses a concept of a set point, a reference value of MAP that can be lowered by a drug through TPR and assumes that a difference between current MAP and the set point acts as the driving force of exerting negative feedback to PP and HR. This idea of the set point being lowered by a drug is found in the literature in the case of losartan which has the same angiotensin II AT1 receptor-blocking mechanism as telmisartan [8] and enalapril which is an angiotensin-converting enzyme inhibitor [9]. The objective of the current report was to improve upon empirical models to achieve better predictive capabilities using non-invasive markers with known physiological and drug action mechanisms, which can be readily applicable in real clinical situations.

Materials and Methods Author for correspondence: Kyungsoo Park, Department of Pharmacology, Yonsei University College of Medicine, 50-1 Yonsei-ro, Seodaemun-gu, Seoul 03722, South Korea (e-mail [email protected]).

Data. The data were acquired from a PK interaction study between rosuvastatin and telmisartan using a randomized, open-label, multiple-

© 2017 Nordic Association for the Publication of BCPT (former Nordic Pharmacological Society)

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dose, 2 9 2 crossover design conducted in healthy male volunteers in 2012 at Severance Hospital, Seoul, Korea [10]. In that study, rosuvastatin 20 mg (Crestor from Astrazeneca, Cambridge, England) and Telmisartan 80 mg (Micardis from Boehringer Ingelheim, Germany) were used. Each drug was administered once daily for 6 days either alone or in combination in a crossover fashion. Blood samples for concentration measurements were taken before the first dose and the fifth dose and at 0, 0.25, 0.5, 0.75, 1, 1.5, 2, 2.5, 3, 4, 5, 6, 8, 12, 16, 24, 48 and 72 hr after the last dose. Cardiovascular observations consisting of systolic and diastolic blood pressure and HR were measured using digital blood pressure cuffs before dosing from the first to the fifth dose and at 0, 2, 4, 8, 12, 24, 48 and 72 hr after the last dose. Administration of the last dose and data collection up to 24 hr after the last dose were conducted while subjects were hospitalized. For 20 subjects who completed treatment with telmisartan alone, only PKPD data up to 24 hr after the last dose were included in this analysis because blood pressure measurements after 24 hr were obtained in the outpatient clinic after discharge, and thus, the measurement conditions were different. Subject demographics are reported in table 1. Models. All model parameters were estimated using NONMEM software (ICON, Dublin, Ireland) version 7.3. Estimation used ADVAN6, FOCE with interaction, NSIG = 3 and TOL = 3. PsN version 4.2 was used for summarizing data, stepwise covariate model building and performing visual predictive checks (VPCs). R (RStudio) and MATLAB R2013a were used for additional data exploration, manipulation and simulation. PK model. Using one- and two-compartment models for drug disposition with first-order elimination, different absorption models were tried: first-order absorption, zero-order absorption, combined zero- and first-order absorption, transit compartment absorption and two parallel first-order absorption [11]. Weight was incorporated into disposition parameters using theorybased allometric relationships [12] standardized to 70 kg [13] as shown below: CL = POPCL ðWT=70Þ3=4

ð1Þ

V = POPV ðWT=70Þ

ð2Þ

POPCL and POPV are the population standard values. CL stands for clearance and V for central volume of distribution in subjects with Table 1. Demographic characteristics of the study subjects. Characteristic Age (y) Mean (S.D.) Range Weight (kg) Mean (S.D.) Range Height (cm) Mean (S.D.) Range Smoking (no.) Smokers Non-smokers Alcohol use (no.) Yes No Caffeine use (no.) Yes No

Subjects (N = 20) 26.8 (4.00) 22–39 73.13 (7.09) 62.4–86 175.0 (5.3) 161.8–185.4 6 14 14 6 13 7

the same weight. These allometric relationships were also applied to intercompartmental clearance and peripheral volume of distribution of the two-compartment model. Exponential error was assumed for interindividual variability, with correlation between CL and V being allowed, and proportional error model was used for residual variability. Then, model selection was carried out based on Akaike information criterion (AIC) and internal evaluation using a VPC.

PD models. Analysis variables. Mean arterial pressure (MAP), HR and PP were used as analysis variables to characterize blood pressure changes after telmisartan administration, taking into account feedback mechanisms of the cardiovascular system as described below. Model for MAP. Mean arterial pressure was calculated from SBP and DBP using the following relation: MAP ¼ 1=3 SBP þ 2=3 DBP

ð3Þ

Assuming the mean venous pressure is negligible, MAP can be described by: MAP ¼ CO TPR

ð4Þ

where CO and TPR denote cardiac output and TPR, respectively. As noted in Equation (4), MAP is proportional to TPR. According to the Hagen–Poiseuille equation, TPR is determined by fluid viscosity and vessel radius and length [14]. If vessel length and viscosity are assumed to be not affected by drug administration, TPR can be solely regarded as a function of vessel radius affected by constriction or dilatation of the vessel. Using CO = SVHR, with SV denoting stroke volume, Equation (4) can be rewritten as: MAP ¼ HR PP C TPR

ð5Þ

In the above, PP is set to the difference between observed SBP and DBP, and C denotes vessel compliance (C = SV/PP). Accordingly, in this work, MAP was modelled as the product of HR, PP and CTPR (=CTPR), where CTPR was introduced as a lumped variable based on the assumption that vessel compliance does not change significantly during the course of treatment. Then, assuming that TPR has a natural tendency to return to its set point whenever perturbation acts on it, CTPR was described as: dðCTPR Þ ¼ kout CTPR0 1 Edrug þ BR kout CTPR dt

ð6Þ

where kout represents the fractional turnover rate of CTPR, CTPR0 the initial value at t = 0 obtained as CTPR0 = MAP0/(HR0PP0) with MAP0, HR0 and PP0 being initial values of MAP, HR and PP, and Edrug drug effect. An additional term, BR, represents CTPR input rate attributed to baroreflex mechanism that acts on a much faster timescale. This term was introduced to assume that when no drug is present, CTPR is maintained at the mean level of CTPR0 with the timevarying fluctuation of BR. Mathematical characterization of BR will be discussed in a later section. Then, from Equation (6), it follows dðCTPR Þ ¼ kout ðSPCTPR CTPRÞ þ BR dt SPCTPR ¼ CTPR0 1 Edrug

ð7Þ ð8Þ

where SPCTPR denotes the set point of CTPR that can be lowered by a drug, and assumes that a difference between current CTPR and the set point acts as the driving force of exerting negative feedback to PP and HR. Drug effect was modelled using log-linear model in Equation (9)


MODEL FOR TELMISARTAN’S CARDIOVASCULAR EFFECT or Emax model in Equation (10) as a function of drug concentration at its effect site (Ce). It was assumed that drug access to the AII receptor is sufficiently rapid so that Ce can be predicted from the time course of plasma concentration Cp. Edrug ¼ keff logð1 þ CpÞ

ð9Þ

Cp C50 þ Cp

ð10Þ

Edrug ¼ Emax

Model for heart rate and pulse pressure. Model structure, Circadian rhythm and Negative feedback. While HR and PP are not known to be significantly affected by drug effect [15], they are far from static variables. Two principle forces act on them to cause systematic fluctuations, which are circadian rhythm and negative feedback. Similarly to CTPR, HR and PP were assumed as being subject to homeostatic principles. dðHR Þ ¼ kout ðSPHR ð1 þ Circ dt dðPP Þ ¼ kout ðSPPP ð1 þ Circ dt

HR Þ

HRÞ

ð11Þ

PP Þ

PPÞ

ð12Þ

SPHR ¼ HR0 ð1 þ FBÞ

ð13Þ

SPPP ¼ PP0 ð1 þ FBÞ

ð14Þ

SPHR and SPPP denote the set points of HR and PP, Circ_HR and Circ_PP denote circadian rhythms of HR and PP, and FB denotes the magnitude of negative feedback (see Equation (17)). The rate of set point recovery (i.e. kout) was assumed to be shared by all three variables (i.e. CTPR, HR and PP) because it is mediated by a common autonomic control system. Thus, SPHR and SPPP were assumed to be modulated by circadian rhythm and negative feedback. Circadian rhythm. It is well known that cardiovascular variables are subject to a 24-hr diurnal rhythm [16]. For example, three harmonic oscillators of periods 8, 12 and 24 hr have been identified [17]. Another study proposed a two-harmonic oscillator model with periods 12 and 24 hr [18]. It has been reported that MAP, HR and PP are governed by a common master clock, possibly in the suprachiasmatic nucleus, which causes oscillators to fluctuate in parallel [19]. In our model, circadian rhythm was described using two cosine functions: HR

2pðt Phase1 1Þ 2pðt Phase2 1Þ þ cos ¼ Amp cos 24 12 ð15Þ

Circ

PP

require some time. We assumed that this feedback process occurs at a rate equal to the rate of set point recovery (i.e. kout). Hence, the following system of equations is proposed. dðFBÞ ¼ kout ðDMAP FBÞ dt DMAP ¼

MAP SPMAP SPMAP

SPMAP ¼ SPHR SPPP SPCTPR

ð17Þ ð18Þ ð19Þ

Formulation of Baroreflex mechanism. As for CTPR, it is known from the literature that circadian fluctuation of CTPR is roughly a mirror image of that in HR and PP [19]. As it is unlikely that a separate circadian clock exists that drives CTPR in just the opposite direction, we assumed that this was due to a baroreflex mechanism that rapidly adjusts TPR in response to changes in HR and PP. BR in Equation (6) was thus modelled as follows (see Appendix for derivation):

where

Circ

141

2pðt Phase1 Þ 2pðt Phase2 Þ þ cos ð16Þ ¼ Amp cos 24 12

To avoid over-parameterization, a common amplitude (i.e. Amp) was assumed for the two cosine oscillators. Also, based on known circadian patterns of PP and HR, a relative phase lag of 1 hr was assumed for HR compared to PP [19]. Negative feedback. It is known that blood pressure is physiologically regulated within narrow limits by various feedback systems such as the carotid baroreflex system and the renin–angiotensin–aldosterone system (RAAS) [20]. The former is known to mediate processes acting within seconds to minutes while the latter is known to take longer time. Based on Equation (5), it is reasonable to assume that percent deviation of CTPR from its set point acts as a driving force of negative feedback on HR and PP. This negative feedback, however, would

BR ¼ CTPR

1 dðHRÞ 1 dðPPÞ þ HR dt PP dt

ð20Þ

Note that BR is proportional to the sum of normalized instantaneous rate of change of HR and PP but opposite in sign. For example, if HR and PP each increases instantly by 1%/sec., CTPR will respond by decreasing 2%/sec. This model successfully generates circadian variation in CTPR without having to incorporate another set of cosine functions. Admission effect. When the value of analysis variables at t = 120 hr (i.e. the beginning of the last dosing interval), which was observed immediately after patient admittance into clinical trial centre, was compared with those observed at t = 144 hr (i.e. the end of the last dosing interval), MAP was elevated and PP was depressed at t = 120 hr compared to those at t = 144 hr. In addition, the variance of MAP at t = 120 hr was significantly higher than that at other times, suggesting an additional source of variability around t = 120 hr. We conjectured that this elevation arose from a psychological effect associated with admission into the clinical trial centre. Hospital admission seems to cause MAP and PP to shift in opposite directions. This could be due to complementary regulation between MAP and PP. Accordingly, this admission effect was multiplied to MAP and PP as 1 þ Adm eAdmt and (1 - Adm eAdmt), respectively, where Adm represents both initial amplitude of admission effect and its rate of decay. Covariate effects. Covariate effects were tested and reflected into the model by stepwise covariate modelling (SCM) performed at the significance level of p < 0.01 for forward addition and p < 0.001 for backward deletion, followed by model refinement of excluding a covariate whose 95% confidence interval of the associated coefficient estimate included the null value. Tested covariates were age, smoking status, alcohol use and caffeine use according to subject demographics reported in table 1. Pharmacokinetic parameters were assumed to be related to total body-weight and scaled using theory-based allometry. Because height was not available in the data set, it was not possible to include body composition in the size metric. Modelling process. Population PKPD model was built sequentially. After the final PK model was determined, PD model building was performed by modelling all three analysis variables simultaneously. In doing so, two different approaches were used, the IPP (individual PK parameters) approach conditional on the empirical Bayes estimates of the PK model parameters and the PPP&D (population PK parameters


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Table 2. Parameter estimates of the final PK model (two-compartment disposition, first-order elimination, first-order absorption with lag time). Parameter1 Structural parameter CL (L/hr) V2 (L) Q (L/hr) V3 (L) Ka (/hr) Tlag (hr) Variance parameter (CV, %) PPVCL (%) q CL-V2 PPVV2 (%) q CL-Q q V2-Q PPVQ (%) PPVka (%) Residual variability (CV, %) RUV (%)

of the predicted percentiles was used to help determine the agreement between observed and predicted percentiles.

Population estimate (%RSE) 22.04 67.87 19.1 835 4.53 0.25

(14.2%) (10.6%) (13.2%) (7.0%) (40.4%) (0.8%)

62.65 0.95 38.79 0.80 0.95 46.24 146.3

(9.2%) (10.2%) (12.9%) (14.5%) (13.7%) (12.8%) (21.9%)

24.55 (6.6%)

Results PK model. A two-compartment disposition model with first-order absorption with lag time and first-order elimination adequately described the plasma telmisartan observations. No covariate was found to be statistically significant. This is most likely because the data were collected from young, healthy volunteers with a narrow range of covariates. The parameter estimates for the final PK model are shown in table 2. The VPC shown in fig. 1 shows the model adequately predicts the distribution of observed concentrations except around the peak where the median predictions are about 25% less than the observations. PD models.

1

CL: clearance, V2: central volume, Q: intercompartmental clearance, V3: peripheral volume, Ka: absorption rate constant, Tlag: absorption lag time; PPV: population parameter variability, RUV: residual unexplained variability.

and data) approach including individual PK data conditional on the population estimates of the PK model parameters [21], with correlation between MAP, HR and PP being taken into account to avoid unrealistic combination of parameters during simulation. After the basic model was built using the aforementioned procedure, the final model was constructed by adding significant covariates to the model. Model evaluation. The final model was evaluated using a VPC given 1000 data sets simulated from the final model. Specifically, 2.5%, median and 97.5% percentiles of observations were compared with 90% prediction intervals at each percentile to assess whether observations are similar to predictions. The 95% confidence interval

Models for MAP, HR and PP. Due to a single dose level of telmisartan utilized in our study and its narrow concentration range, classical Emax model parameters were difficult to estimate. The final model chosen, therefore, assumed a loglinear relationship between telmisartan plasma concentration and drug effects as in Equation (9). The estimation results are summarized in table 3. The drug effect parameter (Keff) was estimated as 0.050. As an illustration, according to Equation (9), 1000 ng/mL concentration leads to 34.5% (=0.050ln (1 + 1000)100) drop of SPCTPR while 2000 ng/mL concentration leads to 38.0% (=0.050ln (1 + 2000)100) drop. The estimated half-life of fractional turnover for CTPR, HR and PP was 10.3 hr. (=ln (2)/kout). Interindividual variability of MAP0, PP0 and HR0 was not high, yielding a CV of 4.65%, 7.76% and 12.6% for MAP0,

Fig. 1. VPC of the final PK model (line: 2.5%, median and 97.5% percentiles of observations, shaded area: 90% CI of 2.5%, median and 97.5% of predicted values).


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143

Table 3. Parameter estimates of final PD model. Parameter1 Structural parameter Amp (%) Phase1 (24 hr) (hr) Phase2 (12 hr) (hr) MAP0 (mmHg) HR0 (/min) PP0 (mmHg) Keff Kout (/hr) Adm (%) Variance parameter PPVMAP0 (CV%) q MAP0_HR0 PPVHR0 (CV%) PPVPP0 (CV%) PPVPHASE1 or 2 (hr) Residual variability RUVHR (CV %) RUVPP (CV %) RUVMAP (CV %)

Population estimate (%RSE) 16.33 7.44 6.60 100.2 65.68 47.12 0.050 0.067 4.89

(35.52%) (18.53%) (20.34%) (1.30%) (3.06%) (2.48%) (10.17%) (28.78%) (28.05%)

4.65 0.42 12.6 7.76 0

(14.38%) (29.42%) (14.35%) (23.99%) (FIX)

7.79 (8.78%) 16.05 (6.58%) 6.28 (7.92%)

1 Amp: common amplitude of circadian oscillator. Phase1 (24 hr), Phase2 (12 hr): phase of the 24 and 12-hr circadian oscillators, respectively. MAP0, HR0, and PP0: mesor of the baseline of MAP, HR and PP, respectively. Keff: scale factor for drug effect. Kout: fractional turnover rate constant related to homeostatic processes. Adm: admission effect represented as a proportion of MAP, and PP at 120 hr.

PP0 and HR0, respectively. Positive correlation of 0.42 between MAP0 and HR0 may be because both are affected by the underlying sympathetic tone. Model for circadian rhythm. Circadian rhythm parameters were best described by a common amplitude of 16.33% and phases of 7.44 hr for the 24-hr period and 6.60 hr for the 12hr period, where per cent values of amplitude represent those relative to the mesor and phase 0 corresponds to 8 am. Admission effect. Admission effect reported in table 3 is an empirical parameter introduced to account for higher pre-dose values of blood pressure and lower values of PP at admission to clinical trial centre, which was estimated to be 4.9% at 120 hr. Covariate effect. For PK parameters, of the tested covariates, age was incorporated into absorption lag time as a linear relation during SCM. However, the 95% confidence interval of the estimated coefficient included the null value of zero. Hence, age was removed during the model refinement step. Failure to discover significant covariates is likely to be due to the homogeneous study population. For PD parameters, none of them was found to be significant.

Fig. 2. VPC of the final model (CI of 2.5%, median and 97.5% prediction percentiles in bands, 2.5%, median and 97.5% percentiles of observations shown as lines).

Modelling process. When testing IPP and PPPD approaches, the two approaches produced very close results. However, the IPP approach was up to approximately 10 times faster, with less numerical difficulty while the PPPD approach was slower, often resulting in unsuccessful covariance steps in NONMEM runs. Accordingly, all PD analyses were conducted using the IPP approach. Model evaluation. When evaluated by VPC, fig. 2 shows overall good agreement between observations and model prediction of MAP, HR and PP.


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Fig. 3. Telmisartan concentration and PD variables (pulse pressure, heart rate, mean arterial pressure, total peripheral resistance scaled by vessel compliance [CTPR]) simulated from the final model (table 3) with feedback excluded (FB = 0) (red) and feedback included (blue) given the multiple dose of 80 mg.

Discussion During PK model development, the misfit observed in the median prediction around the peak concentration could be improved by adopting a two-depot absorption model. However, although this modification led to a significant decrease in OFV, parameter precision decreased significantly. Moreover, when the two-depot absorption model was used, it failed to yield a significant drop of OFV as compared with the case of one-depot absorption model reported in table 2. Hence, this model was not chosen as the final PK model. Empirical PKPD modelling of SBP, DBP or MAP data only, not using HR or PP data together, was attempted using effect compartment and turnover models based on the published literature [4]. However, these models based on blood pressure data only did not describe the data well and had numerical difficulty estimating model parameters, leading to covariance step failure or high standard errors (data not shown). Accordingly, a mechanistic modelling approach using known physiological relationships between MAP, HR and PP was developed in the expectation that the use of physiological mechanisms would help to better describe the observed SBP and DBP. The assumption of drug effect acting on SPCTPR instead of CTPR itself used in Equation (8) thereby affecting DMAP in Equation (18) through SPMAP in Equation (19) seemed more physiological because if the set point was not lowered by the drug, the homeostatic mechanism would lead to a counteracting response that would attempt to restore MAP to the original baseline. On the other hand, if the set point itself is lowered by the drug, the homeostatic mechanism would instead work to lower the blood pressure so as to move the blood pressure nearer to the new target set point.

Certain mechanistic details expected from physiology were omitted or simplified. Firstly, telmisartan–receptor interaction has not been taken into account. Instead, plasma concentration of telmisartan was assumed to immediately affect the pharmacodynamic response. Secondly, given that aldosterone turnover is known to be very rapid, with its plasma half-life known to be less than 20 min. [22], a submodel of aldosterone turnover was omitted altogether. Thirdly, circadian rhythm of blood pressure was described by assuming that HR and PP are governed by a common circadian rhythm differing only in their acrophases. Despite these simplifications, this work has shown that mechanistic modelling is possible even with the limited amount of pharmacodynamic data acquired from a bioequivalence study. Previous modelling work has shown that HR, cardiac output and TPR are subject to negative feedback by MAP as expected from well-understood cardiovascular physiology [23]. In that work, pre-treatment observations were collected and used for the estimation of net feedback effect. Previous work [7] also used the difference in MAP relative to its reference value for feedback on HR, similarly to our work. We were able to demonstrate a significant feedback effect of MAP on HR and PP. To supplement our estimation results, we performed simulations with the final model as shown in fig. 3. The figure shows that when feedback is present, PP and HR are elevated soon after blood pressure begins to decrease whereas without feedback such elevation is not seen and blood pressure decrease in MAP becomes unreasonably larger than what is observed in the data, indicating the important role of feedback in blood pressure regulation Next, we performed simulations with double the dose (i.e. 160 mg) as reported in fig. 4. In the initial period of multiple dosing, PP and HR are slightly



145

A

B

Fig. 4. Telmisartan PD variables (pulse pressure, heart rate, mean arterial pressure, total peripheral resistance scaled by vessel compliance [CTPR]) simulated from the final model (table 3) for feedback included given multiple dose of 80 mg (blue) and 160 mg (red). Panel B is a reproduction of Panel A with later times for t > 120 hr being zoomed in.

higher for 160 mg due to stronger feedback effects. However, as MAP approaches steady-state, such discrepancy gradually disappears. This is due to adaptation of feedback to the new set point levels. Indeed, if such adaptation did not occur, there would be unnecessary expenditure of biological energy fed into the feedback system. As expected, drops of MAP and CTPR are greater for 160 mg. A slight discrepancy in the pattern of circadian rhythm from the known pattern seems to exist. Based on previous reports, a rise in PP, HR and MAP is

expected from 8 a.m. onwards that culminates in the first peak around noon and a second peak at around 6 p.m. [24]. In our simulation, however, there is a temporary drop in PP, HR and MAP in the morning (fig. 4B). We believe this to be due to data limitations – after all, our data consisted of sparse measurement time grid and what is more, no measurements of HR and BP were taken during night-time when the subjects were asleep. With such limited samples, it is difficult to reconstruct the circadian pattern with high precision. Yet, our model still


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successfully depicts the basic circadian pattern of a rise in the daytime and fall in the night-time. The key findings of our model can be summarized as follows: (i) drug effect acts to lower the set point of CTPR. This then leads to concomitant drop in MAP; (ii) baroreflex causes CTPR to offset any abrupt changes in either PP or HR such that MAP remains within a physiological range; (iii) a negative feedback that acts on a slower timescale than baroreflex modulates HR and PP in response to changes in MAP; and (iv) circadian rhythm affects both HR and PP with former lagging behind the latter by about an hour. This propagates via baroreflex to CTPR such that circadian fluctuation shows a mirror image to that of PP and HR. The integrated model well describes the observed time courses of MAP, HR and PP. We expect that the proposed model can be applied to other angiotensin II receptor antagonists that do not have active metabolites. For those that have active metabolites such as losartan and candesartan, the model can still be applied by modifying the drug effect model, with no change in the other parts of the model. For example, in the case of log-linear drug effect model in Equation (9), it can be modified as Edrug ¼ keff1 logð1 þ CpÞ þ keff2 logð1 þ CmÞ with Cp and Cm denoting parent drug and metabolite plasma concentrations, respectively, and keff1 and keff2 denoting the associated effect parameters. In conclusion, this work demonstrated the feasibility of using measurements of SBP, DBP and HR with a physiologically based mechanistic model to describe the cardiovascular effects of telmisartan in human beings. The model may have applications for describing these linked cardiovascular observations after other interventions. Acknowledgements This work was supported by a grant from the Brain Korea 21 PLUS Project for Medical Science, Yonsei University.

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Appendix

147

PP = A(4) HR = A(5)

NONMEM code for PD model.

TPR = A(6) FB = A(7)

$PROBLEM PKPD model of telmisartan ; Specify circadian rhythms $INPUT ID AMT TIME MDV CLI V2I QI V3I KAI ALAG1I ADDL II CMT DV

CIRC = AMP*(COS(2*3.14*(T-PHASE1)/24) + COS (2*3.14*(T-PHASE2)/12))

; CLI V2I QI V3I KAI ALAG1I are empirical Bayes estimates of the PK model parameters $DATA finalpkpd160111.csv IGNORE=(CMT.EQ.7) $SUBROUTINE ADVAN6 TOL=3

CIRC2 = AMP*(COS(2*3.14*(T-PHASE1 - LAG)/24) + COS (2*3.14*(T-PHASE2-LAG)/12)) ; Specify drug effect model EDRUG = KEFF*LOG(1 + CP)

$MODEL COMP(ABS, DEFDOSE)

; Specify set points

COMP(CENTRAL)

SP_PP = PP0*(1 + FB)

COMP(PERIPH)

SP_HR = HR0*(1 + FB)

COMP(PP)

SP_TPR = TPR0*(1 - EDRUG)

COMP(HR) COMP(CTPR)

; hereafter CTPR is denoted as TPR

for notational simplicity COMP(FB)

MAP = HR*PP*TPR SP_MAP = SP_PP*SP_HR*SP_TPR ; Specify differential equations for CP DADT(1) = -KAI*A(1)

$PK ; Specify PK model parameters

DADT(2) = KAI*A(1) + K32*A(3) - K23*A(2) - K20*A(2) DADT(3) = K23*A(2) - K32*A(3)

K20 = CLI/V2I K23 = QI/V2I

; Specify differential equations for PP and HR

K32 = QI/V3I

DPPDT = KOUT*SP_PP*(1 + CIRC) - KOUT*A(4)

S2 = V2I

DHRDT = KOUT*SP_HR*(1 + CIRC2) - KOUT*A(5)

ALAG1 = ALAG1I

DADT(4) = DPPDT

; Specify PD model parameters

DADT(5) = DHRDT

MAP0 = THETA(1)*EXP(ETA(1))

; Specify baroreflex

HR0 = THETA(2)*EXP(ETA(2))

BR = - TPR/PP*DPPDT - TPR/HR*DHRDT

PP0 = THETA(3)*EXP(ETA(3)) ; Specify differential equation for TPR

LAG = THETA(4)

DADT(6) = KOUT*SP_TPR - KOUT*A(6)+BR

KEFF = THETA(5) KOUT = THETA(6)

; Specify DELTA (=DTPR) for feedback effect

AMP = THETA(7)

DELTA = (MAP-SP_MAP)/SP_MAP

PHASE1 = THETA(8) + ETA(4) PHASE2 = THETA(9) + ETA(4)

; Specify differential equation for feedback effect

ADM = THETA(10)

DADT(7) = KOUT*(DELTA - A(7)) ; Specify admission effect $ERROR

IF (TIME.GE.120) THEN ADMEFF = ADM*EXP(-ADM*(TIME - 120))

CP2 = A(2)/V2I CT2 = A(3)/V3I

ELSE

IPP = A(4)

ADMEFF = 0

IHR = A(5)

ENDIF

ITPR = A(6) ; Specify initial conditions

FB2 = A(7)

TPR0 = MAP0/(HR0*PP0) IMAP = IPP*IHR*ITPR

A_0(4) = PP0

SP2 = TPR0*(1 - KEFF*LOG(1 + CP2))

A_0(5) = HR0 A_0(6) = TPR0

IF (CMT.EQ.4) THEN

A_0(7) = 0

IPRED = IPP*(1 - ADMEFF) Y=IPRED*(1 + EPS(1) )

$DES ; Specify PK & PD variables

ENDIF

CP = A(2)/V2I

IF (CMT.EQ.5) THEN

CT = A(3)/V3I

IPRED = IHR


DONGWOO CHAE ET AL.

148 Y=IPRED*(1 + EPS(2) )

As we are dealing with physiological perturbations, both % d (PP) and %d (HR) can be assumed to be significantly smaller than 1 (i.e. 100% change). That is, %d (PP)