Modeling Hemodynamic Profiles by Telemetry in the Rat

Methods

Statistical Evaluation

Rationale for and Characteristics of the Model

In control SHR, circadian rhythms of hemodynamic profiles were found to be more prominent for HR (minimum, 280 beats per minute [bpm]; maximum, 350 bpm, when lights were turned on and off, respectively) than for BP (where differences ranged within 10 mm Hg). Furthermore, “intrarat” random fluctuations (ie, variability unexplained by circadian rhythms) were wide (SD, 12 mm Hg and 30 bpm for BP and HR, respectively), as shown in Fig 1⇓. This “physiological noise” adds up to the effects caused by the administration of an antihypertensive drug and sometimes tends to mask them. In our experience, when the effect is much greater than noise, the typical pattern of the response curve displays a single asymmetric peak, with a prompt onset followed by a slow decrease toward the baseline.

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Figure 1.

Graphs show 24-hour profiles of systolic blood pressure (SBP) and diastolic blood pressure (DBP) (top) and heart rate (bottom) as recorded in control spontaneously hypertensive rats for 3 consecutive days with the telemetry system. Data represent the mean response of six animals.

The above pattern may be modeled by the following family of four-constant exponential functions (model 1):

where E[y(t)] is the expected value of the effect y(t) recorded at time t since drug administration, g( ) is any monotonic function of t, α is the maximum intensity of the effect (peak), and τ is the time to peak. The shape of the curve depends on the function g( ) and constant θ (θ>0), whereas β (β>0) expresses the width of the peak: for a given g( ) and θ, the larger the β, the narrower the peak.

A suitable choice for g( ) is the Box-Cox transformation

restricting attention to δ<0 or δ=0; in the latter case, g(t+θ)=log(t+θ). For a given g( ), increasing values of θ reduce the asymmetry of the peak but may lead to biased estimates of the response, which is supposed to be null at t=0. Conversely, if the shape constant θ is nearly zero, the expected effect E[y(t)] tends toward zero when time t approaches zero. In any case, E[y(t)] tends toward zero when t→∞. In this article, we considered only the case when δ=0, ie, the log transformation. Lower values of δ (eg, δ=−1, the reciprocal transformation) may apply to peaks with more pronounced asymmetry.

Peak value y_max (α) and time to peak t_max (τ) estimated as constants of the function and other constants that may be derived from the model, such as area under the curve and times required to observe a 50% (t_^1/2) or a 75% (t_^1/4) reduction of maximum effect, can be directly compared with the homologous constants typical of pharmaco-kinetics.

In this context, however, it should be noted that kinetics of reversible pharmacological effects and kinetics of plasma levels of drugs are linked by complex nonlinear relations,¹² so that neither shape of response curves nor kinetic constants are expected to overlap.

The adequacy of the above models to describe the response y(t) to the administration of a drug depends critically on what we consider to be a response. Under the assumption that physiological fluctuations and drug-induced effects are simply additive, the response should correspond to the hemodynamic value y_D(t) recorded at time t after drug administration minus the value y₀(t) expected at the same time t after vehicle administration. In this case, y_D(t) may be estimated by fitting an appropriate model to the 24-hour hemodynamic profile recorded after vehicle. When the above assumption is not tenable or when circadian rhythms can be considered negligible compared with drug effects (throughout the 24-hour recording or at least in the period of maximum effect), y₀(t) may be regarded as constant over t and set equal to the mean values recorded during the day before treatment (or during only a few hours before treatment). These approximations are reasonably accurate if drug administration and most significant responses occur in the morning when average HR and BP values undergo small, spontaneous variations and the recorded peak may be ascribed mainly or entirely to drug effects. Furthermore, because of their rigid structure, the above models are rather insensitive to fluctuations far from the peak.

A more complex eight-constant model (model 2) derived from the juxtaposition of two components (each defined as in model 1) may apply to drugs that induce bradycardia followed by tachycardia, such as the nonselective adenosine receptor agonist NECA:

where the constants α₁, β₁, θ₁, and τ₁ and α₂, β₂, θ₂, and τ₂ (of the first and second peaks, respectively) have the same meaning as in model 1, provided that the second effect occurs after the end of the first without overlapping.

Fitting Procedure and Analysis of Residuals

Least-squares estimates of the constants of the above models were obtained by means of proc nlin,¹³ which resorts to the iterative Gauss-Newton algorithm as modified by Marquardt.¹⁴ This procedure allows the definition of bounds within which the estimate must be restrained; eg, the estimates of peak width (β) and time to peak (τ) should be positive, and hypotensive effect (α) should be negative. Furthermore, the procedure allows the definition of a grid of initial estimates for the constants, from which the best combination (the combination that leads to the lowest residual sum of squares) is selected as the starting point for the iterative procedure. Bounds and grid are particularly useful when low doses are tested and hemodynamic effects are weak.

The mean response curve for each drug dose was expressed as the “mean constant curve,”¹⁵ ie, a curve with constants that are the mean of the individual kinetic constants estimated in each rat. This curve indicates the mean location and height of the peak of the effect and the mean pattern of drug response. Mean constant curves are commonly used in pharmacokinetics and auxology because different subjects may have similar kinetics as regards serum concentration of a drug and growth velocity of a somatic trait but different locations of maxima and minima of the kinetic function.

Clearly, any model gives a rather approximate representation of the “true” shape of the kinetics of the response; therefore, fitted values can show, at fixed times, a systematic tendency to underestimate or overestimate the measured values. To detect these tendencies, we calculated for each rat the differences (the so-called residuals) between observed and fitted values.¹⁶ The average of residuals for all rats at a given time estimates the extent of the systematic departure of the model from the true kinetics of the response at that time. The plot of mean residuals versus time is a valuable tool to compare the adequacy of different models and to describe the pattern of circadian rhythms after drug administration.

Results

The typical circadian hemodynamic profiles in Fig 1⇑ are not influenced by vehicle administration, except for a short-lasting peak in HR profile owing to injection (+50 bpm for about 10 minutes).

Fig 2⇓ shows the kinetics of the hemodynamic effects induced by the adenosine A₁ agonist CCPA (0.03 and 0.3 mg/kg IP). CCPA induced hypotension and bradycardia, which are dose-related for both peak effect and half-life (t_^1/2). Regarding time to peak (t_max), the dose-related effect is observed only for HR (Table 1⇓).

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Figure 2.

Scatterplots show the change in systolic blood pressure (SBP) (a, b) and heart rate (HR) (c, d) 24-hour profiles induced by the A₁ adenosine agonist 2-chloro-N⁶-cyclopentyladenosine at 0.03 mg/kg IP (a, c) and 0.3 mg/kg IP (b, d). The continuous line represents the mean constant curve of seven spontaneously hypertensive rats. ▴ represents the mean of residuals obtained as the difference between observed and predicted values of each hemodynamic profile.

The function models systolic BP (SBP) profiles adequately until almost complete recovery to baseline (Fig 2a⇑ and 2b⇑). Systematic departures, ranging within 15 mm Hg, appear after 2 hours at the lower dose and after 6 hours at the higher dose.

For HR, fit is satisfactory during the first 6 to 8 hours after CCPA administration (Fig 2c⇑ and 2d⇑). When lights were turned off, HR increased and the function, which models drug effects and not circadian rhythms, was constantly lower than raw HR profiles. In particular, at 0.3 mg/kg, HR values were higher by 25 to 80 bpm (Fig 2d⇑).

Fig 3⇓ shows the kinetics of hypotensive and tachycardic effects induced by the adenosine A_2a agonist 2-hexynyl-NECA (0.003 and 0.03 mg/kg IP). Peak effect and half-life are dose-related, whereas time to peak is unaffected by dose.

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Figure 3.

Scatterplots show the change in systolic blood pressure (SBP) (a, b) and heart rate (HR) (c, d) 24-hour profiles induced by the A_2a adenosine agonist 2-hexynyl-5′-N-ethylcarboxamidoadenosine at 0.003 mg/kg IP (a, c) and 0.03 mg/kg IP (b, d). The continuous line represents the mean constant curve of seven spontaneously hypertensive rats. ▴ represents the mean of residuals obtained as the difference between observed and predicted values of each hemodynamic profile.

At the lower dose, SBP fitting is satisfactory during the 24 hours (Fig 3a⇑); at the higher dose, the analysis of residuals reveals that in the neighborhood of t_^1/2, the recorded hypotensive effects decreased more quickly than the function, even though differences were within 10 mm Hg. Subsequently, wide pressure fluctuations were apparent (Fig 3b⇑).

As regards the HR fitting, at the lower dose the function was close to the HR profile for 1.5 hours until recovery to baseline. The following systematic departures, caused by circadian rhythms, were not modeled by the function (Fig 3c⇑). At the higher dose, the function fits the HR profile adequately up to 3.5 hours after drug administration but overestimates the recorded values by 10 to 30 bpm between 4 and 8 hours (Fig 3d⇑). At both doses, a sharp and transitory peak followed by a fluctuating pattern of HR occurred when lights were switched off. These phenomena, which seem to be unrelated to drug administration, did not affect the model.

Like the other adenosine agonists, NECA (0.1 mg/kg IP) induced hypotension. The SBP decrease showed a delayed time to peak (t_max=2.2 hours) and was longer lasting than that observed with the other compounds (Table 2⇓).

Fig 4⇓ gives the kinetics of the effects induced on HR by NECA (0.1 mg/kg). The initial A₁-mediated bradycardia and the following prolonged period of reflex tachycardia were modeled by a double-exponential function (model 2). Peaks occur at 1.26 and 13.86 hours after drug administration (Table 2⇑), and the fit is quite satisfactory during the whole 24-hour period (Fig 4⇓).

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Figure 4.

Scatterplot shows the change in heart rate (HR) 24-hour profile induced by the nonselective adenosine agonist 5′-N-ethylcarboxamidoadenosine (0.1 mg/kg IP). The continuous line represents the mean constant curve of eight spontaneously hypertensive rats. ▴ represents the mean of residuals obtained as the difference between observed and predicted values of each individual hemodynamic profile.

Discussion

This study describes the elaboration and testing of mathematical functions that permit valid use of the many data gathered by the telemetry system as applied in experimental pharmacology. With this approach, the effects of drugs on BP and HR are measured continuously in unstressed and freely moving laboratory animals, and accuracy is substantially increased. Thus, an analytical model that exploits the potential of telemetry should allow more thorough evaluation of the pharmacodynamic properties of a drug.

So far, studies based on the telemetry technique usually provided a qualitative description of the changes in hemodynamic parameters induced by drugs, focusing on recorded maximum effect (peak) and time to peak (t_max). Fourier analysis was recently applied as the fitting model, but this approach can be used only to evaluate the alterations of circadian rhythms.¹⁷ In the absence of a model, the peak value is often estimated by the mean response values recorded at the mean t_max.^{18 19} Because t_max is different for each subject in the same experiment, this approach leads to underestimation of the maximum effect. On the other hand, if the peak value is estimated by the mean of each peak effect,^{4 5} the estimate tends to be positively biased. In fact, the response is likely to show its apparent maximum when random fluctuation and drug effect have the same algebraic sign. Furthermore, in the absence of a model, the time required to observe a 50% reduction of peak effect is difficult to identify, especially when random fluctuations are large; the use of raw BP or HR values²⁰ or a least-squares fit of values in the neighborhood of the peak²¹ may not be the best choice. For example, the use of raw HR values observed after administration of 2-hexynyl-NECA (0.003 mg/kg IP) led to t_^1/2=1.30 hours (0.60 to 2.01); this value is higher and has a wider confidence interval than the estimate based on our model (Table 1⇑). According to statistical inference, the wider the confidence interval, the less reliable the estimate.

In most studies, different hemodynamic profiles have been compared at each time point. Such an approach involves a huge number of statistical tests, leading to high risk of type I or II errors and contradictory results (ie, close sequences of significant and nonsignificant differences when observations contiguous on the time axis are considered²² ). Conversely, with our approach, estimates of peak effect and t_max, obtained by fitting a rather rigid mathematical function, are based on the whole response profile and not on one single observation and thus are expected to be both unbiased and precise. With this model, other parameters, related either to persistence (t_^1/2 or t_^1/4) or amount (the area under the response curve at fixed times) of the effects, can be easily estimated. In addition, with our model, which summarizes the response curve in a limited number of constants, the effects of drugs can be compared using appropriate univariate or multivariate tests covering hemodynamic profile, peak effects, time to peak, and time to recovery.

After elaboration of mathematical functions, models of hemodynamic profiles were made with three drugs that have different selectivities for the A₁ or A_2a adenosine receptor. These drugs are good pharmacological tools for their different effects on the hemodynamic parameters examined. The A₁ agonist CCPA markedly reduces both BP and HR¹⁰ ; the A_2a agonist 2-hexynyl-NECA decreases BP and induces reflex tachycardia.¹¹ The nonselective agonist NECA produces hypotension and biphasic effects on HR, ie, tachycardia at lower doses and initial bradycardia followed by tachycardia at higher doses.^{8 23 24}

In the present study, the administration of the above drugs to SHR confirmed the expected changes of the hemodynamic profile. Elaboration of the data on an IBM 486 SX personal computer allowed us to calculate not only BP and HR profiles but also the hemodynamic kinetic constants for each drug, thus providing a quantitative estimate of maximum effect, its occurrence, and its persistence (t_^1/2). Furthermore, a dose relation of the drug effects was detected. Biphasic effects on HR produced by NECA might be a consequence of the different stimulation of A₁ and A_2a receptors achieved at different plasma concentrations.

We developed this mathematical function to provide cardiovascular pharmacologists with a sound statistical tool for the analysis of continuous recordings of hemodynamic parameters obtained by telemetry in laboratory animals.

In this study, we applied mathematical functions only to BP and HR profiles, but it may be useful to evaluate other cardiovascular parameters (eg, arterial blood flow or resistance). We administered compounds by the intraperitoneal route; however, this analysis may be generalized to other administration routes. Finally, this model may be implemented with commercially available programs for nonlinear curve fitting, such as proc nlin (SAS), spss, allfit, and pharmfit. The technical background is the same as that required for fitting pharmacokinetic models.

For investigation of antihypertensive agents and, more generally, hemodynamic effects of new drugs, an in vivo screening is useful for the assessment of their duration of action, efficacy, and potency. In this context, the impact of the telemetry system is arousing interest² and will surely be used more extensively in the near future. The mathematical models we have described aim to exploit the large amount of information provided by the telemetry technique and are therefore expected to be useful in characterizing the hemodynamic profiles of drugs of interest in cardiovascular pharmacology.