Estimating Ke0 From Rising and Falling Concentration-Effect Data

Why Rising and Falling Data Matter

A plasma concentration and its observed response need not correspond to an instantaneous concentration–response curve. When the response lags behind plasma, the same plasma concentration can be paired with different responses while concentrations are rising and falling. An effect compartment represents this delay with a latent effect-site concentration, $C_e(t)$, that moves toward the plasma concentration, $C_p(t)$, at a rate governed by $k_{e0}$.

Estimating $k_{e0}$ means finding the rate that makes the effect-site trajectory, passed through a concentration–response model, agree with the observed responses. Data from both limbs are useful because they show how the effect-site state lags behind changing plasma concentrations in both directions. A loop by itself does not give a unique rate: its shape also depends on the plasma time course, response relationship, initial state, and observation error.

Definition: In the first-order effect-compartment model, $C_e(t)$ follows $$ \frac{dC_e(t)}{dt}=k_{e0}\bigl(C_p(t)-C_e(t)\bigr), \qquad k_{e0}>0. $$ Here $C_p(t)$ and $C_e(t)$ have concentration units, and $k_{e0}$ has units of inverse time. For a response observation $Y_i$ at time $t_i$, a static response model relates the predicted response to the modeled effect-site concentration, for example $\widehat{Y}_i=E_0+E_{\max}C_e(t_i)/(EC_{50}+C_e(t_i))$.

What the Estimate Assumes

The model states that effect-site concentration approaches plasma concentration through a single first-order process. To interpret a fitted $k_{e0}$, assume that the plasma profile is adequately observed or represented between observations; the effect-site initial condition is specified; and the same equilibration rate applies throughout the modeled period. Also assume that the response is an instantaneous function of $C_e$ under the chosen response model. If response measurements have independent errors with equal variance, minimizing the sum of squared response residuals is the usual least-squares criterion.

These assumptions matter because $C_e(t)$ is not directly observed in this model. The fit uses plasma observations to generate a candidate effect-site trajectory, then uses the response observations to evaluate that trajectory. A wrong response curve, an unaccounted direct effect of plasma concentration, a changing drug effect over time, or a mistaken initial condition can therefore distort the estimated rate. If the response model’s parameters are not known, they may be estimated alongside $k_{e0}$; the resulting estimate then depends on the joint fit rather than on an isolated calculation.

For a simple numerical calculation, suppose plasma concentration is treated as constant at $C_{p,j}$ during an interval of duration $\Delta t_j$. Solving the differential equation over that interval gives

C_{e,j+1}=C_{p,j}+\bigl(C_{e,j}-C_{p,j}\bigr)e^{-k_{e0}\Delta t_j}.

This recurrence makes the fitting idea explicit: for each candidate $k_{e0}$, propagate $C_e$ across the observed intervals, predict responses at the sampling times, and compare predictions with observations. In applications, the chosen representation of the plasma profile between samples must be stated; the constant-within-interval assumption is used in the calculations below.

Fitting the Rate to Both Limbs

Let $Y_i$ be the measured response at time $t_i$, and let $g(C_e;\theta)$ be the selected response function, with response parameters $\theta$. For a candidate value of $k_{e0}$, the effect-compartment equation determines $C_e(t_i)$. Under equal-variance response errors, the least-squares objective is

Q(k_{e0},\theta)=\sum_{i=1}^{n}\left[Y_i-g\bigl(C_e(t_i;k_{e0});\theta\bigr)\right]^2.

Choose the candidate rate, and any response parameters being estimated, that minimize this objective over the allowed parameter values. The sum should include observations from rising and falling plasma periods. This is not the same as estimating a separate concentration–response curve on each limb: the effect-compartment model seeks one rate and one response relationship that account for both.

Worked Example: Selecting a Rate From Paired Data on Both Limbs

Consider hypothetical plasma concentrations held constant over four successive one-hour intervals: $2,\ 8,\ 4,$ and $1\ \mathrm{mg/L}$. Suppose $C_e(0)=0\ \mathrm{mg/L}$, and the response model is known to be $E=100C_e/(2+C_e)$, with $E$ expressed as a percentage and concentrations in $\mathrm{mg/L}$. For illustration, the paired responses are generated by the model at $k_{e0}=0.5\ \mathrm{h}^{-1}$; displayed responses are rounded, while calculations use the unrounded values.

The plasma profile first rises from $2$ to $8\ \mathrm{mg/L}$, then falls to $4$ and $1\ \mathrm{mg/L}$. Applying the recurrence with $e^{-0.5}=0.606531$ gives effect-site concentrations of approximately $0.7869,\ 3.6251,\ 3.7726,$ and $2.6817\ \mathrm{mg/L}$ at the four interval ends. The corresponding responses are approximately $28.24\%,\ 64.44\%,\ 65.35\%,$ and $57.28\%$.

For each trial rate, propagate the same plasma profile and compare its predicted effect-site concentrations with the concentrations implied by the paired responses. Since the response parameters are fixed here, invert the response equation:

C_e=\frac{2E}{100-E}.

For example, $E=28.24\%$ implies $C_e=2(28.24)/(100-28.24)\approx0.7871\ \mathrm{mg/L}$. Repeating this inversion for all four responses gives the effect-site targets listed below, subject to rounding.

Trial $k_{e0}$ ($\mathrm{h}^{-1}$)	Predicted $C_e$ values ($\mathrm{mg/L}$)	Sum of squared $C_e$ residuals ($(\mathrm{mg/L})^2$)
0.4	0.6594, 3.0794, 3.3829, 2.5973	0.4730
0.5	0.7869, 3.6251, 3.7726, 2.6817	approximately 0
0.6	0.9024, 4.1047, 4.0575, 2.6780	0.3246

For $k_{e0}=0.4\ \mathrm{h}^{-1}$, for instance, the squared-residual sum is approximately $(0.7869-0.6594)^2+(3.6251-3.0794)^2+(3.7726-3.3829)^2+(2.6817-2.5973)^2=0.4730\ (\mathrm{mg/L})^2$. Among these candidates, $0.5\ \mathrm{h}^{-1}$ best matches the data. A continuous search could refine the minimum if the observations were not generated exactly from one of the trial rates. This is a fit to both the rising and falling portions, not an estimate from one limb alone.

From a Single Interval to a Rate

The interval solution also shows how a rate can be calculated directly when the effect-site concentrations are known or inferred at two times and plasma concentration is constant between them. Rearranging the recurrence yields

k_{e0}=-\frac{1}{\Delta t}\ln\left(\frac{C_e(t+\Delta t)-C_p}{C_e(t)-C_p}\right).

The ratio inside the logarithm must be positive, and the observed change must be compatible with movement toward $C_p$. This calculation is a useful check, but noisy response observations usually make a joint fit across several time points more stable than separate interval estimates.

Worked Example: Checking the Rate on Rising and Falling Intervals

Suppose the concentration–response relationship is $E=100C_e/(2+C_e)$, and the parameters are known. On a rising interval, plasma concentration is held at $5\ \mathrm{mg/L}$ for one hour. The response changes from $33.33\%$ to $50.00\%$, which implies $C_e=1$ then $2\ \mathrm{mg/L}$, using $C_e=2E/(100-E)$.

Substitution into the interval formula gives $k_{e0}=-\ln[(5-2)/(5-1)]/1=-\ln(0.75)=0.2877\ \mathrm{h}^{-1}$. On a falling interval, suppose plasma is held at $0\ \mathrm{mg/L}$, while inferred effect-site concentration changes from $6$ to $4.5\ \mathrm{mg/L}$ over one hour. Then $k_{e0}=-\ln(4.5/6)/1=-\ln(0.75)=0.2877\ \mathrm{h}^{-1}$ again.

The agreement is what a single first-order rate predicts in these idealized intervals. If the rates calculated separately from noisy observations differ, that does not by itself establish that the model is wrong; it signals that the full data should be fitted together and that the assumed plasma profile, response relationship, and observations should be checked.

Choosing the Fitting Scale

When the response relationship and its parameters are known, one can invert it to obtain effect-site concentration targets and fit $k_{e0}$ on the concentration scale. Alternatively, fit the rate directly by minimizing response residuals. These approaches need not produce the same estimate when observations contain error. Inversion can magnify response errors, especially near a response ceiling, and least squares on inferred concentrations gives equal weight to concentration residuals rather than to response residuals.

For example, in the previous hypothetical data, the $k_{e0}=0.6\ \mathrm{h}^{-1}$ trajectory predicts effect-site concentrations of about $0.9024,\ 4.1047,\ 4.0575,$ and $2.6780\ \mathrm{mg/L}$. Passing these through the response curve gives predicted responses of about $31.09\%,\ 67.24\%,\ 66.983\%,$ and $57.25\%$. Compared with the generated observations, the response-scale squared residual sum is approximately

(28.24-31.09)^2+(64.44-67.24)^2+(65.35-67.00)^2+(57.28-57.25)^2 \approx 18.7\ \text{percentage-points}^2.

That value is not directly comparable with the concentration-scale sum, which has units of $(\mathrm{mg/L})^2$. The appropriate fitting scale depends on the observation-error model: if response errors are the modeled errors, response-scale fitting is generally the natural choice. If the effect-site targets are measurements with a defensible error model, fitting on the concentration scale may be appropriate.

Diagnostics, Pitfalls, and Limits

A fitted value is more credible when its predictions track observations across both limbs without systematic discrepancy. Plot observed and predicted responses against time, and inspect residuals—the observed response minus the predicted response—against time and plasma concentration. A persistent run of positive residuals followed by negative residuals, or a pattern that differs between rising and falling observations, suggests the chosen model or inputs may not describe the data adequately. These checks diagnose lack of fit; they do not identify its cause on their own.

Common causes include a poor representation of plasma concentration between sample times, an incorrect effect-site initial condition, an unsuitable concentration–response relationship, or a response that changes for reasons beyond effect-site concentration. Sparse observations, observations only near equilibrium, or a narrow range of plasma concentrations may also leave little information about the rate. A numerical minimum is not sufficient evidence that the rate is well determined.

Using only one limb. A rising-only fit can miss whether the same model accounts for the falling observations. Include both limbs when available.
Treating response as effect-site concentration. Response must be connected to $C_e$ through a stated response model; it is not generally a concentration measurement.
Ignoring the response scale. Squared residuals in percent response and squared residuals in $\mathrm{mg/L}$ target different error structures and have different units.
Overinterpreting the rate. A fitted $k_{e0}$ is a parameter of the specified model and data analysis. It should not automatically be treated as a direct measurement of a particular physiological process.

Graphical loop summaries can describe the observed hysteresis, while interval calculations can provide a quick check under constant plasma concentration. Joint fitting is more useful when the goal is one rate that explains paired data over an entire changing concentration profile. None of these methods removes dependence on the response model, the plasma input representation, or the information content of the observations.

Key takeaway: Estimate $k_{e0}$ by propagating the effect-site model from the plasma history and fitting its response predictions to paired observations from rising and falling periods. State the response and input assumptions, use a fitting scale consistent with the error model, and check the fitted trajectory against the data.

Check Your Understanding

Work through each calculation, then compare with the solutions.

With constant $C_p=4\ \mathrm{mg/L}$, $C_e(0)=0$, and $C_e(2\ \mathrm{h})=2\ \mathrm{mg/L}$, calculate $k_{e0}$.
For a one-hour interval with $C_p=3\ \mathrm{mg/L}$, $C_e(0)=1\ \mathrm{mg/L}$, and $C_e(1\ \mathrm{h})=2\ \mathrm{mg/L}$, calculate $k_{e0}$.
Why can observations at the same plasma concentration on rising and falling limbs help assess an effect-compartment model?
Suppose $E_0=10$, $E_{\max}=90$, $EC_{50}=3\ \mathrm{mg/L}$, and an observed response is $55$ in the response scale used by the model. What effect-site concentration is implied by the $E_{\max}$ model?

Solutions. (1) The interval ratio is $(4-2)/(4-0)=0.5$, so $k_{e0}=-\ln(0.5)/2=0.3466\ \mathrm{h}^{-1}$. (2) The ratio is $(3-2)/(3-1)=0.5$, so $k_{e0}=-\ln(0.5)=0.6931\ \mathrm{h}^{-1}$. (3) If response differs at the same plasma concentration, an instantaneous response-to-plasma relationship alone cannot describe both observations; the effect-site state can differ because it reflects the preceding concentration history. (4) The response fraction above baseline is $(55-10)/90=0.5$. Thus $C_e=EC_{50}(0.5)/(1-0.5)=3.00\ \mathrm{mg/L}$.