Pharmacokinetic And Pharmacodynamic Response Models

Why Separate Pharmacokinetics From Pharmacodynamics?

A genetic variant can be associated with a drug outcome through different biological routes. It may change how quickly a drug is absorbed, distributed, metabolized, or eliminated, thereby changing the concentration achieved at a given dose. Alternatively, it may change how a person responds to a given concentration, for example by altering a drug target or a downstream pathway. These are related but distinct questions: what concentration does the body experience? and what response occurs at that concentration?

Pharmacokinetics (PK) describes the time course of drug concentrations in the body. Pharmacodynamics (PD) describes the relationship between drug concentration and an effect, such as symptom improvement or an adverse reaction. The distinction matters in pharmacogenomics because a genotype associated with a clinical outcome does not, by itself, reveal whether the pathway is altered exposure, altered response, or both.

Definition: A pharmacokinetic genetic effect is a genotype-associated change in a drug concentration profile or a summary of exposure, such as area under the concentration-time curve (AUC). A pharmacodynamic genetic effect is a genotype-associated difference in drug response at a specified concentration or exposure. Efficacy and toxicity are separate response outcomes and can have different concentration-response relationships.

Pharmacokinetic Models of Exposure

A useful starting point is a linear, one-compartment model at periodic steady state. For an oral dose \(D\) given every \(\tau\) hours, let \(F\) be the fraction of the dose reaching systemic circulation and let \(CL\) be clearance, measured in volume per unit time. Under linear PK, with stable dosing and complete dose records, the AUC over one dosing interval and the average concentration over that interval are:

AUC_{\tau}=\frac{FD}{CL}, \qquad C_{\mathrm{avg}}=\frac{AUC_{\tau}}{\tau} =\frac{FD}{CL\tau}.

Here, \(D\) is measured in mass, \(F\) is dimensionless, and \(CL\) has units such as liters per hour. Thus \(AUC_{\tau}\) has units of concentration multiplied by time, while \(C_{\mathrm{avg}}\) has units of concentration. In this model, lower clearance at the same dose and bioavailability produces higher average exposure. A genotype may affect clearance, bioavailability, or other PK processes; an observed genotype association with concentration is not automatically evidence that clearance alone is the mechanism.

The formula summarizes average exposure, not every feature of the concentration-time curve. Peak and trough concentrations can matter clinically, and two regimens with the same average concentration can have different peaks and troughs. The model also presumes linear PK over the relevant range: doubling the dose doubles exposure, and clearance does not change with concentration. These assumptions need to be plausible for the drug and dose range being studied.

Worked Example: A Genotype Difference in Average Exposure

Suppose a hypothetical drug is given orally at 300 mg every 12 hours. Assume \(F=0.80\), linear PK, stable dosing at steady state, and no meaningful differences in adherence or sampling between genotype groups. For this illustration, genotype group 0 has clearance \(10\) L/h and group 1 has clearance \(8\) L/h. Calculate AUC over a dosing interval and average concentration in each group.

For group 0:

AUC_{\tau,0} =\frac{(0.80)(300\ \mathrm{mg})}{10\ \mathrm{L/h}} =24\ \mathrm{mg\cdot h/L}, \qquad C_{\mathrm{avg},0} =\frac{24\ \mathrm{mg\cdot h/L}}{12\ \mathrm{h}} =2.00\ \mathrm{mg/L}.

For group 1:

AUC_{\tau,1} =\frac{(0.80)(300\ \mathrm{mg})}{8\ \mathrm{L/h}} =30\ \mathrm{mg\cdot h/L}, \qquad C_{\mathrm{avg},1} =\frac{30\ \mathrm{mg\cdot h/L}}{12\ \mathrm{h}} =2.50\ \mathrm{mg/L}.

The ratio of average concentrations is \(2.50/2.00=1.25\): group 1 has 25% higher average concentration under the stated model and dosing conditions. This is a hypothetical PK contrast, not evidence that any real genotype changes clearance by this amount. The calculation also does not establish a difference in efficacy or toxicity; those require response models or outcome data.

Pharmacodynamic Models of Efficacy

A common concentration-response curve is the Emax model. For an efficacy outcome measured so that larger values mean greater benefit, a simple version is:

E(C)=E_0+\frac{E_{\max}C}{EC_{50}+C}.

In this expression, \(C\) is concentration, \(E_0\) is the expected response at zero concentration, \(E_{\max}\) is the maximum increase above \(E_0\) attainable under the model, and \(EC_{50}\) is the concentration producing half of that maximum increase. The equation assumes a monotone, saturating response with Hill coefficient fixed at 1. It is a model for the specified efficacy measure; it is not a universal shape for all drugs or outcomes.

A genotype can be associated with efficacy because it changes the concentration supplied to this curve, because it changes a curve parameter such as \(EC_{50}\) or \(E_{\max}\), or through both routes. A change in \(EC_{50}\) is a change in potency: a lower \(EC_{50}\) means a lower concentration is needed to produce a given fraction of the maximum modeled effect. A change in \(E_{\max}\) concerns the maximum modeled response, not the concentration needed to reach a fraction of it.

Worked Example: A Genotype Difference in Efficacy at the Same Concentration

Consider a hypothetical efficacy score measured in points, with higher values indicating greater improvement. At concentration \(C=2.00\) mg/L, suppose both genotype groups share \(E_0=10\) points and \(E_{\max}=80\) points, but group 0 has \(EC_{50}=4.00\) mg/L and group 1 has \(EC_{50}=2.00\) mg/L. Assume the same score definition and the Emax model with Hill coefficient 1 apply in both groups.

For group 0:

E_0(2.00) =10+\frac{80(2.00)}{4.00+2.00} =10+\frac{160}{6} =36.7\ \text{points}.

For group 1:

E_1(2.00) =10+\frac{80(2.00)}{2.00+2.00} =10+\frac{160}{4} =50.0\ \text{points}.

At the same concentration, the modeled response is \(50.0-36.7=13.3\) points higher in group 1. In this example the groups differ in \(EC_{50}\), not in exposure: concentration was held fixed. The result represents a hypothetical conditional PD contrast under the stated model, not proof that genotype causes a response difference in a clinical population.

Modeling Toxicity Separately

A higher concentration can increase therapeutic benefit, toxicity, both, or neither, depending on the drug and outcome. Efficacy and adverse reactions therefore should not be treated as a single response unless a specific combined outcome is defined. One possible model for a binary toxicity outcome is logistic regression:

\operatorname{logit}\{P(Y_{\mathrm{tox}}=1\mid C,G)\} =\alpha+\beta C+\gamma G, \qquad \operatorname{logit}(p)=\log\!\left(\frac{p}{1-p}\right).

Here \(Y_{\mathrm{tox}}=1\) denotes the specified toxicity event, \(G\) is a genotype indicator (for example, 0 or 1), \(\alpha\) is the intercept, and \(\beta\) describes the concentration association on the log-odds scale. The coefficient \(\gamma\) represents a genotype contrast conditional on the modeled concentration. This specification assumes a linear concentration term on the log-odds scale and no genotype-by-concentration interaction. Those are modeling choices, not biological necessities. An interaction term can be considered when the concentration-toxicity relationship itself is hypothesized to differ by genotype.

Worked Example: A Conditional Genotype Association With Toxicity

Suppose a hypothetical toxicity model is \(\operatorname{logit}\{P(Y_{\mathrm{tox}}=1\mid C,G)\}=-4+0.80C+1.00G\), with concentration \(C\) measured in mg/L and \(G=0\) or \(1\). Compare modeled toxicity probabilities for the two genotype groups at \(C=2.00\) mg/L. Assume the logistic model is appropriate for this binary event and that the comparison is within the concentration range represented by the data.

For \(G=0\), the log odds are \(-4+0.80(2.00)+1.00(0)=-2.40\). Converting log odds to probability:

P(Y_{\mathrm{tox}}=1\mid C=2.00,G=0) =\frac{1}{1+\exp(2.40)} \approx 0.0832.

For \(G=1\), the log odds are \(-4+0.80(2.00)+1.00(1)=-1.40\), giving:

P(Y_{\mathrm{tox}}=1\mid C=2.00,G=1) =\frac{1}{1+\exp(1.40)} \approx 0.1978.

At the same modeled concentration, the toxicity probabilities are about 8.32% and 19.78%, a difference of 11.46 percentage points. The conditional odds ratio for group 1 versus group 0 is \(\exp(1.00)\approx2.72\), under this model. This is not a risk ratio, and the hypothetical calculation does not show that the genotype caused toxicity.

Connecting Exposure and Response

PK and PD components can be combined to describe how genotype may relate to an outcome. A useful conceptual sequence is genotype, concentration profile, and response. A genotype association with concentration is evidence about a PK-related pathway. A genotype association with response after accounting for concentration is a conditional association that may reflect PD differences, but it can also be affected by measurement error, model misspecification, and selection or confounding in the clinical data.

Interpretation: A genotype-outcome association alone does not identify whether the genotype changes exposure, efficacy at a given exposure, toxicity at a given exposure, or more than one of these. Assess these pathways using concentration and response information, clearly stated outcome definitions, and models whose assumptions match the data.

The distinction can be summarized by asking what is held fixed in the comparison. A PK comparison asks whether concentration or exposure differs by genotype under a specified dose and regimen. A PD comparison asks whether response differs by genotype at the same concentration or exposure. An overall clinical comparison may include both pathways and is not equivalent to either conditional comparison.

This separation does not by itself justify a causal mediation interpretation. For example, concentration may be measured after treatment begins and may be influenced by adherence, disease severity, co-medications, or clinical decisions. Conditioning on concentration can therefore change the comparison being estimated and may introduce bias if relevant common causes of concentration and response are not addressed. The estimand and assumptions should be explicit before calling a conditional genotype coefficient a direct effect.

A Short Derivation: Why Clearance Changes Average Exposure

Under the linear steady-state oral-dose model, the amount of drug reaching systemic circulation in each interval is \(FD\). Clearance removes drug at a rate proportional to concentration, with elimination rate \(CL\times C\). Over a full dosing interval at steady state, the amount entering the systemic circulation equals the amount eliminated. Integrating the elimination rate over the interval gives:

FD=\int_0^\tau CL\,C(t)\,dt =CL\int_0^\tau C(t)\,dt =CL\cdot AUC_\tau.

Because \(CL\) is constant under the model, rearranging yields \(AUC_\tau=FD/CL\). Dividing by the interval length \(\tau\) gives \(C_{\mathrm{avg}}=FD/(CL\tau)\). Thus the inverse relationship between clearance and average exposure follows directly from mass balance under the assumptions of linear PK, constant clearance, and steady dosing. If those assumptions fail, this formula may not adequately describe exposure.

Common Mistake and Practical Checks

Common mistake: “The genotype is associated with higher drug concentration, so it makes the drug more effective.” That conclusion does not follow from a PK association alone. Higher exposure may increase benefit, increase toxicity, have little effect near a response plateau, or affect the two outcomes differently. The correction is to report the concentration contrast as a PK result and evaluate efficacy and toxicity with their own response models or outcomes.

Before interpreting a pharmacogenomic response model, check that the concentration metric and units are specified, the dose and sampling conditions are comparable, and the assumed PK and PD forms are plausible over the observed range. Also distinguish conditional from overall comparisons: a genotype coefficient adjusted for concentration answers a different question from a genotype association that includes pathways through concentration.

Check Your Understanding

Use the models and distinctions in this tutorial to answer the following questions.

Under the linear steady-state model, what happens to average concentration if clearance decreases while dose, bioavailability, and dosing interval remain fixed?
In the Emax model, what does a lower \(EC_{50}\) mean when \(E_{\max}\) and \(E_0\) are unchanged?
Why does a genotype association with higher exposure not, by itself, establish greater efficacy?
In a logistic toxicity model with a genotype indicator, what is the interpretation of \(\exp(\gamma)\) when the model includes no genotype-by-concentration interaction?
What assumptions support the derivation \(AUC_\tau=FD/CL\), and give one reason the formula might not describe observed data well?