1.3 Dose-response relationships

Dose-response curve characteristics

Dose-response curves plot the relationship between how much drug you give and how much of an effect you see. They're the foundation for figuring out the right dose of any drug: enough to work, but not so much that it causes harm.

Graded vs. quantal dose-response curves

These are two fundamentally different ways of measuring drug response, and mixing them up is a common mistake.

Graded dose-response curves measure the magnitude of response in a single system (one patient, one tissue sample) as you increase the dose. The response rises continuously until it hits a ceiling. Think of measuring how much blood pressure drops as you increase the dose of an antihypertensive.

Quantal dose-response curves measure the proportion of a population that shows a defined all-or-nothing response at each dose. Instead of asking "how much effect?" you're asking "how many people responded?" These curves are used to determine values like $ED_{50}$ and $LD_{50}$.

Median effective dose ($ED_{50}$)

The $ED_{50}$ is the dose required to produce a therapeutic response in 50% of the population. You determine it from a quantal dose-response curve by plotting the percentage of responders against the log dose.

$ED_{50}$ is a direct measure of potency. A lower $ED_{50}$ means the drug is more potent because you need less of it to get the job done. For example, morphine has an $ED_{50}$ of roughly 5 mg for analgesia, which is far lower than many other opioids, reflecting its high potency.

Median lethal dose ($LD_{50}$)

The $LD_{50}$ is the dose that causes death in 50% of a test population. It's determined the same way as $ED_{50}$, but the endpoint is mortality rather than therapeutic effect.

A lower $LD_{50}$ means the drug is more toxic. Caffeine, for instance, has an $LD_{50}$ of about 150–200 mg/kg in rodents. While $LD_{50}$ values are determined in animal studies (not humans, for obvious reasons), they remain a standard benchmark for comparing relative toxicity.

Therapeutic index calculation

The therapeutic index (TI) quantifies a drug's safety margin:

$TI = \frac{LD_{50}}{ED_{50}}$

A higher TI means a wider gap between the dose that works and the dose that kills, so the drug is safer to use. Diazepam, for example, has a TI around 100, meaning its lethal dose is roughly 100 times its effective dose.

• Drugs with a TI greater than 10 are generally considered safe.
• Drugs with a TI less than 2 (like warfarin or lithium) require careful dosing and close monitoring because the effective and toxic doses are dangerously close together.

One limitation: the TI uses population medians, so it doesn't capture individual variation. The certain safety factor ($\frac{LD_1}{ED_{99}}$) is sometimes used as a more conservative measure.

Dose-response relationship factors

Several properties of a drug and its receptor system shape the dose-response curve. Understanding how potency, efficacy, affinity, and receptor occupancy interact is essential for comparing drugs and optimizing therapy.

Potency of drugs

Potency refers to the amount of drug needed to produce a given effect. A more potent drug achieves the same response at a lower dose.

Potency depends on two things:

• The drug's affinity for its receptor (how tightly it binds)
• The efficiency of the drug-receptor interaction in triggering a downstream response

You compare potency between drugs using $ED_{50}$ or $IC_{50}$ (the concentration that inhibits 50% of a response). Morphine is more potent than codeine because morphine's $ED_{50}$ is much lower for the same analgesic effect.

Efficacy vs. potency

Efficacy is the maximum response a drug can produce, no matter how high you push the dose. Potency is how much drug you need to start getting there.

These are independent properties. A drug can be highly potent (works at low doses) but have low efficacy (its ceiling effect is modest). Buprenorphine is a classic example: it's more potent than morphine (lower $ED_{50}$), but its maximum analgesic effect is lower because it's a partial agonist.

Efficacy is determined by factors like receptor coupling efficiency and the presence of spare receptors in the tissue.

Affinity vs. efficacy

Affinity is the strength of binding between a drug and its receptor. High-affinity drugs bind tightly and occupy receptors at low concentrations.

But strong binding doesn't guarantee a strong response. A drug can have high affinity yet produce no activation at all (that's what antagonists do). Conversely, a drug with moderate affinity can still produce a large response if it activates the receptor efficiently.

• Morphine: high affinity, high efficacy (binds well and strongly activates opioid receptors)
• Naloxone: high affinity, zero efficacy (binds tightly but produces no activation; it's an antagonist)

Receptor occupancy effects

Receptor occupancy is the fraction of receptors bound by a drug at a given concentration. You might expect that response scales linearly with occupancy, but it often doesn't.

Because of signal amplification in intracellular pathways, a maximum response can sometimes be achieved when only a small fraction of receptors are occupied. This is the basis of the spare receptor concept (covered below). Factors that influence the occupancy-response relationship include:

• Receptor density on the cell surface
• Receptor reserve (how many "extra" receptors exist beyond what's needed)
• Signal amplification through second messenger cascades

Beta-2 adrenergic receptors in the lungs are a good example: bronchodilation can reach its maximum with relatively low receptor occupancy because of a large receptor reserve.

Agonists in dose-response relationships

Agonists bind to receptors and activate them, producing a biological response. Their dose-response behavior depends on their intrinsic efficacy and the receptor system they act on.

Full vs. partial agonists

Full agonists can produce the maximum possible response of the receptor system. At sufficient concentrations, they drive the system to its ceiling. Morphine is a full agonist at mu-opioid receptors.

Partial agonists produce a submaximal response even when they occupy every available receptor. They have lower intrinsic efficacy than full agonists. Buprenorphine is a partial agonist at mu-opioid receptors: even at saturating doses, its analgesic effect plateaus below what morphine can achieve.

An important clinical consequence: if a partial agonist is given alongside a full agonist, it competes for receptor binding but produces a weaker response. In this context, the partial agonist effectively acts as an antagonist, reducing the full agonist's effect. This is why buprenorphine can precipitate withdrawal in opioid-dependent patients.

Agonist dose-response curves

Agonist dose-response curves typically have a sigmoidal (S-shaped) profile when plotted on a log-dose axis:

1. At low doses, the response increases gradually (few receptors occupied).
2. In the mid-range, the curve rises steeply (small dose increases produce large response gains).
3. At high doses, the curve plateaus as the maximum response is reached.

The steepness of the curve (quantified by the Hill coefficient) reflects how efficiently receptor activation translates into response. A steeper curve means a more switch-like, cooperative response. The plateau height represents the drug's efficacy.

Spare receptor concept

Some receptor systems have far more receptors than needed to produce a maximum response. These extra receptors are called spare receptors.

If a tissue has spare receptors, a full agonist can achieve 100% of the maximum response while occupying well under 100% of the receptors. Acetylcholine receptors in smooth muscle are a classic example: maximal contraction occurs with only a fraction of receptors occupied.

The practical consequence is increased sensitivity. Because spare receptors exist, even low concentrations of agonist can produce a near-maximal effect, shifting the dose-response curve to the left.

Receptor reserve impact

Receptor reserve is essentially the quantitative measure of spare receptors: the excess receptor population beyond what's needed for a maximal response.

The size of the receptor reserve has real consequences for drug behavior:

• A large receptor reserve can make even a partial agonist appear to behave like a full agonist, because the system compensates for lower intrinsic efficacy with extra receptors.
• Receptor reserve varies between tissues. Beta-adrenergic receptors in the heart may have a different reserve than those in the lungs, which is why the same drug can produce different maximal effects in different organs.
• Disease states or chronic drug exposure can alter receptor density, changing the receptor reserve and shifting drug sensitivity.

Antagonists in dose-response relationships

Antagonists bind to receptors without activating them. Their role is to block agonist action. The type of antagonism determines how the agonist's dose-response curve is affected.

Competitive vs. non-competitive antagonists

Competitive antagonists bind to the same site as the agonist (the orthosteric site). Because they're competing for the same binding pocket, increasing the agonist concentration can overcome their blockade. Naloxone at opioid receptors is a classic competitive antagonist.

Non-competitive antagonists bind to a different site (an allosteric site) or bind irreversibly to the orthosteric site. They reduce the system's ability to generate a maximum response, and increasing agonist concentration cannot fully overcome their effect. Memantine at NMDA receptors is an example.

Antagonist dose-response curves

Antagonists don't have their own "response" curves in the traditional sense. Instead, you observe their effects by looking at how they shift the agonist's dose-response curve.

• Competitive antagonists produce a parallel rightward shift of the agonist curve. The maximum response stays the same, but you need more agonist to reach it. Atropine shifting the acetylcholine curve is a textbook example.
• Non-competitive antagonists reduce the maximum achievable response (the plateau drops) and may also shift the curve rightward.

The magnitude of the rightward shift depends on the antagonist's concentration and its affinity for the receptor.

Schild plot analysis

Schild plot analysis is the standard method for quantifying the affinity of a competitive antagonist. Here's how it works:

1. Measure the agonist dose-response curve without the antagonist.
2. Repeat the measurement in the presence of increasing concentrations of the antagonist.
3. For each antagonist concentration, calculate the dose ratio (DR): the factor by which the agonist $ED_{50}$ shifts rightward.
4. Plot $\log(DR - 1)$ on the y-axis against $\log[\text{antagonist concentration}]$ on the x-axis.

For a true competitive antagonist, this plot yields a straight line with a slope of 1. If the slope deviates significantly from 1, the antagonism may not be purely competitive.

The x-intercept of the Schild plot gives you the $pA_2$ value.

$pA_2$ value determination

The $pA_2$ is the negative logarithm of the molar concentration of antagonist that produces a 2-fold rightward shift of the agonist dose-response curve (i.e., $DR = 2$).

$pA_2 = -\log[A]_{DR=2}$

A higher $pA_2$ means the antagonist has higher affinity for the receptor (it takes less antagonist to shift the curve). Atropine, for example, has a $pA_2$ of about 8.8 at muscarinic receptors, reflecting very high affinity.

The $pA_2$ value is independent of which agonist you use in the experiment, making it a reliable way to compare antagonist affinities across different studies.

Dose-response in drug development

Dose-response studies run through every stage of drug development, from early lab work to clinical trials. They answer the most fundamental question in pharmacology: how much drug should you give?

Importance of dose-response studies

Dose-response data help identify two critical boundaries:

• Minimum effective dose (MED): the lowest dose that produces a meaningful therapeutic effect
• Maximum tolerated dose (MTD): the highest dose patients can take before adverse effects become unacceptable

The range between these two defines the therapeutic window. Dose-response studies also inform the recommended starting dose, dose titration schedules, and dosing adjustments for specific patient populations (e.g., elderly patients, those with renal impairment).

In vitro vs. in vivo dose-response

In vitro studies use cell lines, isolated tissues, or biochemical assays to characterize a drug's potency, efficacy, and receptor selectivity under controlled conditions. They're fast and relatively inexpensive, providing early evidence of a drug's mechanism of action.

In vivo studies use animal models to evaluate how the drug behaves in a living system, incorporating absorption, distribution, metabolism, and excretion (pharmacokinetics) alongside the pharmacodynamic response.

In vitro results don't always predict in vivo outcomes. A drug might be highly potent in a cell assay but poorly absorbed or rapidly metabolized in an animal. Both types of data are needed before moving into human trials.

Dose optimization strategies

Finding the right dose is an iterative process. Common strategies include:

• Dose titration: starting at a low dose and gradually increasing until the desired effect is reached or side effects emerge
• Extended-release formulations: smoothing out plasma drug levels to maintain the drug within the therapeutic window over a longer period
• Pharmacogenetic-guided dosing: adjusting doses based on genetic variations in drug-metabolizing enzymes (e.g., CYP2D6 polymorphisms affecting codeine metabolism)
• Therapeutic drug monitoring (TDM): measuring plasma drug concentrations to ensure they stay within the target range, especially for drugs with narrow therapeutic indices

Safety margin considerations

The safety margin is the gap between the dose that produces therapeutic effects and the dose that causes toxicity. A wide safety margin gives clinicians more flexibility in dosing; a narrow one demands precision.

Drugs with narrow safety margins (warfarin, lithium, digoxin) require:

• Careful initial dose selection
• Frequent monitoring of drug levels or clinical markers
• Patient education about signs of toxicity

Safety margin data directly shape clinical trial design. Phase I trials, for example, use dose-escalation protocols informed by preclinical dose-response and toxicity data to identify the safe dosing range in humans before efficacy testing begins.