Concentration-Time Plots

Concentration-Time Plots

Constructing and Understanding Concentration-Time Plots

To help understand the journey of a drug through the body, pharmacologists often use a concentration-time plot. This is a graph constructed by measuring drug concentration in the plasma at regular time intervals after a dose of drug is administered. The x-axis shows time (usually reported in hours) and the y-axis shows plasma concentration (reported in mg/L, mmol/L, g/dL, %w/v or any other appropriate units). The resulting graph shows how the drug concentration changes over time: an early rise in concentration as drug is absorbed into the blood followed by a steady decrease as drug is eliminated from the blood through excretion and metabolic transformation. Since, the effects of a drug (both therapeutic and toxic) are tied closely to plasma concentrations, these plots give very useful information about the drug’s behaviour in the body.

The figure below shows a typical concentration-time plot. As mentioned before, drug effects are closely linked to plasma concentration and for many drugs, there is a defined threshold plasma concentration where therapeutic effects begin to occur. This threshold is called the minimum effective concentration (MEC). Below the MEC, there is insufficient drug in the blood—and hence insufficient drug at the side of action—to cause the desired therapeutic effects. This is the "subtherapeutic" range (highlighted yellow). Above the MEC, there is sufficient drug in the blood and the patient enjoys therapeutic benefit. This is the "therapeutic range" (highlighted in blue). However, there is a second threshold concentration where the drug starts to cause toxic effects. This threshold is the minimum toxic concentration (MTC). Above the MTC, the drug concentration has become so high that the patient experiences adverse effects. This is the "toxic range" (highlighted red).

Interrogating Concentration-Time Plots: therapeutic range, duration of action, c_max, t_max, t_1/2

Therapeutic Range

The therapeutic range is the range of plasma concentrations where therapeutic benefit is achieved without toxicity. The lower and upper bounds of the therapeutic range are the MEC and MTC respectively. Drugs with a narrow therapeutic range may be broadly considered more hazardous. Drugs with narrow therapeutic range can more easily reach into the toxic range; accurate dosing can help to achieve therapeutic plasma concentrations but any small variations (eg in tablet content, dehydration, fasting etc) can cause toxicity or loss of drug effectiveness even with consistent dosing. This is discussed more below. Conversely, drugs with wide therapeutic range can be broadly considered more safe as it is easy to stay above MEC and below MTC.

c_max and t_max

c_max and t_max are important parameters related to the maximum plasma concentration achieved after dosing a drug parenterally (eg peroral, transdermal, sublingual).

c_max is the maximum (peak) plasma concentration achieved following drug administration. t_max is the amounth of time it takes for drug to reach the peak concentration, c_max. Both c_max and t_max are useful parameters for describing the absorption profile of a drug. A high c_max indicates extensive absorption into the bloodstream and a short t_max indicates fast absorption into the bloodstream. Conversely, a low c_max and long t_max indicates inefficient and slow absorption (which is not necessarily a bad thing).

Area Under the Curve

The area under the curve (AUC) of the concentration-time graph represents the total exposure to the drug over time. AUC can be useful for comparing bioavailability bewteen different drugs or different formulations of the same drug because it reflects the total amount of drug that reaches the systemic circulation. AUC has distinct importance from c_max because two curves can have vastly different peak concentrations but, if one curve is broad and the other narrow, they could have similar AUC.

Duration of Action

The duration of action of a drug is simply the length of time that it has biological action. As we will see when discussing drug half-life later, drugs persist in the body much longer than they have actions on the body. Since drugs only act when the plasma concentration is above the MEC, duration of action will be the interval between when plasma concentration rises above MEC during the absorption phase and falls below MEC during the elimination phase.

Getting this information from the concentration-time graph is only possible for drugs with close relationship between plasma concentration and action. A limited number of drugs are sequestered in tissues where they have their actions and can continue having action even after plasma concentration has fallen to zero. Examples include bisphosphonates (incorporates into bone matrix) and amiodarone (accumulates in heart-associated fat). Other drugs have active metabolites which extend the duration of action even after the parent drug has been mostly eliminated by metabolic transformation. Examples include fluoxetine and cariprazine.

Half Life

The plasma half-life (t_1/2) of a drug is a measure of persistence in the body. It expresses the amount of time for the plasma concentration of drug to decrease by half (eg from 10 g/L to 5 g/L) in the terminal part of the concentration-time curve (ie after the peak). Since the terminal part of the concentration-time curve approximates an exponential curve (in first order kinetics), t_1/2 will be the same no matter where it is measured (ie the decrease in plasma concentration of 10 g/L → 5 g/L will take the same amount of time as 1 g/L → 0.5 g/L). It should be remembered that long half-life drugs don’t necessarily act longer as duration of action also dependends on the MEC. Drug may persist in the body for several half-lives even after plasma concentration has fallen below the MEC and therapeutic action has ended.

Undesirable Outcomes

The curve shown above depicts the ideal situation; plasma concentration reaches high enough to achieve effectiveness (above the MEC) without going so high that the threshold of toxicity is exceeded (below the MTC). The following two sections desribe situations in which toxicity can occur: overdose and narrow therapeutic range.

Overdose

Overdose occurs when the administered dose is higher than the safe dosage range and the plasma concentration therefore goes above the MTC causing toxic effects.

Narrow Therapeutic Range

For some drugs, the gap between the MEC and MTC is very small. These drugs are said to have a narrow therapeutic range. Drugs with a narrow therapeutic range can easily reach toxic plasma concentration even at moderate doses. For these drugs, careful dosing is required and patients may need to have regular plasma concentration measurements to ensure toxic concentrations are not reached.

Types of Elimination Kinetics

The time course of drug concentration in the body is governed by elimination kinetics, a critical factor in determining safe and effective dosing regimens. The elimination process determines how quickly a drug is removed from the systemic circulation, impacting both its therapeutic efficacy and potential for toxicity. Remember that in pharmacokinetics, elimination involves both excretion (eg in the urine by the kidneys) as well as biotransformation processes (eg metabolism in the liver) The mathematical models used to describe these processes are foundational to the field of pharmacokinetics.

First-Order Kinetics

First-order kinetics describes the elimination process for the vast majority of drugs at therapeutic concentrations. In this model, the rate of elimination is directly proportional to the drug concentration remaining in the body. In other words, as the amount of drug in the body decreases due to elimination processes, the rate of elimination also decreases. Eg if a drug has half-life of 4 hours, its concentration will decrease from 100 mg/dL to 50 mg/dL in the first 4 hours, and then from 50 mg/dL to 25 mg/dL in the next 4 hours, and then 25 mg/dL to 12.5 mg/dL in the next 4 hours etc.

The linearity implies that the body’s elimination pathways (eg metabolic enzymes, renal trnasporters) are mostly available and not saturated by drug or other substrates. There is always enough functional enzyme/transporters to continuously clear drug. For this reason, the rate-limiting variable is the concentration of drug.

The concentration at any given time (C(t)) is dependent on the initial concentration (C(0)), the elimination rate constant (k_e) and the elapsed time (t):

$$ C(t) = C_0 \times e^{-k_e \times t} $$

Zero-Order Kinetics

Zero-order kinetics describes the elimination process for a small number of drugs. In this model, the rate of elimination is constant and does not depend on the concentration of drug currently remaining in the body. This means that the amount of drug eliminated per unit time is always fixed regardless of how much drug is present. Half life is not applicable in zero-order kinetics. Eg if a drug exhibits zero-order kinetics it may take 4 hours for concentration to decrease from 100 mg/dL to 75 mg/dL, and then another 4 hours to decrease from 75 mg/dL to 50 mg/dL, and then another 4 hours to decrease from 50 mg/dL to 25 mg/dL.

This occurs when the elimination pathways are saturated and cannot eliminate more drug. Examples of drugs that exhibit zero-order kinetics include ethanol, phenytoin and high concentrations of salicylates (such as aspirin) and fluoxetine.

The concentration at any given time (C(t)) is dependent on the initial concentration (C(0)), the zero-order elimination rate (k₀) and the elapsed time (t):

$$ C(t) = C_0 - k_0 \times t $$

Michaelis-Menten Kinetics

Michaelis-Menten kinetics is the model that marries first- and zero-order processes, as it is able to describe elimination rate when elimination pathways are saturated and unsaturated.

The Michaelis-Menten equation introduces the Michaelis constant (K_m) which represents the drug concentration at which the rate of elimination is half of the maximum rate. The relationship between the rate of elimination (V), the maximum elimination rate (V_max), Michaelis constant (K_m) and the concentration is given by the equation:

$$ V = V_{max} \times \frac{C}{K_m + C} $$

This equation has a few important implications. Let us consider the situations when drug concentration is very low and very high.

When drug concentration is very low with respect to K_m, we may omit the +C term from the denominator. Elimination rate, V, can them be approximated by...

$$ V \approx V_{max}\times \frac{C}{K_m} = C\times \frac{V_{max}}{K_m} $$

In other words, at low drug concentration, the elimination rate is directly proportional to current concentration. This is otherwise known as first-order kinetics.

When drug concentration is very high with respect to K_m, the C / K_m+C simplifies to 1 and the influences of K_m and concentration become negligible. Elimination rate, V, can them be approximated by...

$$ V \approx V_{max} $$

In other words, at high drug concentration, the elimination rate is dependent only on the maximum elimination rate. This is otherwise known as zero-order kinetics..

The Michaelis-Menten model effectively marries the concepts of first-order kinetics (at low drug concentrations) and zero-order kinetics (at high drug concetration) and also decsribes elimination behaviour at intermediate drug concentrations. It predicts that (in principle) all drugs exhibit both first-order and zero-order kinetics at the low/high extremes of their concentration ranges. In practise, most drugs are either totally ineffective or totally lethal at these concentrations extremes.

Steady-State Kinetics

Steady state is the goal of repetitive drug administration. Steady state is reached when the rate of drug input is equal to the rate of drug elimination and so the plasma drug concentration remains fairly stable.

The time required to reach steady state is dependent solely on the drug’s half-life. As more half-lives pass, average plasma concentration gets closer to the true steady-state concentration. After four to five half-lives, the average plasma concentration is 90-97% of the true steady state concentration. Therefore, we often simply say it takes about five half-lives to reach steady state. This principle is fundamental to clinical drug monitoring and the adjustment of maintenance doses.

The average steady state concentration (c_ss) is related to the bioavailability (F), the dose (D), clearance (Cl) and the dosing interval or time between doses (τ) according to the following equation:

$$ c_{ss} = \frac{F \times D}{Cl \times \tau} $$

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