Pharmacokinetics Dosing Simulator

What is a pharmacokinetics simulator?

A pharmacokinetics simulator predicts how the concentration of a drug in the blood rises and falls over a course of doses, from the dose, the dosing interval and how quickly the body removes the drug. This one uses the one-compartment model with first-order elimination, which sits behind most dosing calculations. Each dose adds to whatever is left of the doses before it, so the levels climb until the drug eliminated in each interval matches one dose. At that steady state the average concentration is Css = F × D / (CL × τ): the part of the dose that reaches the blood, divided by the clearance and the interval.

Here k = ln 2 / t½ is the elimination rate constant and CL = kV is the clearance, the volume of plasma cleared of drug each hour. The doses can go in as an IV bolus, as an IV infusion of set length or by mouth with first-order absorption, and the curve is drawn against a therapeutic window: the range between the lowest concentration that works and the level above which toxic effects become likely. This is a teaching model, not a dosing tool. Real patients are dosed from measured levels and clinical judgement.

Using the simulator

Choose the route, then set the maintenance dose, the dosing interval τ, the elimination half-life and the volume of distribution. An infusion also takes its infusion time, and an oral dose its absorption rate constant ka and bioavailability F. Play runs the regimen and the scrubber moves through it by hand. With the graph focused, the left and right arrow keys step from one dose to the next.

The main graph shows the concentration over the whole run, with a mark under each dose and the stretch from four to five half-lives shaded. The grey dashed line is the steady-state average. The graph below overlays one dosing interval at steady state on the interval after the first dose, so the gap between the two curves is the accumulation. The readouts give the steady-state levels, the accumulation factor, the time to steady state, a loading dose and the doses that fit the window.

How repeated doses build up to steady state

With first-order elimination every dose decays on its own, whatever else is in the body, so the concentration at any moment is the sum of what remains of every dose so far. This is superposition, and it is how every point here is computed. After an IV bolus a fraction r = e^(−kτ) of each dose is still there when the next arrives, so just after the nth dose the level is C0(1 + r + r² + … + r^(n−1)), where C0 = D/V is the level one dose gives. The geometric series adds up to a steady-state peak of Cmax = C0 / (1 − e^(−kτ)), and the trough is that peak times r.

The factor 1/(1 − e^(−kτ)) is the accumulation factor R. It depends only on the interval measured in half-lives. Dose once every half-life and r = 0.5, so R = 2: the steady-state peak is twice the first peak, and the trough equals the first peak. Dose more often and R grows; dose rarely and it falls towards 1, because each dose has almost gone before the next.

How fast the plateau is approached depends on the half-life alone. After dosing for a time t, the trough before the next dose has covered 1 − e^(−kt) of the way to steady state: half after one half-life, 90 percent after 3.32, 93.75 percent after four and 96.9 percent after five. That is the rule that steady state takes four to five half-lives. A bigger dose raises the plateau and a shorter interval smooths it, but neither gets there sooner.

Worked example: 250 mg every 6 hours with an 8 hour half-life

The simulator opens on an IV bolus of 250 mg every 6 h, with a half-life of 8 h, a volume of distribution of 35 L and a window of 10 to 20 mg/L. Those are close to theophylline in a non-smoking adult, with the British National Formulary’s 10 to 20 mg/L range, though theophylline is not given as a bolus.

• Elimination: k = ln 2 / 8 = 0.086643 h⁻¹ and CL = 0.086643 × 35 = 3.0325 L/h.
• The first dose gives C0 = 250 / 35 = 7.143 mg/L, below the 10 mg/L lower limit.
• Each interval is 0.75 half-lives, so r = e^(−0.086643 × 6) = 0.5946 of each dose is left when the next is given.
• Accumulation factor: R = 1 / (1 − 0.5946) = 2.467.
• Steady-state peak: Cmax = 7.143 × 2.467 = 17.62 mg/L. Trough: Cmin = 17.62 × 0.5946 = 10.48 mg/L. Both sit inside the window.
• Average: Css = 250 / (3.0325 × 6) = 13.74 mg/L, and the area under one interval is AUC = 13.74 × 6 = 82.44 mg·h/L, the same as the area under one whole dose, D/CL.
• Time to steady state: 90 percent after 3.322 × 8 = 26.58 h, and four to five half-lives is 32 to 40 h.
• Loading dose: LD = 17.62 × 35 = 616.7 mg, the steady-state peak times the volume, which is the same as D × R.
• Doses that fit: anything from 250 × 10 / 10.477 = 238.6 mg to 250 × 20 / 17.619 = 283.8 mg, worked from the unrounded trough and peak, keeps the steady state inside 10 to 20 mg/L at this interval.

Play the run and the early doses fall short: the first peak is only 7.143 mg/L, and every trough dips below 10 mg/L until the one just before the seventh dose, at 36 h, which reaches 10.01 mg/L. The fifth trough, at 30 h, is 9.698 mg/L. That day and a half spent partly below the window is what a loading dose is for.

Loading doses

A loading dose is a larger first dose that puts the levels where the steady state will be, so the drug works from the first dose. For repeated IV boluses the dose that does it exactly is the maintenance dose times the accumulation factor, LD = D × R. Tick Start with a loading dose and the curve is at steady state from the first dose: 616.7 mg and then 250 mg every 6 h peak at 17.62 mg/L every time, and the two curves on the lower graph lie on top of each other.

The other common form is LD = Ctarget × V / F for any target concentration. Aiming at the average, 13.74 mg/L, gives 13.74 × 35 = 480.9 mg, which starts the levels mid-window rather than at the peak. For an infusion, D × R run in over the usual infusion time reaches the steady-state peak at the end of the first infusion, and from then on the curve is the steady state exactly. For an oral dose it gives very nearly the steady state once the first dose is absorbed, and less before that, while the level climbs from nothing.

Fitting the regimen to the therapeutic window

Two things have to fit. The average sets where the levels sit, and it depends only on the dose rate D/τ, the clearance and, by mouth, F. The swing sets whether they fit, and for an IV bolus the peak-to-trough ratio is Cmax / Cmin = e^(kτ), which depends only on the interval in half-lives. The window has a ratio of its own, 20/10 = 2 here, so the longest interval that can fit is τmax = ln(20 / 10) / k = 8 h, exactly one half-life, and the only dose that fits at that interval is V × (20 − 10) = 350 mg, which runs from exactly 10 to exactly 20 mg/L.

At 6 h the swing is e^(0.086643 × 6) = 1.682, narrower than the window, which is why a range of doses fits. Lengthen the interval to 12 h and the swing becomes 2.83: no dose fits, and the readout says none, because a dose large enough to hold the trough at 10 mg/L takes the peak past 20 mg/L. Doubling the dose cannot help, since it doubles the peak and the trough alike. That is why drugs with a narrow window and a short half-life are given often, by continuous infusion or as modified-release tablets.

Infusions and oral doses

An infusion delivers each dose at a constant rate over its infusion time T. The level rises while it runs and falls after it stops, so the peak comes at the end of each infusion: Cmax = D (1 − e^(−kT)) / (CL T (1 − e^(−kτ))), with the trough e^(−k(τ − T)) times that, the form Sawchuk and Zaske used for gentamicin in 1976. The opening 250 mg run in over 1 h every 6 h peaks at 16.88 mg/L and falls to 10.94 mg/L. Stretch the infusion to the whole interval and it becomes a continuous infusion: the swing vanishes and the level settles at the average, 13.74 mg/L.

An oral dose has to be absorbed first. With first-order absorption at rate constant ka, a single dose follows the Bateman function, C = F D ka (e^(−kt) − e^(−ka t)) / (V (ka − k)). At steady state the peak comes at t′max = ln[ka(1 − e^(−kτ)) / (k(1 − e^(−kaτ)))] / (ka − k) after each dose, as Gibaldi and Perrier give it. Switch the opening regimen to Oral, with ka = 1 h⁻¹ and F = 1, and the peak falls to 15.22 mg/L, 1.692 h after each dose, while the trough rises to 11.45 mg/L. The average stays at 13.74 mg/L, because absorption changes when the drug arrives and not how much of it does. F does change it: at F = 0.5 every level halves, and the average is 6.87 mg/L.

When absorption is slower than elimination, with ka below k, the slow fall after each peak is set by absorption rather than elimination. This is called flip-flop kinetics, and it also slows the approach to steady state, which then takes roughly four to five absorption half-lives. The simulator says so when it happens. Many modified-release tablets behave this way by design.

Half-life, clearance and volume

The three are tied by t½ = 0.693 V / CL, and each controls something different. Clearance sets the average for a given dose rate. The volume sets how far a single dose lifts the level, D/V, and so the size of the swing. The half-life follows from the two and sets both the time to steady state and how far the levels fall between doses. Double the volume at the same clearance, which here means doubling both the volume and the half-life, and the average stays where it was while the swing narrows and steady state takes twice as long.

Clearance is where the patient comes in. A drug cleared by the kidneys, such as gentamicin or digoxin, has a clearance that falls with kidney function, which is why dosing guides lean on the creatinine clearance from the eGFR and creatinine clearance calculator, and some cytotoxic drugs are dosed by body surface area instead of weight. Digoxin also shows why the upper limit matters: above its narrow range it slows conduction through the heart, causing the kind of blocked beats the ECG rhythm and heart block simulator draws. The fall between doses is the same first-order decay as a radioactive sample’s, and the half-life calculator gives the fraction left after any time.

What this model leaves out

• A distribution phase. Many drugs fall fast at first as they spread into the tissues, then more slowly as they are eliminated, which needs two or more compartments. Digoxin levels mean little until distribution is over, at least six hours after a dose.
• Saturable elimination. Phenytoin, alcohol and high-dose aspirin saturate the enzymes that remove them at the levels they are used at, so the half-life lengthens as the level rises and a small rise in dose can cause a large rise in level.
• A changing patient. Kidney and liver function, body weight and interacting drugs change clearance and volume over days, and carbamazepine speeds up its own clearance over the first weeks of treatment.
• Protein binding. The model follows the total concentration. Only the unbound drug acts, and the bound fraction changes with albumin and disease.
• Real absorption. Oral doses can start after a lag, be slowed by food, absorb in stages or be partly destroyed by first-pass metabolism, which F stands in for.
• Missed and late doses. Every dose here is the same size and on time. A missed or doubled dose changes the levels for several half-lives afterwards.
• Variation between people. The same regimen gives different levels in different patients, several-fold for drugs such as theophylline, which is why narrow-window drugs are measured rather than predicted.

Common mistakes

• Expecting a bigger dose to reach steady state sooner. The time depends only on the half-life. A bigger maintenance dose raises the plateau; only a loading dose gets there faster.
• Using the half-life where k belongs. The exponent is kτ with k = ln 2 / t½. Writing e^(−8 × 6) instead of e^(−0.086643 × 6) turns a 59 percent carry-over into almost none.
• Leaving F out for an oral dose. The average is F × D / (CL × τ), so with F = 0.5 it is half the IV figure.
• Checking only the average against the window. The average can sit mid-window while the peak and trough fall outside it. Lengthen the interval at the same daily dose and the average does not move, but the swing grows.
• Treating the accumulation factor as a constant. R is 2 only when the interval equals the half-life. It is 2.467 at the opening 6 h interval and 1.143 at three half-lives.
• Mixing units. Milligrams over litres give mg/L, which is the same as µg/mL. A dose in micrograms or a half-life in minutes left unconverted shifts the answer by a factor of a thousand or sixty.