ModelRisk example · Pharmaceuticals and biotech

Exposure Variability: Dose Selection with PK/PD

Which dose keeps most patients between the effective level and the safety limit?

Complexity: ★★★★★
Free Excel workbook (PH-07)
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From the model’s saved run

Which regimen?

The typical patient (by formula) against the simulated patients.

Regimen	Typical in window?	Trough effective	Peak above the limit	In the window	Hours at 80% inhibition
1 100 mg every 12 h	no	48%	1%	48%	8.6
2 150 mg every 12 h	yes	71%	12%	59%	30.0
3 200 mg every 12 h	yes	82%	41%	42%	46.1
4 100 mg every 8 h	yes	85%	7%	78%	27.8

Window: trough at least 0.8 mg/L and peak at most 4.0 mg/L. Hours at 80% inhibition: of the 72 simulated.

Concentration over the last dosing hours — 10th to 90th percentile

Line chart: 90th percentile, Median, Mean, 10th percentile, Effective level, Safety limit.

The business question

A biotech company's oral kinase inhibitor has finished its Phase I studies. In a typical patient it is absorbed with a rate constant of about 1 per hour, distributed in about 50 litres and cleared at about 5 litres an hour, with 80% bioavailability — but clearance varies between patients by about 40%, volume by 25% and absorption by 50%.

Efficacy needs a trough above 0.8 mg/L; above 4 mg/L the drug prolongs the QT interval. The drug inhibits its target with an IC50 of 0.5 mg/L, and the biology suggests that more than 80% inhibition matters. The team must choose one regimen for Phase III. For the typical patient 150 mg twice a day fits the window well.

How many patients would it actually suit?

Who it is for: Clinical pharmacologists, pharmacometricians, clinical development teams choosing a Phase III dose.
Main outputs: For each regimen: the chance that a patient's trough is effective, that the peak exceeds the safety limit, and that both are right. Hours above 80% target inhibition · the typical patient against the population · the concentration over a dosing interval across patients · what drives exposure.

How the model works

Three sheets after this one, read from left to right. Each one feeds the next.

Inputs sheet
Typical clearance, volume and absorption rate, and how much each varies between patients (ω, on the log scale); bioavailability; the minimum effective concentration, the safety limit, the IC50 and the inhibition target; each regimen's dose and interval; the solver's time step.
Patient sheet
VoseNormal draws the patient's deviations η from the typical values: CL = typical × exp(η). The model is written as two differential equations — the gut and the central compartment — with concentration and inhibition as outputs. VosePKPDModel solves it for three days of doses under each regimen and returns the concentration over the last twelve hours, the peak and the hours above each threshold.
Results sheet
For each regimen: the typical patient's trough and peak (by formula), and across patients the chance of an effective trough, of a peak above the limit, and of both being right; the hours of 80% inhibition; the spread of concentrations through a dosing interval.

ModelRisk techniques

PK model VosePKPDModel
Patient variability VoseNormal
Four regimens, same patients
Steady-state check
Fan chart and tornado

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The results in detail

Exposure across patients

Trough and peak concentrations (mg/L): the typical patient against the spread.

Trough concentration — cumulative probability

S-curve (cumulative probability) of Regimen 1, Regimen 2, Regimen 3, Regimen 4 from the saved simulation.

Regimen	Trough typical	Trough 10th pct	Trough 90th pct	Peak typical	Peak 90th pct
1 100 mg every 12 h	0.77	0.29	1.64	1.84	2.73
2 150 mg every 12 h	1.15	0.43	2.45	2.77	4.10
3 200 mg every 12 h	1.53	0.57	3.27	3.69	5.46
4 100 mg every 8 h	1.45	0.67	2.78	2.40	3.73

What drives exposure

Lowest and highest mean trough of regimen 2 over ten equal groups of each patient parameter (VoseSimSensitivityMean).

What drives the trough concentration (mg/L)

Tornado chart: the inputs ranked by how far the mean output moves across each input’s range.

The ranking as a table 3 inputs

Input (ranked by influence)	Lowest mean	Highest mean	Rank
Clearance variation	0.32	3.07	1
Volume of distribution variation	1.06	1.58	2
Absorption rate constant variation	1.28	1.47	3

How to read these results

For the typical patient, 150 mg twice a day sits comfortably inside the window — a trough of 1.15 mg/L and a peak of 2.77 — and so does 200 mg.
But patients differ: clearance varies by about 40% between them, and it decides the trough almost entirely: the mean trough is about 3.1 mg/L in the tenth of patients who clear the drug slowest and 0.3 mg/L in the tenth who clear it fastest.
Across patients, 150 mg twice a day suits only about 59%: 29% have a trough below the effective level and 12% a peak above the safety limit. Raising the dose to 200 mg helps the first group and harms the second — 41% exceed the limit — and only 42% are in the window.
100 mg three times a day, the same daily dose as 150 mg twice a day, puts about 78% of patients in the window: smaller doses given more often keep the lowest concentration higher and the highest lower. The price is a third daily dose. A modified-release formulation could give the same profile twice a day.
Time above 80% target inhibition rises with the dose — about 30 of the 72 hours at 150 mg twice a day, 46 at 200 mg. If the biology says inhibition matters more than the trough, the window should be set around it.
Even the best regimen leaves about one patient in five outside the window. Clearance is the lever: dosing by a patient's clearance — measured, or predicted from liver function and other covariates — would narrow the spread.

See it move. Open the workbook, change an input on the Inputs sheet and run the simulation: every table and chart on this page updates in Excel.

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Inputs and assumptions

Every assumption in the model, with its uncertainty and where it comes from. The figures are illustrative.

Pharmacokinetics 7 inputs

Typical values from the Phase I population model, and the variation between patients (ω).

Assumption	Unit	Value
Clearance, typical	L / hour	5.0
Clearance, variation between patients (ω)About ±40%: liver enzymes differ between patients.	log scale	0.40
Volume of distribution, typical	L	50
Volume, variation between patients (ω)	log scale	0.25
Absorption rate constant, typical	per hour	1.00
Absorption, variation between patients (ω)	log scale	0.50
BioavailabilityShare of an oral dose that reaches the blood.	of the dose	80%

Therapeutic window and target 4 inputs

Assumption	Unit	Value
Minimum effective concentration (trough)	mg / L	0.80
Safety limit (peak)QT prolongation above this.	mg / L	4.00
Concentration for 50% target inhibition	mg / L	0.50
Target inhibitionInhibition = concentration ÷ (IC50 + concentration).	share	80%

Regimens 8 inputs

Doses given at once, every interval from time 0, until the trough is read at 72 hours.

Assumption	Unit	Value
Regimen 1: dose	mg	100
Regimen 1: interval between doses12 hours: twice a day; 8 hours: three times a day. It must divide the 72 hours.	hours	12
Regimen 2: dose	mg	150
Regimen 2: interval between doses	hours	12
Regimen 3: dose	mg	200
Regimen 3: interval between doses	hours	12
Regimen 4: dose	mg	100
Regimen 4: interval between doses	hours	8

Solver 3 inputs

Assumption	Unit	Value
Time step of the numerical solutionSmaller is more accurate and slower; the check on the Patient sheet compares with the exact formula.	hours	0.005
Regimen shown in the concentration chart	1 – 4	4
Solve the typical patient only (1 = yes)A test: every η is set to 0, and the Patient sheet compares the solver with the exact formula. Back to 0 to simulate.	1 / 0	0

Units and conventions

Concentration: mg/L in plasma; time in hours from the first dose.
Steady state: Three days of dosing: the trough at 72 hours, the peak over the whole three days.
In the window: Trough at least the minimum effective concentration, and peak at most the safety limit.
η: A patient's deviation from the typical value on the log scale: CL = typical CL × exp(η).

Key assumptions and limitations

Model: One compartment, first-order absorption and elimination, linear kinetics; doses taken exactly on time.
Variability: Clearance, volume and absorption vary independently between patients; no covariates such as weight or renal function.
Solver: VosePKPDModel integrates the equations with a fixed time step. With the typical-patient switch on, its trough and peak are compared with the closed-form steady state.
Data: The drug and all figures are invented for this example.

How to use it

Enter your population PK estimates and their between-patient variability on the Inputs sheet.
Enter the therapeutic window and up to four regimens to compare.
Run a simulation from the ModelRisk ribbon (10,000 samples with a fixed seed are set. Each solves the model four times, so a run takes several minutes — use 1,000 samples while you explore).
Choose the regimen with the best balance; consider how many patients each one leaves outside the window.

Simulate this model with ModelRisk’s classic engine: it does not run in Turbo mode yet.

ModelRisk functions used

VosePKPDModel: Solves the PK/PD differential equations with repeated doses and returns concentrations, peaks and time above thresholds (array).
VoseNormal: Each patient's deviation η of clearance, volume and absorption rate.
VoseInput, VoseOutput: Mark the uncertain inputs and the outputs.
VoseSimMean, VoseSimPercentile, VoseSimProbability: Statistics of the simulation on the Results sheet.
VoseSimSensitivityMean: The tornado chart: every VoseInput ranked by how far the mean trough moves across its range.

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An illustrative example: the figures are invented for teaching and do not describe a real company, project or asset. Last updated 9 October 2026.