From Experimental Measurements to a PBPK Input
A dissolution experiment does not arrive in a PBPK model as a ready-made input. It provides measurements under particular conditions: a particle-size distribution, solubility in a specified medium, or a concentration-time profile. The model needs quantities that correspond to its states and equations, with units and assumptions that remain valid when the physiological conditions change. The central task is therefore to translate the measurements without losing the mass balance or implying more generality than the data support.
This tutorial focuses on constructing that input. It does not treat a measured in vitro release curve as automatically interchangeable with an in vivo dissolution rate. Instead, use the curve to estimate or check a model representation, document the experimental conditions, and test predictions against a profile that did not determine the input.
Representing the Particle-Size Distribution
Particle size matters because dissolution rate often depends on exposed solid surface area. Record what the particle-size measurement represents: for example, a number-weighted, volume-weighted, or mass-weighted distribution. Do not treat these weightings as equivalent. If particles have a common density and similar shape, volume fractions can be used as mass fractions; otherwise, the conversion requires additional information about density, shape, and composition.
For spherical particles in size bin \(i\), let \(M_i\) be the solid mass, \(d_i\) the representative diameter, and \(\rho\) the solid density. The particle mass is proportional to \(\rho d_i^3\), while its surface area is proportional to \(d_i^2\). For a collection of equal-sized spheres in that bin, total area is therefore \(A_i=6M_i/(\rho d_i)\). This relation makes clear why replacing a broad distribution with one average diameter can misrepresent total area.
This derivation assumes spherical particles, a common density, and a representative diameter for each bin. Real particles may be irregular, porous, agglomerated, or partially wetted. In those cases, the computed area is an effective geometric approximation, not necessarily the area accessible to fluid.
Worked Example: Calculating Surface Area by Size Bin
A hypothetical 100 mg dose has a volume-weighted particle-size distribution with 20% at 10 μm, 50% at 30 μm, and 30% at 90 μm. Assume the particles have the same density, \(1.2\ \mathrm{mg/mm^3}\), and are spherical. Calculate the estimated area in each bin and the total area.
Because density is common across bins, the volume fractions are also the mass fractions under these assumptions. The bin masses are \(20\), \(50\), and \(30\) mg. Convert the diameters to millimetres: \(10\ \mathrm{\mu m}=0.010\ \mathrm{mm}\), \(30\ \mathrm{\mu m}=0.030\ \mathrm{mm}\), and \(90\ \mathrm{\mu m}=0.090\ \mathrm{mm}\).
For the 10 μm bin, \(A_1=6(20\ \mathrm{mg})/[(1.2\ \mathrm{mg/mm^3})(0.010\ \mathrm{mm})]=10{,}000\ \mathrm{mm^2}\). For the 30 μm bin, \(A_2=6(50)/[(1.2)(0.030)]=8{,}333\ \mathrm{mm^2}\). For the 90 μm bin, \(A_3=6(30)/[(1.2)(0.090)]=1{,}667\ \mathrm{mm^2}\).
The estimated total area is \(10{,}000+8{,}333+1{,}667=20{,}000\ \mathrm{mm^2}\). The smallest bin contains only 20% of the mass but contributes half of the estimated area. This is why a single mean diameter can be a poor substitute for a size-resolved input.
Matching Solubility to the Model State
A solubility value is interpretable only with its conditions and measurement definition. Record at least the medium composition, pH, temperature, solid form, equilibration procedure, and whether the reported concentration represents free drug or total drug in solution. Where micelles or other solubilizing components are present, total apparent solubility can exceed free-drug solubility. Use the value that corresponds to the dissolved state represented by the model; do not insert a total apparent value as a free-drug saturation concentration in a model that represents free and solubilized drug separately.
Solubility can vary with pH and medium. A value measured in one buffer should not silently become a universal constant across compartments. If a model uses a single solubility value, state the approximation and the conditions under which it is intended to apply. When measurements are available at several conditions, retain those condition-specific values and map them to the corresponding model conditions rather than averaging them without justification.
Worked Example: Checking a Solubility-Based Capacity
In a hypothetical test, the measured total apparent solubility is \(0.25\ \mathrm{mg/mL}\) in a medium containing solubilizing components. The dissolution compartment contains \(250\ \mathrm{mL}\) of that medium. Calculate the maximum dissolved-state amount implied by this value, and state when the calculation is appropriate.
The capacity calculation is concentration multiplied by volume: \(0.25\ \mathrm{mg/mL}\times250\ \mathrm{mL}=62.5\ \mathrm{mg}\). Thus, if the model’s dissolved state is defined to include the drug represented by this total apparent-solubility measurement, its equilibrium dissolved capacity under these conditions is 62.5 mg.
This does not mean that 62.5 mg must dissolve, nor does it establish the dissolution rate. If the model instead defines its saturation concentration as free drug only, this apparent-solubility value is not the appropriate input unless the model also represents the solubilized drug consistently. The medium, temperature, pH, and measurement definition must match the model application.
Converting Profiles into a Mass-Conserving Input
A concentration-time profile can inform a dissolution model, but concentration alone is not mass. To compare it with dose or solid mass, multiply by the relevant well-mixed volume, taking account of sampling, replacement, dilution, and any other losses or additions. A measured concentration may also reflect precipitation or other processes, so a decline in concentration does not necessarily mean that the drug has disappeared from the modeled system.
One common rate representation is a surface-area-driven dissolution rate. Let \(R_{\mathrm{dis},i}\) be the nonnegative dissolution rate from solid size bin \(i\), \(k_d\) a dissolution mass-transfer coefficient with units of length per time, \(A_i\) the exposed area, \(C_s\) the applicable saturation concentration, and \(C\) the modeled dissolved concentration. A simple form is:
With area in \(\mathrm{mm^2}\), concentration in \(\mathrm{mg/mm^3}\), and \(k_d\) in \(\mathrm{mm/min}\), the rate has units of \(\mathrm{mg/min}\). This equation is a model assumption, not a direct consequence of any one profile. Its use requires compatible definitions of \(C_s\) and \(C\), appropriate mixing, and a justified representation of wetted surface area. If the model includes precipitation and redissolution, those processes must be represented as transfers among states, rather than by allowing dissolved mass to vanish.
For a closed dissolution compartment with solid mass \(M_{\mathrm{solid}}\), dissolved mass \(M_{\mathrm{diss}}\), and precipitated mass \(M_{\mathrm{precip}}\), the total is conserved when dissolution, precipitation, and redissolution are internal transfers. If drug can also leave the compartment, track cumulative transferred mass \(M_{\mathrm{out}}\) explicitly:
The earlier material on mass-conserving dissolution, precipitation, and redissolution establishes the importance of representing these processes as transfers. For input construction, the practical implication is straightforward: check the accounting over time, including the drug remaining in every modeled state.
Worked Example: Turning a Profile into Interval Rates
A hypothetical dissolution test starts with 100 mg of drug in a well-mixed volume. After correcting the measured concentrations for sampling and converting them to dissolved amounts, cumulative dissolved masses are 0, 20, 36, and 48 mg at 0, 10, 20, and 30 minutes. Calculate the interval-average dissolution rates and check the mass balance at 30 minutes. Assume there is no precipitation or transfer out during this test.
For each interval, the average rate is the increase in dissolved mass divided by elapsed time. From 0 to 10 minutes, the rate is \((20-0)\ \mathrm{mg}/10\ \mathrm{min}=2.0\ \mathrm{mg/min}\). From 10 to 20 minutes, it is \((36-20)/10=1.6\ \mathrm{mg/min}\). From 20 to 30 minutes, it is \((48-36)/10=1.2\ \mathrm{mg/min}\).
At 30 minutes, 48 mg is dissolved and \(100-48=52\) mg remains solid. Their sum is \(48+52=100\) mg, equal to the dose. The interval rates are summaries of this experiment; they are not automatically transferable as constant rates to another particle distribution, medium, or volume. A mechanistic model can use them to inform parameter estimation, while the mass balance and experimental conditions constrain the interpretation.
Independent Profile Checking and Failure Modes
A profile used to estimate an input cannot, by itself, provide an independent test of that input. Reserve a separate profile, run, or relevant condition for checking predictions. An informative check compares observed and predicted quantities on the same scale, at the same times, and with matching experimental corrections. Depending on the intended application, the check may use a different batch, particle-size distribution, medium, or dose. The choice should test a meaningful prediction rather than merely repeat the calibration conditions.
Before interpreting agreement, check units, dose accounting, and the experimental protocol. For a predicted cumulative dissolved amount, negative values or amounts above the available dose indicate an error in the model or its implementation. For a concentration profile, verify that volume changes and sample removal are handled consistently. Examine the time course of residuals, not only an overall summary: systematic early or late deviations may reveal a mismatch in the input representation even when the average error is small.
Worked Example: Checking Against an Independent Profile
Suppose a hypothetical model input was estimated using one dissolution run. Without refitting, the model predicts cumulative dissolved amounts of 19, 35, and 49 mg at 10, 20, and 30 minutes in a separate run. The independent run measured 20, 36, and 48 mg at those times. Calculate the prediction residuals as observed minus predicted, and their root mean squared error (RMSE).
The residuals are \(20-19=1\) mg, \(36-35=1\) mg, and \(48-49=-1\) mg. The RMSE is \(\sqrt{[1^2+1^2+(-1)^2]/3}=\sqrt{1}=1.00\ \mathrm{mg}\). The predictions are close to these three observations, with errors of at most 1 mg at the checked times.
This result describes agreement for the specified independent profile and time points; it does not establish that the input is valid for every formulation or physiological condition. At 30 minutes, the observed amount is 48 mg and the predicted amount is 49 mg, both below the 100 mg dose. If the modeled states include only solid and dissolved drug for this comparison, their corresponding remaining solid amounts are 52 and 51 mg, respectively. Both comparisons respect the dose balance.
Choosing a Representation and Documenting Its Limits
A size-resolved, mechanistic representation is useful when particle sizes, area, and changing saturation conditions are relevant to the prediction. A simpler empirical release curve can be useful for interpolation within the conditions where it was measured, but it does not necessarily transfer when dose, medium, particle size, or fluid volume changes. A single representative particle size reduces complexity, but may conceal the contribution of fine particles to surface area. The appropriate choice depends on the intended predictions and on the information the experiments can support.
| Representation | Useful when | Main limitation |
|---|---|---|
| Size-resolved particle bins | Particle-size effects and changing surface area matter | Needs suitable size data and assumptions about shape, density, and wetting |
| Single representative size | A compact approximation is adequate for the intended use | Can misrepresent area for a broad distribution |
| Empirical profile or interval rates | Describing or interpolating a measured condition | May not transfer to different conditions without a justified model |
Document the raw measurement, units, conversion, model state it informs, and assumptions made at each step. In particular, state how a reported particle-size weighting was interpreted, whether a solubility value is free or apparent, how sampling affected profile calculations, and which data were held back for independent checking. These records make it possible to distinguish a poor model structure from a data-conversion or unit error.
Common Mistakes and Practical Checks
- Using a particle-size mean as if it determined total area. Area depends on the distribution and, for spherical bins, is proportional to mass divided by diameter. Preserve the bins when their contributions differ materially.
- Mixing free and apparent solubility. A total solubilized concentration is not automatically the free-drug saturation concentration. Match the measurement to the model’s state definition.
- Treating an observed release curve as a universal rate law. A profile is conditional on its test setup. Use it to inform a model, and assess transfer with data not used to construct the input.
- Ignoring sampling or precipitation in a mass calculation. Concentration multiplied by current volume may not equal cumulative dissolved mass when samples were removed, volume changed, or drug precipitated.
- Checking only a fitted profile. Fit quality is not an independent predictive check. Reserve an appropriate separate profile and compare predictions without refitting.
Check Your Understanding
Answer the questions, then compare with the solutions below.
- Under what assumptions can a volume-weighted particle-size distribution be used as a mass-weighted distribution?
- A model represents free drug and micelle-bound drug separately. Which solubility measurement should be used as its free-drug saturation concentration: free solubility or total apparent solubility? Explain.
- A 60 mg dose has 18 mg dissolved, 7 mg precipitated, and 5 mg transferred out of the compartment. How much solid drug remains if the mass balance holds?
- Why should a profile used for independent checking not also be used to estimate the input?
Solutions. (1) This conversion is justified when particles have a common density and the measured volume fractions represent the same solid material; shape or composition differences may require further corrections. (2) Use the free-drug solubility, because the model distinguishes free from micelle-bound drug; total apparent solubility includes more than the free state. (3) The balance gives \(60-18-7-5=30\) mg of solid remaining. Checking all states gives \(30+18+7+5=60\) mg. (4) Reusing the profile for estimation and checking does not provide an independent test: the input was selected to agree with those data. A separate profile evaluates predictive agreement without that direct reuse.