Two Distinct Noise Sources in Concentration-Time Data

Two different locations for randomness

A concentration-time record is not the same thing as the concentration trajectory that generated it. A pharmacokinetic model may describe an underlying state that changes between sampling times, while an assay reports a value at a particular time. Randomness can enter at either stage: it can perturb the evolving state, or it can perturb the reported assay result conditional on the state.

These are different scientific claims. Process noise describes random changes in the state as it evolves. Observation error describes random deviations in the measurement conditional on the state at the sampling time. The two may both be plausible in a study, but neither should be used as a label for all unexplained scatter.

Definition: Let $X_t$ denote the latent pharmacokinetic state at time $t$, such as a concentration or a vector of compartment amounts and concentrations. Process noise is randomness in the evolution of $X_t$, conditional on its initial value and the model parameters. Observation error is randomness in an assay result $Y_j$ conditional on the state $X_{t_j}$ at the assay time $t_j$. A general observation model is $Y_j=h(X_{t_j})+\varepsilon_j$, where $h$ maps the state to the assay scale and $\varepsilon_j$ is the measurement deviation.

For example, if clearance fluctuates within a subject in a way represented as random forcing of the state dynamics, that is a process-noise explanation. If the underlying concentration is fixed at the time of sampling but the assay reports a value above or below it, that is an observation-error explanation. Between-subject differences in fixed parameters are not automatically either one: the classification depends on the model and the scientific source being represented.

Process noise in a concentration state

A common mathematical representation of process noise is an SDE,

dX_t=b(X_t,t)\,dt+G(X_t,t)\,dW_t.

Here $b$ is the drift, which specifies the systematic direction and rate of change, and $G$ scales the random forcing. $W_t$ is a standard Brownian motion (or a vector of such processes); the state is adapted to the information available through time. In a usual well-posedness setup, the coefficients are measurable in time, locally Lipschitz in the state, and satisfy a linear-growth bound, which gives a unique nonexplosive solution under standard conditions. A zero diffusion coefficient $G$ leaves deterministic state evolution; a nonzero coefficient allows random state paths.

The key interpretation is not just that the state is uncertain. It is that randomness enters its evolution. A process perturbation can change the state at one time and thereby affect later states. By contrast, an assay error changes the reported value, not the underlying trajectory. The diffusion scale has units of state per square root of time, so its interpretation depends on the units used for time and concentration.

Theorem: Suppose $k>0$, $q$ is constant, $C_0=c_0$ is fixed, and the concentration state satisfies $dC_t=-kC_t\,dt+q\,dW_t$. Then $$ C_t\sim N\!\left(c_0e^{-kt},\frac{q^2}{2k}(1-e^{-2kt})\right). $$ Thus the state itself has nonzero variance at every $t>0$ when $q\ne 0$.

Derivation. Multiply the SDE by the integrating factor $e^{kt}$. Itô's product rule gives $d(e^{kt}C_t)=q e^{kt}\,dW_t$, and hence

C_t=c_0e^{-kt}+q\int_0^t e^{-k(t-s)}\,dW_s.

The integral has a deterministic integrand. It is Gaussian with mean zero and, by the Itô isometry, variance $\int_0^t e^{-2k(t-s)}\,ds=(1-e^{-2kt})/(2k)$. Multiplying by $q$ gives the stated mean and variance. For $t>0$, $1-e^{-2kt}>0$, so the variance is positive if $q\ne0$. $\square$

Worked Example: Quantifying Process Variation

Suppose a concentration state follows $dC_t=-0.2C_t\,dt+0.3\,dW_t$, starts at $C_0=10$ mg/L, and time is measured in hours. Find the mean and standard deviation of the state after 2 hours.

The theorem applies with $k=0.2$, $q=0.3$, $c_0=10$, and $t=2$. The mean is

E[C_2]=10e^{-0.2(2)}=10e^{-0.4}\approx 6.703\ \text{mg/L}.

The variance is

\operatorname{Var}(C_2)=\frac{0.3^2}{2(0.2)}(1-e^{-2(0.2)(2)}) =0.225(1-e^{-0.8}) \approx0.1239\ (\text{mg/L})^2.

Therefore $\operatorname{SD}(C_2)=\sqrt{0.1239}\approx0.352$ mg/L. This spread is in the latent state, not an assay result. The calculation also exposes a limitation of additive Gaussian process noise: it assigns some probability to negative concentrations, so it may be a poor model when that possibility is scientifically unacceptable.

Observation error conditional on the state

An assay model describes the distribution of a result when the state at sampling is held fixed. For an additive, homoscedastic Gaussian error model, write $Y_j=h(X_{t_j})+\varepsilon_j$, where $\varepsilon_j\sim N(0,R)$ and the error is independent of the latent state. Then the conditional distribution is

Y_j\mid X_{t_j}=x\sim N(h(x),R).

This follows by fixing $X_{t_j}=x$: the quantity $h(x)$ is then a constant, and adding it to a normal error with mean zero and variance $R$ shifts the mean but does not change the variance. Consequently,

E[Y_j\mid X_{t_j}=x]=h(x),\qquad \operatorname{Var}(Y_j\mid X_{t_j}=x)=R.

The Gaussian additive model is an assumption, not part of the definition of observation error. A proportional-error model, a log-scale model, or an assay model with a mean-dependent variance may be more appropriate for particular data. The observation model must be stated on the scale on which the assay is analyzed.

Worked Example: Interpreting an Assay Result at a Fixed State

At a particular sampling time, suppose the latent concentration is $4.0$ mg/L, the assay reports concentration on the original scale, and its error is $N(0,0.5^2)$ mg/L. Find the conditional probability that the assay result lies between 3.5 and 4.5 mg/L.

Conditional on the state, $Y\sim N(4.0,0.25)$. Standardizing the two limits gives

P(3.5<Y<4.5\mid C=4.0) =P\!\left(\frac{3.5-4.0}{0.5}<Z<\frac{4.5-4.0}{0.5}\right) =P(-1<Z<1)\approx0.6827,

where $Z\sim N(0,1)$. This probability concerns assay variation around a fixed state. It does not say that the concentration itself varies over time, and this observation error does not alter the state after the sample is collected.

What an observed discrepancy can and cannot tell you

Suppose a model predicts 5.0 mg/L at a sampling time and the assay reports 5.4 mg/L. That difference alone does not establish whether the latent state was higher than predicted, the assay was high relative to the state, or both. The distinction depends on the proposed mechanisms and on information in the study design and data. A single deviation is not a label for its own cause.

The following contrast is useful when specifying a model. It describes where each source enters, not a method for estimating the sources from data.

Question	Process noise	Observation error
Where does randomness enter?	In the state dynamics between and at observation times	In the assay result conditional on the state
What is random?	The evolving latent state $X_t$	The deviation of $Y_j$ from $h(X_{t_j})$
Does it change the underlying trajectory?	Yes; random forcing changes the state path	No; it changes the reported value, not the state
Typical model location	The diffusion term of an SDE	The conditional distribution of $Y_j\mid X_{t_j}$

Worked Example: Classifying a Concentration Discrepancy

A deterministic model predicts 5.0 mg/L at 8 hours, and the assay result is 5.4 mg/L. Consider two distinct proposed explanations, each in isolation.

Process-noise explanation. The state trajectory was randomly perturbed, so the latent concentration at 8 hours could be 5.4 mg/L even if the deterministic reference prediction is 5.0 mg/L. The discrepancy is attributed to a difference in the state, not to the assay conditional on that state.

Observation-error explanation. The latent concentration at 8 hours is 5.0 mg/L, and the assay deviation is $5.4-5.0=0.4$ mg/L. If the assay standard deviation were 0.2 mg/L, this is $0.4/0.2=2$ standard deviations above the conditional mean. That makes it a relatively large positive assay deviation under a Gaussian model, but it does not make the result impossible.

Both accounts can describe the same recorded number, but they make different claims about the unobserved state. With only this discrepancy, the source is not identified. Independent scientific knowledge and an appropriate study design are needed to assess the competing explanations.

Diagnostics, alternatives, and limitations

Check that each proposed source corresponds to a credible mechanism. A process-noise term should represent random state evolution, and its scale should have meaningful units. An observation-error distribution should describe assay deviations conditional on the state and respect the assay scale. If the assay error is stated as constant in absolute units despite clear concentration-dependent assay variability, its conditional variance assumption may be inappropriate.

Other warning signs include treating all observed scatter as assay error when the model requires random changes in the state, or adding process noise simply because predictions are imperfect. Systematic bias, an incorrect structural model, and between-subject parameter differences are not automatically process noise or observation error. In particular, a persistent mismatch in one direction may indicate a biased assay or a misspecified model rather than mean-zero random error.

A deterministic ODE is the natural alternative when state evolution is treated as fully specified conditional on initial conditions and parameters. An SDE is appropriate when random state evolution is part of the scientific model. For either choice, the assay can still have its own conditional error model. These alternatives are not distinguished by whether a concentration-time plot looks scattered; they are distinguished by the location and mechanism assigned to the randomness.

Neither source is directly observed just because measurements are available. Assay error is conditional on a latent state, and a state path is generally not observed continuously. Consequently, the distinction can be hard to establish with sparse data. The definitions clarify what a model asserts, but do not guarantee that available observations can separate competing assertions. The introductory material on sparse sampling explains why limited observation times can obscure variability sources.

Key takeaway: Process noise randomizes the pharmacokinetic state as it evolves. Observation error randomizes the assay result conditional on that state. Always say which quantity is random and at what stage of the data-generating process.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

A state follows an SDE with a nonzero diffusion term. Which quantity does that term randomize: the latent state or the assay result?
If $Y\mid X=x\sim N(x,0.3^2)$, give the conditional mean and variance when $x=2.0$.
A measured value is above a deterministic prediction. Can that observation alone determine whether process noise or observation error caused the difference? Explain.
Give one reason an additive Gaussian process-noise model may be unsuitable for a concentration state.

Solutions. (1) It randomizes the evolving latent state; assay error is specified separately in the conditional observation model. (2) The conditional mean is 2.0 and the conditional variance is $0.3^2=0.09$. (3) No. The state may differ from the prediction, the assay may deviate from the state, or both mechanisms may be present; the one observed difference does not identify its source. (4) An additive Gaussian state model can assign positive probability to negative concentrations, which may be scientifically impossible or unacceptable for the intended application.