Concentration Units And Diffusion Scale Identifiability

Why Concentration Units Matter for Diffusion

A diffusion parameter is tied to the scale in which the state is expressed. If a concentration is reported in a different unit, the numerical size of a diffusion coefficient may change even though the physical process has not. A separate problem arises when the assay scale itself is unknown: the observed fluctuations may then identify only a combination of the physical diffusion scale and the calibration factor.

The distinction is important in pharmacokinetics. A known conversion between units changes the numerical parameter in a predictable way. An unknown calibration factor, by contrast, can make distinct physical diffusion scales produce exactly the same distribution of observed data. Whether that ambiguity occurs depends on how the diffusion depends on concentration.

Definition: A concentration rescaling is a transformation \(X=qC\), where \(C\) is a concentration state and \(q>0\) is a fixed scale factor. A calibration factor \(a>0\) relates an assay-scale observation \(Y\) to the physical concentration by \(Y=aC\). A parameter is structurally identifiable from these observations if distinct parameter values cannot produce the same probability law for the observations under the stated model and design.

Transforming an SDE to New Concentration Units

Suppose the physical concentration satisfies the Itô SDE

dC_t=b(C_t,t)\,dt+g(C_t,t)\,dW_t.

Here \(b\) has units of concentration per unit time. Since Brownian increments scale as the square root of time, \(g\) has units of concentration per square root of time. For a known conversion \(X_t=qC_t\), with time units unchanged, the transformed equation is

dX_t=q\,b(X_t/q,t)\,dt+q\,g(X_t/q,t)\,dW_t.

The argument \(X_t/q\) matters: the coefficient functions must be evaluated at the concentration expressed in the original units. The entire diffusion term is multiplied by \(q\), but a parameter inside \(g\) need not itself be multiplied by \(q\) if the function’s state dependence also changes under substitution.

Theorem: If \(C_t\) solves the Itô SDE above and \(X_t=qC_t\) for a fixed \(q>0\), then \(X_t\) solves the transformed SDE above, with initial condition \(X_0=qC_0\). The result holds up to the lifetime of the solution; for a global statement, assume the original and transformed equations have solutions on the time interval considered.

Proof. The map \(f(c)=qc\) has derivative \(f'(c)=q\) and second derivative \(f''(c)=0\). Itô’s formula therefore gives

dX_t=df(C_t)=q\,dC_t =q\,b(C_t,t)\,dt+q\,g(C_t,t)\,dW_t.

Since \(C_t=X_t/q\), substituting this identity into both coefficients gives the claimed equation. The zero second derivative is why no additional Itô correction appears. \(\square\)

Worked Example: Converting an Additive Diffusion from mg/L to µg/L

Suppose concentration, measured in mg/L, follows \(dC_t=-0.4C_t\,dt+0.03\,dW_t\), with time in hours. Express the model in µg/L, where \(X_t=1000C_t\).

The drift transforms to \(1000[-0.4(X_t/1000)]=-0.4X_t\). The additive diffusion transforms to \(1000(0.03)=30\). Thus

dX_t=-0.4X_t\,dt+30\,dW_t.

The diffusion coefficient is \(0.03\) mg/L per square-root hour in the first equation and \(30\) µg/L per square-root hour in the second. These are the same physical variability in different units. In this additive model, the concentration state is allowed to take negative mathematical values; the calculation is a scale-transformation example, not a claim that such a model is always a suitable concentration model.

State-Dependent Diffusion and Scaling

The transformation can look different when diffusion depends on concentration. Suppose the drift is linear and the diffusion is a homogeneous function of degree \(p\): \(g(c)=\sigma h(c)\), where \(h(rc)=r^p h(c)\) for positive \(r\) in the model’s state domain. Then

dC_t=-kC_t\,dt+\sigma h(C_t)\,dW_t \quad\Longrightarrow\quad dX_t=-kX_t\,dt+\sigma q^{1-p}h(X_t)\,dW_t.

Indeed, the transformed diffusion is \(q\sigma h(X_t/q)=q\sigma q^{-p}h(X_t)\). For a power function \(h(c)=c^p\) on a positive state domain, the coefficient becomes \(\sigma q^{1-p}X_t^p\). The diffusion parameter’s units are concentration to the power \(1-p\), per square-root time. In particular, an additive diffusion (\(p=0\)) scales by \(q\), while a proportional diffusion (\(p=1\)) has an unchanged numerical \(\sigma\).

Worked Example: Rescaling a Square-Root Diffusion

Let \(C_t\) be measured in mg/L and suppose its diffusion term is \(0.06\sqrt{C_t}\,dW_t\), with time in hours. Convert to µg/L using \(X_t=1000C_t\).

Here \(p=1/2\), so the transformed coefficient multiplying \(\sqrt{X_t}\) is \(\sigma q^{1-p}=0.06\sqrt{1000}\). Since \(\sqrt{1000}\approx31.6228\),

0.06\sqrt{1000}\approx1.8974, \qquad 1000(0.06)\sqrt{X_t/1000}=1.8974\sqrt{X_t}.

Thus the transformed diffusion term is approximately \(1.8974\sqrt{X_t}\,dW_t\). Its units are the square root of µg/L per square-root hour. Multiplying \(0.06\) directly by \(1000\) would be wrong: the state inside the square root also changes units.

Unknown Calibration and Absolute Diffusion Scale

Now suppose the assay reports \(Y_t=aC_t\), but the positive calibration factor \(a\) is unknown. This resembles a change of units, but the scale is no longer available as known information. Consider the model

dC_t=-kC_t\,dt+\sigma C_t^p\,dW_t, \qquad Y_t=aC_t,

where \(k>0\), \(\sigma>0\), and \(p\) are known except that the physical diffusion scale \(\sigma\) and calibration \(a\) are both unknown. For a positive concentration state and \(C^p\) diffusion, the observed-scale equation is

dY_t=-kY_t\,dt+\lambda Y_t^p\,dW_t, \qquad \lambda=\sigma a^{1-p}.

Thus the observed diffusion coefficient involves \(\sigma\) and \(a\) through \(\lambda\). For \(p\ne1\), this combination does not determine the two parameters separately.

Theorem: Suppose the model above has a unique solution on the observation interval, and the observation law is considered conditional on the observed initial value \(Y_0\). If \(p\ne1\), the conditional law of the observed process depends on \(a\) and \(\sigma\) only through \(\lambda=\sigma a^{1-p}\). Consequently, the absolute physical diffusion scale \(\sigma\) is not structurally identifiable from that law when \(a\) is unrestricted and unknown.

Proof. Substituting \(C_t=Y_t/a\) into the SDE for \(C_t\) and multiplying both sides by \(a\) gives

dY_t=-kY_t\,dt+\sigma a^{1-p}Y_t^p\,dW_t.

For any \(r>0\), define \(a'=ra\) and \(\sigma'=\sigma r^{p-1}\). Then

\sigma'(a')^{1-p} =\sigma r^{p-1}(ra)^{1-p} =\sigma a^{1-p}.

The observed SDE coefficients are therefore unchanged. With the same observed initial value, uniqueness gives the same conditional process law for every such pair \((a',\sigma')\). If \(p\ne1\), choosing \(r\ne1\) changes \(\sigma\) while preserving that law, which proves non-identifiability of \(\sigma\). \(\square\)

The condition on the initial value is consequential: an independently known absolute distribution for the physical \(C_0\) could provide scale information and break this particular ambiguity. The theorem describes what the conditional observed trajectory identifies without such an external anchor.

Worked Example: Two Physical Scales, the Same Assay-Scale Diffusion

Take additive diffusion (\(p=0\)) and an OU model \(dC_t=-0.4C_t\,dt+\sigma\,dW_t\). Suppose the assay-scale coefficient is measured in response units per square-root hour and is \(\lambda=0.12\).

The observation equation has coefficient \(\lambda=a\sigma\). For \(a=4\) response units per mg/L, \(\sigma=0.03\) mg/L per square-root hour gives \(a\sigma=4(0.03)=0.12\). But \(a=6\) and \(\sigma=0.02\) give \(a\sigma=6(0.02)=0.12\) as well. Both parameter pairs produce the same observed SDE \(dY_t=-0.4Y_t\,dt+0.12\,dW_t\).

For example, over a one-hour interval, the conditional OU transition variance is

\operatorname{Var}(Y_{t+1}\mid Y_t) =\lambda^2\frac{1-e^{-2k}}{2k} =0.12^2\frac{1-e^{-0.8}}{0.8} \approx0.00991.

It is the same for both parameter pairs. More data from this same observation model can estimate the observed-scale coefficient more precisely, but cannot distinguish these two physical scales without additional calibration information.

When Calibration Does—and Does Not—Confound Diffusion

The proportional case illustrates why the ambiguity is not automatic. If \(p=1\), then \(\lambda=\sigma a^0=\sigma\). A multiplicative assay calibration changes the concentration values but leaves the relative diffusion coefficient unchanged. For a positive process, this is also visible in the log-scale equation:

d\log Y_t=\left(-k-\frac{\sigma^2}{2}\right)dt+\sigma\,dW_t.

The unknown multiplier contributes the constant \(\log a\) to \(\log Y_t\), not to its increments. Thus, under the stated model and observation assumptions, the diffusion scale can be identified from the process’s relative fluctuations even when the absolute concentration level is not.

Worked Example: Proportional Diffusion Under Unknown Calibration

Suppose \(dC_t=-0.5C_t\,dt+0.2C_t\,dW_t\), with an unknown positive assay factor \(a\). For \(Y_t=aC_t\), the transformed model is

dY_t=-0.5Y_t\,dt+0.2Y_t\,dW_t.

The diffusion coefficient remains \(0.2\), whatever \(a\) is. Applying Itô’s formula to \(\log Y_t\) gives drift \(-0.5-0.2^2/2=-0.52\) and diffusion coefficient \(0.2\). Hence

d\log Y_t=-0.52\,dt+0.2\,dW_t.

The calibration shifts the log-concentration level by \(\log a\), but does not change its increments. This example assumes direct observation of the assay-scale process without an additional unknown observation-noise scale.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to write the full observed-scale equation before interpreting any fitted diffusion parameter. Substitute the calibrated state into both drift and diffusion, then simplify the parameters that appear in the observed coefficients. If distinct physical parameter values leave all those coefficients unchanged, the ambiguity is structural, not a small-sample estimation problem.

Several kinds of information can break a calibration–diffusion scale ambiguity: assay standards that estimate \(a\); a known concentration at a reference time; or a known absolute-scale observation component under an explicitly specified measurement model. Such information must genuinely anchor the scale. Merely collecting more observations under the same uncalibrated model does not separate parameters whose observed law depends only on their product or another single combination.

This analysis compares two different approaches to scale handling. With a known unit conversion, transform the SDE and report the corresponding coefficient in the new units. With unknown calibration, treat the factor as an unknown parameter and examine the induced observation law; do not silently set it to one. The conclusion also depends on the model: proportional diffusion is invariant to a multiplicative rescaling, while additive diffusion is not. More generally, the homogeneous power \(p\) determines the combination \(\sigma a^{1-p}\).

The conclusions have limits. A model with concentration-dependent drift beyond the linear form may introduce additional calibration dependence, so the full transformed drift must also be checked. An independently known initial concentration, known dose-to-concentration relationship, or measurement-error model with known absolute scale may supply an anchor. Conversely, if the observation process has its own unknown scale, it can introduce further ambiguities. Identifiability must be assessed for the stated model and observation design, not inferred from the diffusion term alone.

Common Pitfalls

Multiplying every diffusion parameter by the unit conversion. This is correct for additive diffusion, but state dependence must also be transformed. For \(g(C)=\sigma C^{1/2}\), the coefficient scales by \(q^{1/2}\), not \(q\).
Assuming an unknown assay factor always destroys diffusion-scale identifiability. Under proportional diffusion, the calibration cancels from the diffusion coefficient; the answer depends on the state dependence.
Calling a precise estimate identifiable. A large sample can narrow uncertainty about the observable combination \(\lambda\), while leaving infinitely many \((a,\sigma)\) pairs compatible with it.
Treating known units and unknown calibration as the same problem. A known conversion is a deterministic parameter transformation. Unknown calibration adds an unknown quantity that may be confounded with the physical diffusion scale.

Key takeaway: Under \(X=qC\), the diffusion becomes \(qg(X/q,t)\). For homogeneous diffusion \(g(C)=\sigma C^p\), this produces the scale \(\sigma q^{1-p}\). If \(q\) is unknown, additive and sublinear diffusion can confound physical diffusion scale with calibration; proportional diffusion is invariant to that multiplicative calibration under the stated model.

Check Your Understanding

Work through each question, then compare with the solutions below.

If \(X=10C\) and \(dC_t=b(C_t,t)\,dt+g(C_t,t)\,dW_t\), what are the transformed drift and diffusion coefficients?
For \(g(C)=\sigma C^{1/2}\), by what factor does the coefficient multiplying \(X^{1/2}\) change under \(X=qC\)?
In the additive OU model with unknown \(Y=aC\), which combination of \(a\) and \(\sigma\) appears in the observed diffusion coefficient?
Why does an unknown multiplicative calibration not alter \(\sigma\) when diffusion is proportional to concentration?
Name one kind of external information that can break the additive-diffusion calibration ambiguity.

Solutions. (1) The equation is \(dX_t=10b(X_t/10,t)\,dt+10g(X_t/10,t)\,dW_t\). (2) The factor is \(q^{1-1/2}=\sqrt q\), since \(q\sigma(X/q)^{1/2}=\sigma\sqrt q\,X^{1/2}\). (3) The observed coefficient is \(a\sigma\). (4) For \(p=1\), \(\sigma a^{1-p}=\sigma\), so the calibration cancels. (5) An assay standard that estimates \(a\), or a known absolute concentration at a reference time, can supply an absolute-scale anchor.