Affine Linear SDEs for Pharmacokinetic States

Why Use an Affine Linear State Model?

A pharmacokinetic model may have several states, such as amounts in a depot, central compartment, and peripheral compartment. Writing each stochastic equation separately is useful for interpretation, but a matrix form makes it easier to check dimensions and to distinguish deterministic inputs from continuous process noise. The affine linear form also provides a common specification for models with linear transfer and elimination, constant inputs, additive noise, or noise proportional to the states.

The word affine means that each coefficient is a linear function of the state plus a constant term. The word linear in this model name does not mean that every term in the SDE is homogeneous: the drift vector $b$ and diffusion vectors $g_j$ allow nonzero constants.

Definition: Let $X_t\in\mathbb{R}^p$ be a column vector of $p$ pharmacokinetic states, and let $W_t=(W_{1,t},\ldots,W_{m,t})^\mathsf{T}$ contain $m$ independent standard Brownian motions. An affine linear Itô SDE with constant coefficients is $$ dX_t=(AX_t+b)\,dt+\sum_{j=1}^{m}(G_jX_t+g_j)\,dW_{j,t}. $$ Here $A,G_j\in\mathbb{R}^{p\times p}$ and $b,g_j\in\mathbb{R}^{p}$. Thus the state and each drift or diffusion vector have $p$ components, while there are $m$ scalar Brownian drivers.

The state components may be amounts, concentrations, or another set of chosen model states, but the choice must be consistent throughout the equations. The vector $AX_t+b$ is the drift: it describes the local deterministic rate of change conditional on the current state. For driver $j$, the vector $G_jX_t+g_j$ is its diffusion coefficient. It has one entry for each state and multiplies the scalar increment $dW_{j,t}$.

The matrices are not interchangeable. $A$ specifies deterministic transfer and loss; $G_j$ specifies how driver $j$ affects each state and how that effect depends on the current state. The vector $b$ represents constant drift input, such as a constant infusion over an interval. A nonzero $g_j$ represents state-independent stochastic forcing by driver $j$.

Dimensions and Compartment Interpretation

For a two-state model with $m=2$ Brownian drivers, $X_t$ is $2\times1$, $A$ and each $G_j$ are $2\times2$, and $b$ and each $g_j$ are $2\times1$. The diffusion can also be displayed as a $p\times m$ matrix whose $j$th column is $G_jX_t+g_j$:

\Gamma(x)= \begin{bmatrix} \vert & & \vert\\ G_1x+g_1 & \cdots & G_mx+g_m\\ \vert & & \vert \end{bmatrix} \in\mathbb{R}^{p\times m}.

This is a useful dimension check: multiplying $\Gamma(X_t)$ by the $m$-vector of Brownian increments produces a $p$-vector of state increments. If model errors or measurement errors are represented elsewhere, they should not be silently folded into these continuous-time process-noise terms.

In a compartment model, an off-diagonal entry $A_{ik}$ can describe transfer from state $k$ into state $i$, while negative diagonal entries commonly represent total first-order loss from a compartment. The exact signs depend on how the balances are written. Each column of $A$ describes the effects of one state on all drift components; it is not, in general, a list of independent compartment-specific elimination rates.

Worked Example: A Depot State With Input and Multiplicative Noise

Let $X_t$ be the amount in an absorption depot, measured in milligrams. Suppose the depot has first-order absorption at rate $k_a=0.25$ per hour and receives a constant input $R=2$ milligrams per hour over the interval considered. Let the process noise be proportional to the depot amount, with coefficient $0.10$ per square-root hour. Write the model and identify its dimensions.

The drift rate is input minus absorption: $$ -0.25X_t+2. $$ The single Brownian driver $W_t$ contributes diffusion $0.10X_t$. Consequently, $p=m=1$, and the scalar coefficients are $A=-0.25$, $b=2$, $G_1=0.10$, and $g_1=0$:

dX_t=(-0.25X_t+2)\,dt+(0.10X_t)\,dW_t.

The drift has units of milligrams per hour. Since $dW_t$ has units of square-root hours, the diffusion coefficient has units of milligrams per square-root hour, as required for the stochastic increment to have units of milligrams. This specification is affine linear, but its state is not guaranteed to remain nonnegative merely because it represents an amount.

Worked Compartment Specifications

Worked Example: Two Compartments With Two Brownian Drivers

Let $X_{1,t}$ and $X_{2,t}$ be amounts in central and peripheral compartments, in milligrams. Use transfer rates $k_{12}=0.30$, $k_{21}=0.10$ per hour, and central elimination rate $k_{10}=0.20$ per hour. A constant input of $R=5$ milligrams per hour enters the central compartment. Let two independent Brownian drivers have the following diffusion matrices, with units chosen so each diffusion coefficient is amount per square-root hour:

A= \begin{bmatrix} -(k_{10}+k_{12}) & k_{21}\\ k_{12} & -k_{21} \end{bmatrix} = \begin{bmatrix} -0.50&0.10\\ 0.30&-0.10 \end{bmatrix}, \qquad b=\begin{bmatrix}5\\0\end{bmatrix},

G_1= \begin{bmatrix} -0.02&0.04\\ 0.02&-0.04 \end{bmatrix}, \qquad G_2= \begin{bmatrix} 0.01&0\\ 0&-0.03 \end{bmatrix}, \qquad g_1=g_2=\begin{bmatrix}0\\0\end{bmatrix}.

The model has $p=2$ states and $m=2$ independent scalar Brownian drivers. Its drift is $$ AX_t+b= \begin{bmatrix} -0.50X_{1,t}+0.10X_{2,t}+5\\ 0.30X_{1,t}-0.10X_{2,t} \end{bmatrix}. $$ At the state $x=(20,5)^\mathsf{T}$ milligrams, the first diffusion column is $$ G_1x= \begin{bmatrix} -0.02(20)+0.04(5)\\ 0.02(20)-0.04(5) \end{bmatrix} = \begin{bmatrix}-0.2\\0.2\end{bmatrix}. $$ In particular, the two entries are $-0.2$ and $0.2$, respectively. The second column is $$ G_2x= \begin{bmatrix} 0.01(20)\\ -0.03(5) \end{bmatrix} = \begin{bmatrix}0.2\\-0.15\end{bmatrix}. $$ Thus, at this state, the $2\times2$ diffusion matrix has columns $(-0.2,0.2)^\mathsf{T}$ and $(0.2,-0.15)^\mathsf{T}$. The entries describe the coefficients multiplying $dW_{1,t}$ and $dW_{2,t}$; they are not themselves Brownian increments.

Worked Example: Separating Additive and State-Dependent Noise

Consider a central amount $X_{1,t}$ and peripheral amount $X_{2,t}$, with drift $AX_t$, where $$ A=\begin{bmatrix}-0.4&0.1\\0.2&-0.2\end{bmatrix}. $$ Suppose a single driver produces constant diffusion $g_1=(0.3,0.1)^\mathsf{T}$ and state-dependent diffusion with $$ G_1=\begin{bmatrix}0.01&0\\0&0.02\end{bmatrix}. $$ Then $p=2$ and $m=1$. The full model is $$ dX_t= \begin{bmatrix}-0.4X_{1,t}+0.1X_{2,t}\\0.2X_{1,t}-0.2X_{2,t}\end{bmatrix}dt+ \begin{bmatrix}0.01X_{1,t}+0.3\\0.02X_{2,t}+0.1\end{bmatrix}dW_{1,t}. $$ At $x=(10,5)^\mathsf{T}$, the drift is $(-3.5,1.0)^\mathsf{T}$, since $-0.4(10)+0.1(5)=-3.5$ and $0.2(10)-0.2(5)=1.0$. The diffusion vector is $(0.4,0.2)^\mathsf{T}$, since $0.01(10)+0.3=0.4$ and $0.02(5)+0.1=0.2$. The constants $g_1$ add to the noise coefficient; they do not add to the drift.

Regularity and an Immediate Covariance Result

A finite-dimensional affine coefficient has useful regularity. This does not guarantee that the model is biologically appropriate, but it rules out coefficient blow-up at finite states and supports the standard existence and uniqueness result for SDEs.

Result: For constant finite matrices $A,G_1,\ldots,G_m$ and vectors $b,g_1,\ldots,g_m$, the drift $a(x)=Ax+b$ and the combined diffusion $\Gamma(x)=[G_1x+g_1,\ldots,G_mx+g_m]$ are globally Lipschitz and have at most linear growth. In particular, with an initial state $X_0$ that is square-integrable and measurable at time zero, and a usual filtration supporting the independent Brownian drivers, the global Lipschitz existence and uniqueness theorem gives a unique nonexplosive strong solution.

To verify the coefficient conditions, use the Euclidean norm for vectors and its induced operator norm for matrices. For any $x,y\in\mathbb{R}^p$,

|a(x)-a(y)|=|A(x-y)|\leq \|A\|\,|x-y|, \qquad \|\Gamma(x)-\Gamma(y)\|_{\mathrm F}^2 =\sum_{j=1}^{m}|G_j(x-y)|^2 \leq\left(\sum_{j=1}^{m}\|G_j\|^2\right)|x-y|^2.

Here $\|\cdot\|_{\mathrm F}$ is the Frobenius norm. Also, $|Ax+b|^2\leq 2\|A\|^2|x|^2+2|b|^2$, and $\|\Gamma(x)\|_{\mathrm F}^2\leq 2\sum_j(\|G_j\|^2|x|^2+|g_j|^2)$. These are the required linear-growth bounds. Together with the stated initial-state and Brownian assumptions, they verify the hypotheses of the global Lipschitz theorem. This establishes well-posedness for the specified affine SDE; it does not establish positivity or pharmacological plausibility. $\square$

The independent Brownian drivers also give a simple expression for the instantaneous covariance of the state increments.

Result: At state $x$, define $h_j(x)=G_jx+g_j$. The instantaneous covariance-rate matrix is $$ Q(x)=\sum_{j=1}^{m}h_j(x)h_j(x)^\mathsf{T}. $$ It is a $p\times p$ positive semidefinite matrix; its $(i,k)$ entry is the instantaneous covariance rate between state increments $i$ and $k$.

For a derivation, write the stochastic increment in component $i$ as $\sum_j h_{ij}(X_t)dW_{j,t}$, where $h_{ij}$ is entry $i$ of $h_j$. Independent standard Brownian motions satisfy $d[W_j,W_\ell]_t=0$ for $j\ne\ell$ and $d[W_j,W_j]_t=dt$. The finite-variation drift has zero quadratic covariation with the Brownian terms. Therefore,

d[X_i,X_k]_t =\sum_{j=1}^{m}h_{ij}(X_t)h_{kj}(X_t)\,dt, \qquad \frac{d[X]_t}{dt}=Q(X_t).

For any $v\in\mathbb{R}^p$, $v^\mathsf{T}Q(x)v=\sum_j(v^\mathsf{T}h_j(x))^2\geq0$, proving positive semidefiniteness. This result concerns local quadratic variation, not the full distribution or a long-term covariance of $X_t$. $\square$

Assumptions, Diagnostics, and Limitations

Before interpreting a fitted or proposed model, check both its dimensions and its stochastic assumptions. The displayed form uses independent standard scalar Brownian drivers. If a model instead begins with correlated Brownian drivers, their covariance must be accounted for, for example by expressing them as a linear transformation of independent drivers and adjusting the diffusion columns accordingly. Counting correlated drivers as independent without such an adjustment changes the implied covariance.

Check matrix sizes. $AX+b$ and every $G_jX+g_j$ must each have $p$ entries. There must be exactly $m$ diffusion columns for $m$ scalar drivers.
Check units. Drift coefficients and inputs must give state units per time. Diffusion coefficients must give state units per square-root time.
Check the compartment balance. A transfer leaving one state and entering another should appear with compatible rates and signs in the corresponding drift components.
Check the driver interpretation. A shared Brownian driver can affect several states simultaneously; separate drivers encode separate independent noise sources under this specification.

A common mistake is to read $G_jX+g_j$ as a vector of noise realizations, or to multiply each component by an unrelated Brownian motion. It is instead the coefficient vector for the single increment $dW_{j,t}$. For example, with $p=2$ and $m=1$, the two entries of $G_1X+g_1$ both multiply the same $dW_{1,t}$. Another mistake is to infer nonnegative states from nonnegative starting amounts. General additive or linear Gaussian-type diffusion can cross a zero boundary; global existence does not prevent that. A model that requires nonnegative states needs additional structural conditions or a different specification.

The affine linear form is useful when drift is based on first-order transfer and when a linear state dependence is a reasonable approximation for process noise. It does not represent saturable elimination or other nonlinear drift without approximation, and its constant coefficients do not describe time-varying rates unless the model is extended to time-dependent coefficients. The matrix form is a specification, not a guarantee that its noise mechanism is scientifically justified. The introductory material on writing a pharmacokinetic SDE distinguishes state, drift, diffusion, and Wiener notation; here the focus is on organizing them consistently in the affine multistate form.

Check Your Understanding

Answer each question, then compare with the brief solutions.

If $X_t\in\mathbb{R}^3$ is driven by four independent Brownian motions, what are the dimensions of $A$, $b$, $G_j$, $g_j$, and $\Gamma(x)$?
For $h(x)=\begin{bmatrix}0.1x_1\\0.2x_2\end{bmatrix}$ with one driver, find $Q(x)$ at $x=(2,3)^\mathsf{T}$.
Explain why a constant infusion belongs in $b$, while a constant additive noise coefficient belongs in $g_j$.
Does global Lipschitz well-posedness imply that every amount remains nonnegative? Give the reason.

Solutions. (1) $A,G_j\in\mathbb{R}^{3\times3}$, $b,g_j\in\mathbb{R}^3$, and $\Gamma(x)\in\mathbb{R}^{3\times4}$. (2) $h(x)=(0.2,0.6)^\mathsf{T}$, so $Q(x)=h(x)h(x)^\mathsf{T}=\begin{bmatrix}0.04&0.12\\0.12&0.36\end{bmatrix}$. (3) Infusion is a deterministic rate and therefore contributes to the $dt$ coefficient; additive process noise multiplies a Brownian increment and therefore contributes to a diffusion vector. (4) No. The existence theorem controls solution existence and uniqueness, not the direction of sample paths at the boundary; a diffusion can cross zero unless additional structural conditions prevent it.