Axial Dispersion in the Dispersion Liver Model

Why Axial Dispersion Changes the Liver Model

The parallel-tube model describes blood moving through the liver along paths with no axial mixing between neighbouring locations. Real flow can depart from that idealization: transport along the organ’s length may include both forward convection and axial spreading. The dispersion liver model represents that spreading with a dispersion term while retaining a spatially distributed elimination process. Its concentration is therefore a profile along the flow direction, not a single well-mixed liver concentration.

The focus here is the model’s transport-and-elimination balance and the meaning of its boundary conditions. The equations require a consistent concentration basis and assumptions about the distribution of flow and elimination. The resulting profile can then be used to calculate the outlet concentration and the amount eliminated.

Definition: Let $z$ measure distance along a liver of length $L$, from inlet $z=0$ to outlet $z=L$. Let $C(z)$ be the blood concentration, $u$ the mean forward blood velocity, $D_{\mathrm{ax}}$ the axial dispersion coefficient, and $k$ a first-order elimination rate per unit volume. The steady dispersion model is $$ D_{\mathrm{ax}}\frac{d^2C}{dz^2}-u\frac{dC}{dz}-kC=0. $$ Here $D_{\mathrm{ax}}$ has units of length squared per time, $u$ has units of length per time, and $k$ has units of inverse time.

Deriving the Axial Balance

The balance follows from conservation of drug mass. In a one-dimensional description with constant cross-sectional area, the forward axial flux per unit area is the convective flux minus the dispersive flux:

J(z)=uC(z)-D_{\mathrm{ax}}\frac{dC}{dz}.

The minus sign means dispersion carries drug down a concentration gradient. At steady state, the change in this net flux along the organ equals the local loss to elimination. If elimination is first order, the loss per unit volume per unit time is $kC$, giving $dJ/dz=-kC$. Substituting the expression for $J$ and treating $u$ and $D_{\mathrm{ax}}$ as constant gives the stated equation.

Write the total axial flux.
Convection carries drug forward at rate $uC$ per unit area; dispersion contributes $-D_{\mathrm{ax}}\,dC/dz$.

Apply steady conservation.
The net flux decreases downstream by the local elimination rate: $dJ/dz=-kC$.

Substitute and rearrange.
$u\,dC/dz-D_{\mathrm{ax}}\,d^2C/dz^2=-kC$, or $D_{\mathrm{ax}}\,d^2C/dz^2-u\,dC/dz-kC=0$.

For a consistent clearance formulation, suppose $C$ is blood concentration, $CL_{\mathrm{int}}$ is intrinsic clearance referenced to unbound blood concentration, and $f_{u,b}$ is the unbound fraction in blood. If elimination is linear and distributed uniformly over liver volume $V_L$, then

k=\frac{f_{u,b}CL_{\mathrm{int}}}{V_L}.

This expression has units of inverse time. It assumes local unbound concentration is $f_{u,b}C$ and that intrinsic clearance is proportional to that concentration. If a model instead uses plasma concentration or defines intrinsic clearance on another basis, its binding and blood-to-plasma terms must be converted consistently rather than inserted unchanged.

Dimensionless Axial Coordinate and Boundary Conditions

Use the dimensionless coordinate $x=z/L$, so $x=0$ is the inlet and $x=1$ is the outlet. The liver volume is $V_L=AL$, where $A$ is cross-sectional area, and the flow rate is $Q=uA$. Define the dispersion number $D_N=D_{\mathrm{ax}}/(uL)$ and the dimensionless elimination term $r=kL/u$. With the clearance definition above, $r=f_{u,b}CL_{\mathrm{int}}/Q$. The balance becomes

D_N\frac{d^2C}{dx^2}-\frac{dC}{dx}-rC=0.

Two boundary conditions are needed because the balance is second order. At the inlet, the imposed inflow is the total forward flux. If $C_{\mathrm{in}}$ is the concentration entering the liver, that flux per unit area is $uC_{\mathrm{in}}$. Matching it to the internal flux at $x=0$ gives $uC(0)-(D_{\mathrm{ax}}/L)C'(0)=uC_{\mathrm{in}}$, where the prime denotes differentiation with respect to $x$. Dividing by $u$ yields the inlet condition below.

C(0)-D_NC'(0)=C_{\mathrm{in}}.

At the outlet, the usual condition is zero dispersive flux across the downstream boundary. It does not set the total outflow to zero: blood still carries drug out by convection. Zero dispersive flux means $dC/dz=0$ at $z=L$, or $C'(1)=0$.

$$ C'(1)=0. $$

These conditions represent a liver supplied by an imposed inflow and discharging into an outlet where there is no additional axial spreading beyond the modelled region. They are often called Danckwerts-type boundary conditions. The inlet concentration within the model, $C(0)$, need not equal $C_{\mathrm{in}}$: the imposed quantity is the inlet flux, which includes both convection and dispersion.

Assumptions: The one-dimensional model assumes steady state, constant forward flow and cross-sectional area, a constant axial dispersion coefficient, and a concentration profile that is adequately represented by a cross-sectional average. Elimination is linear and distributed along the length, with $k$ constant. The inlet flux is specified, and no dispersive flux crosses the outlet. Changes in flow, binding, intrinsic clearance, or geometry along the liver require a modified balance or additional state variables.

Worked Solutions

Worked Example: Solving a Dimensionless Concentration Profile

Consider a hypothetical liver with $D_N=1$, $r=2$, and $C_{\mathrm{in}}=1.00\ \mathrm{mg/L}$. Solve for $C(x)$ and the outlet concentration using the stated boundary conditions.

The equation is $C''-C'-2C=0$. Trying $C=e^{mx}$ gives the characteristic equation $m^2-m-2=0=(m-2)(m+1)$, so the general solution is $C(x)=a e^{2x}+b e^{-x}$. Its derivative is $C'(x)=2a e^{2x}-b e^{-x}$.

The outlet condition $C'(1)=0$ gives $2ae^2-be^{-1}=0$, hence $b=2ae^3$. At the inlet, $C(0)-C'(0)=C_{\mathrm{in}}$, so $(a+b)-(2a-b)=-a+2b=1.00\ \mathrm{mg/L}$. Substituting the outlet relation yields $a(4e^3-1)=1.00\ \mathrm{mg/L}$. Therefore,

a=\frac{1.00}{4e^3-1}=0.01260\ \mathrm{mg/L},\qquad b=\frac{2e^3}{4e^3-1}(1.00)=0.5063\ \mathrm{mg/L}.

The profile is $C(x)=0.01260e^{2x}+0.5063e^{-x}\ \mathrm{mg/L}$, with coefficients shown to four significant figures. Using the unrounded coefficients, the outlet concentration is $C(1)=ae^2+be^{-1}=0.2794\ \mathrm{mg/L}$. The boundary check is $C'(1)=0$, as required. Since the outlet has no dispersive flux, its total flux is convective; relative to the inlet flux, the fraction not leaving in the outlet stream is $1-C(1)/C_{\mathrm{in}}=0.7206$. Under this steady-state model, that fraction corresponds to the fraction eliminated.

Worked Example: Translating Physiological Parameters

Suppose, hypothetically, $Q=90\ \mathrm{L/h}$, $CL_{\mathrm{int}}=45\ \mathrm{L/h}$, $f_{u,b}=0.20$, $V_L=1.5\ \mathrm{L}$, $L=0.15\ \mathrm{m}$, and $D_{\mathrm{ax}}=0.45\ \mathrm{m^2/h}$. Compute the model’s rate and dimensionless parameters, taking $C$ as blood concentration.

The distributed elimination rate is $k=f_{u,b}CL_{\mathrm{int}}/V_L=(0.20)(45\ \mathrm{L/h})/(1.5\ \mathrm{L})=6.00\ \mathrm{h^{-1}}$. Convert flow to $0.090\ \mathrm{m^3/h}$. Since $A=V_L/L=0.0015\ \mathrm{m^3}/0.15\ \mathrm{m}=0.0100\ \mathrm{m^2}$, the mean velocity is $u=Q/A=9.00\ \mathrm{m/h}$.

Thus $D_N=D_{\mathrm{ax}}/(uL)=0.45\ \mathrm{m^2/h}/[(9.00\ \mathrm{m/h})(0.15\ \mathrm{m})]=0.333$. Also $r=kL/u=(6.00\ \mathrm{h^{-1}})(0.15\ \mathrm{m})/(9.00\ \mathrm{m/h})=0.100$. Independently, $f_{u,b}CL_{\mathrm{int}}/Q=(0.20)(45)/90=0.100$, confirming the conversion. The dimensionless equation for these hypothetical parameters is $0.333C''-C'-0.100C=0$, subject to the inlet-flux and outlet-gradient conditions.

Worked Example: Checking an Inlet Flux Condition

At the inlet of a hypothetical liver, suppose $D_N=0.25\ \mathrm{(dimensionless)}$, the internal profile has $C(0)=0.80\ \mathrm{mg/L}$ and $C'(0)=0.40\ \mathrm{mg/L}$ per unit $x$. What inlet concentration corresponds to this profile?

Use the inlet condition, not the assumption that the two concentrations are equal: $C_{\mathrm{in}}=C(0)-D_NC'(0)=0.80-(0.25)(0.40)=0.70\ \mathrm{mg/L}$. The gradient is expressed per unit dimensionless coordinate, so multiplying it by $D_N$ gives concentration units. This result means the net imposed flux equals the internal convective flux minus the dispersive contribution at the inlet. It does not by itself establish that the rest of the profile solves the differential equation; the equation and the outlet condition must also be satisfied.

Diagnostics, Alternatives, and Common Pitfalls

A computed profile should satisfy the differential equation throughout the domain and both boundary conditions. A useful independent check is the integrated mass balance. In dimensionless coordinates, the total flux per unit area is $u[C-D_NC']$. The inlet condition makes that equal to $uC_{\mathrm{in}}$, while the outlet condition makes it $uC(1)$. Integrating the steady balance therefore gives

u\{C_{\mathrm{in}}-C(1)\} =\int_0^L kC(z)\,dz.

This identity says that inlet flux minus outlet flux equals integrated elimination. A negative predicted concentration, an outlet slope inconsistent with zero, or a failure of this balance can indicate a sign error, an incorrectly applied boundary condition, or numerical error. Strong changes in the gradient close to a boundary also warrant checking that the spatial grid resolves the profile adequately.

The parallel-tube model is the main spatial alternative in this sequence: it represents forward flow without axial dispersion and uses a specified inlet concentration. The dispersion model adds axial spreading, so its inlet boundary condition matches total flux rather than simply setting $C(0)=C_{\mathrm{in}}$. The well-stirred model takes a different approach, representing the liver with one concentration rather than an axial profile. These models answer related clearance questions under different mixing assumptions; a profile from one should not be treated as interchangeable with another’s state variable.

A frequent mistake is to impose both $C(0)=C_{\mathrm{in}}$ and the flux-based inlet condition while retaining the second-order dispersion equation. That generally over-constrains the problem and ignores the dispersive component of inlet flux. The correction is to specify the incoming total flux through $C(0)-D_NC'(0)=C_{\mathrm{in}}$, then apply the zero-dispersive-flux condition at the outlet. Another error is to set $C(1)=0$ at the outlet; this would remove convective drug outflow and is not the stated physical boundary condition.

The model is an idealized representation, not a direct description of every anatomical feature. A single axial coordinate may fail when flow is strongly heterogeneous, exchange is limited, elimination varies regionally, or concentration-dependent processes make elimination nonlinear. In such cases, fitting a dispersion coefficient cannot necessarily compensate for a structurally inadequate model. The dispersion number is a parameter in the formulation; interpreting its limiting behavior and implications is a separate question.

Key takeaway: The dispersion model combines convection, axial dispersion, and distributed elimination. Its inlet condition specifies the total incoming flux, whereas its outlet condition sets the dispersive flux to zero. Both conditions are essential to define a physically interpretable concentration profile.

Check Your Understanding

Answer each question, then compare with the solutions below.

Starting from $J=uC-D_{\mathrm{ax}}\,dC/dz$ and $dJ/dz=-kC$, write the steady differential equation.
For $D_N=0.20$, $C(0)=1.10\ \mathrm{mg/L}$, and $C'(0)=0.50\ \mathrm{mg/L}$ per unit $x$, find $C_{\mathrm{in}}$.
What does the outlet condition $C'(1)=0$ say about total outlet flux?
If $CL_{\mathrm{int}}$ is referenced to unbound blood concentration, state the expression for $k$ under uniform distributed linear elimination.
Name one diagnostic check for a numerically computed profile and explain what a failure could indicate.

Solutions. (1) Differentiation gives $dJ/dz=u\,dC/dz-D_{\mathrm{ax}}\,d^2C/dz^2=-kC$, so $D_{\mathrm{ax}}\,d^2C/dz^2-u\,dC/dz-kC=0$. (2) $C_{\mathrm{in}}=C(0)-D_NC'(0)=1.10-(0.20)(0.50)=1.00\ \mathrm{mg/L}$. (3) The dispersive part of the outlet flux is zero; total outflow remains convective, equal to $uC(1)$ per unit area. (4) $k=f_{u,b}CL_{\mathrm{int}}/V_L$. (5) Check both boundary conditions or the integrated mass balance; failure can indicate a sign or boundary-condition error, inadequate numerical resolution, or an incorrect implementation of the elimination term.