Deriving the Diffusion Equation from Conservation

From Local Mass Balance to a Spatial Model

A spatial model should explain how material moves, not merely prescribe how concentration changes. For a diffusing molecule in tissue, the starting point is a local accounting statement: the amount in a small region changes when more material flows in than flows out. Fick’s law supplies a constitutive rule for that flow, relating flux to the concentration gradient. Combining these two statements gives the diffusion equation.

The distinction matters in quantitative systems pharmacology. Conservation specifies what must be true for mass to balance, while Fick’s law approximates how transport occurs. Together they predict spatial smoothing: high-concentration regions lose material toward lower-concentration regions, and the rate of that redistribution depends on diffusivity.

For example, if \(c\) is measured in nmol/cm³ and distance in cm, then \(\nabla c\) has units nmol/cm⁴. Because \(\mathbf{J}\) has units nmol/(cm²·min), \(D\) has units cm²/min. Diffusivity is therefore not a speed; it sets the scale of how rapidly spatial concentration differences relax.

Deriving the Diffusion Equation

Consider any fixed control volume \(V\), with outward-pointing unit normal vector \(\mathbf{n}\) on its surface \(\partial V\). If there are no sources, sinks, or other processes that add or remove the species inside \(V\), the rate of change of its total amount equals the negative of the outward flux through the surface. The minus sign is necessary: outward flow reduces the amount inside.

The divergence theorem converts the surface flux into a volume integral: \(\int_{\partial V}\mathbf{J}\cdot\mathbf{n}\,dA=\int_V\nabla\cdot\mathbf{J}\,dV\). For a fixed volume, differentiation can be brought inside the integral. Thus,

This balance holds for every sufficiently small control volume. If the fields are smooth, the integrand must therefore vanish locally. This gives the local conservation equation. Substituting Fick’s law then gives the diffusion equation.

The last expression is the appropriate form when \(D\) may vary with position. If \(D\) is constant in space, it can be taken outside the divergence, yielding \(\partial c/\partial t=D\nabla^2c\). In one spatial dimension, these become \(\partial c/\partial t=\partial_x(D\,\partial_xc)\) and, for constant \(D\), \(\partial c/\partial t=D\,\partial^2c/\partial x^2\).

Assumptions and Scope

The derivation separates general accounting from the transport assumptions used to close the model. Conservation applies regardless of the detailed mechanism, provided the chosen species is being tracked consistently. Fick’s law is a particular constitutive approximation. Before using the resulting equation, check that its assumptions are reasonable for the tissue, molecule, and length and time scales of interest.

• Continuum description: concentration and flux are defined as fields that vary smoothly at the scale of interest. A model may be unsuitable if the relevant structure is so small that individual cells, binding sites, or molecules must be represented explicitly.
• Diffusion-dominated transport: flux is taken to be proportional to the local concentration gradient. The equation as derived includes no bulk flow, active transport, or other distinct transport mechanism.
• Stationary medium and no internal sources or sinks: the control volume does not move or deform, and there is no local production, degradation, or binding term in this balance. Adding such processes changes the mass balance; they are not hidden inside the diffusion term.
• Isotropic diffusivity: the scalar \(D\) gives the same transport response in every direction. Anisotropic tissues may require a direction-dependent transport law.
• Regularity: the derivation uses fields smooth enough for the derivatives and divergence theorem to apply. At abrupt material interfaces, the balance must be interpreted piecewise with appropriate flux accounting.

A diffusion equation also needs an initial concentration profile and conditions describing transport at the tissue boundary to determine a particular solution. Those conditions do not follow from the local derivation; they represent the physical setup. Here the focus is the local equation itself.

Variable Diffusivity and the Importance of Divergence Form

When diffusivity varies spatially, the local flux depends on both the local diffusivity and the local gradient. The conservation equation acts on the whole flux, so the correct one-dimensional form is \(\partial_x(D\,\partial_x c)\). Expanding this derivative gives two contributions: one from concentration curvature and one from the spatial change in diffusivity. Replacing it with \(D\,\partial_{xx}c\) would omit the second contribution.

How Diffusivity Sets the Rate of Smoothing

The diffusion equation smooths a concentration pattern because the flux carries material down its gradient. The constant-\(D\) equation makes the rate dependence precise. Consider a sinusoidal spatial variation with wavelength \(\lambda\), represented by \(c(x,t)=\bar{c}+a(t)\cos(kx)\), where \(k=2\pi/\lambda\), \(\bar{c}\) is the spatial mean, and \(a(t)\) is the amplitude. Substitution into \(\partial_t c=D\partial_{xx}c\) gives an ordinary differential equation for the amplitude.

Thus the decay rate is \(Dk^2\), with units of inverse time. Higher diffusivity makes a given spatial pattern decay faster. At the same diffusivity, shorter-wavelength variations decay faster because they have larger \(k\). This is a statement about the relaxation of spatial differences, not about loss of total material. If no material crosses the outer boundary, total amount is conserved even as the profile becomes more uniform.

Diagnostics, Alternatives, and Limitations

Several checks can reveal an incorrectly constructed diffusion model. First, check dimensions: \(\partial_t c\) must have units of concentration per time, and \(D\nabla^2c\) has those units because \(D\) is length²/time. Second, check the sign: at a smooth local maximum, \(\nabla^2c\) is nonpositive, so constant-\(D\) diffusion does not increase the concentration there at that instant. At a smooth local minimum, the opposite sign applies. Third, check that the flux points down the gradient. A missing minus sign in Fick’s law would reverse the physical direction and produce sharpening rather than smoothing.

The main alternatives answer different modeling needs. A well-mixed compartment model assumes no resolved spatial gradient within each compartment; it can be efficient when spatial detail is unnecessary, but cannot represent a continuous profile. A diffusion model resolves gradients using Fickian transport. If bulk fluid motion is important, a flux containing only the Fickian term is incomplete; a transport model must also represent the effect of flow. If tissue structure makes transport direction-dependent, a scalar diffusivity may likewise be inadequate. These alternatives should be selected from the biology and scale, not from convenience alone.

A common mistake is to write \(\partial_t c=D\nabla^2c\) whenever diffusivity is present. That expression assumes \(D\) is spatially constant. With variable \(D\), conservation gives \(\partial_t c=\nabla\cdot(D\nabla c)\), and expanding the divergence produces an additional gradient-of-\(D\) term. Another mistake is to interpret diffusion as a source or sink because concentrations change locally. Local concentration can rise or fall through redistribution while the total amount remains constant when the net boundary flux is zero.

Finally, Fick’s law is a model assumption rather than a consequence of conservation. A poor fit to observed spatial profiles, persistent directional transport, or changes that cannot be explained by gradient-driven flux may indicate that the constitutive law or the continuum assumptions need revision. A diffusion equation by itself also does not specify the initial profile or what happens at tissue boundaries; those are separate parts of a complete spatial model.