Why Check the Units of an ODE?
A concentration-based tissue model describes how a tissue concentration changes over time. Before interpreting its simulated profile, check the equation itself: the left-hand side is a concentration derivative, so every term on the right-hand side must have the same units. A term with incompatible units cannot be added to the others, even if the equation produces numerical output in software.
Dimensional analysis is a structural check, not a test of whether the biology is correct. A dimensionally consistent equation may still have an incorrect mechanism, parameter value, sign, or compartment assignment. But an inconsistent equation is not a valid expression for the stated rate of change, and identifying the mismatch can prevent errors from being mistaken for unusual pharmacokinetics.
For an ODE, the independent variable matters. A rate reported per minute is not numerically interchangeable with a rate reported per hour. Likewise, quantities can have the same broad dimension but still be represented in incompatible units: mg/L and µg/mL are both concentrations, but their numerical values must be reconciled before they are used together.
The checks here assume the model variables are differentiable in time over the interval being considered, and that every quantity has a declared unit. They also assume that concentrations combined by addition are expressed on compatible concentration bases. A dimensionless partition factor can relate concentration values, but it does not supply a missing unit conversion between unlike units.
The Derivative and Each Additive Term
A derivative is a rate of change: its units are the units of the changing quantity divided by the units of the independent variable. This follows from the difference quotient. If \(C_t\) is measured in mg/L and time in h, then \((C_t(t+\Delta t)-C_t(t))/\Delta t\) has units (mg/L)/h, and its limit has the same units.
Square brackets here mean “units of.” For example, \([C_t]=\mathrm{mg/L}\) and \([t]=\mathrm{h}\) imply \([dC_t/dt]=\mathrm{mg/L/h}\). The same rule applies if concentration is measured in µmol/L and time in min: the derivative is in µmol/L/min.
Addition imposes a second requirement. If \(a+b\) appears in an equation, \(a\) and \(b\) must have matching units. Thus, in an ODE, first check the units of every complete term on the right-hand side, not just the units of parameters in isolation. A coefficient may carry units that make a product valid, even when the coefficient itself is not a rate.
For instance, a first-order coefficient \(k\) multiplying a concentration difference must have units of inverse time. Then \(k(C_1-C_2)\) has units of concentration per time. A source term added directly to the derivative must already have concentration-per-time units. A dimensionless partition factor \(K\) does not change the units of a concentration divided by \(K\).
A Systematic Audit
A reliable audit separates the equation into its terms and records the units of each factor. Do not rely on the parameter name alone: a parameter called “rate” might be defined in different ways in different models. Use its declared units and the exact algebraic form in which it appears.
Write the units of the state variable, the independent variable, and every parameter. Record which concentration units are being used.
Divide the state-variable units by the time units. This gives the required units for the full right-hand side.
Multiply or divide the units of its factors as the equation specifies. Check that differences and sums combine compatible quantities.
Correct a parameter definition, equation, or unit conversion before simulation. Do not patch a mismatch by changing a numerical value without documenting the unit conversion.
This procedure checks dimensional homogeneity: all terms in an equation have compatible units. The check applies regardless of whether the equation is linear or nonlinear. For example, a concentration-dependent term may require a coefficient with units different from an inverse-time coefficient; the product still must reduce to concentration per time.
Worked Example: Finding the Units of a Concentration Derivative
Suppose a hypothetical tissue concentration \(C_t\) is measured in mg/L and time \(t\) in h. Over a short interval, \(C_t\) rises from 8.0 mg/L to 26.0 mg/L in 3.0 h. Find the average rate over the interval and verify its units.
The average rate is the concentration change divided by the elapsed time:
Substituting the hypothetical values gives
The result is an average rate, not necessarily the instantaneous derivative at every time in the interval. Its units nevertheless match those required of \(dC_t/dt\): concentration divided by time. A model using hours must therefore return a rate in mg/L/h when the state is in mg/L.
Checking an Exchange-and-Source Tissue ODE
Consider a hypothetical concentration-based tissue equation with two exchange contributions, first-order loss, and a source:
Here \(C_t\), \(C_b\), and \(C_i\) are concentrations expressed in mg/L; \(K_{bt}\) is dimensionless; \(k_{bt}\), \(k_{it}\), and \(k_{\mathrm{met}}\) are first-order coefficients in h\(^{-1}\); and \(R_t\) is a source rate in mg/L/h. The equation presumes that the concentration quantities in each difference are compatible. Under these stated assumptions, \(C_b/K_{bt}\) and \(C_t\) are both in mg/L. Multiplying either concentration difference by a coefficient in h\(^{-1}\) yields mg/L/h. The loss term has the same units, as does \(R_t\). Thus every term can be added and the right-hand side matches the derivative.
Worked Example: Auditing and Evaluating Every ODE Term
Use the equation above with hypothetical values \(C_b=12.0\) mg/L, \(C_t=4.0\) mg/L, \(C_i=2.0\) mg/L, \(K_{bt}=2.00\), \(k_{bt}=0.300\) h\(^{-1}\), \(k_{it}=0.100\) h\(^{-1}\), \(k_{\mathrm{met}}=0.0400\) h\(^{-1}\), and \(R_t=0.500\) mg/L/h. First check the units, then calculate the derivative.
For the first term, division by the dimensionless partition factor leaves a concentration. Therefore,
The second exchange term is
The loss term is
Each term has the required units. Adding them, including the source, gives
At this state, the model predicts an increasing tissue concentration at 0.740 mg/L/h. The positive derivative follows from the stated values and signs; dimensional analysis alone does not determine whether those signs or parameter values are biologically appropriate.
Diagnosing a Mismatch and Repairing It
A common error is to insert a coefficient with the wrong dimensions into a familiar-looking term. Suppose an attempted exchange term is written as \(\lambda(C_b-C_t)\), with \(C_b-C_t\) in mg/L. For the product to be mg/L/h, \(\lambda\) must have units h\(^{-1}\). If \(\lambda\) is instead specified in L/mg/h, then the product has units (L/mg/h)(mg/L) = h\(^{-1}\), not mg/L/h. That term cannot represent the required concentration rate as written.
The remedy depends on the model definition. If \(\lambda\) was intended to be a first-order coefficient, its units should be h\(^{-1}\). If it was intended to have units L/mg/h, the equation needs an additional concentration factor or a different formulation, justified by the intended mechanism. Dimensional analysis identifies the inconsistency but does not choose the biological model.
Worked Example: Correcting a Time-Unit Error
A hypothetical first-order tissue coefficient is \(0.200\) min\(^{-1}\), but the ODE is solved with time in hours and concentration in mg/L. At \(C_t=5.00\) mg/L, calculate the loss rate and show how to express it consistently in hours.
Using minutes, the magnitude of the loss rate is
There are 60 min in 1 h, so the same rate expressed per hour is \(60.0\) mg/L/h. The coefficient must also be converted:
The hour-based calculation is then
If the numerical value 0.200 were entered as though it were h\(^{-1}\), the computed magnitude would instead be 1.00 mg/L/h, a factor of 60 too small. The fix is to convert the coefficient or use minutes consistently; the unit label is part of the parameter value.
What Dimensional Analysis Can and Cannot Tell You
A unit audit is especially useful before simulation, when the equation can be inspected term by term. It can reveal an omitted rate coefficient, a source expressed in the wrong units, a concentration difference formed from incompatible units, or a parameter recorded per minute while model time is in hours. In implementation, repeating the audit against the code helps identify a mismatch between the written equation and the quantities actually passed to the solver.
Dimensional analysis is complementary to other checks, not a replacement for them. Numerical testing can reveal unexpected behavior over selected parameter values, but plausible-looking output does not establish dimensional consistency. Conservation or mass-balance checks address whether transfers and losses are accounted for consistently; they ask a different question from whether every term has the units of a concentration derivative. Similarly, biological review assesses whether the compartments, mechanisms, and parameter values make sense. A strong model review uses these checks together.
Dimensional consistency is necessary but not sufficient. Two competing equations can both be dimensionally valid while describing different mechanisms. A coefficient can have correct units and still have an implausible value. And a unit check cannot establish that an estimated concentration profile is identifiable or that the model fits observations adequately. Its contribution is narrower and essential: it rules out equations whose terms cannot mathematically be combined as stated.
Common Mistakes
- Checking parameters instead of complete terms. A coefficient need not itself have concentration-per-time units. Check the product or ratio in which it appears.
- Treating a dimensionless factor as a unit conversion. A dimensionless partition factor does not convert mg/L to µg/mL. Convert units explicitly before combining concentrations.
- Ignoring the model time unit. A coefficient in min\(^{-1}\) cannot be used numerically as h\(^{-1}\). Convert it or change the time unit consistently.
- Assuming a successful simulation proves the equation is valid. Software can calculate an expression with mismatched units because it often stores only numbers. Check the model independently.
- Taking dimensional consistency as biological validation. Correct units do not prove that a mechanism, sign, or parameter value is correct.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- If \(C_t\) is in µmol/L and time is in min, what are the units of \(dC_t/dt\)?
- A term is \(k(C_1-C_2)\), where both concentrations are in mg/L. What units must \(k\) have for the term to be a concentration rate measured in mg/L/h?
- A coefficient \(a\) is dimensionless, and \(C_t\) is in mg/L. Does \(aC_t\) have the units required for an ODE derivative when time is in hours? Explain.
- A first-order coefficient is \(0.0500\) min\(^{-1}\). Express it in h\(^{-1}\).
- Can a dimensionally consistent tissue ODE be assumed to represent the correct biological mechanism? Why or why not?
Solutions. (1) µmol/L/min, because the concentration units are divided by minutes. (2) h\(^{-1}\), since (h\(^{-1}\))(mg/L) is mg/L/h. (3) No. \(aC_t\) remains mg/L, so it lacks the division by time; an appropriate time-based coefficient or other rate formulation is needed. (4) \(0.0500\ \mathrm{min}^{-1}\times60\ \mathrm{min/h}=3.00\ \mathrm{h}^{-1}\). (5) No. Dimensional consistency is necessary, but it does not establish that the mechanisms, signs, or parameter values are biologically correct.