Nondimensionalization for Identifying Fast and Slow Processes

Why Nondimensionalize a QSP Model?

A QSP model may combine processes whose rates differ by orders of magnitude: a signaling state may change quickly while a circulating mediator turns over slowly. The equations express these processes in physical units, but the rate constants can be hard to compare directly, particularly when their units differ. Nondimensionalization rewrites the system in terms of scaled states and time so that its coefficients are dimensionless and their relative sizes are visible.

A small dimensionless rate coefficient can indicate that a process changes little over the chosen time scale; a large one can indicate rapid change on that scale. These are candidates for slow and fast processes, not automatic instructions to remove a state or reaction. The interpretation depends on the selected scales, the model's nonlinearities, the state range of interest, and the question the model must answer.

Definition: For a state $X_i(t)$, choose a characteristic state scale $X_i^*>0$ with the same units as $X_i$, and define $x_i=X_i/X_i^*$. Choose a characteristic time scale $T>0$, and define dimensionless time $\tau=t/T$. If $dX_i/dt=f_i(X,\theta)$, then the scaled equation is $$ \frac{dx_i}{d\tau}=\frac{T}{X_i^*}f_i(X_1^*x_1,\ldots,X_n^*x_n,\theta). $$ The resulting coefficients and parameter combinations are dimensionless when the original equation is dimensionally consistent.

Deriving the Scaled Equations

Since $X_i=X_i^*x_i$ and $t=T\tau$, differentiating gives $dX_i/dt=(X_i^*/T)(dx_i/d\tau)$. Substituting this expression into the original ODE and multiplying by $T/X_i^*$ yields the definition above. The choice of scales does not alter the physical model: it changes the variables and the numerical form of its coefficients, not the trajectories in physical units.

For a first-order loss term $-kX_i$, where $k$ has units of inverse time, the scaled contribution is $-(Tk)x_i$. The dimensionless coefficient $Tk$ compares the selected observation time $T$ with the process's characteristic time $1/k$. For a second-order term such as $-kX_iX_j$, $k$ has units of inverse concentration per time; the scaled coefficient also involves a state scale, $TkX_j^*$. Therefore, coefficients should be compared only after the relevant state scales and units have been handled.

A practical choice is to scale each state by a typical positive magnitude in the intended application, such as a baseline concentration, a reference steady-state level, or a meaningful range. Choose $T$ to match a characteristic observation time or one well-defined process time. Record the choices and their units. If a state is expected to be zero or changes sign, select a nonzero scale that reflects the size of its deviations rather than dividing by its baseline.

Worked Example: A Coupled Cytokine–Effector Cascade

Consider a hypothetical linear cascade in which cytokine concentration $C$ is produced at a constant rate, while an effector concentration $A$ is driven by $C$: $$ \frac{dC}{dt}=u-k_C C,\qquad \frac{dA}{dt}=k_{\mathrm{act}}C-k_A A. $$ Assume $C$ and $A$ are measured in nM, $t$ in hours, $u=1.0\ \mathrm{nM\,h^{-1}}$, $k_C=0.50\ \mathrm{h^{-1}}$, $k_{\mathrm{act}}=0.20\ \mathrm{h^{-1}}$, and $k_A=0.050\ \mathrm{h^{-1}}$. These are hypothetical values.

Choose $C^*=u/k_C=1.0/0.50=2.0\ \mathrm{nM}$, the steady-state scale for the first equation; choose $A^*=k_{\mathrm{act}}C^*/k_A=(0.20)(2.0)/0.050=8.0\ \mathrm{nM}$; and choose $T=1/k_C=2.0\ \mathrm{h}$. Define $c=C/C^*$, $a=A/A^*$, and $\tau=t/T$.

For cytokine, $dc/d\tau=(T/C^*)(dC/dt)$. Substitution gives $$ \frac{dc}{d\tau}=\frac{2.0}{2.0}\left[1.0-(0.50)(2.0)c\right]=1-c. $$ For the effector, the corresponding substitution is $$ \frac{da}{d\tau}=\frac{2.0}{8.0}\left[(0.20)(2.0)c-(0.050)(8.0)a\right]=0.10(c-a). $$ Thus cytokine turnover has dimensionless coefficient $1$, while the effector's loss coefficient is $Tk_A=(2.0)(0.050)=0.10$. The characteristic times are $1/k_C=2.0\ \mathrm{h}$ and $1/k_A=20\ \mathrm{h}$, respectively, and their rate ratio is $k_A/k_C=0.050/0.50=0.10$. On this scaling, $C$ is a candidate faster process than $A$. Because $C$ drives $A$ but $A$ does not feed back into $C$, the interpretation is especially direct here; a small ratio still does not by itself establish that an approximation is adequate for a particular output.

Reading Rate Ratios Carefully

For two first-order processes with rates $k_f$ and $k_s$, their characteristic times are $1/k_f$ and $1/k_s$. If both act on states scaled with compatible choices, their relative rate is $$ \frac{k_s}{k_f}=\frac{1/k_f}{1/k_s}. $$ A ratio much less than one means the process associated with $k_s$ is slow relative to the one associated with $k_f$, by these characteristic-rate definitions. The word “much” has no universal cutoff: whether a separation is useful depends on the desired accuracy, outputs, and time window.

This comparison is straightforward for constant first-order rates, but nonlinear processes need more care. In a Michaelis–Menten loss term, the amount of loss per unit state depends on concentration. A coefficient in the scaled equation measures the size of the term under the chosen scales; the term's actual contribution can be smaller or larger across the state range. For a coupled system, one process may also alter the state that controls another process. Inspect the dimensionless equations over the states and conditions relevant to the analysis rather than reading one coefficient in isolation.

Worked Example: A Nonlinear Clearance Term

Consider a hypothetical mediator $C$ with linear and saturable clearance: $$ \frac{dC}{dt}=q-k_C C-\frac{V_{\max}C}{K+C}. $$ Let $q=0.40\ \mathrm{nM\,h^{-1}}$, $k_C=0.20\ \mathrm{h^{-1}}$, $V_{\max}=0.60\ \mathrm{nM\,h^{-1}}$, and $K=2.0\ \mathrm{nM}$. Choose $C^*=K$ and $T=1/k_C=5.0\ \mathrm{h}$, then set $x=C/K$ and $\tau=t/T$.

Using $dx/d\tau=(T/K)(dC/dt)$, the dimensionless equation is $$ \frac{dx}{d\tau}=\frac{q}{k_CK}-x-\frac{V_{\max}}{k_CK}\frac{x}{1+x} =1-x-1.5\frac{x}{1+x}. $$ The input group is $q/(k_CK)=0.40/[(0.20)(2.0)]=1.0$, and the saturable-clearance group is $V_{\max}/(k_CK)=0.60/[(0.20)(2.0)]=1.5$. These groups are dimensionless.

At $x=1$, the linear loss contributes $x=1$ per unit dimensionless time, whereas the saturable loss contributes $1.5(1)/(1+1)=0.75$. In physical units, the loss terms there are $(0.20)(2.0)=0.40\ \mathrm{nM\,h^{-1}}$ and $(0.60)(2.0)/(2.0+2.0)=0.30\ \mathrm{nM\,h^{-1}}$. The coefficient 1.5 alone therefore does not mean saturable clearance is 1.5 times faster at every concentration: its state-dependent contribution is $1.5x/(1+x)$, which approaches zero as $x$ approaches zero and approaches 1.5 only as $x$ becomes large.

At $C=K=2.0\ \mathrm{nM}$, the instantaneous fractional loss rate from these two terms is $k_C+V_{\max}/(K+C)=0.20+0.60/4.0=0.35\ \mathrm{h^{-1}}$. This corresponds to a local loss time of about $1/0.35=2.86\ \mathrm{h}$, not a constant timescale for the whole nonlinear model. It is a local interpretation at the stated concentration.

Scale Choice, Units, and Model Interpretation

A useful check is to repeat the time scaling with another positive $T$. For first-order rates $k_1$ and $k_2$, the coefficients become $Tk_1$ and $Tk_2$, but their ratio remains $$ \frac{Tk_1}{Tk_2}=\frac{k_1}{k_2}. $$ Thus a common change in time scale changes the displayed coefficients but not the relative rate. State scaling can likewise change individual coefficients, especially for nonlinear or cross-state terms, so a ratio is meaningful only when its construction and scales are stated.

Consider hypothetical rates $k_f=6.0\ \mathrm{day^{-1}}$ and $k_s=0.25\ \mathrm{day^{-1}}$. With $T=1/k_f=1/6.0\ \mathrm{day}$, the dimensionless coefficients are $Tk_f=1$ and $Tk_s=(0.25/6.0)=0.04167$, so the fast-to-slow rate ratio is $6.0/0.25=24.0$. If time is expressed in hours, the rates are $6.0/24=0.25\ \mathrm{h^{-1}}$ and $0.25/24=0.0104167\ \mathrm{h^{-1}}$; their ratio is $(6.0/24)/(0.25/24)=24.0$. Unit conversion changes the numerical rate constants, not the physical comparison.

Nondimensionalization complements, rather than replaces, other model checks. Sensitivity measures describe how outputs change as inputs or parameters vary; a parameter with high output sensitivity is not necessarily associated with a fast process. Conversely, a fast process can have little influence on a chosen output. Rate ratios are a structural and scale-based way to identify candidate timescales, not a substitute for checking predictive consequences.

Worked Example: Comparing Rates Under Two Time Scales

Use the hypothetical rates $k_f=6.0\ \mathrm{day^{-1}}$ and $k_s=0.25\ \mathrm{day^{-1}}$. First choose $T=1/k_f=1/6.0\ \mathrm{day}$. Then $Tk_f=(1/6.0)(6.0)=1$, and $Tk_s=(1/6.0)(0.25)=0.04167$. The slow-to-fast coefficient ratio is $0.04167/1$, approximately $0.0417$, corresponding to the rate ratio $k_s/k_f=0.25/6.0=0.04167$.

Now choose $T=1/k_s=1/0.25=4.0\ \mathrm{days}$. The coefficients are $Tk_f=(4.0)(6.0)=24$ and $Tk_s=(4.0)(0.25)=1$. Their ratio is $1/24=0.04167$, the same slow-to-fast comparison. The first scaling emphasizes the fast process; the second makes the slow process's coefficient one. Both express the same 24-fold rate separation. Neither scaling proves that one state can be removed or that the model's outputs are insensitive to the fast transient.

Diagnostics, Limitations, and Common Mistakes

Before interpreting a dimensionless system, check that every state scale is positive and has the same units as its state, every time scale has units of time, and every term in each original ODE has the same units. Verify the chain-rule factor $T/X_i^*$ in each equation. Then inspect the complete scaled terms, including state-dependent factors, rather than comparing coefficients that have different meanings.

Inconsistent units: Comparing a rate per day with one per hour without converting units can produce a meaningless ratio. Convert both to the same time unit or use an explicitly dimensionless ratio.
Misleading state scales: A very small or irrelevant reference scale can make coefficients appear large. Choose scales tied to the model's intended conditions and explain them.
Overinterpreting a small coefficient: A small coefficient suggests slow change relative to the chosen time unit, but state-dependent terms, strong coupling, or an initial transient can still matter for the outputs.
Assuming a single timescale: Nonlinear processes can change their effective rates across concentrations, and parameters may vary across doses or patient conditions. Reassess the scaled terms across the intended range.
Equating fast with removable: A fast process may determine an early response or peak. A rate ratio alone does not show that removing the process preserves the predictions of interest.

A common mistake is to say, “The coefficient is 0.01, so this state can be deleted.” The coefficient is dimensionless only relative to the declared time and state scales; it describes a term's size in the scaled equation, not the effect of deleting a state on every output. The correction is to identify which physical process the term represents, evaluate its contribution over relevant states and initial conditions, and test whether the model's intended predictions remain adequate under any proposed simplification.

Nondimensionalization is most useful when it makes the model easier to interpret, exposes combinations of parameters that control behavior, or guides which processes deserve closer investigation. It does not resolve parameter identifiability, guarantee a timescale separation across all conditions, or validate a reduced model. Those questions require additional analysis suited to the model and its application.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $X^*=5\ \mathrm{nM}$, $T=2\ \mathrm{h}$, and $dX/dt=-0.10X$, what is the dimensionless loss coefficient?
For rates $k_1=0.8\ \mathrm{h^{-1}}$ and $k_2=0.04\ \mathrm{h^{-1}}$, calculate the slow-to-fast rate ratio and state what it suggests.
In a term $V_{\max}C/(K+C)$, why is $V_{\max}/(k_CK)$ not by itself the ratio of saturable to linear clearance at every $C$?
What should be checked if a dimensionless equation contains a coefficient that seems unexpectedly large?

Solutions. (1) With $x=X/X^*$ and $\tau=t/T$, the coefficient is $Tk=(2)(0.10)=0.20$, so $dx/d\tau=-0.20x$. (2) The ratio is $k_2/k_1=0.04/0.8=0.050$; the process with rate $k_2$ is a candidate slower process, with a characteristic rate 20 times smaller, but this alone does not justify reduction. (3) The scaled saturable term includes the state factor $x/(1+x)$, so its contribution varies with $C$; compare the full term with the linear loss over the relevant concentration range. (4) Check time and state scales, units, the chain-rule factor, and whether the coefficient multiplies a state-dependent factor. Then reassess its magnitude under the model conditions of interest.