Propagating Parameter Uncertainty Through a Tissue Model

Why Propagate Uncertainty Through a Tissue Model?

A tissue concentration prediction depends on more than the central values selected for a model. Uncertainty in tissue volume, blood flow, and tissue-to-plasma partitioning can change both the speed and extent of tissue uptake. Propagating parameter uncertainty means evaluating how uncertainty in those inputs is reflected in the model output. The aim here is to make that connection in a single, transparent perfusion-limited example, not to treat an output from one set of central inputs as if it were certain.

The calculations below concern epistemic uncertainty: uncertainty about the appropriate values of fixed model inputs. They do not describe the range of tissue concentrations among different people. That distinction matters: varying a population’s physiology and varying one uncertain input for a reference individual answer different questions.

Definition: In this tutorial, \(V_t\) is tissue volume in L, \(Q_t\) is tissue blood flow in L/h, and \(K_{t:p}\) is the dimensionless tissue-to-plasma partition coefficient. \(C_a\) is a constant arterial plasma concentration in mg/L, \(C_t(t)\) is tissue concentration in mg/L at time \(t\) in h, and \(R(t)=C_t(t)/C_a\) is the dimensionless tissue-to-plasma concentration ratio. For positive \(V_t\), \(Q_t\), and \(K_{t:p}\), input uncertainty is propagated by evaluating how the model output changes as these inputs vary over specified plausible values or ranges.

The Perfusion-Limited Tissue Model

Assume a single, well-mixed tissue compartment with constant volume and blood flow. Blood entering the tissue has concentration \(C_a\); tissue concentration is uniform; and tissue efflux is represented by the concentration \(C_t/K_{t:p}\). There is no metabolism, active transport, or other loss in this example. The arterial concentration is held constant, and the initial tissue concentration is zero. Under these assumptions, the tissue mass balance is:

V_t\frac{dC_t}{dt} = Q_t\left(C_a-\frac{C_t}{K_{t:p}}\right), \qquad C_t(0)=0.

This is a perfusion-limited model: exchange is treated as sufficiently rapid that blood flow, rather than an additional membrane-transfer resistance, governs tissue delivery. The partition coefficient describes the equilibrium relationship used for tissue efflux; it is not itself the transient concentration ratio at every time.

Dividing by \(V_t\) and collecting the terms involving \(C_t\) gives a linear ordinary differential equation. With constant inputs, its solution and the corresponding ratio are:

\frac{dC_t}{dt}+\frac{Q_t}{V_tK_{t:p}}C_t = \frac{Q_t}{V_t}C_a, \qquad C_t(t)=K_{t:p}C_a\left(1-e^{-x}\right), \qquad R(t)=K_{t:p}\left(1-e^{-x}\right), \qquad x=\frac{Q_t t}{V_tK_{t:p}}.

For a brief justification, multiply the differential equation by the integrating factor \(e^{at}\), where \(a=Q_t/(V_tK_{t:p})\). The left side becomes the derivative of \(e^{at}C_t(t)\). Integrating from 0 to \(t\), using \(C_t(0)=0\), yields:

e^{at}C_t(t) = \frac{Q_tC_a}{V_t}\int_0^t e^{as}\,ds = \frac{Q_tC_a}{V_t}\frac{e^{at}-1}{a} = K_{t:p}C_a(e^{at}-1).

Multiplication by \(e^{-at}\) gives the stated solution. The exponent \(x\) is dimensionless: \(Q_t t/V_t\) has units \((\mathrm{L/h})(\mathrm{h})/\mathrm{L}=1\), and \(K_{t:p}\) is dimensionless. At time zero, \(C_t=R=0\); as time increases, \(R\) approaches \(K_{t:p}\), so \(C_t\) approaches \(K_{t:p}C_a\).

Worked Example: A Central Prediction and Its Units

Consider hypothetical inputs \(V_t=8.00\) L, \(Q_t=2.00\) L/h, \(K_{t:p}=2.00\), and \(C_a=1.50\) mg/L. Predict tissue concentration at \(t=4.00\) h.

First calculate the dimensionless time term:

x=\frac{(2.00\ \mathrm{L/h})(4.00\ \mathrm{h})}{(8.00\ \mathrm{L})(2.00)} =\frac{8.00}{16.0}=0.500.

Then calculate the concentration ratio and tissue concentration:

R(4.00)=2.00(1-e^{-0.500}) =2.00(1-0.606531)=0.786939, $$ $$ C_t(4.00)=(1.50\ \mathrm{mg/L})(0.786939) =1.18041\ \mathrm{mg/L}.

Thus the predicted tissue concentration is approximately 1.180 mg/L. The ratio is dimensionless, while the concentration retains the units of \(C_a\). Keeping extra digits until the final step also avoids a mismatch between a rounded concentration and a separately reported ratio.

How the Inputs Change the Prediction

The solution makes the roles of the three inputs visible. At a fixed positive time, greater blood flow increases \(x\) and advances tissue uptake. Greater tissue volume decreases \(x\), slowing the approach to the limiting concentration. A larger partition coefficient raises that limiting concentration and also changes the time scale in \(x\). The direction of the partition effect on the transient ratio is positive as well.

These directions can be checked by differentiating \(R=K_{t:p}(1-e^{-x})\), with \(x=Q_t t/(V_tK_{t:p})\). For positive inputs and \(t>0\):

\frac{\partial R}{\partial V_t} = -\frac{K_{t:p}xe^{-x}}{V_t}<0, \qquad \frac{\partial R}{\partial Q_t} = \frac{K_{t:p}xe^{-x}}{Q_t}>0, $$ $$ \frac{\partial R}{\partial K_{t:p}} = 1-(1+x)e^{-x}>0.

The final inequality follows because \(e^x>1+x\) for \(x>0\); multiplying by \(e^{-x}>0\) gives \((1+x)e^{-x}<1\). At \(t=0\), \(x=0\), \(R=0\), and all three derivatives are zero. These derivatives describe local changes around a chosen input set; they are not, by themselves, a full description of uncertainty over a wide input range.

Worked Example: Local Sensitivity at the Central Inputs

Use the hypothetical inputs from the previous example at \(t=4.00\) h. At these values, \(x=0.500\), \(e^{-x}=0.606531\), and \(R=0.786939\). Calculate the local changes in \(R\) per unit change in each input.

For tissue volume:

\frac{\partial R}{\partial V_t} = -\frac{(2.00)(0.500)(0.606531)}{8.00\ \mathrm{L}} = -0.0758\ \mathrm{L}^{-1}.

For blood flow:

\frac{\partial R}{\partial Q_t} = \frac{(2.00)(0.500)(0.606531)}{2.00\ \mathrm{L/h}} = 0.303\ \mathrm{h/L}.

For the dimensionless partition coefficient:

\frac{\partial R}{\partial K_{t:p}} = 1-(1+0.500)(0.606531) = 0.0902.

Near these inputs, a small increase of 0.10 L in \(V_t\) corresponds to an approximate decrease in \(R\) of \(0.0758\ \mathrm{L}^{-1}\times0.10\ \mathrm{L}=0.00758\). This is a local approximation, not the exact change for an arbitrary finite adjustment. Because \(C_t=C_aR\) and \(C_a=1.50\) mg/L is fixed, the corresponding local concentration change is approximately \((1.50\ \mathrm{mg/L})(-0.00758)=-0.0114\) mg/L. Units and the direction of change are consistent with a larger volume slowing uptake.

Propagating Specified Input Ranges

A direct way to show the implications of specified uncertainty ranges is to evaluate the model at their limiting combinations. This yields a prediction interval for the stated input bounds—not a confidence interval and not a probability statement. For this approach, assume all input values are positive, the arterial concentration and time are fixed, and the ranges can be combined freely. In particular, using every lower or upper bound together is appropriate only if those combinations are considered plausible; if inputs must move together, the rectangular range can include impossible combinations.

Since \(R\) decreases as \(V_t\) increases and increases as \(Q_t\) or \(K_{t:p}\) increases for \(t>0\), the minimum and maximum over a rectangular set of positive input ranges occur at identifiable corners. The monotonicity results justify the corner evaluations rather than requiring an arbitrary selection of scenarios.

Worked Example: Bounds From Volume, Flow, and Partition Ranges

Suppose the same hypothetical tissue has ranges \(V_t=7.00\) to \(9.00\) L, \(Q_t=1.60\) to \(2.40\) L/h, and \(K_{t:p}=1.80\) to \(2.20\). Keep \(C_a=1.50\) mg/L and \(t=4.00\) h. Assume every combination inside this rectangular range is considered plausible.

The minimum concentration occurs at the largest volume, lowest flow, and lowest partition coefficient: \(V_t=9.00\) L, \(Q_t=1.60\) L/h, and \(K_{t:p}=1.80\). Thus:

x_{\min\ output} = \frac{(1.60)(4.00)}{(9.00)(1.80)} = 0.395062, $$ $$ R_{\min} = 1.80(1-e^{-0.395062}) \approx 1.80(1-0.6736) \approx 0.5875, \qquad C_{t,\min}\approx(1.50)(0.5875)=0.881\ \mathrm{mg/L}.

The maximum occurs at the smallest volume, highest flow, and highest partition coefficient: \(V_t=7.00\) L, \(Q_t=2.40\) L/h, and \(K_{t:p}=2.20\). Thus:

x_{\max\ output} = \frac{(2.40)(4.00)}{(7.00)(2.20)} = 0.623377, $$ $$ R_{\max} = 2.20(1-e^{-0.623377}) \approx 2.20(1-0.5360) \approx 1.021, \qquad C_{t,\max}\approx(1.50)(1.021)=1.531\ \mathrm{mg/L}.

For these hypothetical bounds, the model therefore gives a tissue concentration range of approximately 0.881 to 1.531 mg/L at 4.00 h. This range is conditional on the stated model and the assumption that all combinations in the input ranges are allowable. It is not a probability interval: the calculation has not assigned probabilities to values within those ranges.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check whether the output behaves as the model says it should. At \(t=0\), \(R\) must be zero; for positive time, \(R\) must be positive and below \(K_{t:p}\); and at long times it must approach \(K_{t:p}\). Increasing \(V_t\) at fixed \(Q_t\), \(K_{t:p}\), and time must not increase \(R\). Increasing \(Q_t\) or \(K_{t:p}\) must not decrease it. A calculation that violates one of these checks may contain an algebra, unit, or implementation error.

Also check units before interpreting results. \(Q_t/V_t\) has units of inverse time, so multiplying by \(t\) produces a dimensionless quantity. A negative volume, flow, or partition coefficient is outside the model’s physical domain. Finally, a valid mathematical output does not establish that the parameter ranges are physiologically plausible or that constant arterial concentration is a good description of the exposure.

Three approaches answer related but distinct questions. A single central-input prediction is concise, but hides sensitivity to uncertain inputs. Local derivatives show the direction and approximate magnitude of small changes near one input set; they can mislead when changes are large or the response is strongly curved. Evaluating stated bounds, as above, makes a transparent range under explicit assumptions, but does not describe how likely different outputs are. More comprehensive distribution-based analyses can summarize output uncertainty when input distributions and their dependence are specified; they require those assumptions and are not equivalent to simply combining marginal extremes.

The one-compartment model also has substantive limitations. It assumes rapid tissue equilibration, constant flow and volume, a fixed partition coefficient, constant arterial input, and no local elimination. It cannot represent a changing arterial concentration or distinguish subregions with different perfusion. If these assumptions are not credible, careful uncertainty calculations within this model do not correct the structural mismatch.

Common pitfall: It is incorrect to call a range formed from minimum and maximum parameter values a “95% interval” merely because it is wide, or to assume that central input values must give the midpoint of the output range. The range has no probability coverage unless a probability model has been specified and used. Because the tissue model is nonlinear, the output at central inputs need not equal the midpoint of outputs at extreme inputs.

Check Your Understanding

Answer each question using the perfusion-limited model and the assumptions stated above. Brief solutions are included for checking.

If \(t=0\), what are \(R\) and the three local derivatives with respect to \(V_t\), \(Q_t\), and \(K_{t:p}\)?
At positive time, which input changes increase the predicted concentration when the other inputs are held fixed: increasing tissue volume, blood flow, or partition coefficient?
For \(C_a=1.00\) mg/L, \(V_t=10.0\) L, \(Q_t=2.00\) L/h, \(K_{t:p}=2.00\), and \(t=5.00\) h, calculate \(x\), \(R\), and \(C_t\).
Why is a concentration range obtained from input bounds not automatically a probability interval?
Name two model assumptions that could make this tissue prediction unsuitable for a particular exposure.

Solutions. (1) At \(t=0\), \(x=0\), so \(R=K_{t:p}(1-e^0)=0\). The derivatives contain \(x\), or reduce to \(1-(1+0)e^0=0\), so all three are zero. (2) Increasing \(Q_t\) or \(K_{t:p}\) increases the prediction; increasing \(V_t\) decreases it. These directions apply at positive time. (3) \(x=(2.00\times5.00)/(10.0\times2.00)=0.500\). Therefore \(R=2.00(1-e^{-0.500})=0.786939\), and \(C_t=(1.00\ \mathrm{mg/L})(0.786939)=0.786939\) mg/L. (4) Bounds specify allowable values, not probabilities or a probability distribution over those values. (5) Examples include constant arterial concentration, fixed flow and volume, rapid perfusion-limited equilibration, and no tissue metabolism; any two are sufficient.