Application of Kirchhoff’s Laws to Eliminate the Need for Solving Differential Equations

More than a century and a half ago, Gustav Kirchhoff presented publications concerning applications of Ohm’s Laws for electric currents. His work has been identified as Kirchhoff’s Laws and their application to electrical processes in parallel and in series designated as Kirchhoff’s Loop Rules that can be found in elementary Physics textbooks (3, 4). Here, we reference book material that can be downloaded for free on-line (openpress.usask.ca/physics155/). Application of the first Law demonstrates that when two electrical resistors are in parallel, the total conductance (the inverse of resistance) is equal to the sum of the individual conductances. Application of the second Law demonstrates that when two resistors are in series, the inverse of the total conductance is equal to the sum of the inverse conductance for each resistor. Applications of Kirchhoff’s Laws are consistent with Eq. 1 as stated above. We reasoned that processes in pharmacology, pharmacokinetics, and chemistry also are consistent with Eq. 1 and that Kirchhoff’s Laws are applicable to deriving the kinetic relationships both for rate constants (Eq. 2) and clearances (Eq. 3) independent of differential Eqs. (1). Thus, there is now a way to derive clearance equations independent of the differential equation-based process, which has been followed since clearance was first recognized as the defining parameter in drug dosing a half-century ago. That is, it is now no longer necessary to derive the kinetic relationship in terms of rate constants and amounts, set the differential equation equal to zero at steady state, and then multiply the rate constant by a volume of distribution term and divide the amount by volume of distribution, to define clearance. Furthermore, for chemistry reactions, it is not necessary to derive the kinetic relationships and define the rate constants for first-order processes using differential equations.

As we report (1), the application of Kirchhoff’s Laws to clearance can be summarized in Eq. 6 for parallel processes and Eq. 7 for processes in series.

$$_}=_\;\mathrm\;\mathrm\;\mathrm\;1}+_\;\mathrm\;\mathrm\;\mathrm\;2}+\ldots$$

(6)

$$\frac_}}=\frac_\;\mathrm\;\mathrm\;\mathrm\;\mathrm\;1}}+\frac_\;\mathrm\;\mathrm\;\mathrm\;\mathrm\;2}}+\ldots$$

(7)

Equations 6 and 7 allow derivation of total clearance in Eq. 3. However, Kirchhoff’s Laws may also be applied to rate constants (and the total rate constant in Eq. 2) and can be derived via Eqs. 8 and 9, independent of solving differential equations for first-order processes.

$$k_}=k_\;\mathrm\;\mathrm\;\mathrm\;1}+k_\;\mathrm\;\mathrm\;\mathrm\;2}+\ldots$$

(8)

$$\frac}}=\frac\;\mathrm\;\mathrm\;\mathrm\;\mathrm\;1}}+\frac\;\mathrm\;\mathrm\;\mathrm\;\mathrm\;2}}+\ldots$$

(9)

A rate-defining process is a parameter that describes an elimination or movement process for which it is possible under certain conditions that the total clearance or total rate constant may be equal to this value. For example, a rate-defining clearance process for hepatic elimination could be hepatic blood flow, i.e., the rate at which drug arrives to the liver is the maximum value that hepatic elimination can be. Thus, for a very high hepatic clearance (CLH) drug, total CLH would equal hepatic blood flow. To exemplify a rate-defining rate constant process, for a series of chemical reactions occurring in a beaker, the elimination rate constant for the parent drug could be the minimum value rate-defining process for all subsequent metabolic steps. In contrast, basolateral hepatic efflux transport cannot be a rate-defining process since it is not possible for clearance or rate of elimination to be solely equal to this parameter.

Application of Kirchhoff’s Laws to Determine Rate Constants for In-Series Metabolic Steps

We now examine a hypothetical drug metabolism scenario where the drug (\(D\)) is metabolized to a first metabolite (\(M1\)) and partially excreted unchanged in the urine, which is then further metabolized to a second in-series metabolite (\(M2\)) and partially excreted in the urine, which is then further metabolized in series to a third metabolite (\(M3\)).

The advantage of utilizing Kirchhoff’s Laws for describing the rate of \(M2\) elimination is that the coefficient of proportionality (the measured elimination rate constant for \(M2\)) is determined based on the measured outcome and the rate-determining driving force parameters (the individual rate constants). Thus, for the in-series metabolism scenario pictured above

$$\frac\;\mathrm\;\mathrm\;\mathrm}=\frac+\frac}+\frac}$$

(10)

Assuming that the values for overall elimination of the drug and the metabolites by metabolism plus urinary excretion are kd = 0.07 h−1, km1 = 0.12 h−1, and km2 = 0.42 h−1, the measured \(M2\) elimination rate constant according to Eq. 10 will be 0.040 h−1. Therefore, if we wanted to predict the measured rate of the reaction (mass/time) for the elimination of \(M2\) in Eq. 2, it would be 0.040 multiplied by the amount of drug \(D\) we added to the beaker that reaches \(M2\). Prior to this publication, how would we have estimated this parameter? We demonstrate in the Appendix the process we would follow. That is, derive the differential equation for the measured concentrations of \(M2\) (CM2) as a function of time. We carried out the derivation using Laplace transforms. We then solved the equation for CM2 as a function of time. We then determined the mean residence time of \(M2\); the inverse of which would be the measured rate of elimination of \(M2\). This value is 0.040 h−1, the same value reached here using Kirchhoff’s Laws. As far as we can tell, we are the first to recognize that Kirchhoff’s Laws can be used to simply define measured reaction rates for in-series processes without carrying out the derivation in the Appendix.

Application of Kirchhoff’s Laws to In-Series Rate-Defining Steps for In Vivo Hepatic Clearance

A major advance in our application of Kirchhoff’s Laws to hepatic clearance (1) was the recognition that for hepatic elimination, the in-series rate-defining clearance processes could be (a) clearance entering, which is hepatic blood flow (QH) and (b) clearance leaving, which is hepatic clearance, i.e., the product of the fraction unbound to blood proteins (fuB) and the sum of the intrinsic ability of the liver to metabolize and to excrete unbound drug into the bile, independent of blood flow (CLint). As we point out in Pachter et al. (1), the CLint term is the sum of two parallel elimination processes (CLint metabolism + CLint biliary excretion). Thus, in a general approach

$$\frac_}=\frac_}}+\frac_}}$$

(11)

where CLentering is hepatic blood flow and CLleaving is metabolism of unbound drug

$$\frac_H}=\frac1+\frac\cdot_}}$$

(12)

Solving Eq. 12 gives

$$_H=Q_H\cdot\frac\cdot_}}_}}$$

(13)

For the past 50 years in the pharmacokinetic literature, Eq. 13 had previously been considered as the well-stirred model of hepatic elimination, based on the differential equation and steady-state derivation of Rowland, Benet, and Graham (5) for hepatic blood flow and intrinsic clearance plus the addition of the protein binding term by Wilkinson and Shand (6). Here, Eq. 13 was derived making no assumptions related to the mechanistic characteristics of hepatic elimination. It is organ model independent. Based on this recognition, we now understand why all steady-state isolated perfused rat liver clearance data, the only experimental studies that directly test the various hepatic disposition models, appear to preferentially fit what was previously believed to be the well-stirred model (7). It is because, in fact, Eq. 13 is a model-independent relationship and not the well-stirred model of hepatic disposition as the field has regarded it to be. Even in early 2022, we (8) also continued to mistakenly consider Eq. 13 to be the well-stirred model, since we had participated in the initial steps of its derivation (5). Therefore, there appears to be no value to predicting whole body clearance values utilizing different mechanistic models of hepatic organ elimination (e.g., parallel tube, dispersion, well-stirred), as is presently employed in some physiologically based pharmacokinetic (PBPK) approaches. To clarify, this manuscript is not questioning nor debating the pros and cons of the various hepatic disposition models. The thesis of this manuscript is focused on the methodologies employed to analyze pharmacokinetic data and obtain correct measures of pharmacokinetic parameters, i.e., the differential equation approach versus the approach presented here based on Kirchhoff’s Laws. The derivation of Eq. 13, independent of differential equations, is an excellent example of the advantage of the Kirchhoff’s Law approach.

Adding Rate Defining Basolateral Transport to Hepatic Clearance: The Extended Clearance Equation

In Pachter et al. (1), we noted that previously Eq. 13 had been commonly used in clearance predictions when hepatic metabolism was the only relevant process, although assuming it was the well-stirred model; however, the incorrect hepatic clearance equation was being employed when the possibility of basolateral transporters could be the rate-defining process. This is an area of strong interest, since lipid-lowering statins (HMG CoA reductase inhibitors) and other large molecular weight acids are found to be substrates of organic anion transporting polypeptides (OATPs) and inhibition of these transporters can significantly affect the pharmacokinetics of such drugs. Thus, when basolateral transporters are clinically significant, there are two entering clearances in series in Eq. 11, and Eq. 12 equivalent becomes

$$\frac_H}=\frac+\frac\;\cdot\;(PS}_}-_})}+\frac\;\cdot\;_}}$$

(14)

where PSinflux is basolateral influx clearance and PSefflux is basolateral efflux clearance. Kirchhoff never considered simultaneous reversible steps in his derivations, and one may ask why the difference in basolateral transport is assumed in Eq. 14, rather than separating out the two processes? Recall in the above discussion following Eq. 9 that Kirchhoff’s Laws applied to pharmacology, pharmacokinetics, and chemistry hold for combining rate-defining processes. Thus, we cannot ignore PSefflux, as it will certainly affect hepatic clearance, but it cannot be a rate-defining process by itself, as it is a negative parameter, and drug has to enter the liver prior to being effluxed out. Therefore, the rate-defining basolateral hepatic transport process is the difference between influx and efflux clearances. Of course, if basolateral transporters are not clinically relevant, then Eq. 14 reverts to Eq. 12. This is true since passive diffusion into and out of an organ is never a rate-defining process. There are no experimental studies where a diffusion rate constant or clearance has been identified as a rate-defining process. However, more frequently when basolateral transporters are relevant, it is assumed that the hepatic blood flow entering clearance is much larger than the other two processes, so under such conditions Eq. 14 would ignore the blood flow process, resulting in Eq. 15

$$\frac_H}=\frac\cdot(PS}_}-_})}+\frac\cdot_}}$$

(15)

which when solved for CLH yields

$$_H=\frac\cdot_}\cdot(_}-_})}_}+(_}-_})}=\frac\cdot(_}-_})}_}-_})}_}}}$$

(16)

Equation 16 is the correct extended clearance concept equation, rather than the equation universally reported in the literature (e.g., references 9,10,11,12,13,14). When CLint is much larger than \((_}-_})\), the difference in basolateral transporter clearances becomes rate defining and when CLint is much smaller than \((_}-_})\), CLint becomes rate defining. In the presently employed equation found throughout the literature, there are two erroneous outcomes: (a) PSefflux must be zero for transporters to become the rate-defining process, and (b) it is impossible for CLint to become the rate-defining process independent of basolateral transporter clearances, as we previously detailed (1).

It is important to recognize that \(_}\) and \(_}\) are the summation of both passive and active permeability clearances. If there is no active transport, or under the very unlikely condition that active transport in both directions is equal, or if \(_}\) is greater than \(_}\), these cannot be rate-defining processes for clearance and thus they would not be included in Kirchhoff’s Law derivation. Thus, both in the liver and in the kidney (\(_}-_})\), the difference must be positive for these parameters to be rate defining. One should also recognize that in vitro attempts to predict changes in membrane permeability will always be measures of the differences in the permeability clearances, rather than the individual permeability clearances themselves.

Rate Defining vs. Rate Limiting

In Pachter et al. (1), we described the use of Kirchhoff’s Laws in terms of “rate-limiting processes,” where we now refine the nomenclature to “rate-defining processes.” We do this because rate-limiting processes may be viewed as the upper and lower limits of the overall relationship. Thus, for Eq. 12, we may view hepatic blood flow (QH) as the upper limit of hepatic clearance, which it is. Correspondingly, we may then view hepatic metabolism (fuB·CLint) as the lower limit of hepatic clearance, which it is not. It is well recognized that hepatic clearance is not greater than QH; however, it may not be well recognized that hepatic clearance will often be less than fuB·CLint. For example, if QH is 1500 ml/min and fuB·CLint is 750 ml/min, CLH calculated by Eq. 13 will be 500 ml/min as also recently detailed (15). Therefore, hepatic metabolism (fuB·CLint) should be considered a rate-defining process, rather than a rate-limiting process.

Kirchhoff’s Laws vs. Differential Equation Derivations of Clearance Following Oral Dosing

The marked advantage of Kirchhoff’s Laws is that the equations are directly derived in terms of the parameter of relevance, here organ clearance. This is in contrast to the traditional way in which pharmacokinetic relationships have been historically derived using differential equations that consider amounts and rate constants, which are then converted to concentrations/clearances by dividing/multiplying by volume of distribution. This difference between the Kirchhoff’s Laws’ approach and the traditional differential equation approach can be exemplified by analyzing oral absorption. Consider first-order absorption (absorption rate constant ka) into a 1-compartment body model (disposition rate constant kd) where bioavailability is F and all drug is eliminated by renal processes unchanged so that clearance has no effects on first pass gut and liver elimination process. Solving the differential equation for the amount of drug in the systemic circulation (Asystemic circulation) as a function of time, t, gives

$$A_}=\frac_}}\cdot(e^-e^)$$

(17)

Converting Eq. 17 into a concentration (C) relationship by dividing by the systemic volume of distribution (V) yields

$$C_}=\frac_}}\cdot(e^-e^)$$

(18)

Integrating Eq. 18 over all time to determine the area under the curve (AUC) and dividing the available oral dose by this area yield Eq. 19 that has been universally taught and believed to be the clearance after oral dosing.

$$_\;\mathrm\;\mathrm\;(\mathrm\;\mathrm)}=\frac_}}_}=k_d\cdot V=_\;\mathrm}$$

(19)

Thus, based on the resulting equation from the differential equation approach, our field teaches that after oral dosing the absorption process has no effect on the measured clearance (it is identical to the clearance determined following iv bolus dosing). This is believed to be true independent of the value of F and true independent of the clearance affecting hepatic (and gut) bioavailability.

Now, let us examine the Kirchhoff’s Laws derivation for oral absorption in terms of clearance measures. Rewriting Eq. 11

$$\frac_\,(}^^}\mathrm)}}=\frac_}}+\frac_}}$$

(20)

where CLleaving for our 1-compartment body model is CLiv bolus, which equals kd·V, and CLentering would be clearance from the gut, a parameter that is never measured but could be envisioned as the product of the absorption rate constant (determined from the inverse of the mean absorption time) and the volume of distribution of drug in the gut (which is frequently taken to be 250 ml for all drugs in bottom-up attempts to predict first pass gut metabolism or intestinal drug-drug interactions). But let us leave CLentering as CLgut so that Eq. 20 becomes

$$\frac_}}=\frac_}}+\frac_}}$$

(21)

that when solved gives

$$_}=\frac_}}_}=\frac_}\cdot_\;\mathrm}}_}+_\;\mathrm}}=\frac_\;\mathrm}}_\;\mathrm}}_}}}$$

(22)

There are important considerations for Eq. 22. First, it demonstrates that the clearance measured after oral dosing is not the iv bolus clearance unless CLgut >  > CLiv bolus, which is unlikely to be true in many cases. Yes, the absorption rate constant will most frequently be greater than the disposition rate constant, but we are comparing clearances, not rate constants, and we suspect that the volume of distribution in the systemic circulation will be markedly greater than the volume of distribution of drug in the gut. Yet, it is well known that the absorption rate constant can be less than the elimination rate constant in the well-recognized flip-flop models. Also, in the derivation of Eq. 19 using differential equations, we are not required to make any assumption about bioavailability (F) since CLafter oral dosing is known if we also give an iv bolus dose. The value of bioavailability does not affect the value of clearance. However, this is not true for the Kirchhoff’s Laws derivation, since even when we have given an iv bolus dose to determine CLiv bolus, we cannot determine CLgut unless we know F. Therefore, what is the difference between the differential equation derivation and the Kirchhoff’s Laws derivation? We carried out the differential equation derivation in terms of rate constants and then converted the systemic amount to systemic concentration by the systemic volume of distribution, not recognizing that the absorption rate constant from the gut is multiplied by the systemic volume of distribution rather than the gut volume of distribution. Note in Eq. 18 that the concentration driving absorption is not the gut concentration but the same systemic concentration driving elimination.