Analytical method for calculation of deviations from intended dosages during multiinfusion
 Maurits K. Konings^{1}Email author,
 Roland A. Snijder^{1},
 Joris H. Radermacher^{1} and
 Annemoon M. Timmerman^{1}
Received: 4 October 2016
Accepted: 27 December 2016
Published: 17 January 2017
Abstract
Background
In this paper, a new method is presented that combines mechanical compliance effects with Poiseuille flow and pushout effects (“dead volume”) in one single mathematical framework for calculating dosing errors in multiinfusion setups. In contrast to existing numerical methods, our method produces explicit expressions that illustrate the mathematical dependencies of the dosing errors on hardware parameters and pump flow rate settings.
Methods
Our new approach uses the Ztransform to model the contents of the catheter, and after implementation in Mathematica (Wolfram), explicit expressions are produced automatically. Consistency of the resulting analytical expressions has been examined for limiting cases, and three types of invitro measurements have been performed to obtain a first experimental test of the validity of the theoretical results.
Results
The relative contribution of various factors affecting the dosing errors, such as the Poiseuille flow profile, resistance and internal volume of the catheter, mechanical compliance of the syringes and the various pump flow rate settings, can now be discerned clearly in the structure of the expressions generated by our method. The invitro experiments showed a standard deviation between theory and experiment of 14% for the delay time in the catheter, and of 13% for the time duration of the dosing error bolus.
Conclusions
Our method provides insight and predictability in a large range of possible situations involving many variables and dependencies, which is potentially very useful for e.g. the development of a fast, bedside tool (“calculator”) that provides the clinician with a precise prediction of dosing errors and delay times interactively for many scenario’s. The interactive nature of such a device has now been made feasible by the fact that, using our method, explicit expressions are available for these situations, as opposed to conventional timeconsuming numerical simulations.
Keywords
Background
If a patient needs intravenous administration of medication, it is important for the clinician to understand the basic pharmacokinetics of these medications. However, in recent years, ample evidence has been found [1–4], including a number of clinical cases and in vivo studies [5–9], indicating that physical effects related to the infusion hardware are equally important to understand [10, 11]. Especially in critical care, on the ICU and the OR, where multiple medications are typically delivered through one single lumen of a thin catheter, these physical effects may cause ambiguous and counterintuitive discrepancies between the intended dose and the dose that has been actually delivered. This is particularly true if the dosing rate is adjusted ad hoc [12]; for example, fastacting and critical inotropic drugs are often titrated, based on the mean arterial blood pressure (MABP).
There are three major factors that can produce significant deviations from the intended medication dosing rate scheme: (i) the length of the catheter causes a delay in the administration of the medication into the patient, and therefore, a mixture, corresponding to previous medication dosing rates, may still be present inside the catheter, i.e. the contents inside the catheter constitutes a “memory” in which the effects of previous medication dosing rates may be stored. This has been called “dead volume effect” in the literature [7, 13]. (ii) The syringes that are used in clinical practice are far from ideal, because these syringes have a significant mechanical compliance primarily due to the compressibility of the rubber plunger inside the syringe [14]. Therefore, whenever the flow rate setting of one of the infusion pumps in a multiinfusion setup is altered, pressure changes within the entire system cause a change in the deformation of compressible and expandable parts (i.e., the other syringes) within the system, and hence may cause a deviation from the intended flow rates. (iii) The low velocities of the fluids in catheters ensure that the flow inside these lines will be laminar (low Reynolds number), and hence exhibit a Poiseuille flow profile, in which the fluid particles near the central longitudinal axis of the catheter travel faster than fluid particles near the wall of the catheter, thus giving rise to a “mixing effect” of its own.
Several infusion simulation studies have shown the importance of these problems by demonstrating the influence of the physical effects of dead volume, compliance and the Poiseuille effect on drug delivery in isolation: There are a number of studies that describe the “dead volume effect” in isolation from the syringe compressibility (“compliance”) effect [7]. In simple cases, calculation of the “dead volume effect” is straightforward: If the actual volumetric flow rate \(u_{cath}\) inside a single lumen catheter is assumed to be constant, and the internal volume \(V_{cath}\) of the catheter (i.e., the internal volume as measured starting from the mixing point, at which the medications from all syringes come together, up to the distal catheter tip inside the vasculature of a patient) is known, then calculation of the “delay time” \(t_{delay}\), i.e. the time needed for a droplet of medication to travel through the dead volume before it reaches the blood stream of the patient, is simply \(t_{delay} = V_{cath}/u_{cath}\).
In clinical practice, however, the situation is often more complex: the flow rate may vary during the delay time due to changes in the pump flow rate settings, and the actual flow rate also typically differs from the pump flow rate setting value temporarily, due to the effects of mechanical compliance as mentioned above. These compliance effects have been studied in isolation as well [2, 3, 15]. Its basic mechanism is a “capacitor” effect, which has been quantified using the electric analog of a system containing capacitors, and the calculations of the dosing errors have been performed using the Laplace transform.
The third major factor, the mixing effect due to the Poiseuille profile [16], can be described as a convolution, as will be explained in more detail in “Appendix”. It causes a spreading out of the dosing error in time, in which the first arrival of the dosing error occurs sooner then it would have without Poiseuille mixing effect.
As a result, due to the combination of the effects (i), (ii), and (iii) mentioned above, nontrivial deviations from the intended medication dosages scheme may occur, as will be explained below.
In an earlier paper, we have shown that the “dead volume effect” on one hand, and syringe compliance (“capacitor”) effect on the other hand, produce opposite deviations from the pump flow rate settings in the actual drug output concentrations, making the net result hard to predict and often counterintuitive [4].
In this paper, however, we focus on the mathematical method of calculating dosing errors in multiinfusion setups in complex situations. The aim of the new method described in this paper is to obtain analytical expressions for the deviations from the intended dosages during multiinfusion that result from the combination of the effects (i) and (ii) described above. These analytical expressions contain dependencies on parameters that represent the physical variables in a multiinfusion setup, such as: intended flow rates of the various pumps, compliances of the syringes, resistances of the various tubes, height differences within the multiinfusion setup, etc.
As opposed to conventional numerical simulations, our objective therefore is to obtain explicit analytical expressions for these deviations, in which physical characteristics of the multiinfusion setup are represented explicitly as variables.
We see the need for a tool for clinicians and medical physicist that provides understanding of the role that various physical parameters (i.e., characteristics of the multiinfusion hardware) play in the emergence of a dosing error. In order to make these roles explicit, we will use our new mathematical approach, presented in the “Method: analytical model, and invitro setup” section of this paper, to explicate these roles in the form of direct mathematical relationships between physical hardware parameters and dosing errors.
For our mathematical model we used the strong symbolic calculation capabilities of the Mathematica package (Mathematica 10, Wolfram^{®} Inc., USA) to process Laplacetransforms and Ztransforms analytically. This will be explained in “Method: analytical model, and invitro setup” section. Furthermore, we have performed invitro measurements in order to verify our mathematical results experimentally.
Method: analytical model, and invitro setup
New method for incorporating the memory effect of the catheter into an analytical model
First, standard techniques [17] (involving the Laplace Transform) are used to calculate pressures and total flow rates (see Fig. 3a), in which the differences between the “Red”, “Blue” and “Green” solutions are disregarded; i.e. in this “colorblind” calculation, only the total flow rate inside the catheter \(u_{cath}(t)\) is calculated. The mechanical compliance of the catheter is very small with respect to the mechanical compliance of the syringes [18], and therefore the mechanical compliance of the catheter is neglected in our model. As a result, the flow rate entering the tube at time t and the flow rate leaving the tube (entering the blood stream) at the same time t are both equal to \(u_{cath}(t)\), disregarding the different constitutions that the fluids entering, and leaving, respectively, may have in terms of the partial fluids R, G, and B.
Secondly, our new analytical method is introduced (see Fig. 3b), which enables incorporation of the “memory effect” of the catheter into the model. In order to make the analytical approach possible throughout the entire calculation up to the end of point (d) in the figure, a formulation in the Zdomain (using the Ztransform, which is a discrete variant of the Laplace transform) is introduced. This analytical method uses the results form the standard (Laplacebased) method [see (a)] as an input. This input has the form of general expressions for the various total flow rates as function of time. The Ztransform formulation in our method [see (b)] therefore does not replace the Laplacebased method from (a), but is used after it.
Third, the Zdomain formulation from (b) enables an easy, analytical, incorporation of the Poiseuille mixing effect (c) into the model, because the convolutionlike nature of the Poiseuille mixing effect is reduced to a mere multiplication in the Zdomain. The mathematical details of the Poiseuille mixing effect are described in “Poiseuille mixing effect” section.
Finally (d), we derive expressions for the volume, and the first and second moment, of the dosing error as function of hardware parameters. These moments can be calculated directly within the Zdomain. From the first and second moment, key characteristics concerning timing and duration of the dosing error are derived.
Throughout the entire process, we have used the symbolic calculation capabilities of the Mathematica package (Mathematica 10, Wolfram^{®} Inc., USA) to process Laplacetransforms and Ztransforms analytically, as can be seen in “Results” section.
Starting point: Laplace transform and Kirchhoff’s laws
In this paper, we focus on the multiinfusion of aqueous solutions, in which all solutions have a viscosity close to the viscosity of water. Therefore, in our mathematical model, all fluids are assumed to have approximately the same viscosity. For the calculation of the pressures and total flow rates, we use an electric circuit model that is analogous to the infusion setup, in which current sources (the infusion pumps), resistances (infusion lines and catheter), and capacitors (mechanical compliance of the syringes) are present, and in which Kirchhoff’s laws are applied on the voltages (pressures) and currents (flow rates) in the Laplace domain. The output \(U_{cath}(s)\) of the calculation is the Laplace transform of the total flow rate \(u_{cath}(t)\) inside the catheter. This \(u_{cath}(t)\) equals the total flow rate entering the tube at time t as well as the total flow rate leaving the tube (entering the blood stream) at the same time t. Let \(\Gamma \) denote the complete set of these changes in pump flow rate setting values, together with the physical hardware parameters of the setup, which are present in the form of explicit parameters in the analytical expressions for \(U_{cath}(s)\). Using the Mathematica package (Mathematica 10, Wolfram^{®} Inc, USA), we were able to perform an inverse Laplace transform in order to retrieve an analytical expression for \(u_{cath}(t, \Gamma )\) from the \(U_{cath}(s, \Gamma )\), in which the above mentioned hardware parameters \(\Gamma \) are present as explicit variables. This result is rendered in Eq. (33) in “Results” section.
Key parameters of dosing errors
The dosing errors that are produced after changes in pump flow rate setting values in a multiinfusion setup are of a temporary nature, i.e. “dead volume” and “mechanical compliance” (points (i) and (ii), respectively, as mentioned in the introduction) give rise to only a temporary deviation of the actual dose rates (entering the blood stream) from the intended ones (see e.g. Fig. 2). As a result, each of these dosing errors has the shape of a “bolus”.
There are two basic mechanisms that contribute to the dosing errors, and the effect of each of these two mechanisms is restricted to a particular time interval: During the time interval \(0< t < t_{delay}\), the dosing error entering the blood stream is caused by the pushout effect, as explained in Fig. 2. During the time interval \(t > t_{delay}\), however, the dosing error entering the blood stream is caused by the effects of mechanical compliance, because the “pushout effect” is by definition restricted to the time interval \(0< t < t_{delay}\). The effects of mechanical compliance are particularly strong directly after changing the pump flow rate settings, and thus the effects of mechanical compliance (in the form of deviating mixtures) are entering the catheter at point \(\mathcal {M}\) at \(t=0\), and hence entering the patient at point \(\mathcal {P}\) directly after \(t = t_{delay}\).
In the following, we will use \(t^{POIS}_{delay}\) instead of \(t_{delay}\), because, as will be demonstrated below, the Poiseuille profile of the flow affects the \(t_{delay}\), resulting in a modified delay time that we will refer to as \(t^{POIS}_{delay}\). Furthermore, we will consider a situation with two pumps (“Red” and “Green”) only, and will change the flow rate setting of the “Green” pump only (at \(t=0\)), leaving the flow rate setting of the “Red” pump (denoted as \(u_{pump}^{(R)}\)) constant. This does not affect the applicability of our method in more general cases. The flow rate setting of the “Green” pump changes from \(u_{old}^{(G)}\) to \(u_{final}^{(G)}\) at \(t=0\).
Contents of the catheter without poiseuille mixing effect
In this subsection, we derive a generic expression for the ZTransform of the contents \(\{a_k^{(R)}\}\) of the catheter, without incorporating the Poiseuille mixing effect. The Poiseuille mixing effect will be incorporated in the model later on.
Poiseuille mixing effect
In “Appendix”, it is derived that the Poiseuille flow profile causes a mixing effect that features a simple linear relation between concentration and distance along the line from \(\mathcal {M}\) to \(\mathcal {P}\) (see Fig. 4b), which is consistent with earlier work by e.g. Taylor [19] and Hutton and Thornberry [20].
Calculation of moments of the dosing error distribution after T\(_{delay}\)
We will now use the “theorems of moments” from Ztransform theory to provide expressions for key characteristics of the \(\psi _{patient}^{(R)}(\tau )\) that enters the patient.
Spectrometric invitro setup for feasibility tests
Results
General result from laplace transform as a starting point
Standard values in simulations and invitro measurements
Variable  Standard value 

\(R_{cath}\)  1145 Pa/(ml/h) 
\(R_1\)  23 Pa/(ml/h) 
\(R_2\)  23 Pa/(ml/h) 
\(C_1\)  1.5 10\(^{5} \;\) ml/Pa 
\(C_2\)  1.5 10\(^{5} \;\) ml/Pa 
\(u_{pump}^{(R)}\)  0.5 ml/h 
\(u_{old}^{(G)}\)  12 ml/h 
\(u^{(G)}_{downstep}\)  6 ml/h 
Typical example: resulting dosing error during syringe exchange
We now consider the specific case in which the “Green” syringe is exchanged, during which the green pump is stopped and green line has been clamped (with a Kocher), but the red line is not clamped, and the settings of the red pump remain unchanged. Let \(T_{restart}\) denote the time between clamping the green line and the reopening the green line, i.e. \(T_{restart}\) denotes the time duration of the entire procedure of exchanging the green syringe.
Analytical results for reducing the flow rate of the green pump, without Poiseuille mixing effect
We now consider the more general case in which the pump flow rate setting of green pump, i.e. of the fast pump, is lowered with an amount \(u^{(G)}_{downstep}\), without clamping any line. The volume \(Q^{(R)}_{dosingerror}\) does not depend on the specific distribution of the dosing error over time, and hence yields the same result as in Eq. (35).
If, in Eq. (41), the \(u^{(G)}_{downstep}\) would be very small with respect to \(u_{final}\), then this expression for \(\sigma \) approaches the familiar \(\sigma \approx 2 C R_{cath}\).
Analytical results for reducing the flow rate of the green pump, incorporating the Poiseuille mixing effect
If the catheter has a large internal volume but the time \(C R_{cath}\) is short and the flow rates are low, then the \(\sigma ^{POIS}\) in Eq. (42) reduces to: \(\sigma ^{POIS} \approx V_{cath} / u_{final}\).
If, however, in Eq. (42), the \(u^{(G)}_{downstep}\) would be very small with respect to \(u_{final}\), and the time \(C R_{cath}\) would be very large and the \(V_{cath}\) would be small, then this expression for \(\sigma \) approaches \(\sigma ^{POIS} \approx 4 C R_{cath}\), which is two times larger than the \(\sigma \approx 2 C R_{cath}\) that was calculated before when omitting the Poiseuille mixing effect. This factor two is a result from the fact that, in a Poiseuille flow, the velocity at the centerline of the catheter equals two times the average velocity.
Results of invitro experiments, and comparison with theoretical predictions
 (i)
measurement of the flow rate of the Green fluid as function of time, immediately after a change in pump flow rate setting value of the green pump, corresponding to \(u^{(G)}_{downstep} = \) 6 ml/h; see Fig. 10.
 (ii)
measurement of \(t_{delay}\) for a set of three different syringes and three resistances, corresponding to various values of C and \(R_{cath}\); see Fig. 11. The experimental measurement of the \(t_{delay}\) showed a standard deviation between theory and experiment of 16.2 s, which is 14% of the total range of variations in \(t_{delay}\).
 (iii)
measurement of \(\Delta t_{central}\) as function of various values of the relative resistance of the catheter with respect to its standard value R0, in which \(\Delta t_{central}\) was defined as the measured \(t_{central}\) minus the standard value of \(t_{central}\) for \(R_{cath}=R0\). Three different values of \(R_{cath}/R_0\) have been used; and the measurement \(\Delta t_{central}\) of was repeated three times for each value of \(R_{cath}/R_0\). See Fig. 12. The experiment measuring the \(\Delta t_{central}\) showed a reasonable agreement between theory and measurements, in which standard deviation between theory and experiment was 8.4 s, which is 13% of the total range of variations in \(\Delta t_{central}\).
Discussion
General findings from the expressions derived
In many of the resulting expressions as presented in “Results” section, the relative contribution of various factors affecting the dosing errors, such as the poiseuille mixing effect, can be discerned clearly in the structure of these expressions. The characteristic “time constant” \(R_{cath} C\) is clearly recognisable in the expressions. Furthermore, the ratio between the \(u^{(G)}_{downstep}\) (the size of the change in pump flow rate setting value) and the \(u_{final}\) (the stabilized final flow rate) appears as an important factor in determining the nature of the dosing error bolus, particulary how the dosing error will spread out in time (the “width” or “duration” (\(2\sigma \)) of the dosing error bolus). The contribution of the Poiseuille mixing effect is visible in the resulting expressions. Of particular clinical importance is the fact that the Poiseuille flow profile, once fully developed, causes a significant reduction of the time that is needed for a newly administered medication to reach the patient. This may come as an unexpected effect for the clinician, as may also the fact that the value of \(2\sigma \) (the “spread out”) is increased at the same time. The results in this paper may help to determine the magnitude of this “spreadout” effect as function of the hardware parameters (most notably, the characteristic \(R_{cath} C\) time), and e.g. the length and resistance of the catheter.
Limitations of the method, and possible extensions
A number of limitations can be identified in our method in its present form; most apparent is the assumption that when a time interval of duration \(t^{POIS}_{delay}\) has lapsed after having changed a pump flow rate setting value, the actual flow rate has already stabilized and reached the value \(u_{final}\). This needs not to be true in a general case. However, using the same reasoning as presented in this paper, our method can be extended to include nonstable flow rates at \(t=t^{POIS}_{delay}\) as well. Another limitation may arise by the fact that in our method we did not examine other elements (other than just syringes, infusion lines, catheters, and pumps) that may be present in an infusion setup, such as a nonreturn valve (preventing flow back from the catheter into a line towards a pump), or filters. The nonreturn valves may be incorporated into the method by restricting flow rates in the catheter to positive values or zero, whereas filters may be modeled on basis of a resistance and a mechanical compliance, as indicated in the literature (e.g. [21]). Another complicating factor may be the use of high viscosity fluids in the infusion setup. In our method, as it stands now, all fluids in the system were assumed to be of the same viscosity. Incorporation of deviating (high) viscosities into our method would entail an extension of the laminar flow profile used in this paper, because differences in viscosity of mixing liquids may produce destabilization of laminar flow. An easy, but useful, extension of our method is the incorporation of the possibility that the height at which the syringes are positioned in a setup near the patient, is altered during the period that a patient receives medication using the multiinfusion setup. Since a change in height may be modeled as the addition of an extra pressure source within the model during the process of infusion, this results in an extra bolus of dosing error, and, hence, in the addition of some extra terms and factors in the equations in “Results” section. Finally, the parabolic flow profile of a laminar flow does not come into existence instantaneously at the mixing point; it has been calculated [20] how long, c.q. what distance, it takes for the flow profile to approach the parabolic shape. All of these complicating factors need to be incorporated into the model to make it more realistic. As far as we can anticipate now, we do not expect these extra factors to be incompatible with our general approach outlined in this paper; however, further research is needed.
Potential use of the results in clinical practice
Healthcare professionals working with infusion technology in critical care have expressed the desire for a realtime tool that visualizes the multiinfusion drug therapy, e.g. continuously calculates predictions to indicate when the drugs will be entering the blood stream of the patient and in what dose. Such a tool may also visualize the causal consequences of an intervention (i.e., change in a pump flow rate setting value), before a clinician decides to proceed with such an intervention. In order to develop such a tool in the future, a fast and generic model will be necessary, combining all the relevant physical effects. The desirability of such a visualization tool, and the mathematical modeling that is a prerequisite for the development of such a tool in the future, is what prompted us to go beyond the stateoftheart and to develop the fully analytical method described in this paper. We envision three types of developments in which the results from this paper may be useful: (i) an interactive tool, running synchronized with the multiinfusion system on a smartphone device or on a bedside display, e.g., next to the vital signs display monitors. Such a tool could then be used in several cases, such as inotropic titration, or to visualize the effects of a syringe changeover, or even the consequence of changing the height of a pump during infusion. (ii) Another application of the method presented in this paper could be actual computer control of an infusion system. It has been shown that a computercontrolled pump with “knowledge” about the dead volume and the mixing effect within the dead volume can be useful in preventing overshoot [16]. Moreover, a control system with a feedback approach has also been attempted, where the mean arterial blood pressure was used to control the administration of a fastacting vasodilator [22, 23]. In both cases, however, increasingly complex situations and infusion setups were encountered where only the incorporation of all the interdependent physical effects, as described in this paper, would provide a sufficiently accurate prediction in order to make computer control feasible. (iii) The model can potentially also be used as an analysis and design tool, prior to investing in potential new infusion hardware. It is known that flow characteristics are influenced by valves [24, 25], syringes [26], infusion lines and catheters [27–29], and filters [21]. By using the method from this paper, benefits of these components can be compared against potential tradeoffs.
Conclusions
We have developed a new method that combines mechanical compressibility (compliance) effects with poiseuille flow and pushout effects (“dead volume”) in one single mathematical framework for calculating dosing errors in multiinfusion setups.
In contrast to existing numerical methods, our method produces explicit expressions that indicate the mathematical dependencies of the dosing errors on hardware parameters and pump flow rate settings.
The results from the invitro experiments show a reasonable to good agreement between measurements and theoretical results.
The relative contribution of various factors affecting the dosing errors, such as the poiseuille mixing effect, resistance and internal volume of the catheter, mechanical compliance of the syringes and the various pump flow rate settings, can now be discerned clearly in the structure of the expressions generated by our method.
This enables insight and predictability in a large range of possible situations involving many variables and dependencies, which is potentially very useful for e.g. the development of a fast, bedside tool “calculator” that provides the clinician with a precise prediction of dosing errors and delay times interactively for many scenario’s. The interactive nature of such a device has now been made feasible by the fact that, using our method, explicit expressions are available for these situations, as opposed to conventional timeconsuming numerical simulations. Other potential applications of our method involve analysis and design tools for new infusion hardware, and interactive devices connected to the infusion hardware that counteract impending dosing errors using predictive calculations.
Abbreviations
ICU: intensive care unit; OR: operation room; MABP: mean arterial blood pressure.
List of symbols
 \(V_{cath}\) :

internal volume of the catheter
 \(t_{delay}\) :

time needed to travel through the catheter
 \(u_{cath}\) :

volumetric flow rate inside the catheter
 \(\mathcal {M}\) :

mixing point where fluid enters the catheter
 \(\mathcal {P}\) :

tip of catheter where fluid enters the blood stream
 L :

length of the catheter
 N :

large dimensionless number, representing the number of voxels inside the catheter
 \(\gamma \) :

L/N, i.e. length of a single voxel inside the catheter
 \(t=0\) :

point in time at which a change in pump flow rate setting value takes place
 \(a_k^{(R)}\) :

fraction of the volume of voxel k that contains “red” solution
 \(u_{patient}^{(R)}(t)\) :

partial flow rate of the “red” solution that enters the blood stream at time t
 \(u_{pump}^{(R)}\), \(u_{pump}^{(G)}\) :

pump flow rate setting of the “red”, and “green”, pump, respectively
 \(\Gamma \) :

complete set of flow rate settings and physical hardware parameters
 \(\beta _{patient}^{(R)}(t)\) :

dosing error of “red” solution, i.e. deviation from intended partial flow rate, as function of t
 \(t_{delay}^{POIS} = {{1}\over {2}}t{delay} \) :

time needed for the tip of the Poiseuille flow to travel through the catheter along the central longitudinal axis of the catheter
 \(u_{old}^{(G)}\) :

pump flow rate setting of the “green” pump before \(t=0\)
 \(u_{final}^{(G)}\) :

pump flow rate setting of the “green” pump after \(t=0\)
 \(\tau \) :

\(t  t_{delay}^{POIS}\)
 \(\beta _{pushout}^{(R)}(t)\) :

\(\beta _{patient}^{(R)}(t)\) for \(0< t < t_{delay}^{POIS}\)
 \(\psi _{patient}^{(R)}(\tau )\) :

\(\beta _{patient}^{(R)}(t) / (u_{final}^{(G)} + u_{pump}^{(R)} )\) for \(t_{delay}^{POIS} < t\)
 \(\lambda _{tot}(t)\) :

distance that the fluid would have travelled without the Poiseille mixing effect
 \(u_{\mathcal {M}}^{(R)}(t)\) :

the actual partial flow rate of “red” fluid entering the catheter at the mixing point \(\mathcal {M}\) at time t
 \(u_{\mathcal {M}}^{(R)diff}(t)\) :

\(u_{\mathcal {M}}^{(R)}(t)  u_{pump}^{(R)}\)
 \(w_i\) :

weight factor for contribution of original voxel nr i to the voxel at the tip \(\mathcal {P}\)
 \(b_{2j}\) :

\(a_j^{(R)diff}\)
 \(\psi _k^{(R)patient}\) :

discrete form of \(\psi _{patient}^{(R)}(\tau )\)
 \(\psi _k^{(R)POIS}\) :

part of \(\psi _k^{(R)patient}\) that is calculated in the Zdomain
 \(\Psi ^{(R)POIS}(z)\) :

Ztransform of \(\psi _k^{(R)POIS}\)
 Q :

total volume of the \(\psi _k^{(R)POIS}\)part of the dosing error
 \(t_{central}\) :

see Fig. 5c
 \(\sigma \) :

see Fig. 5c
 \(R_{cath}\) :

resistance of the catheter
 \(R_1\) :

resistance of the “green” feeding line
 \(R_2\) :

resistance of the “red” feeding line
 \(C_1\) :

mechanical compliance of the “green” syringe
 \(C_2\) :

mechanical compliance of the “red” syringe
 \(u_{downstep}^{(G)}\) :

change in pump flow rate setting of the “green” pump at \(t=0\)
 \(\vartheta _{first}\), \(\vartheta _{second}\), a, b, c :

see Eq. (34)
Declarations
Authors' contributions
MKK has invented the new mathematical approach, performed the calculations, and largely wrote the manuscript, including all mathematical derivations. RAS has revealed the importance of Poiseuille flow characteristics in the evolution of dosing errors, and has initiated the incorporation of this effect into the model. Furthermore, RAS has placed the calculations within meaningful clinical context, and contributed significantly to the “Background” and “Discussion” sections and the invitro experiments. JHR has performed the invitro experiments, and has contributed to early stages of the modelling. AMT has initiated the study described in this manuscript, identified its targets, defined its clinical purpose, designed the invitro experiments, and coauthored the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This research was partly funded by the EMRP project Metrology for drug delivery. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.
Competing interests
The authors declare that they have no competing interests.
Data availability statement
Please contact author for data requests.
Ethics approval and consent to participate
Not applicable: No human or animal tissues involved.
Funding
This research was partly funded by the EMRP project Metrology for drug delivery. The EMRP has not been involved in the design of this particular study, and has not been involved in writing the manuscript, or in the collection, analysis, interpretation of the data.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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