## Abstract

Intravenous drug infusion driven by syringe pumps may lead to substantial temporal lags in achieving steady-state delivery at target levels when using very low flow rates (“microinfusion”). This study evaluated computer algorithms for reducing temporal lags *via* coordinated control of drug and carrier flows.

Novel computer control algorithms were developed based on mathematical models of fluid flow. Algorithm 1 controlled initiation of drug infusion and algorithm 2 controlled changes to ongoing steady-state infusions. These algorithms were tested *in vitro* and *in vivo* using typical high and low dead volume infusion system architectures. One syringe pump infused a carrier fluid and a second infused drug. Drug and carrier flowed together *via* a manifold through standard central venous catheters. Samples were collected *in vitro* for quantitative delivery analysis. Parameters including left ventricular max dP/dt were recorded *in vivo*.

Regulation by algorithm 1 reduced delivery delay *in vitro* during infusion initiation by 69% (low dead volume) and 78% (high dead volume). Algorithmic control *in vivo* measuring % change in max dP/dt showed similar results (55% for low dead volume and 64% for high dead volume). Algorithm 2 yielded greater precision in matching the magnitude and timing of intended changes *in vivo* and *in vitro*.

Compared with conventional methods, algorithm-based computer control of carrier and drug flows can improve drug delivery by pump-driven intravenous infusion to better match intent. For norepinephrine infusions, the amount of drug reaching the bloodstream per time appears to be a dominant factor in the hemodynamic response to infusion.

## Abstract

Computer-executed algorithms can coordinate the output of pumps infusing drug and carrier fluid to reduce delivery lag time *in vitro* and *in vivo.*

Syringe pumps are used to infuse highly concentrated short-acting titratable medications at very low flow rates, with low flow rate carrier fluid propelling drugs into the patient’s blood stream. Low flows protect the patient from fluid overload.

Because of the low flow rates relative to dead volumes of infusion systems, steady-state drug delivery at target levels may lag behind the clinician’s intent.

Computer-executed algorithms can coordinate the output of pumps infusing drug and carrier fluid to reduce delivery lag time

*in vitro*and*in vivo*.

CRITICALLY ill or anesthetized patients commonly receive short-acting, titratable medications by IV infusion. To reduce the amount of fluid administered to patients, many institutions use syringe pumps to deliver drugs prepared in highly concentrated form (*e.g.*, norepinephrine, 4 mg/ml or higher; nitroglycerine, 1 mg/ml or higher)^{1 } at very low flow rates (“microinfusion”). Inert carrier fluids, also at low flow rates, propel the drugs into the patient’s blood stream.

Low total flow rates, when taken relative to infusion system dead volumes, lead to long time constants (dead volume divided by total fluid flow).^{2 } Under these low flow conditions, laboratory studies show that achieving steady-state delivery at target levels will likely lag behind the clinician’s intent.^{3–5 } These temporal lags can be substantial, as long as 20 to 30 min,^{3,5 } but are often unappreciated in the clinical setting. They also have the potential for clinically significant dosing errors. The critically ill pediatric patient may be particularly vulnerable because low flow rates interact with relatively high infusion system dead volumes.^{5,6 } We now present an approach to overcome the problem of these delivery lag times.

We hypothesized that delivery of infused drugs to target levels could be improved by creating algorithms (described in appendices 1 and 2) based on physical principles and executed by computers, for coordinated control of drug and carrier pumps. Drug delivery would then better match the clinician’s intent in terms of timing and magnitude, with the goal of rapid and precise dosing. Two basic clinical situations were targeted: initiation of a new drug infusion and dose titration of an ongoing infusion. We developed two algorithms and tested them both *in vitro* and *in vivo*.

## Materials and Methods

### Development of Algorithms to Drive Infusion Pumps by Computer Control

Conceptually, initiation of drug into a flowing carrier results in a diffusion gradient as the drug (now mixed with carrier fluid) travels along the fluid path. We used Taylor dispersion theory^{7 } to model the drug concentration along the fluid path, characterizing the diffusion wavefront over time. The model predicted drug delivery at the next incremental time step and adjusted drug and carrier flows to minimize target error. Further detail can be found below and in appendix 1.

The differential equations of the drug infusion initiation model were solved computationally *via* a forward difference numerical approximation; the output included drug delivery values (mass of drug per unit time) at the distal tip of the catheter. Parameters of the model included a diffusion coefficient of drug in aqueous solution and the length and radius of the fluid path. The latter two parameters are dependent on the infusion set and connectors used, and these parameters’ product is the dead volume used with the algorithm. Given that the model is based on the flow characteristics of straight, uniform diameter tubes, we sought to determine the “empiric dead volume” for a straight uniform tube that would approximate the behavior of the real-world infusion system used in the laboratory experiments. This empiric dead volume used with the model was arrived at *via* a calibration procedure and differed slightly from measured dead volume. A small difference was expected given the connectors, right angle bends and changes in lumen diameter of real-world infusion systems such as those used in the laboratory experiments.

Based on the model, we developed an algorithm that allows the rapid delivery of drug to achieve the target in steady state when initiating an infusion (the term steady state refers to drug delivery at the intended rate over time, which can be calculated for the *in vitro* experiments based on the known flow rates and the known concentration of drug or tracking dye). The algorithm is designed to enhance delivery while minimizing excess fluid administration (for a detailed description of both the algorithm and the model, see appendix 1). In brief, the algorithm maintains increased carrier and drug flows until drug delivery is calculated to have reached the desired level. At that time, the algorithm begins rapidly ramping down carrier and drug flows together toward their final desired rates in concert and in a precise manner as calculated by the algorithm to maintain drug delivery at the desired steady-state level. This algorithm was used to generate control instructions in the form of coordinated drug and carrier flow settings over time. We compared algorithm-controlled initiation of drug delivery with the conventional approach of initiating drug delivery (“turning on the drug pump” while maintaining a constant carrier flow).

We also conceptualized and implemented a second algorithm (described in appendix 2) based on maintenance of a fixed ratio between drug and carrier flows that allows pumps to precisely deliver the intended amount of drug when dose changes are made from an ongoing infusion at steady-state delivery. The second algorithm requires that drug is already at a constant concentration along the fluid path. This simplifying condition allows the algorithm to coordinate rapid and precise changes in drug dosing without having to rely on mathematical models of diffusion during drug flow. The second algorithm does not apply to drug initiation, where there is initially no drug in the fluid path. In the case of drug initiation, changing concentration of drug along the fluid path is unavoidable due to diffusion, and another algorithm (described in appendix 1) that takes diffusion into account *via* mathematical modeling is needed.

These algorithms were tested *in vitro* as well as *in vivo* in animal experiments.

### Development of a System Allowing External Control of Syringe Pumps

Testing the control algorithms required design and creation of a novel system allowing external regulation of pump flows by a computer executing the algorithms. We invented an interface system between a computer and laboratory-grade syringe infusion pumps capable of receiving external, electronic, instructions regulating pump output (U.S. Patent Application 2575933 A2, “Prediction, visualization, and control of drug delivery by infusion pumps”).^{*} Pump system output was internally tested and confirmed in every *in vitro* experiment as empiric drug delivery reached predicted steady-state levels as calculated based on the known flow rates and the concentration of methylene blue (MB).

*In Vitro* Studies

All components of the *in vitro* test infusion system consisted of standard clinical materials as previously described.^{4,8 } For simulation of infusions, we connected the 16-gauge lumen of a 7-French triple lumen central venous catheter (dead volume 0.39 ml, MSO-12703-PHS; Arrow International, Reading, PA) or a 9-French introducer (dead volume 3.26 ml, MSO-09903-PHS; Arrow International) to a four-port stopcock manifold (dead volume from end closest to the patient to third upstream position is 1.17 ml, W21122, Arrow International) with the addition of a new venting method which mitigates compliance and pump start-up factors.^{8 } One externally controllable pump (NE-500 Programmable OEM Syringe Pump; New Era Pump Systems, Inc., Farmingdale, NY) loaded with a 60-ml syringe delivered a normal saline carrier. A second externally controllable pump (same model as above), also loaded with a 60-ml syringe, drove the infusion of either MB used as a visibly detectable model drug or norepinephrine. The drug infusion was connected to the third upstream position of the manifold.

Experimental drug infusion was initiated by closing the vent and opening the manifold stopcock port. Fluid leaving the tip of the catheter was collected at 1-min intervals as previously described.^{3,8 } For experiments using MB as the tracking dye, samples were transferred to microtiter plates containing a standard curve constructed from serial dilutions of the same stock of MB used for the infusion. Quantitative measurements were made by absorbance at 668 nm.

For experiments studying the delivery of norepinephrine, the concentration of norepinephrine in each sample was also determined by spectrophotometric methods.^{9 } Metaperiodate (6 μl of 2% NaIO_{4} in ddH_{2}O, #S1878; Sigma-Aldrich, St. Louis, MO) and ethanol (9 μl, 100%) were added to the samples and the absorbance at 490 nm was measured. A standard curve was constructed by diluting norepinephrine from the same stock as was used in the experiments.

We compared the delivery kinetics of norepinephrine and MB *in vitro*. Norepinephrine and MB exhibited similar delivery profiles under control conditions as well as when tested under algorithm-controlled infusion conditions (fig. 1).

*In Vivo* Studies

#### Animal Surgical Preparation.

All surgical procedures were approved by our local Institutional Animal Care and Use Committee (Steward St. Elizabeth’s Medical Center, Boston, Massachusetts). Three adolescent Yorkshire swine (36 to 41 kg) were fasted for 12 h preceding the experiment, but had access to water. The animals were anesthetized and catheterized as previously described.^{10 } In brief, a femoral artery (16 gauge, Arrow CS-04300) and right external jugular vein pulmonary artery catheter (746HF8; Edwards Life Sciences, Irvine, CA) were placed. A 9-French introducer was placed in the contralateral femoral artery and used to pass a Millar pressure–volume conductance catheter (Ventri-cath 507; Millar Instruments, Houston, TX) retrograde into the left ventricle. A triple lumen catheter (MSO-12703-PHS; Arrow International) or a 9-French introducer sheath (MSO-09903- PHS; Arrow International) was placed in one of the femoral veins and connected to a four-stopcock manifold (W21122; Arrow International). Anesthesia was maintained with midazolam (0.25 mg kg^{−1} h^{−1}), fentanyl (12 to 25 μg kg^{−1} h^{−1}), and ketamine (5 mg kg^{−1} h^{−1}) and stabilized for 30 min before norepinephrine infusion. We chose to use norepinephrine as the biologically active drug because its receptor activation properties should be associated with fewer confounding cardiovascular side effects than other catecholamines such as epinephrine. In addition to contractility, heart rate, blood pressure, cardiac output, and central venous and pulmonary artery pressures were all measured and recorded. We report data for max dP/dt because this physiologic variable is less subject to confounding interactions and cardiovascular compensatory mechanisms.

For the low dead volume experiments (two animals), drug delivery was controlled in the order specified in table 1. The rest interval between experimental runs was 20 min during which time hemodynamics returned to baseline. Therefore, each animal served as its own control. The high dead volume experiments (one animal) were intended to support the biological relevance of the *in vitro* findings while sparing the use of animal life for experimental purposes. At the end of the day, animals were sacrificed using Euthasol (0.1 ml/kg, Virbac Corporation, Fort Worth, TX) as previously described^{10 }.

### Statistical Analysis

The time delay from the start of MB flow until MB delivery reached 50% of predicted steady-state delivery (T50) was measured, and the differences between T50 measurements for the low and high dead volume infusion systems were compared by the Mann–Whitney test. Differences were considered significant for *P* values less than 0.05. For the *in vitro* measurements, T50 values were derived by linear interpolation using values from the 1-min collection intervals.

For the analysis of *in vivo* data, T50 was calculated as the time delay from the start of norepinephrine flow until max dP/dt reached 50% of the plateau value for max dP/dt. All max dP/dt data were normalized between 0% change (pretreatment) and 100% change (max dP/dt value at steady-state drug delivery). Differences between T50 measurements for the low and high dead volume infusion systems were compared by the Mann–Whitney test, with *P* values less than 0.05 considered significant. Data are reported as mean ± SD and range.

Power analyses were performed before data collection and based on previous T50 data according to the method by Machin *et al.*,^{11 } which assumes that the data groups are independent of each other. Previous *in vitro* data suggested that the T50 with a high dead volume infusion system was 5.7 min longer than with a low dead volume infusion, with maximum SD of 0.6 min.^{10 }*In vivo* data from this previous study showed that the T50 was 6.7 min longer for the high dead volume infusion system, with a maximum SD of 1.3 min. To be conservative, the power calculations used differences between groups that were one half of those observed in the previous study. Power calculations showed that two repetitions were needed *in vitro* and five repetitions were needed *in vivo* to achieve 90% power and to have less than 5% chance of a type I error.

*In vitro* measurements were repeated at least three times and *in vivo* measurements for the low dead volume infusion system were repeated five times in the first two animals. The order of each experiment is shown in table 1, and the animals were rested 20 min in between each experiment. The *in vivo* measurements for the large dead volume infusion system were performed in one animal and the data were not adequately powered to show statistical significance. Rather, it was meant to be confirmatory of the trends observed *in vitro*.

## Results

*Comparison of MB and Norepinephrine Characteristics* In Vitro

The delivery kinetics of MB as the model drug were compared with the delivery kinetics of norepinephrine under two conditions, conventional initiation of an infusion (“turning on the drug pump” at a constant carrier flow) and algorithm-driven initiation of an infusion. Figure 1 focuses on the comparison of the delivery profiles and demonstrates substantially similar profiles for these two drugs under the conditions tested.

### Initiation of an Infusion

As described above, the delivery kinetics of MB as the model drug were studied *in vitro* under two conditions, conventional initiation of an infusion and algorithm-driven initiation of an infusion. For the two catheter lumens studied (triple lumen 16-gauge and 9-French introducer), algorithm-driven infusion initiation reached steady-state delivery significantly faster than initiation by the conventional initiation method (figs. 2 and 3 and table 2). For the algorithm-driven infusions, the total amount of additional fluid required to more rapidly achieve steady-state delivery was 0.8 ml (16-gauge lumen) and 3.43 ml (9-French introducer) under the conditions tested.

For *in vivo* experiments, the delivery system was identical to the delivery system studied *in vitro*. With max dP/dt as the experimental endpoint in the anesthetized pig model, contractile effects of a newly initiated norepinephrine infusion achieved steady state faster when algorithm-controlled pumps drove both drug and carrier flows (figs. 2B and 3B and table 2).

### Adjustments to the Dose of an Ongoing Infusion at Steady State

For *in vitro* studies, the experimental infusion of MB was allowed to achieve steady-state drug delivery through a 16-gauge catheter lumen. We modeled a clinician’s intent to first halve the steady-state rate of drug delivery and then to double the initial steady-state rate of drug delivery (fig. 4A). Algorithm-controlled adjustments of delivered drug dose more closely matched the clinician’s intent than did conventional adjustments (fig. 4A). The profile of max dP/dt measurements in the anesthetized pig model closely resembled the *in vitro* findings (fig. 4B). An additional set of *in vivo* measurements compared max dP/dt effects for algorithm *versus* conventional control when doubling and then shortly thereafter returning to baseline the rate of a norepinephrine infusion delivered through a 9-French introducer lumen. With algorithm control, max dP/dt increased promptly and then returned to baseline rapidly as expected. Contractility changed very little when dose delivery control was attempted by conventional means (fig. 5).

## Discussion

Safe and effective drug administration by continuous infusion depends on delivering the intended drug dose at the desired time while minimizing unintended consequences. We aimed to test whether computer control could enhance the likelihood of delivery according to intent. The results demonstrate that, compared with conventional methods (for example, initiating drug flow by turning on the drug pump while maintaining a constant carrier flow), algorithm-based coordinated control of carrier and drug flows can improve drug delivery by pump-driven IV infusion to better match intent. Furthermore, at least in the case of norepinephrine, the amount of drug reaching the bloodstream per time appears to be a dominant factor in the hemodynamic response to infusion.

Traditional drug infusions driven by large volume pumps deliver dilute drug mixtures with high carrier flows. An unintended consequence is the obligate large fluid volume absorbed by the patient. Low flow fluid management strategies relying on concentrated drug solutions minimize fluid volume delivery to protect patients with congestive heart failure, renal dysfunction, respiratory failure, or increased intracranial pressure from fluid overload. These are sometimes called “microinfusion” strategies.

We have previously demonstrated *in vitro* that there may be a time gap between a clinician’s intent and actual drug delivery at the intended steady-state dose when using low flow, fluid volume conservative, approaches to initiate drug delivery by continuous IV infusion. A key factor in this discordance is the time constant, which is the dead volume divided by total fluid flow.^{1 } Consequently, we aimed to improve on conventional means to initiate or adjust low flow drug delivery by infusion to meet clinical expectations and needs. The approach centers on developing algorithms that can be executed by a computer to coordinate the output of the infusion pumps driving delivery of drug and carrier fluid.

We now show that using two such algorithms to control drug and carrier flows *in vitro* can more closely match actual drug delivery to the intended steady-state goal. We also show that using the algorithms *in vivo* leads to physiologic responses that mirror the *in vitro* findings, better approximating a clinician’s intent. The *in vivo* findings suggest that controlling drug delivery can significantly and predictably alter the time course of physiologic response.

When adjusting the dose delivered by an ongoing drug infusion, the caregiver’s conventional approach is to simply raise or lower the drug pump’s output rate (“turn up or turn down the drug pump”). Such situations are common, particularly in the operating room environment where clinical needs can change rapidly. We tested conventional adjustments to a volume conservative infusion at steady-state delivery. We show that actual dose delivery changes occur slowly, if at all. Thus actual delivery may not match the intent to adjust dosing. With the dose adjustment algorithm, delivery *in vitro*, and physiologic response *in vivo*, more closely match intent.

The time course of physiologic response to the norepinephrine infusion results from the combination of delivery kinetics through the intravascular catheter and circulatory pharmacokinetics. After the drug enters the animal’s circulation, for any given level of drug delivery, we would expect little difference in physiologic response (contractility change) between algorithm-driven and conventional delivery conditions; the circulatory pharmacokinetics and receptor activation should be similar between the two conditions. What is the response to the drug once it reaches the circulation? If the response is very slow in both algorithm-driven and conventional conditions, we would conclude that circulatory pharmacokinetics is the dominant factor in determining the time course of the physiologic response. We observed that the physiologic response to the norepinephrine infusion reflects the differences in drug delivery through the IV catheter as measured *in vitro*. This finding is consistent with the concept that the kinetics of drug delivery through the intravascular catheter represents a significant, and perhaps dominant, factor governing the time course of max dP/dt responses. This concept is supported by our previous work as well.^{10 }

We calculated the difference in delivered fluid volume between the algorithm-driven and conventional approaches to managing initiation of drug infusions. In the algorithm-directed acceleration of drug delivery for a new or resumed infusion, coordinated high flow rates for drug and carrier are restricted to brief intervals before ramping down to final flows while maintaining target delivery. This approach maintains fluid volume conservation. Under the chosen experimental conditions (16-gauge catheter lumen or 9-French introducer, with final flows of 10 ml/h [carrier] and 3 ml/h [drug]), the algorithm as tested requires only 0.8 ml (16-gauge lumen) or 3.43 ml (9-French introducer) of additional fluid to reduce the time to achieve the intended steady-state delivery rate *versus* the conventional approach of maintaining fixed carrier (10 ml/h) and drug (3 ml/h) flows.

In the conventional approach, which involves turning on the drug pump at a fixed rate, the algorithm’s rise time can be matched by using a high initial carrier flow. If the carrier flow is later turned down to limit volume delivery, the drug will be underdosed. If the carrier flow is kept high to maintain target delivery, the amount of extra fluid delivered in this conventional approach *versus* the algorithm approach depends on the duration of the infusion and may be substantial.

In both algorithm-directed and conventional adjustments of an ongoing infusion, fluid volume delivered will increase when raising the drug dose. A potential disadvantage of the algorithm’s coordinated pump mechanism to precisely raise the dose of an ongoing fusion at steady state is the obligate increase in fluid delivery that is somewhat greater than results from the conventional approach. The amount of extra fluid is likely to be relatively small but could be clinically significant for some patients, such as neonates or infants. A gradual “ramp down” of carrier fluid flow can likely overcome this limitation with minimal impact on drug delivery.

We have not tested all possible combinations of intravascular catheters, manifolds, fluid flows, tubing, and other devices that may be encountered in medical environments. Although the results must be interpreted with caution given the small N values, the strength of the *P* values lends validity to the conclusion of significance. The experimental confirmation of our approach, with similar findings obtained with two different catheters in both laboratory and animal testing, is consistent with general validity. The rate of onset for the initiation algorithm can be adjusted based on the choice of maximal allowed drug and carrier flow rates used with the algorithm. The concepts we describe may have clinical utility, at least for drugs such as catecholamines and other medications for which rapid delivery onset and dose adjustment may be needed depending on the condition of a patient. Such drugs likely have short biological half lives and short-term effects once they enter the patient’s circulation. The concepts are less likely to be beneficial for infusions of drugs with longer-lasting biological effects once entering the circulation, for example, insulin, milrinone, *etc*. Rapid delivery onset and rapid dose adjustment are less important for this class of medications. The *in vitro* data suggest that norepinephrine and MB have very similar delivery profiles (fig. 1). These molecules have very different sizes and structures, suggesting that the algorithms may be applicable to a variety of drugs.

The technology we describe addresses the case of a single-drug infusion plus carrier and confers no advantage when a patient receives one or more drug infusions *via* a multiple lumen central line allowing dedication of one lumen to each drug (without a carrier fluid). However, for many critically ill intensive care unit patients or patients in the operating room, limited IV access complicates simultaneously monitoring central pressures and administering antibiotics, hyperalimentation, sedative/hypnotic agents, analgesics, one or more vasoactive agents, inotropes, antidysrhythmics, blood products, *etc*. Dedicating a catheter lumen to a single-drug infusion may not be feasible in such circumstances. This is particularly relevant to intensive care unit or intraoperative anesthesia management of the neonate, infant, or small child.

Our previous work demonstrated the impact on delivery of a primary drug when the administration of a coinfused drug starts or stops.^{8 } The technology described in this report does not address the situation where more than one drug flows through the same line. We anticipate that future technological developments will allow optimization of delivery conditions for a carrier fluid with two or more drugs.

In addition, with current manifolds, tubing and catheters, termination of drug delivery by infusion cannot easily be hastened. Increasing the rate of carrier flow to clear the dead volume will produce an initial, predictable, bolus of the drug from its reservoir in the dead volume. If both the carrier and the drug infusion are stopped, the dead volume does not clear. The algorithmic control demonstrated in this report does not address termination of an infusion. We do, however, address this scenario in the previously referenced patent application, describing the use of very low carrier flows during the drug cessation phase to “wash out” drug within the dead volume at a negligible rate of delivery. The model upon which our algorithms are based allows calculation of the remaining concentration of drug along the fluid path during the “washout” period. Such tracking of dead volume drug concentration creates the possibility for warnings regarding a potential bolus if the infusions were to be reactivated during this cessation phase.

In conclusion, algorithms derived from mathematical models of fluid flow can direct infusion pump output in predictable, controlled manner. The algorithms provide a means to closely match the clinician’s intent without relying on clinical experience and intuition to manipulate pump flows. Furthermore, the animal data indicate that the rate of drug delivery to the circulation is at least a significant, if not dominant, factor in the physiologic response to medications administered by infusion. This has broad implications for pharmacokinetic concepts.

## Acknowledgments

The authors thank Harold Demonaco, M.A., Massachusetts General Hospital, Boston, Massachusetts, for ongoing guidance and support of this work.

Support was provided solely from institutional and/or departmental sources.

## Competing Interests

Drs. Peterfreund, Lovich, and Parker have submitted an application for U.S. patent protection for the technology reported in this article. This technology has not been licensed; there is no commercial value at this time. Hence, there are no royalties or any other financial implications. The authors declare no competing interests.

Available at: http://www.google.com/patents/EP2575933A2?cl=en. Accessed October 10, 2014.

## References

*in vitro*analysis of central venous drug delivery by continuous infusion: The effect of manifold design and port selection.

*via*a pediatric central venous infusion system: Quantification of delays in achieving intended doses.

*in vitro*modeling study of microinfusion.

*in vitro*and also

*in vivo*in anesthetized swine.

### Description of Mathematical Model and Algorithm for Reduction of Drug Delivery Onset Delay

This appendix describes the methods we use for predictive modeling and control of drug delivery in intravenous infusion. These methods were used as the basis for the infusion initiation experiments described in this article.

We used the mathematical model (described below) to predict drug delivery for the case of one drug infused with a carrier fluid. One of the algorithms (described after the model) makes use of the values predicted by the model to calculate coordinated adjustments to drug and carrier pump flows, with the goal of achieving reduced time to therapeutic drug levels at drug initiation. The other algorithm (described in appendix 2) that was used in our experiments does not rely on the predictive modeling.

#### Equations

We used the Taylor dispersion effect as the basis of our model. Taylor dispersion deals with longitudinal dispersion of flow in tubes and is also broadly applicable to any kind of flow in which there are velocity gradients. The equation (shown below) takes into account radial diffusion, axial diffusion, and laminar flow:

where D = molecular diffusion coefficient; = mean axial velocity of fluid; *x* = axial distance along fluid path; *t* = time; *R* = radius of tube; = concentration of dye/drug averaged over cross-section of tube (as a function of *x*, axial distance along the tube).

We assume that the Reynolds Number for the very low velocities found in microinfusions is consistent with nonturbulent flow.

The terms in parentheses take into account diffusion and shear and are dispersion (due to shear from laminar flow plus radial diffusion) and molecular axial diffusion (second term).

We implemented the model numerically *via* forward difference models. We tested the model in the laboratory using a tracking dye for which delivery could be determined with quantitative spectrophotometry.

The molecular diffusion coefficient, D (cm^{2}/s), was estimated based on the properties of the drug/dye molecule. For the purposes of all of the later experiments (the included graphs), D was held constant at 0.00001 because this should be a property of the drug/dye molecule used in the experiments, and we only used one type of dye. Values of D for individual drugs may need empirical determination if experimental values are not available.

The mean axial velocity of fluid, , was calculated from the total flow rate and the properties of the fluid path.

The concentration of dye/drug averaged over cross-section of tube (as a function of *x*, axial distance along the tube), , is what we are solving for over the length of the tube for any given time.

We consider the fluid path to have an “empiric dead volume” that is different from the measured dead volume, given that the fluid path is irregular, with changes in diameter between stopcocks, manifold, tubing, *etc*., and changes in angle (*e.g.*, with stopcock ports meeting the manifold at right angles) and, therefore, does not behave in an idealized manner (as a uniform diameter straight tube would).

A calibration procedure for finding the empiric dead volume used with the model consists of systematically examining a series of candidate empiric dead volumes, and finding the one that gives the best fit to the control curve in the least-squares sense. The empiric dead volume was used in place of the measured dead volume for modeling.

In brief, the algorithm used for reduction of drug delivery onset delay for single drug plus carrier maintains increased carrier and drug flows until the desired drug delivery is calculated to have reached the distal end of the fluid path. At that time, the algorithm begins ramping down carrier and drug flows in concert toward their final desired rates. The algorithm calculates this ramping in a precise manner to maintain drug delivery at the desired steady-state level. Thus, carrier and drug flows are changing rapidly even as the drug delivery is maintained constant. The initial, increased, carrier and drug flows can be chosen to balance the intended hastening of delivery with the amount of excess fluid administered (compared with the conventional practice protocol studied).

A detailed description of this algorithm follows.

Terminology used in the following descriptions and in figure 6 is as follows:

Q_{c} = carrier flow; Q_{d} = drug flow; Q_{T} = total flow = Q_{c} + Q_{d}; c_{d} = stock drug concentration; Q_{cmax} = maximum allowable carrier flow; dd = drug delivery (mass of drug per time arriving at patient end of fluid path); ss = steady state (state reached when drug delivery has stabilized); p = Q_{c}/Q_{d} (proportionality between carrier and drug flows); d*t* = time increment or time scale used in stepping through the model calculations. This is generally much less than 1 s (*e.g.*, 0.001 s).

Other than where we say “user sets…” or “user enters…,” all other adjustments below are carried out by the control program, as in our prototype experiments.

Method for reducing time to achieve therapeutic levels of drug delivery for single drug and carrier (summarized in fig. 6):

User enters the characteristics of the infusion system (manifold, intravascular catheter and any intravenous tubing that may be used), so that the dead volume can be considered in the computations.

User enters desired drug dose (drug delivery target, referred to below as dd

_{target}), max allowable carrier flow (Q_{cmax}), desired steady-state carrier flow (Q_{css}). Summarized as:a. user enters dd

_{target}, Q_{cmax}, Q_{css}

Algorithm starts by setting the carrier flow at its maximum level (Q

_{cmax}) and setting the drug flow at a level that achieves a drug concentration along the upstream portion of the fluid path identical to what the final drug concentration will be when carrier and drug flows are reduced to steady-state levels. In mathematical terms:a. Calculate eventual steady-state drug concentration and proportionality between carrier and drug flows. Algorithm will maintain this proportionality from the start.

i. Drug flow at steady state calculated as Q

_{dss}= dd_{target}/ c_{d}ii. Total flow at steady state calculated as Q

_{Tss}= Q_{css}+ Q_{dss}iii. Mixed drug concentration at steady state c

_{ss}= (Q_{dss}/ Q_{Tss}) × c_{d}iv. Proportionality calculated as p = Q

_{css}/ Q_{dss}

b. Start carrier flow at Q

_{c}= Q_{cmax}c. Start drug flow at Q

_{d}= Q_{cmax}/ p

Algorithm calculates drug concentration along fluid path over time using forward difference model based on Taylor dispersion equation

a. Solve for

*c*(*x*,*t*), where*x*is axial distance along fluid path, and*t*is timeb. Calculate drug delivery at the patient end of the fluid path dd =

*c*(*L*,*t*) × Q_{T}, where*L*is the effective length of the tube, and Q_{T}is the total flow (drug flow plus carrier flow)

Algorithm checks at each time increment dt if drug delivery is at target level

a. Is dd = dd

_{target}?i. NO → repeat step 4 check for next time increment

ii. YES → continue with step 6

Algorithm starts decreasing Qc and Qd in a controlled manner. dd has just reached dd

_{target}and Q_{c}and Q_{d}are still at their initial (high) values. Goal is to maintain dd_{target}while ramping down Q_{c}and Q_{d}, and continuing to maintain initial proportion p. Mixed concentration of drug arriving at patient end of fluid path is not yet c_{ss}(value calculated in 3.iii.)a. Algorithm now predicts and evaluates drug concentrations that will arrive at patient one time increment later using the model.

i. Use current fluid velocity (derived from Q

_{T}and architecture of fluid path) to find*x*position of fluid near patient end of tube that would arrive one dt later if current Q_{T}were to be maintained (*x*_{1}=*L*−*u*(*t*_{current}) dt where*L*is the length of the fluid path being used in the Taylor dispersion equation and*u*(*t*_{current}) is the fluid velocity at the current time)ii. Concentration of mixed drug about to arrive comes from solution

*c*(*x*_{1},*t*_{current}) being calculated using the model

b. Algorithm uses that drug concentration to calculate new flows that will maintain dd at dd

_{target}i. Calculate new total flow Q

_{Tnew}= dd_{target}/*c*(*x*_{1},*t*_{current})ii. Calculate new drug flow Q

_{dnew}= Q_{Tnew}/ (1 + p)iii. Calculate new carrier flow Q

_{cnew}= Q_{Tnew}− Q_{dnew}

c. Set Q

_{c}= Q_{cnew}and set Q_{d}= Q_{dnew}

Algorithm checks at each dt increment in time if mixed drug concentration reaching patient is at the calculated steady-state level

a. Is

*c*(*L*,*t*) = c_{ss}?i. NO → repeat step 6 for next time increment

ii. YES → continue with step 8

Maintain Q

_{c}and Q_{d}at new levels. dd = dd_{target}and*c*(*x*,*t*) = c_{ss}for all*x*(along the length of the fluid path)

### Description of Algorithm for Precise Control of Rapid Changes to Ongoing Drug Delivery

This appendix describes the algorithm used to achieve changes to ongoing drug delivery in the experiments described in this article. This algorithm does not rely on the predictive modeling described in appendix 1 but does rely on similar principles of coordination of pump activity. This model assumes that the carrier and drug fluids are incompressible, such that changes in flows made at the pumps are rapidly transmitted to the patient end of the fluid path. The model also assumes that the infusion system has been running at steady state such that the compliance of the system (syringe plungers, tubing, *etc*.) is not a major factor when making changes in flows.

Please see appendix 1 for the terminology used in the following description and in figure 7.

Other than where we say “user sets…” or “user enters…,” all other adjustments below are carried out by the control program.

Method for achieving rapid changes to ongoing drug delivery (after drug delivery has initially reached steady state) for single drug and carrier (summarized in fig. 7):

User enters the characteristics of the infusion system (manifold, intravascular catheter, and any intravenous tubing that may be used), so that the dead volume can be considered in the computations.

User enters steady-state Q

_{c0}and Q_{d0}(initial carrier and drug flows) or these values could come from monitoring the pumps. User also enters desired changes in dd_{target}levels for upcoming times or can enter those target levels in real time (at the times of intended changes). We will refer to the initial dd levels as dd_{target0}and the first desired change in level as dd_{target1}, the second change as dd_{target2},*etc*.Algorithm calculates initial values:

a. Calculate initial steady-state drug concentration and proportionality between carrier and drug flows. Algorithm will maintain this proportionality from the start. Algorithm assumes starting from steady-state drug delivery

i. Initial drug delivery at steady state calculated as dd

_{target0}= Q_{d0}× c_{d}ii. Total flow at initial steady state calculated as Q

_{T0}= Q_{c0}+ Q_{d0}iii. Mixed drug concentration at initial steady state c

_{ss}= (Q_{d0}/ Q_{T0}) × c_{d}iv. Proportionality calculated as p = Q

_{c0}/ Q_{d0}

Algorithm calculates new flows and adjusts pumps when time of desired dd change is reached. Drug concentration is maintained at c

_{ss}by maintaining proportionality p between Q_{c}and Q_{d}a. Time

*t*_{1}is reached → user desires new dd_{target1}b. Calculate new Q

_{T1}= dd_{target1}/ c_{ss}c. Calculate new drug flow Q

_{d1}= Q_{T1}/ (1 + p)d. Calculate new carrier flow Q

_{c1}= Q_{T1}− Q_{d1}e. Set Q

_{c}= Q_{c1}and set Q_{d}= Q_{d1}

Maintain Q

_{c}and Q_{d}at new levels until time of next desired dd change. At that time, repeat step 4 for dd_{target2}, Q_{T2}*etc*. with same p and c_{ss}