Central venous catheters (CVCs) are commonly used for treatment of many chronic and acute conditions: examples include hemodialysis, parenteral nutrition, and chemotherapeutic agent delivery. It is estimated that more than 5 million CVCs are placed annually in the US.^{1} Central venous catheters are also used for vascular access in approximately 80% of incident hemodialysis patients in the US.^{2} Despite their prevalence, CVCs are associated with known risks of morbidity and mortality to patients.^{3} In addition, the patency of the catheter is oftentimes compromised, leading to expensive and time-consuming interventions. As many as 15% of patients who receive CVCs are reported to have some complications.^{1}

A common complication is the formation of thrombus at or near the tip of the catheter.^{4} The thrombus may impair the function of the catheter by obstructing flow. The development of thrombosis has led most hospitals to use routine administration of heparin into the catheter to reduce this risk, a procedure known as *heparin locking*.^{5} Typically, a heparin/saline mixture (with a concentration of 1,000–10,000 U/ml) is injected into the catheter lumen(s) to fill the lumen volume when the catheter is not actively used (*e.g.*, the time between hemodialysis sessions which lasts 48–72 hours). Because heparin is a known anticoagulant, the locking solution is thought to protect against the formation of thrombus at the tip or inside the catheter during the catheter “no-flow” time.

Heparin locking has been in practice for many years.^{6} However, the practice seems to have developed as the default treatment, and it has been criticized as not being grounded in a sufficient level of evidence.^{7},^{8} When compared with a simple saline lock, representing a “null treatment,” randomized controlled trials^{9–12} have shown weak or conflicting evidence for the benefit of heparin locking in preventing CVC occlusion. In addition, there are known risks of heparin locking. Some amount of the locking solution will inevitably “leak” from the catheter into the circulation which places the patient at risk for systemic anticoagulation and hemorrhaging both in adults^{13} and in children.^{14}

In hemodialysis CVCs, the heparin leakage seems to occur in two stages *in vivo*^{15}: 1) an early stage, occurring within 10 minutes of locking, whereby 10–15% of the lock will leak into the circulation and 2) a late stage whereby an additional 10–15% of the lock leaks during the longer interdialytic period of 48–72 hours. *In vitro* experiments of locking suggest that the early-phase leakage is related to the lock fluid instillation itself.^{16–19} The lock solution is forced out through the catheter tip as the internal lumen is filled with the likely mechanism being: 1) oversizing of the locking fluid volume, 2) finite flow velocities developing inside the lumen which transport the locking fluid, or 3) a combination of both. A hydrodynamic effect has been proposed as the mechanism of the late-phase leakage, namely, that a buoyancy force, which arises from the differing densities of the blood (1,050 kg/m^{3}) and the lock solution (1,006 kg/m^{3} for normal 0.9% saline), drives the transport of heparin from the catheter into the circulation.^{20}

However, previous *in vitro* studies reported very large locking fluid leakage rates, as much as 70–80% leakage within an hour,^{17},^{20} rates that are not consistent with *in vivo* data which show ≈10% lock leakage during 48 hours.^{15} The *in vitro* experiments had conditions that were not physiologically realistic and therefore may not have been capable of fully assessing the strength of buoyancy-driven fluid motion *in vivo*. The previous studies placed the heavier fluid inside the catheter which is the opposite of what would occur *in vivo* when a patient is upright and the catheter is oriented vertically; the less dense heparinized saline in the CVC would be above the more dense blood with a CVC *in vivo*. When a less dense fluid is placed above a heavier fluid, buoyancy forces tend to suppress fluid movement.^{21} Agharazii *et al.*^{17} used a 5% glucose/water mixture (density of 1,016 kg/m^{3}) for the locking fluid, whereas Polaschegg^{20} used a soluble polymer and water mixture (density of 1,020–1,030 kg/m^{3}) for the locking fluid in some cases. Both groups used normal water (998 kg/m^{3}) for the fluid surrounding the catheter. Furthermore, Polaschegg^{20} did not report any detectable leakage when the catheter was filled with the lighter fluid, which suggests that the buoyancy force would indeed suppress fluid motion out of the catheter rather than stimulating it when the lighter fluid is on top.

We propose here an alternative hypothesis that the leakage of the heparin from the catheter is driven by advective-diffusive mass transfer because of the flow of blood around the surface of the catheter *in situ*. Note that advection refers to the mass transfer of a chemical species due to the bulk, or macroscopic, fluid motion, whereas diffusion refers to the mass transfer arising from gradients in species concentration which does not require bulk fluid motion. Convection, however, typically refers to the combined mass transfer by both advection and diffusion.

An understanding of lock fluid leakage dynamics is needed to provide evidence-based guidelines that can be translated to formulate quantitative models of lock fluid leakage and systemic anticoagulation risk. Such models can be used to optimize the catheter selection or stimulate new designs for various types of patients, for example, children *versus* adults. Although CVCs are used in the treatment of patients of all ages, the catheter types and calibers available for pediatric applications are much more limited than for adults, thereby placing children at a greater risk of complications.^{22}

#### Materials and Methods

We study the leakage dynamics of heparin in CVCs by mathematical modeling and numerical simulations of mass transfer by advection and diffusion. The hemodynamic conditions of the simulations are meant to mimic that of an *in situ* catheter in a pediatric patient. Three-dimensional unsteady numerical simulations of the hemodynamics near and around the catheter tip in an idealized model of the pediatric superior vena cava (SVC) are performed to assess the leakage dynamics on the time scales associated with the cardiac cycle (~1 second). A one-dimensional (1D) mass transfer mathematical model is developed to study the leakage dynamics during the longer interdialytic period (~24–48 hours). Results from the three-dimensional (3D) simulations are also used to infer appropriate boundary conditions for the 1D mass transfer model.

##### Three-Dimensional Geometric Model

A double-lumen catheter design is investigated through numerical simulation. The external shape of the catheter is cylindrical with a diameter of 8 Fr (2.67 mm). The internal lumens are “D” shaped in cross-section, each with an area of 1.90 mm^{2}. Both lumens of the catheter, arterial and venous, have outlet ports on the sides and the tip. The total surface areas for the arterial and venous ports are 3.4 and 2.9 mm^{2}, respectively. A schematic of the design is shown in Figure 1. The internal lumens are labeled arterial or venous according to the direction of blood flow: blood is drawn from the SVC, moving away from the heart, through the arterial side where it can be sent to a dialyzer and is returned to the SVC, moving toward the heart, through the venous lumen. The volume of each internal lumen is 0.6 ml (600 mm^{3}).

The catheter is placed into an idealized model of the pediatric SVC and the entrance to the right atrium. The diameter of the SVC is 9 mm which is typical of a 1–1.5 year old patient.^{23} The catheter is placed concentric with the SVC, a position that is highly idealized and represents an upper bound for advective flow near the tip as CVCs typically lie close to the SVC or atrial walls which would limit advection. Approximately 25 mm of the catheter tip and internal lumens are included in the 3D model. A diagram of the 3D model showing the position of the catheter in the SVC is shown in Figure 2.

##### Three-dimensional Numerical Simulations

The fluid flow in and around the catheter is simulated by solving the Navier–Stokes equations using ANSYS FLUENT, release 12.1 (ANSYS, Inc., Canonsburg, PA). The blood is assumed to be incompressible and Newtonian with a density of ρ of 1,050 kg/m^{3} and dynamic viscosity μ of 3.5 cP. The fluid in the 3D domain is treated to be uniform and homogeneous with constant density and viscosity rather than a mixture of saline and blood. A semistructured tetrahedral mesh is created using ANSYS Gambit, release 2.4 (ANSYS Inc.). The computational mesh elements are 0.15–0.25 mm in length, and the total mesh size is approximately 4 million elements.

The heparin is assumed to be a passive scalar which obeys the advection–diffusion equation. The concentration of the heparin (*i.e.*, its mass per unit volume of solution) is a function of position **x** and time *t* and is denoted by *c* (**x**, *t*). In this case, passive means that the scalar concentration does not affect the hemodynamic variables, for example, blood velocity and pressure. The mass transfer across the side holes and end-tip hole of the catheter is calculated to quantify the leakage of heparin out of the catheter into the circulation. The advective,

, and diffusive,

, contributions are computed separately for either the arterial or venous lumen, respectively.

The mass diffusivity of heparin in blood, denoted *D*_{m}, is taken to be 3.33 × 10^{−6} m^{2}/s. Although this value is much higher than the true value^{24} (heparin in saline solution at 20°C has a diffusivity ~10^{−10} m^{2}/s), a large diffusivity is chosen so that the Schmidt number is equal to 1. The Schmidt number, *Sc*, is the dimensionless ratio of the mass diffusivity to momentum diffusivity, that is,

. This simplification is made so that thin mass transfer boundary layers need not be spatially resolved in the 3D simulations. The results can be appropriately nondimensionalized and rescaled to extrapolate the mass transfer rates to cases with realistic values of the heparin diffusivity (*Sc* ~10,000). Details of the passive scalar model are given in Appendix 1.

##### Boundary Conditions

A pulsatile velocity is prescribed as the fluid boundary condition at the SVC inlet. The mean flow rate is 900 ml/min and the peak is 1,125 ml/min which are characteristic of a pediatric patient.^{23},^{25} The velocity profile at the inlet is assumed to be that of Poiseuille flow between two concentric cylinders.^{21} A zero concentration boundary condition is prescribed at the inflow for the passive scalar equation. At the outflow boundary in the right atrium, the hydrodynamic pressure is fixed at 10 mm Hg, and a gradient-free condition is enforced for both the fluid velocity and the passive scalar equation. All solid surfaces are rigid and obey the no-slip condition for the flow velocity and an impermeability condition (*i.e.*, zero flux) for the passive scalar. Unsteady flow and concentration fields are computed for one cardiac cycle that lasts 1 second.

##### Initial Conditions

The initial velocity and pressure fields are obtained from a prior simulation with no mass transfer which had been executed during two cardiac cycles to ensure decay of initial transients. For the mass transfer simulation, the interior lumen of the catheter is initialized to a constant heparin concentration of 1,000 U/ml (≈5 kg/m^{3}) which is a commonly used clinical concentration for locking.^{26} Recall that 1 U of heparin is approximately 5 μg in mass. The SVC and the right atrium are initialized with a zero heparin concentration.

##### One-Dimensional Mathematical Model

The long-time transport of heparin along the length of the catheter is posited, based on the lack of flow inside the proximal side of the catheter lumen, to be by diffusion only. The linear 1D diffusion equation is used to describe the interdialytic interval heparin mass transport. The model computes the concentration of heparin in a single internal catheter lumen as a function of position and time, that is, *c* (*x*, *t*) where *x* is the position along the catheter length and *t* is the time. The model requires specification of the catheter length, *L*, initial concentration, *ci*, mass diffusivity, and the convection mass transfer coefficient, *hm*, which parameterizes the advection and diffusion at the catheter tip. The results of the 3D simulations are used to calculate the value of *hm* which is then used as the boundary condition in the 1D model. Details of the 1D model are given in Appendix 2.

#### Results

##### Three-dimensional Model Results

Observation of the simulation results suggests that we divide each internal catheter lumen into the following two regions: 1) a region adjacent to the side holes extending distally to the end/tip of that lumen and 2) a proximal region that contains the remainder of the lumen volume. The two regions are separated for our analysis at a cross-sectional plane 1 mm proximal to the most proximal side hole. The first region is characterized by large flow velocities and, hence, significant advective and diffusive transport. The second region is characterized by nearly stagnant flow, hence, diffusion-dominated transport. For the arterial and venous internal lumens, the first region is approximately 31 and 47 mm^{3} in size, respectively. The near-tip regions represent approximately 5% and 8% of the total volume of the arterial and venous lumens, respectively.

In region 1, flow from the parent vessel enters the internal lumen *via* the catheter side holes. The inflow velocities are relatively large, ≈5–10 cm/s, which are similar to, but less than, the peak flow velocities in the SVC of ≈ 40 cm/s. The inflow is strongest in the proximal side holes. There is a strong outflow through the more distal side holes. Because the internal lumen is rigid and the flow is incompressible, any inflow of blood through the side holes must be balanced by a corresponding outflow in a different side hole.

Region 1 is depleted of heparin quickly. After a single cardiac cycle, the concentration in region 1 is approximately half of its original concentration. The heparin concentration averaged over the region 1 volumes for the arterial and venous lumens is shown in Figure 3. The cardiac cycle–averaged advective transport rates are 12.6 U/s for the arterial side and 6.2 U/s for the venous side. Cycle-averaged diffusive transport rates are 9.4 U/s on the arterial side and 10.1 U/s on the venous side. Note that because of the linearity of the passive scalar equation, the results can be rescaled to some arbitrary level which would correspond to a different initial concentration. For example, one can simply multiply the transport rates by 10 if the initial concentration is 10,000 U/ml rather than 1,000 U/ml. The advective transport rates *versus* time during the 3D simulation are shown in Figure 4.

Figure 3 Image Tools |
Figure 4 Image Tools |

The inflows and outflows of blood *via* the side holes explain the fast rate of advective transport of heparin out of the catheter lumens. The blood entering the catheter *via* the side holes has essentially zero heparin concentration. As it enters, it dilutes the heparin concentration inside the lumen. Likewise, the blood exiting the catheter is of a higher heparin concentration. As it leaves the catheter, the blood flow carries heparin with it and further dilutes the concentration. The flow velocity pattern and the concentration pattern on a two-dimensional plane within the arterial lumen are visualized in Figure 5 (see Video, Supplemental Digital Content 1, http://links.lww.com/ASAIO/A48).

We posit that the advective transport occurring near the tip region is (approximately) independent of the Schmidt number. The advective mass transfer will scale like

, where *ci* is the initial heparin concentration, *u* is the characteristic velocity of the flow through the catheter ports, and *A* is the surface area of the catheter ports, each of which is independent of the mass diffusivity and the Schmidt number. The advective mass transfer rates computed from the 3D simulations at a Schmidt number of 1 are therefore taken to be equal to those for higher Schmidt number species.

Extrapolating to large Schmidt numbers implies that the advective transport in region 1 will dominate and that diffusive transport will be small. Therefore, we assume that the transport in region 1 during the short times of the first few cardiac cycles is by advection only, for a high Schmidt number species such as heparin. The ratio of the initial amount of heparin units in region 1, *Ci*, to the advective transport rate,

, gives a characteristic time,

, which represents the time during which the near-tip region will be depleted of heparin. The initial amount of heparin in region 1 is 1,000 U/ml · 0.031 ml = 31 U on the arterial side and 1,000 U/ml · 0.047 ml = 47 U on the venous side. The advective time scales are therefore estimated as *τ*_{Adv} ~2.4 seconds and ~7.6 seconds for the arterial and venous sides. We round the advective transport time scales to 10 seconds as these are order of magnitude estimates. Note here that the advective time scale is independent of the initial concentration of heparin. Thus, using a higher initial concentration does not slow the rate of advection. Because the initial quantity of heparin would be increased by some amount if the initial concentration is higher, the advective transport rate would also be increased by the same amount, leaving the ratio of the two the same.

A plane surface inside the internal lumen is defined to delineate region 1 from region 2. It is located at 1 mm proximal to the most proximal side holes on either the arterial or venous lumens. The mass transfer coefficient on this surface is computed so that it can be used as an appropriate boundary condition for the 1D model. The mass transfer coefficient is computed by the following expression^{27}:

where Γ represents the planar surface that delimitates regions 1 and 2, and *A*_{Γ} is the area of the planar surface. The mass transfer coefficients computed for the arterial and venous lumens are shown in Figure 6.

##### One-Dimensional Model Results

The relevant parameters for the 1D model are the catheter length, *L* = 12 cm, heparin diffusivity, *D*_{m} = 10^{−10} m^{2}/s, initial concentration, *c*_{i} = 1,000 U/ml, and convection mass transfer coefficient, *hm*. Although the 3D simulations are computed using a Schmidt number of 1, the 1D model can easily be computed with a more realistic Schmidt number for heparin using known mass transfer scaling laws to estimate the leakage during the interdialytic period.

As the convective mass transfer coefficient is dependent on the Schmidt number of the scalar species, it would not be appropriate to directly use the results of the 3D simulations for the value of *hm*, ~1 mm/s. The mass transfer coefficient is usually expressed in nondimensional terms as the Sherwood number, *Sh* = *h*_{m} *l/D*_{m}, where *l* is a length scale of the convective mass transfer, say the diameter of the internal CVC lumen. We therefore assume that the Sherwood number scales with the one-third power of the Schmidt number,^{27} that is,

where κ is a proportionality coefficient which would depend on other relevant flow parameters, for example, geometry or Reynolds number. As heparin is a passive scalar, it does not affect the flow field in any way. Because the length scale *l* would be independent of the Schmidt number, **Equation 2** can be rearranged such that the mass transfer coefficient for an arbitrary Schmidt be estimated as:

where *hm*_{,0} and *h*_{m,1} are the mass transfer coefficients at unity Schmidt number and at an arbitrary Schmidt number, respectively. Taking *h*_{m,0} ~ 10^{−3} m/s and *Sc* ~10,000 gives *h*_{m,1} ~ 10^{−6} m/s. The mass transfer Biot number, Bi*m* (Appendix 2) is calculated as Bi*m* = *hm,0* *L*/*Dm*. For the large Schmidt number case, the Biot number is approximately 1,000 which is in the asymptotically high regime. In the high Biot number limit, the solution of the 1D model becomes independent of the mass transfer coefficient, thus only the order of magnitude estimate of *hm* is needed to establish that it is asymptotically large.

The leakage rates for each lumen, as predicted by the 1D model, are initially approximately 1 U/hour and steadily decline to less than 0.1 U/hour after a 48 hour period. Only a small fraction of heparin leaks, 8.5 U for a single lumen, during a 48 hour period. Figure 7 plots the cumulative heparin leakage *versus* time for a single lumen. Because it is assumed that each lumen initially contains 600 U upon locking, the diffusive leakage is only 1.4% of the total amount of heparin, assuming no leakage during instillation. The concentration profile after 48 hours in a single lumen as predicted by the 1D model is shown in Figure 8. The diffusion penetration length, that is, the zone where the concentration has been reduced by more than 10% of its initial value, is only approximately 1 cm after 48 hours. However, 48 hours is a small length of time considering that the characteristic time scale of the diffusion, *L2*/*Dm*, is approximately 1,000 days! The 1D model predicts that it would take approximately 23 days for 5% of the heparin to leak by diffusion only. Note too that because the 1D model is based on a linear equation, the percent leakages would not be altered by changing the initial concentration of the heparin in the locking fluid.

Figure 7 Image Tools |
Figure 8 Image Tools |

#### Discussion

On the basis of the above results, we propose a two-stage mechanism for heparin leakage from a CVC by advective-diffusive mass transport: 1) an early phase lasting from a few seconds to at most a few minutes which is dominated by advection and 2) a late phase occurring during the longer interdialytic period which is limited by the slow diffusion process.

The early-phase leakage occurs very rapidly because of the relatively quick transport by advection. Given the short time scales associated with the advective transport, *τ*_{Adv} ~10 seconds, nearly all of the heparin originally located in region 1, that is, near the tip and the side holes, would likely be depleted within a few minutes of the initial locking. The near-tip region contains approximately 5–10% of the total mass of the lock solution. A previous report^{15} suggested that there is an initial early-phase leakage of lock solution in patients on hemodialysis lasting not more than 10 minutes amounting to approximately 10–15% of the total lock volume. Although the early leakage has previously been attributed to either the instillation process itself and/or fluid buoyancy forces, the current data suggest that advective transport at the catheter tip accounts for a sizable proportion of the early-phase leak. The advective transport mechanism would occur independently of the instillation process or fluid density: the leakage due to advection would still occur even if the catheter lock fluid instillation was perfectly “leak-free” or if the locking solution was exactly neutrally buoyant.

The transport of heparin along the length of the catheter through region 2 is only by diffusion as there is no bulk flow. Once the initial advection-dominated phase of transport has ended, the total mass transfer process becomes diffusion limited owing to the small diffusivity of heparin. There is a slow leakage of heparin out of the catheter during the longer 48 hour interdialytic period due diffusion transporting heparin along the length of the catheter to the tip where it is removed rapidly by advection. Our 1D model predicts an additional 1–2% leakage during a 48 hour period.

It has been estimated^{15} that the lock leakage occurring during longer interdialytic periods is approximately 10–15% of the total lock. The diffusion mechanism proposed here may partially explain the mechanism responsible for this leak of locking fluid during the late-phase period. It is possible that there are additional hemodynamic processes that exist in parallel to the diffusion process and enhance the total transport of locking solution. For example, the normal daily movement of a patient may cause “sloshing” of fluid within the catheter which could contribute to enhanced mass transfer. In addition, if a patient is lying down such that the catheter is oriented horizontally, the less dense locking fluid may be below the more dense blood, and buoyancy forces may contribute to bulk motion of the locking fluid.

The above results raise important questions regarding the specific mechanisms by which heparin locking would prevent catheter thrombosis. The presumed mechanism is that the heparin concentration inside the catheter somehow remains high enough to impede thrombus formation at the tip and side holes. However, most of the heparin initially located near the tip region is quickly released into the circulation by the fast advective transport. Hence, the long-term concentrations of heparin near the tip would only be an infinitesimal fraction of the initial concentration. It is not known if such low concentrations would have sufficient antithrombin effect such that thrombus formation could be inhibited.

We estimate the long-term concentrations of heparin at the catheter tip using the result from the 3D pulsatile flow simulations, the 1D mass transfer model, and an order of magnitude analysis. At the boundary of the 1D model, in this case the near-tip region, the ratio of the concentration at the boundary tip, *c*_{tip}, to the initial concentration, *ci*, scales according to the relation:

where Bi*m* is the mass transfer Biot number. With the low diffusivity of heparin, the estimate for the Biot number is approximately 1,000. Hence, with *c*_{i} = 1,000 U/ml, then *c*_{tip} ~1 U/ml. The whole near-tip volume, that is, region 1, of the internal lumen is approximated to have the same order of magnitude concentration estimate. This analysis suggests that increasing the initial lock concentration by significant amounts may be necessary to increase the long-term concentration at the catheter tip to protect against CVC occlusion. It is still not known, however, what levels of increased concentrations would be required to offer any antithrombotic benefit. Indeed, two recent clinical studies have shown that increasing the lock concentration from 1,000 to 5,000 U/ml^{26} or to 10,000 U/ml^{28} does not prolong CVC patency. Any increases in lock concentration would need to be balanced against increased risks of bleeding events.

Another possible mechanism to explain locking effectiveness is that the slow diffusive leakage is still enough to systemically anticoagulate the patient and, therefore, impede thrombus formation. Thus, the catheter could act like an “internal drip.” However, this mechanism is inconsistent with our analysis of advection and diffusion at the catheter tip. The late-phase leakage rates are computed as approximately 1 U/hour by the 1D model, yet doses of 10 U/hour/kg are necessary for heparin drips to achieve therapeutic levels.^{29} Such low leakage rates would be insufficient even for an infant patient of 10 kg.

##### Limitations

We recognize a number of limitations in this study. The 3D simulations use idealized conditions such as the vessel anatomy and rigid-wall catheters rather than patient-specific anatomies with distensible boundaries. Thus, the simulations can discern the approximate magnitudes of the heparin mass transfer processes but cannot account for any patient-to-patient variability.

This study also only simulated a single catheter design. Additional studies are necessary to investigate the mass transfer conditions of different designs, which could in turn assist physicians in selecting an optimal catheter type to minimize risks associated with lock fluid leakage.

Furthermore, the fluid of the 3D simulations is a simple blood/heparin mixture rather than a more realistic blood/saline/heparin mixture with variable fluid densities and viscosities. Such additional hemodynamic effects may alter the overall mass transfer rates, particularly during the late-phase leakage, but their relative importance is second order compared with the physical processes studied here.

Last, we focus here on heparin as the locking agent and do not investigate alternative locks, such as trisodium citrate, which have different physical properties, such as diffusivity, from heparin. However, the mass transfer analysis will be largely unchanged if repeated for citrate. We can repeat the 1D analysis by changing the diffusivity^{30} (1.04 × 10^{−9} m^{2}/s at 25°C) and initial concentration (4% mass fraction solution) to match that of trisodium citrate solution in water. In this case, the total leakage by diffusion during a 48 hour period is only 4–5% (*vs.* 1–2% for heparin), and the diffusion length is only 3 cm (*vs.* 1 cm for heparin). Likewise, the Schmidt number for a trisodium citrate solution is still ~1,000. The early-phase leakage would still be advection dominated, and the advective transport rates computed in the 3D analysis would be practically unchanged. It is not known, however, whether the lowered concentrations of citrate would provide the desired antithrombotic effect in the catheter. We do note that the locking fluid density is 1,030 kg/m^{3} at a 4% citrate concentration,^{30} so our current analysis would not be strongly affected by buoyancy forces. The lock solution density at a 30% citrate concentration is 1,220 kg/m^{3}, in which case the transport will likely be strongly influenced by buoyancy.

#### Conclusions

Mathematical modeling and hemodynamic simulations are used here to study the dynamics of lock solution leakage, specifically heparin, in CVCs. The mass transport of heparin is described by an advection–diffusion process which is shown to be a plausible hemodynamic mechanism for heparin leakage. The initial mass transfer phase, lasting on the order of 10 seconds, is dominated by advective transport by blood flow through the catheter ports. The near-tip volume of the internal catheter lumen could be nearly depleted of heparin during this time, accounting for 5–10% of the total lock solution, a result that is comparable with *in vivo* measurements of early-phase lock leakage. The advective mechanism occurs independent of other proposed leakage mechanisms such as spillage due to instillation and buoyancy-driven convection. The late-phase leakage is diffusion limited and therefore very slow. Only an additional 1–2% of heparin exits the catheter during a subsequent 48 hour period by diffusive transport. This is an amount lower than what has previously been reported *in vivo*. However, diffusion may be one of the several important and coexisting processes driving heparin leakage during the late phase. The thermophysical and hydrodynamic properties of anticoagulants ought to be considered when selecting alternatives to heparin for CVC locking.

Considering the initial speed with which advection removes locking solution from the tip, the long-term concentrations at and near the tip are significantly reduced, likely by several orders of magnitude, compared with the initial concentration levels. This may lead one to question the effectiveness of the typically assumed process that heparin concentrations at or near the tip remain high enough to impede thrombus formation. Given this result, we suggest that future research attention be diverted away from seeking a locking fluid possessing “optimal” anticoagulation properties, although we do recognize that antimicrobial locks may offer a benefit against catheter-related bacteremia.^{8}

Instead, we suggest that research be focused in two directions. First, quantitative models of heparin leakage and systemic anticoagulation should be developed which would aid clinicians in their selection of catheter size and type so that bleeding risks are minimized. An understanding of bleeding risks may also allow clinicians to increase anticoagulant concentrations in locking solution, when appropriate, possibly providing benefit against CVC thrombosis. Second, novel CVC designs should be developed^{31} based on a deeper awareness of hemodynamics and mass transfer effects. Designs that minimize lock fluid leakage, say by minimizing flow velocity through side holes, while maintaining low hydrodynamic resistance for dialyzing are a possible way forward.

#### Appendix 1. Advection–Diffusion Equation

The passive scalar advection–diffusion equation that governs the transport of a generic scalar concentration *c* (**x**, *t*) due to fluid advection and molecular diffusion is:

where **u** is the fluid velocity and *Dm* is the mass diffusivity. There are two modes of mass transfer across some surface Γ: advection,

, and diffusion,

. Once a solution to **Equation 5** has been found, the two mass transfer modes are calculated by the surface integrals:

where **n** is the outward unit normal vector and Γ represents the entrance/exits holes of the catheter in contact with the blood. As the passive scalar equation is linear, then any solution of **Equation 5** can be rescaled to some arbitrary level, say

, where *a* is some constant, in which case

will still satisfy **Equation 5**.

#### Appendix 2. One-Dimensional Model Details

The transport of heparin along the length of the internal lumen(s) with a total length of *L* is postulated to obey the one-dimensional diffusion equation:

where *c* = *c* (*x*, *t*) is the heparin mass per unit volume, *x* is the position along the catheter, and *t* is the time. The position of *x* = *L* is taken to be located at the tip of the catheter. The initial condition is that the concentration is initially at a constant value, *ci*. The diffusion is subjected to the boundary conditions of zero flux at the top of the catheter, that is,

and a convection condition at the catheter tip, that is,

where *hm* is the convection mass transfer coefficient, which is assumed constant, and *c*_{∞} is the heparin concentration in the circulation, which is assumed to be zero.

The initial-boundary value problem is nondimensionlized by the following:

where the primes denote dimensionless quantities. The diffusion equation in dimensionless form is:

subjected to the initial condition of

, and boundary conditions of

where

is the mass transfer Biot number. Physically, the Biot number represents the ratio of convective mass transfer at the catheter tip relative to the diffusion along catheter’s length.

In the limit of very large Biot numbers, which can be taken as correct when Bi*m* ≳ 100, the exact solution^{27} to the initial-boundary value problem is

where

and

The large Biot number limit implies, by the boundary condition in **Equation 14**, that

, or that *c* (*L*,*t*) ≈ 0. Under this condition, the tip can be treated as fixed at a constant concentration of zero. The heparin mass transfer rate per unit area by diffusion from the catheter into the circulation is calculated from the expression:

where the double primes denote a per unit area quantity. One can multiply the flux by the cross-sectional area of the internal lumen(s) and integrate in time to obtain the total mass of heparin leaked during a given time interval.