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Dialysis & Kinetics

Slow Continuous Ultrafiltration with Bound Solute Dialysis

Patzer, John F. II; Safta, Stefan A.; Miller, Richard H.

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doi: 10.1097/01.mat.0000196524.36394.0d
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Abstract

Approximately 1% of the US population has some form of liver disease, and about 46,000 people die annually from liver disease and its complications.1,2 Patients with liver disease generally fall into two classifications: fulminant liver failure, defined as sudden onset (less than 8 weeks) with no previous history of liver disease, and chronic liver disease. Both classifications have subclassifications that further define the etiology of the disease. Patients with either classification can progressively worsen until they require orthotopic liver transplantation, which is the only recognized effective treatment.3 In 2003, whereas about 17,000 people were listed for liver transplantation, only 5,671 liver transplants were performed and 1,688 patients died while awaiting transplant.4 A medical need for a treatment modality that can “bridge” patients to transplant or either slow or reverse the progression of acute liver disease clearly exists. Even better would be a new standard-of-care treatment that can be easily used by any hospital at early stages (Parson's encephalopathy grade 1 and 2) of liver failure that would retard or prevent progression to acute liver failure that requires orthotopic liver transplantation.

Albumin, the protein with highest free concentration in the blood, has several roles, one of which is to bind and shuttle sparingly soluble or nonpolar compounds from one site to another. With liver failure, concentrations of insoluble, albumin-binding toxins, such as middle chain fatty acids, aromatic amino acids, free phenols, and unconjugated bilirubin,5 reduce the ability of albumin to perform other roles and, possibly worse, maintain their presence in the blood stream. Such tightly bound toxins are resistant to removal by conventional dialysis, because the thermodynamic driving force for dialysis is the free solute concentration difference across the dialysis membrane, and free solute concentrations in the blood stream are typically in the micromolar range.

Bound solute dialysis (BSD) offers the potential of a new standard of care treatment modality than can be practiced in any hospital setting with readily available equipment. Bound solute dialysis, also referred to as albumin dialysis, has been practiced heuristically in the clinical setting by the molecular adsorbents recirculating system (MARS),5–11 Biologic-DT (liver dialysis unit),12–15 and single-pass albumin dialysis (SPAD)16 approaches using complex systems to remove albumin-bound solutes (toxins) from a liver failure patient's blood stream with some success. Recent analysis of the underlying thermodynamics and mass transport processes involved in BSD17 has found, however, that equivalent toxin removal is possible at albumin dialysate concentrations on the order of 1.0% of that used by MARS8,9 and that BSD can be practiced with conventional dialysis equipment readily available in the hospital setting. A recent case report18 on BSD in removal of bilirubin from a hyperbilirubinemic patient supports the theoretical conclusions: equivalent bilirubin removal was found at 1.85% and 5.0% albumin in the dialysate in single-pass operation.

Many institutions, ours included, practice slow, continuous ultrafiltration with dialysis (SCUF-D) rather than conventional dialysis in treatment of patients with acute liver failure both to remove fluid volume and to correct electrolyte and small molecule balances in the patient blood stream. Slow, continuous ultrafiltration (SCUF) without dialysis can only remove solutes at the same concentrations as in the blood stream; it cannot affect concentrations in the blood stream. Dialysis in addition to ultrafiltration is required to alter blood stream chemical concentrations.

The present work extends the comprehensive, thermodynamic analysis of BSD17 to encompass SCUF-BSD in order to gain greater understanding of the impact of BSD in a clinically relevant setting for treatment of acute liver failure.

Theory

As illustrated in Figure 1, the clinical objective in SCUF is to remove fluid from the patient at approximately the same rate, Qf (mL/min), that the patient gains fluids in order to avoid the excess fluid gain generally associated with liver failure. SCUF uses a counter-current ultrafiltration cartridge (dialyzer) to effect fluid removal from a patient. The ultrafiltrate flow convecting across the membrane carries species at the same concentration as in the plasma, subject to the sieving characteristics of the membrane. Low-molecular-weight species move across the membrane with ultrafiltrate flow at the bulk plasma concentration. Higher-molecular-weight species experience a sieving effect that gradually reduces concentration in the ultrafiltrate flow up to the molecular weight cut-off for the membrane, where the effective concentration in the ultrafiltrate approaches zero. Most ultrafiltration membranes have a molecular weight cut-off that excludes movement of albumin (63 kd) across the membrane.

Figure 1.
Figure 1.:
Schematic diagram of bound solute-albumin binding equilibria and solute transport in SCUF-BSD. Bound solute (A•S) dissociates to free albumin (A•) and free solute (S), with associated free solute concentration, (CS)b, on the blood side. Ultrafiltration produces a convective flow, Qf mL/min, which carries unbound, free solute at the blood concentration (CS)b mmol/l. In addition, free solute can diffuse across the membrane. Free solute from both convective and diffusive processes then combines with free albumin to form bound solute on the dialysate side.

SCUF is typically used to effect moderate reductions in plasma species concentration. Dialysate flows on the order of 1 to 2 l/h dilute the ultrafiltrate that crosses the membrane, lowering species concentrations, thus inducing a secondary diffusive species transport across the membrane from the higher concentrations on the plasma side to the lower concentrations on the dialysate side. Dialysate flow rates comparable to conventional dialysis, 30 l/h, would be more effective in establishing diffusive transport because dialysate-side concentrations would be considerably lower, and higher flow rates lower diffusive mass transfer resistance. However, the logistics of continuously providing 30 l/h of dialysate in the intensive care unit setting are formidable.

Solute-Binder Interaction

Because the ultrafiltration membrane excludes convective transfer of albumin across the membrane, removal of albumin-bound solutes (toxins) is largely inhibited in SCUF and SCUF-D. However, addition of albumin to the dialysate side in SCUF-BSD can significantly impact diffusive transport of albumin-bound solutes through the sequence dissociation of bound solute complex on blood side:

Solute transport across the membrane:

Binding of solute on dialysate side:

where B represents the binder (albumin) and S represents the solute of interest. The dissociation/binding steps, Equations 1 and 3, are lumped in the sense that there may be single or multiple binding sites for the solute on albumin.

Equations that describe solute-binder equilibrium are presented in Appendix A. The present development is directed toward understanding the interaction of a bound solute with a single binding site on albumin. Extension to multiple binding sites and/or other binding moieties such as activated carbon, although more complex, is straightforward.

Dialysis/Ultrafiltration Cartridge

Development of the dimensionless equations that describe diffusive/convective transport of solute across the membrane in the cartridge is presented in Appendix B. In order to focus on the role of solute-binder interactions in removal of solute, the equations were developed with the assumptions: (1) Only free solute crosses the membrane, i.e., the membrane is impermeable to binders such as albumin. (2) The dissociation/association reactions in Equations 1 and 3 are fast in comparison with the movement of solute across the membrane as represented by Equation 2. This implies that the equilibrium relations in Appendix A apply everywhere in the cartridge. (3) Convection in the direction of flow dominates diffusion in the direction of flow. (4) The blood stream and dialysate stream are locally well-mixed. The contributions of radial concentration gradients and concentration polarization due to ultrafiltration are ignored. (5) Ultrafiltration is equally distributed along the length of the cartridge. (6) Local mass transfer coefficients are independent of fluid velocity. This assumption can be relaxed by incorporating flow rate dependence as described by Michaels.19 (7) The combined solute diffusive/ultrafiltration flux across the membrane is described by Villarroel.20 (8) Solute transport across the membrane can be by convection, by bulk free diffusion through the pores, by surface diffusion on the membrane pore surfaces, or by a combination of the three.

Using the scalings in Appendix B, the solute-binder equilibrium interaction on the blood side of the membrane is given by

and on the dialysate side of the membrane by

where Ψ is the fraction of binder that is bound by solute. The dimensionless equation for the free solute concentration in the blood as a function of position in the dialyzer is

with inlet condition

The corresponding equation for free solute concentration in the dialysate is

with inlet condition

The numerator in Equations 5a and 6a reflects the contributions of diffusion and ultrafiltration to solute transport. The denominator in both equations reflects the retardation in solute transport due to solute-binder interaction. With no solute binding, the derivative term in the denominator is zero, and the equations reduce to the normal equations for SCUF-D. The dimensionless parameters found in Equations 4 through 6 are described in Table 1.

Table 1
Table 1:
Dimensionless Parameters in SCUF-BSD

One-Compartment Model

The impact of SCUF-BSD on reduction of total solute concentrations in a patient can be explored with a one-compartment dialysis model as illustrated in Figure 2. The model assumptions include:

Figure 2.
Figure 2.:
One-compartment model for SCUF-BSD. The compartment (patient) gains fluid at net rate Qm mL/min. Fluid (blood) with total albumin concentration (CtA)res (mmol/l) and total solute concentration (CtS)res (mmol/l) is removed from a reservoir of volume Vres (mL) at rate Qb (mL/min) and passed through the dialyzer. Dialysate is passed through the dialyzer countercurrent to the blood at rate Qd (mL/min). Ultrafiltrate is removed from the blood stream at rate Qf (mL/min). Fluid is then returned at rate Qb−Qf to the reservoir. Solute has net generation rate GS (mmol/min). Albumin has net generation rate GA (mmol/min). The volume and concentrations in the reservoir are time dependent.
  • (1) The patient is being administered fluids and may or may not have renal function. The net compartment volume gain (loss), Qm (mL/min), is given by administered fluids less renal loss.
  • (2) Neither the administered fluids nor eliminated fluids contain the solute of interest.
  • (3) The solute of interest is generated at a net rate GS (mmol/min) due to ordinary metabolism or the disease state.
  • (4) Binder (albumin) is gained at a net rate GA (mmol/min), which is the difference between metabolic production plus direct administration, if any, and degradation.
  • (5) Patient blood is passed through the ultrafiltration/dialysis cartridge counter-current to dialysate. The cartridge operation is described by Equations 4 and 5.

As outlined in Appendix C, the dimensionless equations that describe the one-compartment model are

for the reservoir volume as a function of time;

for the reservoir total binder concentration as a function of time; and

with initial condition

for the reservoir total solute concentration as a function of time. The dimensionless parameters found in Equations 7 through 9 are described in Table 1.

Equations 5 and 6 were solved as a function of the dimensionless parameters in Matlab (The MathWorks, Inc., Natick, MA) using Runge-Kutta integration (ODE45), integrating from SYMBOL = 0 to SYMBOL = 1, with a regula-falsi predictor-corrector method to converge the solution to (JOURNAL/asaio/04.03/00002480-200601000-00009/ENTITY_OV0472/v/2021-01-29T102231Z/r/image-pngS)D,in = 0 at ??? = 1. Equations 7 through 9 were also solved as a function of the dimensionless parameters using a Runge-Kutta routine calculating F by solutions to Equations 5 and 6 at each time step.

Symbol
Symbol
Symbol
Symbol

Methods

An experimental system was constructed according to Figure 2. A well-mixed, 250 mL reservoir containing physiologic dialysate buffer solution (Renasol, Minntech Renal Systems, Minneapolis, MN) was prepared with 0.6 mmol/l human serum albumin (66 kd; Sigma-Aldrich Co., St Louis, MO) as the binding agent and 0.54 mmol/l unconjugated bilirubin (B4126, Sigma-Aldrich Co.) as the bound solute. The approximate molar ratio, ϕ, of unconjugated bilirubin to albumin in the reservoir was approximately 0.9. Dialysate consisted of the same buffer containing the designated concentration of human serum albumin according to the experimental design (Table 2).

Table 2
Table 2:
Settings for Full 24 Factorial Experimental Design

The reservoir fluid was recirculated through the lumen (blood) side of a Baxter CT•110G dialysis cartridge (Baxter Healthcare Corporation, Deerfield, IL). The cartridge is comprised of cellulose triacetate hollow fibers, 200 μm lumen diameter and 15 μm wall thickness, with a nominal surface area of 1.1 m2 and priming volume of 70 mL. Fresh, single-pass dialysate was pumped through the extralumenal compartment. A Prisma System (Gambro Renal Products, Lakewood, CO) was used to control reservoir flow rate, dialysate flow rate, and ultrafiltration according to the experimental design (Table 2). The system was maintained at room temperature (22°C). Reservoir aliquots (0.5 mL) were removed at specified time intervals for later analysis of relative unconjugated bilirubin concentration by comparison of transmission of ultraviolet light (540 nm) through a sample at any time divided by the transmission through the sample obtained at time zero (transmission was found to be linear with concentration; data not shown).

A 24-full factorial experimental design21 that spanned the variable ranges of clinical practice at our institution was used to determine unconjugated bilirubin transport as a function of the independent experimental variables with low and high settings as shown in Table 2. The full factorial design, with all possible combinations of high (+) and low (–) settings, comprises 16 experiments. In addition, the centerpoint condition experiment was replicated three times in order to produce an estimate for the unconjugated bilirubin membrane mass transfer constant, KA, and estimates for the standard error of the mean.

Results

The dimensionless groupings that govern the behavior of bound solute dialysis, defined in Table 1, can be divided into two sets. Parameters that define the physiological state of the patient include the solute-binder interaction strength, λ, the total solute/total binder concentration ratio in the patient, ϕ, and the net fluid gain/blood flow rate ratio, ηm. These are dependent parameters that can change through intervention by SCUF-BSD. The independent parameters that can be specified for SCUF-BSD intervention include the mass transfer/blood flow rate ratio, κ, dialysate/blood flow rate ratio, α, dialysate/blood binder ratio, β, and ultrafiltration/blood flow rate ratio, γ, and the dialysate/blood binding strength ratio, λdb. This last parameter is important only if one considers using an alternative binder to albumin, such as charcoal, in the dialysate. It is implicitly contained in the denominator of Equations 4a through 6b.

Ultrafiltration Cartridge

In order to focus on the effects of the primary control parameters on solute removal in SCUF-BSD, Equations 5 and 6 were solved for (constant) typical parameters of total solute/total binder concentration ratio, ϕ = 0.9, total albumin concentration, (CtB)res = 0.6 mmol/l, associated with unconjugated bilirubin and albumin in blood for liver failure patients, and the same binder on both sides of the membrane, and solute-membrane reflection coefficient, σ = 0. The fractional solute removal, Equation C3, was then explored as a function of the mass transfer/blood flow rate ratio, κ, dialysate/blood flow rate ratio, α, dialysate/blood binder ratio, β, and ultrafiltration/blood flow rate ratio, γ, over the ranges indicated in Table 1.

The fractional extraction, F, is calculated as

The fractional extraction times the blood flow rate is equivalent to the term “clearance” as used in conventional hemodialysis. As indicated, F is a function of all of the system parameters, but most strongly, a function of the binding affinity, ranging from about 0.1 for KB ∼ 103 to about 0.0003 for KB ∼ 107 without binder in the dialysate. These low values for fractional extraction reflect the fact that most solute remains bound to albumin in the blood stream as the blood passes through the dialyzer.

An additional dimensionless relation, the fractional relative removal, Frel(κ),

is more useful for interpreting the results. The mass transfer/blood flow rate ratio, κ, is specified through choice of a dialysis cartridge, thereby fixing the mass transfer rate, KA, and by physician specification of blood flow rate, Qb, necessary to meet clinical objectives. The question of interest then becomes how specification of the other operating parameters affects removal of bound solute from the patient. Frel identifies how bound solute removal depends upon specification of the operating parameters of dialysate/blood flow rate ratio, α, dialysate/blood binder ratio, β, and ultrafiltration/blood flow rate ratio, γ, compared to bound solute removal at the maximum dialysate/blood flow rate considered for SCUF-D, α = 0.5, with no albumin in the dialysate, β = 0, and with no ultrafiltration, γ = 0. An alternative interpretation of Frel is the gain in theoretical clearance that can be obtained by using binder in the dialysate and/or ultrafiltration.

The parametric effect of the mass transfer/blood flow rate ratio, κ, on relative removal, Frel, versus dialysate/blood binder ratio, β, at constant ultrafiltration/blood flow rate ratio, γ = 0.03, over a range of dialysate/blood flow rate ratios, 0.1 < α < 0.5 is shown in Figure 3 for a solute-albumin binding constant KB = 106 mL/mmol. Relative removal approaches an asymptote with respect to the dialysate/blood binder ratio, β, that is dependent upon the mass transfer/blood flow rate ratio, κ, but independent of dialysate/blood flow rate ratio, α. Note that the asymptotic improvement in theoretical clearance, Frel, increases with increasing mass transfer/blood flow rate ratio, κ. The approach to an asymptote reflects an interplay between α and β: relative removal is dependent upon α and β until the supply rate of binder in the dialysate (loosely, the product αβ) exceeds the maximum rate of solute transfer, which occurs when the free solute concentration is maintained effectively at zero on the dialysate side. Of final note is the behavior at dialysate/blood binder ratio, β = 0: relative removal is dependent upon the dialysate/blood flow rate ratio, α, but not dependent upon the mass transfer/blood flow rate ratio, κ. This is another reflection of one limitation on removal being related to the capacity of the dialysate to carry solute out of the system. Higher mass transfer/blood flow rate ratios have no advantage if the dialysate cannot remove the increased flux possible with the higher mass transfer rates.

Figure 3.
Figure 3.:
Parametric effect of mass transfer/blood flow rate ratio, κ, on relative removal, Frel, in an ultrafiltration cartridge versus dialysate/blood binder ratio, β, at constant ultrafiltration/blood flow rate ratio, γ = 0.03, and constant total solute/total binder concentration ratio, ϕ = 0.9 over a range of dialysate/blood flow rate ratios, 0.1 < α < 0.5 for a solute-albumin binding constant KB = 106 mL/mmol. The relative solute removal approaches an asymptote with respect to β that is dependent upon κ and independent of α. Note also that relative solute removal is dependent upon β but independent of κ at β = 0.

The asymptotic (maximum) gain in theoretical clearance, (Frel)max, is dependent upon the solute-albumin binding constant, KB, and the mass transfer/blood flow rate ratio, κ, as shown in Figure 4 for constant ultrafiltration/blood flow rate ratio, γ = 0.03, and a range of dialysate/blood flow rate ratios, 0.1 < α < 0.5. The maximum gain in theoretical clearance is lowest for low mass transfer/blood flow rate ratio operation (corresponding to low permeability membranes and/or high blood flow rates). The maximum gain in theoretical clearance, (Frel)max, is also relatively insensitive to the solute-albumin binding strength for low mass transfer/blood flow rate ratio, κ = 1. Greater gains in theoretical clearance are possible at greater mass transfer/blood flow rate ratio (higher permeability membranes and/or lower blood flow rates). The gain in theoretical clearance also increases with increasing solute-albumin binding strength for the higher mass transfer/blood flow rate ratio, κ = 3, with (Frel)max, nearly doubling from about 3 to about 6 over the range of solute-albumin binding constants considered, 104 < KB < 107.

Figure 4.
Figure 4.:
Asymptotic (maximum) gain in theoretical clearance, (Frel)max, versus the logarithm of solute-albumin binding constant, KB, as a function of the mass transfer/blood flow rate ratio, κ, for constant ultrafiltration/blood flow rate ratio, γ = 0.03, range of dialysate/blood flow rate ratios, 0.1 < α < 0.5, and constant total solute/total binder concentration ratio, ϕ = 0.9.

The dialysate/blood binder ratio, β, required to approach within 10% of the asymptotic (maximum) gain in theoretical clearance, (Frel)max, as a function of dialysate/blood flow rate ratio, α, and solute-albumin binding constant, KB, for constant mass transfer/blood flow rate ratio, κ = 2, is depicted in Figure 5. Very tightly bound solutes, such as unconjugated bilirubin, KB 0.65 × 107 mL/mmol, require very little albumin in the dialysate, β < 0.04, to reach the maximum removal rate. Less tightly bound solutes, such as lithocholic acid,22KB 2 × 104, require substantially more albumin in the dialysate. As alluded to in the discussion of Figure 3, the individual curves displayed in Figure 5 are adequately described by the relation αβ = constant (constant = 0.004, 0.03, 0.15, and 0.48 for KB = 107, 106, 107, and 104, respectively), reinforcing that maximal removal is achieved when the supply rate of albumin in the dialysate is sufficient to maintain free solute concentrations near zero in the dialysate. The results depicted in Figure 5 should not be interpreted as the amount of binder in the dialysate required to effectively remove bound solute from a patient as any amount of binder in the dialysate will increase the gain in theoretical clearance.

Figure 5.
Figure 5.:
Dialysate/blood binder ratio, β, required to approach within 10% of the asymptotic (maximum) gain in theoretical clearance, (Frel)max, as a function of dialysate/blood flow rate ratio, α, and solute-albumin binding constant, KB, for constant mass transfer/blood flow rate ratio, κ = 2, and constant total solute/total binder concentration ratio, ϕ = 0.9.

The impact of the ultrafiltration/blood flow rate ratio, γ, on the fractional relative removal, Frel, is depicted in Figure 6, which presents two perpendicular slices through Figure 3, one at blood/dialysate binder ratio, β = 0 and the other at blood/dialysate binder ratio, β = 0.1. For clarity, only the mass transfer/blood flow rate ratio, κ = 2, results are plotted. Fractional relative removal is only weakly dependent on ultrafiltration rate. The slope of Frel versus β is slightly less for mass transfer/blood flow rate ratio, κ = 1, and slightly greater for mass transfer/blood flow rate ratio, κ = 3 (results not shown).

Figure 6.
Figure 6.:
Fractional relative removal, Frel, versus ultrafiltration/blood flow rate ratio, γ, as a function of blood/dialysate binder ratio, β, and dialysate/blood flow rate ratio, α, for constant mass transfer/blood flow rate ratio, κ = 2, and constant total solute/total binder concentration ratio, ϕ = 0.9. Two perpendicular slices through Figure 3, one at blood/dialysate binder ratio, β = 0 and the other at blood/dialysate binder ratio, β = 0.1, are presented. For clarity of Frel dependence on γ, only the mass transfer/blood flow rate ratio, κ = 2, results are displayed.

One-Compartment Model

In order to focus on the effect of SCUF-BSD operating parameters on the removal of total solute from the reservoir, the one-compartment model equations were solved for constant patient parameters of initial total solute/total binder concentration ratio, ϕ 0 = 0.9, rate of net fluid gain equal to the rate of removal by ultrafiltration, ηm = γ, and generation rates, ĜS = ĜA = 0. The first condition is typical of unconjugated bilirubin/albumin concentration ratios in patients with liver failure. The second condition is a clinical treatment decision equivalent to maintaining the patient in an isovolumic state by fluid removal through ultrafiltration at a rate equivalent to fluid gain through intravenous or other means. The third condition was imposed because of a paucity of information on toxin and binder generation rates associated with liver failure.

The parametric effects of the solute-albumin binding constant, KB, and dialysate/blood binder ratio, β, on the rate of total solute removal from the reservoir in the one compartment model are displayed in Figure 7 for constant mass transfer/blood flow rate ratio, κ = 2.0, dialysate/blood flow rate ratio, α = 0.2, and ultrafiltration/blood flow rate ratio, γ = 0.03. The removal rate is highly dependent upon binding strength, with rapid removal of loosely bound solutes, KB 104, and much slower removal of tightly bound solutes, KB 107. Loosely bound solutes are rapidly removed regardless of the presence of albumin in the dialysate as indicated by the close grouping of the β = 0.0, β = 0.05, and β = 0.10 curves. While the removal of tightly bound solutes, KB 107, is significantly enhanced by addition of albumin to the dialysate, the solute removal rate is insensitive to the dialysate/blood binder ratio, β, once some albumin is present in the dialysate (the β = 0.05, and β = 0.10 curves overlap). The most sensitivity in improvement of solute removal rate with respect to the dialysate/blood binder ratio, β, is observed for intermediate binding strengths, KB 105 and KB 106.

Figure 7.
Figure 7.:
Parametric effects of solute-albumin binding constant, KB, and dialysate/blood binder ratio, β, on the rate of solute removal in the one compartment model at constant mass transfer/blood flow rate ratio, κ = 2.0, dialysate/blood flow rate ratio, α = 0.2, and ultrafiltration/blood flow rate ratio, γ = 0.03. KB as indicated on plot; dark solid line: β = 0; dashed line: β = 0.05; light solid line: β = 0.10. Removal rate is very dependent upon binding strength with loosely bound solutes, KB 104, rapidly removed and tightly bound solutes, KB 107, slowly removed. Removal rate is most sensitive to dialysate/blood binder ratio, β, for intermediate binding strength, KB 105.

The parametric effects of dialysate/blood flow rate ratio, α, ultrafiltration/blood flow rate ratio, γ, and dialysate/blood binder ratio, β, on the rate of solute removal in the one-compartment model at constant solute-albumin binding constant, KB = 106, and mass transfer/blood flow rate ratio, κ = 2.0, are displayed in Figure 8. The three pairs of curves at the top of the figure represent solute removal without binder in the dialysate, β = 0, at ultrafiltration/blood flow rate ratios, γ = 0 and γ = 0.03, for dialysate/blood flow rate ratios, α = 0.1, α = 0.2, and α = 0.3, respectively. The behavior is similar to that depicted in Figure 6 for a single pass through the ultrafiltration cartridge: strong dependence on the dialysate/blood flow rate ratio, α, and weak dependence upon the ultrafiltration/blood flow rate ratio, γ. The lower set of curves in Figure 8 depicts the rapid approach to the asymptotic removal rates with addition of binder to dialysate that is depicted in Figure 3 for a single pass through the ultrafiltration cartridge. Solute removal is only weakly dependent of dialysate/blood flow rate ratio, α, ultrafiltration/blood flow rate ratio, γ, and dialysate/blood binder ratio, β, once some albumin is added to the dialysate

Figure 8.
Figure 8.:
Parametric effects of dialysate/blood flow rate ratio, α, ultrafiltration/blood flow rate ratio, γ, and dialysate/blood binder ratio, β, on the rate of solute removal in the one-compartment model at constant solute-albumin binding constant, KB = 106, and mass transfer/blood flow rate ratio, κ = 2.0. The top six curves, three pairs of two each with no albumin in the dialysate, β = 0, reflect the dependency depicted in Figure 6 for a single pass through the ultrafiltration cartridge: strong dependence on the dialysate/blood flow rate ratio, α (values as shown), and weak dependence upon the ultrafiltration/blood flow rate ratio, γ = 0 for the upper curve in each pair and γ = 0.03 for the lower curve. The lower set of curves depicts the rapid approach to the asymptotic removal rates with addition of binder to the dialysate depicted in Figure 3 for a single pass through the ultrafiltration cartridge. Solute removal is only weakly dependent of dialysate/blood flow rate ratio, α, ultrafiltration/blood flow rate ratio, γ, and dialysate/blood binder ratio, β, once some albumin is added to the dialysate.

Finally, the parametric effect of the mass transfer/blood flow rate ratio, κ, on the rate of solute removal in the one-compartment model is presented in Figure 9 for constant solute-albumin binding constant, KB = 106, dialysate/blood flow rate ratio, α = 0.2, and ultrafiltration/blood flow rate ratio, γ = 0.03. Reminiscent of the behavior depicted in Figure 3 for a single pass through the ultrafiltration cartridge, the κ = 1, κ = 2, and κ = 3 results for β = 0 superimpose onto the single upper curve in the figure: removal is independent of the mass transfer/blood flow rate ratio, κ, without binder added to the dialysate. The lower sets of curves depict the rapid approach to the asymptotic removal rates, dependent upon κ, with addition of binder to the dialysate, also reminiscent of the behavior depicted in Figure 3 for a single pass through the ultrafiltration cartridge. Significant enhancement in solute removal is possible with dialysate/blood binder ratio, β, as low as 0.05, for a solute-albumin binding constant, KB = 106.

Figure 9.
Figure 9.:
Parametric effect of the mass transfer/blood flow rate ratio, κ, on the rate of solute removal in the one-compartment model at constant solute-albumin binding constant, KB = 106, dialysate/blood flow rate ratio, α = 0.2, and ultrafiltration/blood flow rate ratio, γ = 0.03. The κ = 1, κ = 2, and κ = 3 results for β = 0 superimpose onto the single upper curve in the figure. This reflects the behavior seen in Figure 3 for a single pass through the ultrafiltration cartridge: removal is independent of the mass transfer/blood flow rate ratio, κ, without binder added to the dialysate. The lower sets of curves depict the rapid approach to the asymptotic removal rates, dependent upon κ, with addition of binder to the dialysate, as also depicted in Figure 3 for a single pass through the ultrafiltration cartridge (dotted line, β = 0.05, solid lines superimposition of β = 0.25 and β = 0.50).

Experiment

The dialysate/blood flow ratio, α, dialysate/blood binder ratio, β, and ultrafiltration/blood flow ratio, γ, are easily specified and controlled. However, the cartridge mass transfer/blood flow ratio, κ, requires knowledge of the unconjugated bilirubin mass transfer coefficient, KA, which is not available. Therefore, the one-compartment model with a representative literature value23 of KB = 0.65×107 mL/mol for unconjugated bilirubin/albumin binding was fit to the centerpoint experimental condition results by minimization of least squared differences between the model and the average experimental data to obtain an estimate of KA = 185 mL/min (at 22°C) for the dialysis cartridge used in the experiments. This compares favorably with the manufacturer reported KA = 414 mL/min for phosphate and KA = 192 mL/min for vitamin B12 at 37°C. The replicate centerpoint experiments were also used to estimate the standard error of the mean error bars that are shown in Figures 10A–10D.

Figure 10.
Figure 10.:
Experimental results (data points with error bars) versus SCUF-BSD one-compartment model predictions (lines) for the four faces of the factorial design hypercube of Table 2. Error bars represent standard error of the mean from the centerpoint replicate experiments. Parameter values: κ+ = 1.85, κ– = 1.03, β+ = 0.04, β– = 0.01 in all panels. A: [γ+ = 0.02, α+ = 0.2] face of hypercube. B: [γ+ = 0.02, α– = 0.1] face of hypercube. C: [γ– = 0.00, α+ = 0.2] face of hypercube. D: [γ– = 0.00, α– = 0.1] face of hypercube.

The 24 factorial experimental design, which was chosen to span the potential operating range associated with clinical practice of SCUF-BSD at our institution, can be thought of geometrically as a hypercube in four dimensions that can be parametrically displayed in two dimensions by depicting cubic planes as a function of two parameters. Because the one-compartment modeling results indicated that SCUF-BSD results would be impacted, in declining order, by β, κ, α, and then γ, the experimental results are displayed separately for the [γ+ α+] face, Figure 10A, the [γ+ α–] face, Figure 10B, the [γα+] face, Figure 10C, and the [γα–] face, Figure 10D, in order to obtain the greatest separation between the experimental results and highlight the impact of β and κ on bound solute removal. The pluses and minuses refer to the experimental settings for the parameters as defined in Table 2. Based upon the experimentally determined KA = 185 mL/min, κ– = 1.03, and κ+ = 1.85 for the flow rates listed in Table 2.

The panels of Figure 10 display the experimental results plotted as data points with standard error of the mean error bar estimates. The lines in each panel are not experimental fits to the data, rather, the lines are predictions of the one-compartment SCUF-BSD model using the experimentally specified values for the model parameters and KB = 0.65×107 mL/mol. There are no adjustable parameters in the model to bring predictions in line with the data. Agreement between the experimental results and the one-compartment model predictions is quite good with the exception of the [κ+ β–] experiment depicted in Figure 10B.

Discussion

Coupling the basic thermodynamics for transfer of bound solutes across a dialysis membrane with the flow dynamics through a dialysis cartridge produces a powerful tool for understanding and predicting the time course of clinical application for detoxification. The development presented here is simplified in that it assumed that equilibrium holds at all times on both sides of the dialysis membrane. This is equivalent to stating that the rate of solute transfer across the membrane is slow in comparison with the rate of disassociation of the bound solute complex in response to the movement of solute across the membrane. The development also assumes that the lumped dialysis mass transfer coefficient, KA, is not dependent upon either blood or dialysate flow rate. This is obviously problematic as dialyzer clearance, which is related to the mass transfer coefficient,17 is known to increase asymptotically as blood flow rate increases at constant dialysate flow rate. However, the change is minimal under the flow ranges considered for SCUF-BSD applications. Given these limitations and uncertainty in the actual unconjugated bilirubin-albumin binding constant, the ability of the SCUF-BSD one-compartment model to predict experimental results, without the need for adjustable parameters, indicates the robustness of the modeling approach.

The model as developed and presented here is for a single solute binding to a single site on a binder, presumably albumin in the blood stream. The number of bilirubin binding sites on albumin is somewhat controversial, ranging from one to three.23–27 Hence, the potential impact of a solute with multiple binding sites or multiple solutes with the same binding site merits discussion. Thermodynamically, solute will preferentially bind to the binding site with highest affinity for the solute over sites with lower affinity if multiple sites are present. Likewise, solute will preferentially dissociate first from sites with lower affinity followed by dissociation from sites with higher affinity. In the context of bound solute dialysis, solute will dissociate on the blood side and bind on the dialysate side. Therefore, in terms of affinity constants, KB, the blood side would have an effectively lower KB than the dialysate side in the early phases of solute removal because solute would be preferentially dissociating from the lower affinity site in the blood and binding to the higher affinity site in the dialysate. Such a scenario makes the ratio of KB between the blood side and dialysate side more favorable for bound solute dialysis than that for a solute with a single binding site. Similar arguments can be made with respect to competitive binding: the solute with the greater binding affinity will be favored to bind over the solute with lesser affinity. Removal thus favors the lower affinity solute before the higher affinity solute.

Suggestions that the solute/binder concentration ratio gradient across the membrane drives the solute transport28 are phenomenological interpretations of data from a specific experimental setup that ignore the basic thermodynamics of solute transport across a membrane. The experimental setup in question is comprised of two closed-loop recycle circuits that monitor the transport of solute and find that transport of solute continues until equilibrium is reached. At equilibrium, the experimentally measured solute/binder concentration ratio on each side of the membrane is approximately equal, which leads to the suggestion that the solute/binder concentration ratio gradient is the driving force. However, simple application of Equation A10 to the equilibrium situation with the thermodynamically correct assumption that the free solute concentration on both sides of the membrane is equal, i.e.,

where the subscript 1 refers to the recycle blood compartment and the subscript 2 to the recycle dialysate compartment, produces the relation

As previously noted, many toxins have a high affinity for binder, such as unconjugated bilirubin for albumin, that greatly reduces the free solute concentration. For example, with albumin and unconjugated bilirubin concentrations representative of liver failure, 0.6 mmol/l and 0.3 mmol/l, respectively, and an unconjugated bilirubin-albumin binding constant of 0.5 ×107 l/mol, the free unconjugated bilirubin concentration is approximately three orders of magnitude lower than the total concentration: 0.0002 mmol/l. This means that the numerator in the center of Equation 13 cannot be experimentally distinguished from the total solute concentration, as indicated by the approximate equivalence on the right hand side of Equation 13. Hence, experimentally and phenomenologically, the solute/binder concentration ratios on each side of the membrane appear to be equivalent when it is actually the free solute concentrations that are equal. Open, single-pass dialysate systems need only a small amount of binder in the dialysate to be effective.

Two major conclusions can be drawn from this work. The first is that a membrane with maximal KA for solute transport should be used in order to maximize the mass transfer/blood flow rate ratio, κ. The membrane mass transfer constant, KA, is a lumped parameter that includes hydrodynamic boundary layer effects, previously discussed, membrane permeability for bulk diffusion of the solutes in question, and the possibility of augmented surface diffusion transport such as that described in the early MARS literature.5,6

Following specification of the dialysis cartridge, the second conclusion is that the concentration of binder in the dialysate supplied to the dialysis cartridge required to achieve benefits from SCUF-BSD is minimal, on the order of 1% to 5% (0.5 to 2.0 g/l albumin) of the equivalent binder concentration in the blood stream, depending upon the dialysate flow rate. The amount of binder required to achieve maximal benefit is dependent upon the toxin of interest as illustrated in Figure 6; however, binder concentrations in the dialysate lower than that required for maximal benefit are still effective. Blood stream free solute concentrations on the order of micromoles per liter represent an inherent limit on the maximal rate transport across the membrane. The purpose of the binder in the dialysate is to maintain a constant, nearly zero, concentration of free solute in the dialysate in order to maintain driving force for free solute transport along the entire length of the cartridge. Only small amounts of binder are required to effectively bind the small amount of free solute that transits the membrane in a single pass through the dialyzer.

As demonstrated clinically by others working with heuristic BSD approaches,7,12,16,18 BSD can successfully remove bound toxins from patients. The present work provides a scientific, thermodynamically-based description for the processes and can be used to design improved approaches that will extend the clinical reach, utility, and economic feasibility of BSD.

Acknowledgments

This work was funded in part by through NIH grant R01 DK0632244 and the Thomas E Starzl Transplantation Institute, University of Pittsburgh. Nicole Burns, Amine Hallab, Laniel Hodges, and Greggory Housler helped perform the reported experiments.

Appendix A: Solute-Binder Equilibrium

Solute binding to albumin on either side of the membrane can described by the equilibrium binding constant relation

where KB is the equilibrium binding constant (mL/mmol), CB·S is the concentration of bound solute (mmol/mL), C is the concentration of free binder (mmol/mL), and CS is the concentration of free solute (mmol/mL). The fraction of binder that is bound to solute, Ψ (dimensionless), is related to the equilibrium binding constant by

The total concentrations of solute and binder are then given by

and

respectively, where the superscript t refers to the total concentration (sum of free and bound concentration) of the respective species. With appropriate subscripts, b for blood and d for dialysate, Equations A3 and A4 apply equally on the blood and dialysate sides of the membrane.

Appendix B: Ultrafiltration Cartridge

Assuming that ultrafiltration, Qf (mL/min), is uniformly distributed along the length of the cartridge results in the two algebraic relations:

for blood flow, Qb (mL/min), and dialysate flow, Qd (mL/min), respectively, as a function of position, z (cm) where z = 0 is the blood flow inlet (dialysate flow outlet) and z = L is the blood flow outlet (dialysate flow inlet), in the ultrafiltration cartridge. L (cm) is the total flow path length in the cartridge. The subscript in refers to the respective stream inlet condition. Because the binder cannot transit the membrane, two additional algebraic relations can be developed to express binder concentration as ultrafiltration progresses:

Equations B3 and B4 assume that the flowing streams are well-mixed with immediate binder redistribution in a fluid element as ultrafiltration takes place. In reality, some degree of concentration polarization will exist with ultrafiltration.

Postulating that only free solute can cross the ultrafiltration membrane, a mole balance for total solute on the blood side of the membrane as a function of position in the dialyzer yields

where

K is the overall mass transfer coefficient for free solute diffusion (cm/min), A is the total membrane area (cm2), and σ is the solute-membrane reflection coefficient. The Peclet number, Pe, is the ratio of convective to diffusive solute transport across the membrane. The first term on the right hand side of Equation B5 reflects the diffusive transport of solute across the membrane with the Villarroel20 correction to the concentration gradient induced by ultrafiltration. The second term reflects convective contribution of ultrafiltration.

The relation between the mass transfer coefficient, K, as used here and other common measures such as dialysance and clearance has been detailed elsewhere.17,29 While K is a weak function of blood and dialysate flow rates in the ranges employed for SCUF, it will be considered a constant for this analysis. This assumption can be relaxed by incorporating flow rate dependence as described by Michaels.19

A similar balance for the dialysate side of the membrane yields

The right hand sides of Equations B7 and B8 are the same because solute transfer from the blood appears in the dialysate and the concentration gradient has the same sign for both streams.

Making the variable substitutions

in Equations B7 and B8 together with using the equilibrium relations A3 and A4 and the dimensionless parameter definitions in Table 1 produces the dimensionless Equations 5 and 6.

Appendix C: One-Compartment Model

The patient is modeled as a single reservoir of fluid with volume Vres (mL), total binder concentration of (CtA)res (mmol/mL), and total solute (bound and free) concentration of (CtS)res (mmol/mL). If the patient has residual renal function, fluid without the solute of interest is excreted at a rate Qr (mL/min). The patient is administered fluids that do not contain the solute of interest at a rate Qa (mL/min). The net fluid gain/loss for the patient is represented by

The patient may generate new solute at net rate GS (mmol/min) due to ordinary metabolism or the disease state. Finally, the patient may also generate new binder at net rate GA (mmol/min) which is the difference between metabolic production and degradation.

Blood flows from the patient to the ultrafiltration cartridge at rate Qb (mL/min) with solute concentration (CtS)b = (CtS)res (mmol/mL). Inside the cartridge, volume is removed from the blood at rate Qf (mL/min) resulting in a blood outlet flow of

(mL/min). Solute is also transferred from the blood to dialysate by convection and diffusion within the cartridge, resulting in an outlet solute concentration (CtS)out (mmol/mL). Defining the fractional removal of solute in the ultrafiltration cartridge as

a mass balance across the cartridge provides the relation

The corresponding relation for binder, which is not removed in the cartridge, is

Assuming incompressibility, the volume of the reservoir as a function of time t (min) is

where Vres,0 is the reservoir volume at initiation of treatment. Because binder cannot pass the cartridge membrane, the concentration of binder in the reservoir is given by

where (CtA)res,0 is the initial reservoir binder concentration. Finally, a mole balance on the total concentration of solute in the reservoir yields the ordinary differential equation

with initial condition

Making the variable substitutions

in Equations C6 through C8 produces the dimensionless relations, 7, 8 and, 9 respectively.

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