Ind. Eng. Chem. Res. 2010, 49, 1359–1369
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Middle-Molecule Uremic Toxin Removal via Hemodialysis Augmented with an Immunosorbent Packed Bed Shu Xia,† Nichole Hodge,† Melvin Laski,‡ and Theodore F. Wiesner*,† Department of Chemical Engineering, Texas Tech UniVersity, Lubbock, Texas 79409-3121, and Department of Internal Medicine, Texas Tech UniVersity Health Sciences Center, Lubbock, Texas 79430
Outcomes for hemodialysis patients are disappointing. Middle molecules too large to cross dialysis membranes, such as parathyroid hormone (PTH), accumulate in the blood and are implicated in a complex collection of symptoms known as the uremic syndrome. We report our investigation into the removal of middle molecules by adding an immunosorptive packed bed, known as a hemoperfuser, to hemodialysis. We established in vitro that PTH can be reduced to normal levels within a typical dialysis session. We then developed a mathematical model of a combined hemodialysis-hemoperfusion system integrated with a patient pharmacokinetics model. We validated the model in vitro and then scaled up it up to clinical practice. We predict that immunosorptive hemoperfusion can reduce PTH levels to the normal range within the clinical scenario as well. However, we also predict that, because of the high synthesis rate of PTH in vivo, the toxin concentration might recover so rapidly as to limit the effectiveness of added immunosorption. Introduction Outcomes for patients on hemodialysis are considered to be disappointing.1 Hemodialysis membranes are optimized for the removal of low-molecular-weight solutes (0 c(0, 0) ) cb(0) ) 1
(3)
The solute concentration on the adsorbent is commonly modeled as (1 - ε)
dq ) kcav(C - Ceq) dt
(4)
where ε is the column porosity, kc is the mass-transfer coefficient, and av is the specific area of adsorbent per unit column volume. Ceq is the liquid-phase concentration of PTH in equilibrium with the concentration in the solid. Receptor-ligand affinities vary from 106 to 1012 M-1. For our PTH-antibody complex, the affinity constant is KA ) 2 × 1010 M-1, near the high affinity end.18 When the affinity of the sorbate to sorbent is very high, as in the case of antigens and their corresponding antibodies, the liquid-phase solute concentration in equilibrium with the solid is small. Thus, we can approximately describe the solid-phase solute concentration by setting Ceq in eq 4 to zero, giving kcav dq C ) dt 1-ε
dCb ) QB(1 - EHD)CHP - (QB + kuVb)Cb + GVb dt
(6) The blood perfusion rate is symbolized by QB, and CHP is the concentration of PTH exiting the adsorber. We now introduce the dimensionless variables cb ) Cb/Cb0 and τ ) jkut, where jku is the rate constant for endogenous solute removal in healthy persons. Therefore, model eq 6 becomes dcb ) acHP - bcb + γ dτ cb(0) ) 1
(7)
where a)
QB(1 - EHD) kuVb
,
b)
QB + kuVb kuVb
,
γ)
G kuCb0
Substituting eq 5 into eq 2 yields
(
)
kcav ∂C ∂C + ν + kd C ) 0 +u ∂t ∂z 1-ε
(9)
We then introduce the following dimensionless quantities
C , Cb0
c(x, τ) ) cb(x - τ)e-βτ
τex
(12)
c(x, τ) ) cb(τ - x)e-βx
τ>x
(13)
Noting that cHP(τ) ≡ c(R,τ), eq 7 becomes a first-order, linear delay differential equation. When the delay, R, is very small compared to the duration of the adsorption period, we can set it to zero (to be justified shortly). Under this assumption, eq 7 and its initial condition simplify to dcb(τ) ) -(b - ae-Rβ)cb(τ) + γ dτ cb(0) ) 1
kuz , x) u
(14)
Equation 14 is a simple first-order, linear nonhomogeneous ordinary differential equation. Its solution is given by
{
cb(τ) )
(
)
γ exp[-(b - ae-Rβ)τ] + b - ae-Rβ γ b - ae-Rβ γ 1exp[-(b - ae-Rβ)τd] + b - ae-Rβ γ γ -br(τ-τd) γ e + -Rβ b b b - ae r r
1-
[(
)
]
kcav ν + kd 1-ε β) , ku
and R )
kuLHP u (10)
where LHP is the length of the hemoperfuser. Equation 9 and its initial and boundary conditions become
0 e τ e τd
τd < τ e τmax
(15)
The dimensionless time, τd, is the duration of the dialysis session, and τmax is the interval between the starts of successive dialysis sessions. The quantity br is the value of the parameter b in the recovery phase br )
(8)
c)
System 11 is a nonhomogeneous kinematic wave equation and has the following solution19
(5)
The idealized process of patient plus extracorporeal circuit is illustrated in Figure 2. The coupling among the dialyzer, the hemoperfuser, and the extracellular compartment is given by the equation Vb
(11)
ku ku
(16)
Two important characteristics of the recovery phase are evident from eq 15: (1) The extracorporeal circuit influences the recovery of solute only through the value of the solute concentration at the end of dialysis (i.e., the beginning of the recovery phase). (2) The dynamics of recovery is governed by the endogenous generation and clearance of the solute (embodied in the parameters γ and br, respectively). In particular, the time constant of recovery is ku-1. For sizing of the adsorption cartridge, the quantity of adsorbed solute in the solid phase is also of interest. This quantity can be obtained by integrating eq 5. We nondimensionalize it and introduce the concentration in the adsorber liquid phase from eqs 12 and 13. Assuming that the initial solute concentrations in both the liquid and solid phases of the adsorber are zero, employing solution 15, and integrating, we obtain a complicated expression for the solid-phase saturation, which is nonetheless in closed form
Ind. Eng. Chem. Res., Vol. 49, No. 3, 2010 n(x, τ) ) exp(-ae-Rβx + bx + Rβ){exp[ae-Rβ(b - 1)τ] - 1} + ηe-βx a - beRβ exp(-ae-Rβx + bx + 2Rβ - bτ)[exp(ae-Rβτ) - ebτ]γ + (a - beRβ)2 γτ , x < τ e τd (17) b - ae-Rβ
(
)
n(x, τ) ) 0,
η)
Table 2. Values of Dimensionless Parameters for the Simulation of in Vitro Experiments dimensionless parameter a)
otherwise
In eq 17, n(x,τ) ) q(x,τ)/qmax, which is the dimensionless solid concentration. qmax is the maximum capacity of the sorbent (pg/ mL of column volume). The coefficient η is a dimensionless mass-transfer coefficient, given by kcavCb0 ku(1 - ε)qmax
b)
R)
QB(1 - EHD) kuVb QB + kuVb kuVb
kuLHP u
description
value
ratio of hemodialyzer clearance to endogenous clearance
0.296
ratio of total clearance to endogenous clearance
0.315
dimensionless hemoperfuser length 1.194 × 10-3
(18)
Equations 15 and 17 describe the time course of solute concentration in the liquid and solid phases, respectively, from the onset of one dialysis session to the onset of the next dialysis session. The complete derivation of the mathematical model, particularly eq 7, is provided in the Supporting Information. Estimation of Model Parameters. For the design of a clinical apparatus, it is important that the model used for scaling up be accurate. To validate the model developed in the preceding section, we investigated how well it could reproduce our experimental concentrations of PTH in the solution. For the experimental scenario, solute generation is zero (γ ) 0), and model eq 15 simplifies to cb(τ) ) exp[-(b - ae-Rβ)τ]
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(19)
The dimensionless parameters were defined in eqs 8 and 10. We modeled the removal of the solute by degradation in the buffer and by adsorption onto surfaces other than the sorbent as a simple first-order process. The removal rate constant, kd, was estimated from a static (no-flow) experiment (data in Supporting Information). The PTH reduction in the flow experiment without adsorbent and the value of the degradation constant kd from the static experiment allowed for the determination of the overall mass-transfer coefficient in our experimental apparatus, K0. Very important for scale-up purposes is an accurate estimate of the concentration-based mass-transfer coefficient, kc. We employed the well-established ChiltonColburn analogy for a packed bed20 for its estimation. All dimensioned parameters for the experiments are summarized in Table 1. A total of 23 parameters were reduced to 5 dimensionless parameters as defined in the Model Formulation section, and the values of the latter are listed in Table 2.
kcav ν + kd 1-ε β) ku
γ)
G kuCb0
ratio of clearance in hemoperfuser to endogenous clearance
1.64 × 103
dimensionless endogenous solute generation rate
0
greatly accelerates the reduction of the PTH concentration compared to that observed with degradation alone. Because of the very low concentration of PTH in the human body (low in the sense of mass transfer), nonspecific adsorption methods (middle curve) are inadequate to realize the clinical goal. Therefore, specific adsorption is necessary when PTH is the solute. Comparison of Model to Experimental Results. In Figure 4, the results of the in vitro experiments are compared with the prediction of the model in eq 15, employing the parameters in Table 2. The two time courses are very similar. In both cases, the PTH concentration drops below the target level of 100 pg /mL within 2 h. A small discrepancy between the two time courses appears after 1 h, with the model prediction approaching a zero steady state whereas the experimental time course reaches a steady state of ca. 100 pg/mL. The discrepancy is due to the assumption of irreversible adsorption in the model. Although the affinity constant of the antibody for PTH is large, it is not infinite. Therefore, the experimental results exhibit a small, nonzero steady state. Assumption of Negligible Delay in the Extracorporeal Circuit. With the addition of delay due to hemoperfusion, we wished to verify that the dimensionless hemoperfuser length R could be neglected in the simulations. Intuitively, one would
Results Flow Experiments. The experimental results for the perfusion system are presented in Figure 3. They indicate that, in the presence of functionalized adsorbent, the PTH concentration can be decreased to a normal level, less than 100 pg/mL, within 2 h. Referring to eq 19, we see that the time constant for the decline in concentration is 1 b - ae-Rβ
(20)
According to the parameters in Table 2, the value of this time constant varies from 3.6 (degradation plus hemoperfusion) to 44 (degradation only). Thus, the addition of specific adsorption
Figure 3. Results of the flow experiments for changes in dimensionless PTH concentration versus time with no beads, nonfunctionalized beads, and functionalized beads.
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Figure 4. Comparison of experimental results to simulation results for the in vitro experiments. Simulation parameters are listed in Tables 1 and 2.
expect that, if the residence time of the blood in the extracorporeal circuit were much smaller than the time scale of hemodialysis, the transportation delay in the extracorporeal circuit could be neglected in modeling blood purification. In our model, this corresponds to the dimensionless hemoperfuser length R being a very small fraction of the purification time. For both the in vitro and clinical scenarios, this is indeed the case (R/τd ) 3.6 × 10-5 and 1.05 × 10-4, respectively). Therefore, we proceeded with the simulations neglecting R. The agreement between model and experiment in Figure 4 was judged adequate for scaling the model up to a clinical scenario. All simulations were carried out in MathCAD 13 (Parametric Technology Corporation, Needham, MA), unless otherwise indicated. Scaleup to Clinical Scenario. For the experiments to be relevant to clinical application, they must be properly scaled from a laboratory to a clinical scenario. The performance of the hemoperfuser is dependent on many parameters that, if investigated individually, would be prohibitively expensive. Fortunately, process scale-up methodology is available to lump parameters and thereby reduce the number of costly in vivo and in vitro experiments. The essence of scaling up our in vitro results is maintaining geometric similarity between our laboratory hemoperfuser and the clinical hemoperfuser. This is done by keeping the dimensionless numbers describing the adsorption process the same in the two scenarios. Application of the π theorem21 to the mathematical model leads the following set of dimensionless groups
{
QBLHP QB(1 - EHD)LHP GLHP kuLHP kdLHP kcavLHP , , , , , u0Vb u0Vb Cb0u0 u0 u0 εu0
}
(21)
The third, fourth and fifth dimensionless values can be combined into Gku-1Cb0-1 and Gkd-1Cb0-1, which can be removed from
the pool of manipulated dimensionless values because of the intrinsic nature of G, kd, and ku. In addition, QB is equal to the product of the superficial flow velocity u0 and the cross-sectional area AHP. Therefore, the above set of dimensionless groups can be simplified to
{
VHP VHP(1 - EHD) kcavtHP , , Vb Vb ε
}
(22)
where tHP is the residence time of the adsorption column. The first dimensionless value of the above set shows the relationship between the volume of the adsorber and the patient extracellular volume, the second dimensionless value adds the properties of the dialyzer, and the last adds the characteristics of the adsorber. If we assume that the modified extracorporeal circuit retains existing hemodialyzers and focus on the design of the packed bed, then the two key dimensionless numbers to be used in scaleup are
{
VHP kcavtHP , Vb ε
}
(23)
As long as these two dimensionless values are similar, the performance of the two columns will be similar. For our in vitro scenario, the values of these dimensionless numbers are
{
VHP kcavtHP , Vb ε
}
) {7.363 × 10-4,4.079}
(24)
The essential results of the scaling procedure on the dimensions of the adsorber are given in Table 3. The resulting hemoperfuser is physically small; it is a disk that is 3.2 cm in diameter and has a length of 1.37 cm.
Ind. Eng. Chem. Res., Vol. 49, No. 3, 2010 Table 3. Comparison of Dimensions of the Hemoperfuser Before and After Scaleup dimension
in vitro experiment (before scaleup)
clinical design (after scaleup)
radius, Rb length, LHP
2.5 cm 0.3 cm
3.2 cm 1.37 cm
Clinical Predictions of PTH Removal by Immunosorption. Using the clinical scenario described in De Francisco et al.,12 we calculated what additional reduction in PTH level would be obtainable by employing the hemoperfuser design scaled up from our laboratory experiments. The combination of values from the De Francisco et al. experiments and our scaled-up hemoperfuser yields the dimensioned parameter set in Table 4 and the reduced dimensionless parameter set in Table 5. Physical properties were changed from those of water to those of blood as part of the scale-up procedure. A comparison of the two simulated scenarios is illustrated in Figure 5. As anticipated, addition of immunosorption significantly lowers the level of PTH. In the dialysis-only case, the simulation predicts that the PTH level will drop from its initial value of 432 pg/mL to approximately 360 pg/mL within the intradialytic period (3.5 h). Addition of immunosorptive hemoperfusion reduces the PTH level to 77 pg/mL by the end of the intradialytic period. In both scenarios, the hormone level will recover to 99% of its predialysis value approximately 18.5 h after the termination of dialysis. Thus, both treatment modalities reduce the PTH levels for approximately 22 h over the 48-h interval between the starts of successive dialysis sessions, each with approximately the same time of recovery for the toxin. The associated PTH concentration profiles adsorbed in the hemoperfuser are given by the two plots of eq 17 that are illustrated in Figure 6. The fractional bed saturation as a function of time (upper pane) and as a function of position (lower pane) is on the order of 10-5 in both cases, far below saturation. Assessment of the Benefit of Adding Immunosorption to Hemodialysis. Many of the deleterious effects of PTH are driven by the high concentration of the hormone in the bloodstream (high in the sense of biological activity). The advantage of adding the adsorber in terms of potential reduction
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Table 5. Values of Dimensionless Parameters for Clinical Predictions dimensionless parameter a)
b)
R)
QB(1 - EHD) k¯uVb QB + kuVb kuVb
kuLHP u
kcav ν + ku 1-ε β) ku
γ0 )
G kuCb0
description
ratio of hemodialyzer clearance to endogenous clearance
0.17
ratio of total clearance to endogenous clearance
0.218
dimensionless hemoperfuser length
2.1 × 10-3
ratio of clearance in the hemoperfuser to endogenous clearance
1.34 × 103
dimensionless endogenous PTH generation rate in patients with 0.036 secondary hyperparathyroidism. dimensionless mass-transfer coefficient defined by eq 18
η
value
1.41 × 10-6
of solute can be obtained from the following ratio (see the Supporting Information for the derivation) cbss b-a ) cbss | Rβ)0 b - ae-Rβ
(25)
The ratio in eq 25 is bounded by (b - a)/b ) 0.22 (Rβ ) ∞, adsorber acts as an infinite solute sink) and (b - a)/b ) 1 (Rβ ) 0, dialysis only). For the clinical removal of PTH (parameters in Table 5), the ratio is 0.23. Referring again to Figure 5, the area between the two curves represents the additional reduction of PTH over the 48-h period. An average hormone reduction (AHR) between the start of successive hemodialysis sessions can be defined as
Table 4. Dimensioned Parameters for Clinical Predictiona parameter
value
kd ku
rate constant for PTH degradation in buffer rate constant for PTH clearance by liver and kidneys in patients with SHPT
0 3.9 × 10-3 min-1
Vb G
15 L 1.687 pg mL-1 min-1
td u0
volume of the extracellular compartment endogenous generation rate of PTH per unit volume of the extravascular compartment initial concentration of PTH circulation rate of blood extraction fraction of PTH in the hemodialyzer length of the dialysis session superficial velocity of blood in the adsorber
Rb LHP av
radius of the adsorption column length of the hemoperfuser specific area per unit volume of adsorbent
3.2 cm 1.37 cm 347 cm2/cm3
µ F kc
viscosity of blood density of blood liquid-film mass-transfer coefficient in the packed bed
3 cP 1.056 g/mL 3.39 × 10-3 cm/s
Cb0 QB EHD
a
description
Where different from the in vitro parameters in Table 1.
source perfusate is blood adjusted for consistency with Cb0 and parameters from Momsen and Schwarz29 p 338, Habener31 calculated using parameters from Momsen and Schwarz29
432 pg/mL 300 mL/min 0.067
De Francisco et al.12 De Francisco et al.12 De Francisco et al.12
3.5 h 0.62 cm/s
Meyer and Hostetter1 typical value for hemoperfusers, Table 2, p 128, Ronco et al.32 scaled up from experiment scaled up from experiment calculated assuming a packed bed of spheres p 62, Fournier16 p 62, Fournier16 calculated from the Chilton-Colburn analogy; Chapter 6, Skelland20
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Figure 5. Comparison of PTH profiles with and without immunosorbent hemoperfusion in the clinical scenario of De Francisco et al.12
AHR )
∫
tmax
0
[CPTH(without adsorbent) - CPTH(with adsorbent)] dt
∫
tmax
0
dt
(26) An analytical expression for AHR can be obtained using the compound solution in eq 15. However, the expression is very complicated, so we calculated the value of AHR using numerical integration. Employing the parameters of Table 4, which were used to generate the PTH trajectories of Figure 5, the value of AHR was found to be 41 pg/mL or 9.7% of the predialysis level. The benefit of added immunosorption decreases with increasing hormone synthesis. We simulated the sensitivity of AHR to the endogenous synthesis rate, the results of which are given in Figure 7. For the range of dimensionless secretion rates 0.2 e γ/γ0 e 5, the average percentage reduction in PTH level ranges from ca. 18% to 3%. The next step in evaluating the clinical efficacy of immunosorption of PTH would be to correlate the AHR with symptomatic improvements in patients. Discussion
Figure 6. Solid-phase concentration profiles in the adsorber: (a) fractional bed saturations as a function of time at the bed inlet and outlet, (b) axial concentration profiles at the beginning, middle, and end of the dialysis session. The saturation value is n ) 1.
Hemoperfusion is typically carried out over activated charcoal coated with a biocompatible material, over ion-exchange resins, or over nonionic macroporous resins.22 Our goal was to improve upon the selectivity of these materials for the treatment of chronic uremia using immunosorption. We formulated a model of combined hemodialysis and immunosorptive hemoperfusion and validated it in vitro. Because of the high affinity of antibody-antigen interactions, irreversible adsorption in the packed bed was assumed, leading to a simple first-order differential equation with a closed-form solution. We found that, at least in the case of the uremic toxin PTH, solute levels can be reduced to normal levels within the time of a typical hemodialysis session.
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Figure 7. Average hormonal reduction as a function of dimensionless secretion rate.
Figure 8. Serum PTH level with and without hemoperfusion over 1 week.
According to Vanholder et al.,23 the European Uremic Toxin Work Group has characterized 90 compounds as uremic toxins. Of these 90 substances, 10 have molecular weights between 500 Da and 12 kDa (the middle molecules). These “middle” molecules are commonly glucuronide conjugates, small peptides, carbohydrate derivatives, advanced glycosylation/glycol-oxidation products, peptide hormones, and metabolites such as atrial natriuretic peptide.3 Twelve of the 90 compounds have molecular weights greater than 12 kDa. These substances consist mainly of proteins of the immune system such as complement proteins and pro-inflammatory cytokines.24 The clearance by hemodialysis becomes less than the clearance of healthy kidneys for solutes of molecular weights greater than approximately 1000 Da. Because many of the uremic toxins are larger than 1 kDa and are also proteinacious, immunosorption is potentially an efficacious treatment for 10-22 compounds associated with uremia.
When immunosorption is considered to remove multiple solutes simultaneously, the question arises as to whether interference among the species might reduce the effectiveness of the technique. The question can be answered only with rigorous multicomponent experiments. Nonetheless, the results of this PTH-only study indicate that a single device that absorbs multiple solutes might be feasible. Figure 6 predicts that even the small volume of sorbent in the clinical scenario is well below saturation at the end of the dialysis session. Multiple types of beads, distinguished by being functionalized against different solutes, can be mixed together in a device no larger than charcoal-based hemoperfusers. The small volumes of sorbents required result from the low concentrations, on the order of picograms per milliliter, of many of the uremic toxins.23 Sepharose is typically used for plasma perfusion rather than hemoperfusion of whole blood.10 This does raise the question
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as to Sepharose’s compatibility with the formed elements of blood. There is a dearth of literature on the biocompatibility of Sepharose with whole blood. Thus, phenomena such as platelet depletion and hemolysis should be evaluated in future ex vivo studies in which whole blood contacts the Sepharose adsorbent. Levels of all uremic toxins will recover after termination of blood purification, whether removed by dialysis or by other means. Normally, kidney-failure patients undergo hemodialysis three times a week. The resulting plasma concentrations of the solutes follow a sawtooth pattern with time.3 For our clinical scenario, PTH concentrations vary periodically as in Figure 8. The time of recovery depends on the purification method only through the level to which the solute is reduced during blood purification. For middle molecules whose endogenous secretion is elevated and whose endogenous clearance is reduced, as is the case with PTH in ESRD, recovery is very rapid. The average hormonal reduction (AHR) in such a scenario might be so small as to limit the benefit of added immunosorption. Immunosorption might be more applicable to middle molecules with transient secretion profiles, such as pro-inflammatory cytokines, than to middle molecules whose endogenous secretion is constitutively elevated and whose endogenous clearance is reduced, as is the case for parathyroid hormone. Release of the cytokines is triggered by dialysis itself, and their concentrations exhibit peaks during the acute-phase inflammatory response, followed by a decline. (See for example, the IL-6 temporal profile in Figure 1 of Schouten et al.25) Removal of the cytokines via immunosorption could interrupt the inflammatory cascade and reduce the cardiovascular sequelae resulting from chronic induction of inflammation. Immunosorption might likewise be advantageous in the cases of high-frequency dialysis. Numerous studies have recently found significant improvement in patient outcome as the frequency of dialysis increases.26 Two examples of high-frequency dialysis are daily, nocturnal dialysis in the home27 and continuous ambulatory peritoneal dialysis (CAPD). In the case of CAPD, the dialysate is passed over an immunosorbent to remove a target toxin, thus increasing the removal of any of the target toxins that have passed through the peritoneal membrane.28 If the dialysis frequency can be increased such that the solute does not return to predialysis levels prior to the start of a new dialysis session, sustained reduction of middle-molecule levels can be achieved with the addition of immunosorbents. Summary Addition of immunosorbent hemoperfusion can rapidly lower the concentration of parathyroid hormone to normal levels within the duration of a typical hemodialysis session. Mathematical modeling predicts that the time for the PTH to recover to predialysis levels is very short, which could adversely affect its clinical efficacy. Acknowledgment We acknowledge the technical assistance of Ms. Betty Lonis and Dr. Jan Simoni in the analysis of parathyroid hormone. We thank Dr. Maybin Simfukwe for providing the dialyzer cartridges. We also thank Christina Foster for preparing Figure 1 and measuring the porosity of the adsorbent. Supporting Information Available: Description and results of the static binding experiment. Complete derivation of the
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ReceiVed for reView April 14, 2009 ReVised manuscript receiVed November 24, 2009 Accepted December 7, 2009 IE900597Z
