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Blood urea clearance with microencapsulated urease
J. theor. Biol. (1995) 175, 295–303
Blood Urea Clearance with Microencapsulated Urease K. B. L, D. K. B R. J. N
Department of Chemical Engineering, McGill University, 3480 University St., Montreal, Quebec, Canada H3A 2A7
(Received on 4 April 1994, Accepted in revised form on 8 February 1995)
The response of a kidney patient to treatment using a microencapsulated urease artificial kidney (MUAK) system was modelled. The model was used to simulate the patient’s response and reactor performance for an initial blood urea concentration of 10 mM and a MUAK void volume of 0.5. The performance of the reactor was strongly dependent on the enzyme activity. An optimal activity of 10 mM sec−1 was achieved in the analysis. After operation for 4 h at a flowrate of 200 ml min−1 , the reduction in blood urea concentration for reactor dimensions of 2×10 cm, 2×20 cm, 4×10 cm and 4×20 cm were 38%, 52%, 60% and 62%, respectively. The effect of flow rate on the performance of urea removal was studied using the reactor dimensions of 4×10 cm (optimal design). The results for flow rates of 100, 200, 300 and 400 ml min−1 predicted blood urea reductions of 38, 60, 70 and 76%, respectively. Although, the conversion of urea in the reactor decreased from 100% to 96% for the respective flows of 100 to 400 ml min−1 , the high turnover of reactor volume at a higher flow rate was responsible for the improved reduction of the patient’s blood urea level. The model has the ability to predict the performance of the MUAK and the patient’s blood urea level simultaneously, at various operating conditions. 7 1995 Academic Press Limited
Berk, 1944). Approximately half a million patients worldwide are being supported by hemodialysis (Nose´, 1990). Although hemodialysis has been effective for the treatment of kidney failure, there are a number of related problems. The conventional artificial kidney is bulky, heavy, complex and expensive, and difficult to handle, limiting the mobility of the patient (Chang, 1978; Krajewska et al., 1988). Thus, several attempts have been made to simplify the machine and reduce its size (Walker et al., 1977; Denti & Biagini, 1977; Shettigar et al., 1983). Reports by Gordon et al. (1970, 1971) led to the development of the commercially available REDY8 portable absorption dialysis system. However, bacteriological (So nderstrup, 1976), toxicological (Moller et al., 1976) and acid-base (Pederson & Christiansen, 1976) problems in the REDY8 system have necessitated improvements or further research into other alternatives. Investigations into the application of the artificial cell as a basis for the construction of an artificial kidney were initiated by Chang (1964). Urease was microencapsulated in ultra-thin non-toxic polymeric
Introduction The function of the human kidneys is to clear the blood of body wastes, processing up to 1.5 l of blood each minute (Cooney, 1976). Normal renal function includes the removal of the end products of metabolism such as urea, uric acid, creatine, ammonium, sulfates, and phenol; regulation of the body chemistry; regulation of the amount of water in the body to prevent edema and hypertension; and the removal of the substances that are not utilized or metabolized in the body, such as drugs and toxins, thus preventing renal poisoning. During malfunctioning of the kidneys, the patient becomes progressively ill: metabolic waste products, excess electrolytes and water accumulate and the chemical balance is upset and may lead to death within days. Intermittent treatment with a mechanical device such as the artificial kidney reduces the accumulation of waste products, normalizing the blood conditions. The conventional artificial kidney (hemodialyzer) system has been in clinical use since 1943 (Kolff & 0022–5193/95/150295+09 $12.00/0
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membranes which could be used for dialysate regeneration or direct blood urea clearance purposes. The delay in the development of this new concept was due in part to the lack of appropriate theoretical models for the characterization of the microencapsulated system. Additional challenges associated with the direct hemoperfusion of microencapsulated urease include thromboembolism, platelet adhesion and the potential for microcapsule rupture releasing enzyme into the blood stream. One alternative that has been suggested is conventional hemodialysis with dialysate recirculated through a microencapsulated urease/ion exchange resin bed. The present analysis would be applicable by taking into account mass transfer across the membrane dialyser. In the present study, a model was developed to demonstrate the applicability of the microencapsulated urease artificial kidney (MUAK) system with time-varying blood urea composition. The main objective was to study the response of a patient to the operations of the MUAK system and to assess its operability at various enzyme activities. The influence of the reactor design on the patient’s response under various conditions was analysed for optimal design and control purposes.
systems in blood are (Ferguson, 1965): (i) bicarbonate/carbonic acid buffer, HCO− 3 + H2 CO3 especially in plasma; (ii) dihydrogen phosphate/dibasic phosphate 2− buffer, H2 PO− 4 +HPO4 , which is of minor importance in blood; and (iii) protein buffer to some extent in plasma. Among these, the bicarbonate buffer system is the most important for controlling blood pH. Thus, the appropriate ionization reactions for this model can be summarized as follows: + NH+ 4 F H +NH3(aq) ;
K1=5.60×10−10 (M) (2)
H2 CO3 F H++HCO− 3 ;
K2=7.94×10−7 (M) (3)
2− + HCO− 3 F H +CO3 ;
K3=6.31×10−12 (M) (4)
H2 O F H++OH−;
K4=1.0×10−14 (M)
NaHCO3 : Na++HCO− 3 ,
(5) (6)
where the Kj ’s ( j=1,..., 4) denote the ionization equilibrium constants for the respective reactions. Microencapsulated urease was reported to lose some activity with a half-life of one week at a body temperature of 37°C (Chang, 1965). The experimental data was used in the following empirical equation for enzyme deactivation.
Theory
j=1+c1 t+c2 t 2+c3 t 3+c4 t 4+c5 t 5,
The conceptual model is shown in Fig. 1 where a patient is directly connected to an extracorporeal shunt chamber (microencapsulated fixed bed reactor). The urea in the contaminated blood diffuses into the microcapsules and undergoes hydrolysis by urease to ammonia and carbon dioxide as follows:
where j is the activity coefficient which is the ratio of the reaction rate between deactivating urease and the fresh enzyme, t is the time in seconds and c1 to c5 are the empirical constants, with values of 2.97×10−6 , 1.84×10−11 , −7.15×10−17 , 1.33× 10−22 and −9.25×10−29 , respectively. The rate of substrate urea reaction, Rs (Moynihan et al., 1989), is therefore modified to account for urease deactivation as follows:
(NH2 )2 CO+H2 O F 2NH3+CO2
(1)
The blood pH (7.4) is considered to be homeostatically regulated in the body. The three chemical buffer
Rs =
Vm ES j, (S+Km+S 2/Ks )(1+P/Ki )
(7)
(8)
where E, S and P denote the concentration of urease (enzyme), urea (substrate) and ammonium ion (product) respectively, and Ks and Ki are the substrate and product inhibition constants. Vm and Km are the pH-dependent maximum reaction velocity and Michaelis constants, respectively which may be expressed as follows (Dixon & Webb, 1964):
F. 1. A schematic representation of a patient-FBR system.
Vm=Vm,o
1 1+Kes1 /H++H+/Kes2
(9)
Km=Km,o
1+Ke1 /H++H+/Ke2 , 1+Kes1 /H++H+/Kes2
(10)
where Vm,o and Km,o are the pH-independent maximum reaction velocity and Michaelis constants, respectively. The parameters Ke1 and Ke2 are the ionization equilibrium constants of the enzyme, whereas Kes1 and Kes2 are the equilibrium constants of the enzyme–substrate complex. The following assumptions were made in the development of mass conservation equations: (a) Mass transport through and within the semipermeable membrane takes place by passive molecular diffusion. (b) External transport takes place by passive molecular diffusion and convection. (c) The enzyme reaction rate is described by a modified Michaelis–Menten form, which accounts for the substrate and product inhibition, and pH-dependent kinetic constants. (d) No interactions exist among the enzyme, the diffusing solutes and the encapsulating membrane. (e) Enzyme is uniformly distributed in the enzyme phase of the microcapsule. (f) The patient’s blood system is assumed to be a single pool. A single compartment model for the patient (Fig. 1) is considered and a differential mass balance for the body fluid pool, which is assumed to be perfectly mixed, takes the form 1Cs,in QB = (CB,out−CB,in )+G, 1t VB
(11)
where VB represents the volume of body fluids, QB is the volumetric flow rate of blood, CB,out is the urea concentration of blood returning to the patient body, CB,in is the urea concentration of blood leaving the body, and G is the urea production rate by the body metabolism. The subscripts ‘‘in’’ and ‘‘out’’ refer to the reactor inlet (body outlet) and outlet (body inlet) concentrations. The medium in the reactor may consist primarily − of nine species (urea, NH+ 4 , NH3 , H2 CO3 , HCO3 , 2− + − + CO3 , H , OH , Na ). The required nine equations can be reduced to three by combining eqns (2)–(5) for the product species. Therefore, specific balance equations within the microcapsule can be expressed as follows: 1Cs,1 =−9 · Ns,1−Rs 1t
(12)
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1CNH+4,1 =fNH+4 {−9 · (NNH+4,1+NNH3 ,1 )+2Rs } (13) 1t 1CHCO−3,1 =fHCO−3 {−9 · (NH2 CO3 ,1+NHCO−3,1 1t +NCO2− )+Rs } 3 ,1
(14)
where fNH+4= fHCO−3=
H+ H +K1
(15)
+
1 H+ K3 1+ + + K2 H
and N is the molar flux. Rs is the rate of urea removal and subscript 1 denotes the enzyme phase. The concentration of Na+ species would be constant throughout the reactor since it is not involved in any of the reactions. At any given time, the concentrations of H+ and OH− species can be obtained from a balance of charges in the enzyme solution and in the bulk as follows: (CH+,1+CNH+4,1+CNa+,1 ) −(COH−,1+CHCO−3,1+2CCO2− )=0 3 ,1 (CH+,b+CNH+4,b+CNa+,b ) )=0, −(COH−,b+CHCO−3,b+2CCO2− 3 ,b respectively. Fickian diffusion is often used to describe the transport of species within the microcapsule. However, the molecular dynamics of diffusing ions may affect the molar flux of other species. Therefore, the Nernst– Planck equation (Bard et al., 1980) will be adopted as follows:
0
−Ni,1=Di,1 9Ci,1+Ci,1 zi
F9C Rg T
1
(19)
where D, F, Rg and T are the coefficient of diffusion, Faraday constant, molar gas constant, and temperature, respectively. C is the potential of a streaming charge zi , and Ci is the concentration of species i in the solution. The boundary conditions are: r=0; r=r1 ;
Ci,1=Ci,2
1Ci,1 =0 1r
(20)
−Ni,1=ki,O (Ci,b−Ci,r=r 1 ) (21)
where, r1 is the inner radius of curvature of the membrane and ki,O is the combined transfer coefficient of species i for the liquid-phase mass transfer and
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298
diffusion through the membrane, which is defined by 1 1 1 = + ki,O ki,L ki,M
(22) +
where, ki,L and ki,M are the liquid phase and membrane wall mass transfer coefficients of species i, respectively. The effectiveness factor h, which represents the influence of mass transfer on the overall reaction process, is defined for the present system by
h=
0 1b > 0 10 1
3 1S1 D r1 s,1 1r
2 b
S Ks
1+
(23)
Pb Ki
The overall material balance for the FBR will be presented to account for each species involved in the reaction system. Figure 1 shows the model fixed bed reactor of length L which is packed with microcapsules. The system is fed with blood at a superficial velocity Us and undergoes removal of urea by passage through the encapsulated packing in a reactor of total volume VT . Utilizing the concept of dispersion coefficient Dz to account for axial mixing, a mass balance on urea, − NH+ 4 , HCO3 , in the bulk liquid within the column yields: 1 1 2Cs,b 1 1Cs,b 3aks,O 1Cs,b − (Cs,b−Cs,r 1 ) = − 1t tr Bos 1Z 2 tr 1Z or2 (24) + NH4 ,b
+ NH4 ,b
3akNH+4 ,0 (CNH+4 ,b−CNH+4 ,r 1 ) or2
1 2CNH3 ,b 1 1CNH3 ,b 1 − tr 1Z tr BoNH3 1Z 2
+
3akNH3 ,0 (CNH3 ,b−CNH3 ,r 1 ) or2
6
7
1 2CHCO−3,b 1CHCO−3,b 1 =fHCO−3 − 1t 1Z 2 tr BoHCO3 1 1CHCO−3,b tr 1Z
−
+
1 2CCO2− 1 1 1CCO2− 3 ,b 3 ,b − 2 2− tr BoCO3 1Z tr 1Z
3akCO2− 3 ,0 (CCO2− −CCO2− ) 3 ,r 1 3 ,b or2
−
7
(26)
where, a is the fraction of reactor volume occupied by the microcapsules, t is the physical time for reaction, tr is the residence time, Z is the vertical coordinate direction in the axial length of the bed, and Bo is the Bodenstein number, and Us L Dz,i
(27)
Vc VT
(28)
Lo Us
(29)
Boi= a=
tr=
The boundary conditions for solving eqns (24)–(26) are given by . Z=0,
1 1Ci =Ci,o+−Ci,f Boi 1Z
(30)
1Ci =0 1Z
(31),
where the position Z=0 is chosen to be the bottom inlet port of the bed. The dispersion coefficient Dz is given by the following correlation (Chung & Wen, 1968): NRe omf Dz = (32) 0.48 0.20+0.011N Re rf
−
−
3akH2 CO3 ,0 (CH2 CO3 ,b−CH2 CO3 ,r 1 ) or2
. Z=1,
1C 1CNH+4,b 1 1 1C =fNH+4 − 1t tr BoNH+4 1Z 2 tr 1Z 2
1 2CH2 CO3 ,b 1 1CH2 CO3 ,b 1 − tr BoH2 CO3 1Z 2 tr 1Z
−
Vm ESb
r=r 1
Sb+Km+
6
3akHCO−3,0 (CHCO−3,b−CHCO−3,r1 ) or2
−
(25)
where NRe is the particle Reynolds number (dp Us nf /mf ); dp is the mean microcapsule diameter; Us the superficial velocity; mf , the dialysate viscosity; o, the bed void fraction, and rf is the dialysate density. The method of orthogonal collocation (Finlayson, 1980) was used to carry out the integration for both the intraparticle concentration profiles, Ci,1 (r) and the bulk liquid concentration profiles, Ci,b (Z). The material balance equations were solved using subroutines from
299
the International Mathematical and Statistical Library (IMSL). For steady-state cases, the DNEQNF (double precision nonlinear equations by finite differences) subroutine was used. This routine applies the Levenberg–Marquardt algorithm and a finite difference approximation to the Jacobian. Transient computations were also made using the Gear method from the DIVPAG (double precision initial value problem by Adam-Moulton and Gear) subroutine. Details of the numerical scheme have been discussed elsewhere (Lee, 1993).
Results and Discussion The operation of a microencapsulated urease artificial kidney for the treatment of a patient was simulated. Generally, eqns (11)–(14) and (24)–(26) must be solved simultaneously to obtain information on the intracapsule, bulk reactor and the body urea concentration profiles. Since the human blood contains a bicarbonate buffer system, the intracapsule and bulk charge balances given by eqns (17) and (18), respectively, will be required for a complete solution. Decoupled solutions could only be obtained with the aid of the equilibrium relations given by eqns (2)–(5) and the reaction rate equation (8) which accounts for the deactivation of the enzyme. Due to the complexity of the problem involved, certain assumptions were made to facilitate the computation without loss of details in the trends of the simulation. It was assumed that the urease is coencapsulated with ion exchange resins for ammonia removal and pH control as previously demonstrated (Levine & LaCourse, 1967; Sparks et al., 1969). The partially cleansed blood is continuously recycled to the patient by blood pumps or by the natural arterial to venous pressure difference. Neglecting ammonium ion inhibition and pH kinetic effects reduces eqn (8) to a newly-modified Michelis–Menten model which accounts for urease deactivation as follows: Rs=
Ae S j (S+Km,0+S 2/Ks )
(33)
where Ae=Vm,0 E is the enzyme activity and j is the enzyme deactivation factor which is empirically determined by eqn (7). Secondly, the urea production rate by body metabolism, G [in eqn (11)], was assumed to be zero. The membrane wall mass transfer coefficient ki,M , was obtained from Mogensen (1972) whereas the liquid film coefficient was calculated from the following (Wilson & Geankoplis, 1966): ki,L=1.09
Us1/3Di2/3 od p2/3
(34)
F. 2. Effect of enzyme activity on urea conversion for the various column length to diameter ratios.
Steady-state calculations were initially made across the reactor for better choice of enzyme activity and reactor dimensions. This was followed by transientstate calculations for reactor start-up operations and the body’s response to the treatment. For steady-state calculations, pH and enzyme activities are fixed, and a steady-state urea concentration within the reactor and the microcapsule are assumed for iteration using the method of collocations. The transient-state calculations were made with an initial assumption that the intracapsule and bulk urea concentrations are zero. The body blood pool was chosen to be 50 l with an initial urea concentration of 10 mM. The microcapsule diameter was chosen as 100 mm. An optimal design for a urease microencapsulated fixed bed reactor involves the proper choice of enzyme activity as a first step. The appropriate enzyme activity was selected by calculating the conversion after one pass through the reactor for a flow of 200 ml min−1 and a reactor volume of about 126 ml. In order to study residence time effects for a constant reactor volume, the length to diameter ratio was varied. The steady state conversion for the various length to diameter ratios was computed by assuming various enzyme activities. The results which are presented in Fig. 2 show that for a constant fixed bed reactor volume, the overall conversion is independent of the length to diameter ratio (residence time) but rather dependent on the enzyme activity. The conversion increased with an increase in enzyme activity from 0.1 to 10 mM sec−1 . A transient calculation was made to
300
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monitor the exit urea concentration with time at various enzyme activities as shown in Fig. 3. At activities greater than 5 mM sec−1 , the gain in conversion was only marginal. For the purpose of this study, an activity of 10 mM sec−1 was chosen as optimal on the basis of cost and level of conversion achieved after one pass through the reactor. At lower activities, conversions of less than 90% were achieved whereas at activities greater than 10 mM sec−1 , the cost of the enzyme could be very high with a maximum gain of only 6% in the conversion. The effect of initial blood urea concentration on the conversion (after one pass through the reactor) was calculated using an optimal enzyme activity of 10 mM sec−1 for various reactor height to diameter ratios, as presented in Fig. 4. The higher the substrate concentration, the lower the conversion. Blood urea concentration of 10 mM was used as a practical representation for computing the reactor exit urea concentration with time, for various length to diameter ratios. The results (shown in Fig. 5) reveal that there is only a marginal difference in the steady-state conversion values for reactor length to diameter ratios between 1 and 10. Beyond a ratio of 10, there is no gain in conversion. The proposed artificial kidney with blood recycle (shown in Fig. 1) was studied by modelling the complete process. It was assumed that the treatment process involved 50 l of body fluid which is completely mixed within the patient. The transients in the artificial
F. 3. A transient profile of the exit urea concentration with time at various enzyme activities. Column height (H)=10 cm; Column diameter (D)=4 cm.
F. 4. Effect of blood urea concentration on the conversion using an enzyme activity of 10 mM sec−1 for various reactor height to diameter ratios.
kidney (reactor) are coupled with those of the body (patient). The first stage in the modelling was to determine the required volume of reactor to reduce the body urea concentration to reasonable levels. This was achieved by calculating the blood urea level in the body with time for various reactor volumes. In the calculations, a reactor voidage of 0.5 and an optimal enzyme activity of 10 mM sec−1 were used. The results, which are presented in Fig. 6, show the body’s response (continuous lines)
F. 5. Reactor exit urea concentration profile with time for various reactor length to diameter ratios. (VT=126 ml, tL=18.8 sec).
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F. 6. The body’s response (continuous line) in comparison to the overall reactor performance (dashed line). The curves labelled 1 to 4 represent reactor volumes of 31.4 (2×10 cm), 62.8 (2×20 cm), 126 (4×10 cm) and 252 ml (4×20 cm), respectively.
F. 8. Effect of flow rate on the patient’s blood urea level (solid line) and the overall urea conversion (dashed line). Curves 1 to 4 represent volumetric flow rates of 100, 200, 300 and 400 ml min−1, respectively.
in comparison to the overall reactor performance (broken). The curves labelled 1–4 represent reactor volumes of 31.4 (2 cm×10 cm), 62.8 (2 cm×20 cm), 126 (4 cm×10 cm) and 252 ml (4 cm×20 cm), respectively. The results show that large reactor volumes, which are normally associated with large quantities of microencapsulated urease, cause substantial reductions in the body urea level. Nevertheless, the performance is little enhanced for reactor volumes exceeding 126 ml (curves labelled 3). Hence, under the conditions used for the calculations, a reactor volume of 126 ml (4 cm×10 cm) would be sufficient for optimal performance. For such a reactor configuration, the body urea level was reduced by about 60%
in 4 h and 80% in 8 h. Figure 7 shows a comparison of the profiles for the reactor’s overall and transient urea conversions at the various reactor volumes considered. The overall conversion in the reactor is defined as the percentage of the initial blood urea level which is hydrolysed, whereas the transient conversion is the percentage of the blood urea entering the reactor which is hydrolysed after one pass. The profiles in Fig. 7 show that the overall urea conversion increases with time—an indication that the body blood urea level is reducing. However, the transient urea conversion is a constant, which is dependent on the volume of reactor or quantity of microencapsulated urease. From the results shown in Figs 7 and 8, a
F. 7. Comparison of the profiles for the reactor’s overall (dashed line) and transient (solid line) urea conversions for the various reactor volumes considered. The definitions of the curves labelled 1–4 are the same as Fig. 6.
F. 9. The urea concentration profiles in a FBR at various times using a flow rate of 200 ml min−1, and a column dimension of 4×10 cm (H/D=2.5).
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reactor dimension of 4×10 cm was chosen as the optimum design and further studied. Using the optimum dimension of 4×10 cm, the effect of flow rate was studied by computing the blood urea level at various volumetric flow velocities. In the calculations, the velocity of blood flow in the recycle system was assumed to be constant. The variation with time of the patient’s blood urea level and the overall urea conversion at the various flow velocities are shown in Fig. 8. In this figure, curves 1–4 represent flow velocities of 100–400 ml min−1 respectively. The slower the flow, the higher the conversion (dashed lines) as a consequence of the associated long residence times. It was expected that the high conversions accompanying lower flow velocities would lead to lower blood urea levels in the patient. However, higher flow velocities result in lower blood urea levels with time, as shown in Fig. 8. At high flow rates, the residence time is short and thus the turnover of the reactor volume is large in comparison with that for lower flow velocities. For example, after 4 h, the blood urea level was about 5.5 mM at a conversion of 99% for a flow of 100 ml min−1 in comparison to 2 mM at 94% conversion for a flow of 400 ml min−1 . During this period, the reactor volume turnover is 190 for a flow of 100 ml min−1 in comparison to 762 for 400 ml min−1 flow rate. For a practical flow of 200 ml min−1 , the blood urea level could be reduced from 10 mM sec−1 to 4 mM and 2 mM in 4 and 8 h respectively. The urea concentration profiles in the fixed bed reactor at various times is shown in Fig. 9 for a sample calculation using a flow rate of 200 ml min−1 , and a reactor capacity of 4×10 cm. The profiles show a continuous decrease in the urea concentration along the column. The results have shown the ability of the model to simultaneously predict the reactor performance and patient’s blood urea level under various conditions. In principle, the model could be used for control and monitoring of a patient’s condition (e.g. Figs 7 to 9). - The reactor start-up and transients are very important for control and operational purposes. The transients in the fixed bed reactor were calculated using a flow rate of 200 ml min−1 and a reactor configuration of 4×10 cm. It was assumed that the reactor was initially void of urea. The urea concentration profiles along the reactor were calculated for two enzyme activities as shown in Figs 10(a) (1 mM sec−1 ) and 10(b) (10 mM sec−1 ). The profiles show the dynamic front of the blood at various times. For example in
F. 10. The urea concentration profiles along the column reactor for two enzyme activities: (a) Ae=1 mM sec−1; (b) 10 mM sec−1. Reactor dimensions: 4×10 cm (H/D=2.5).
Fig. 10(a) (1 mM sec−1 urease activity), the blood urea concentration is zero at a dimensionless column height of 0.3 after 4 sec. This is the incipience of blood into the column and therefore the top 70% is void of urea. The column eventually will experience a complete first pass of blood after 20 sec. Steady state was reached
F. 11. The urea concentration profiles along the column reactor for two enzyme activities: (a) Ae=1 mM sec−1; (b) 10 mM sec−1. Reactor dimensions: 2×10 cm (H/D=5).
after 60 sec, which represents three replacements of the reactor volume. Similarly, for an enzyme activity of 10 mM sec−1 [Fig. 10(b)], the first pass is after 20 sec although the urea concentration level within the column is lower than that of a 1 mM sec−1 activity. However, the steady state was reached only after approximately one pass in 24 sec. Similar results [Figs 11(a) and 11(b)] were obtained for a smaller column of dimensions 2×10 cm (half the optimal size). In these figures, the residence time after one pass is about 4.7 sec. Meanwhile, the steady state is reached after four replacements of the reactor for the 1 mM sec−1 [Fig. 11(a)] as compared to about 1.2 reactor volumes for the 10 mM sec−1 [Fig. 11(b)] enzyme activity. Hence, steady-state conditions are achieved quickly at high enzyme activities.
Conclusion A simulation for the prediction of a patient’s response to treatment using a microencapsulated urease artificial kidney system has been made, taking into account the deactivation of the urease. The model has the ability to predict the performance of the artificial kidney and the patient’s blood urea level simultaneously, at various operating conditions. The results showed that an optimal microencapsulated urease activity of 10 mM sec−1 and reactor height to diameter ratio (H/D) in the range of 1 to 10 are recommended for good performance. An optimal reactor configuration of 4×10 cm (H/D=2.5) was achieved in the simulations. There was an increase in the reduction of blood urea from low to high flow rates, although the reactor performance was better at low flow rates. This was due to the high turnover of the reactor volume at high flow.
REFERENCES B, A. J. & F, L. R. (1980). Electrochemical Methods: Fundamentals and Applications. New York: Wiley. C, T. M. S. (1964). Semipermeable microcapsules. Science 146, 524–525. C, T. M. S. (1965). Semipermeable aqueous microcapsules. PhD Thesis, McGill University, Montreal. C, T. M. S. (1978). Introduction. In: Artificial Kidney, Artificial Liver, and Artificial Cells (Chang, T. M. S., ed.), pp. 3–8. New York: Plenum Press.
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C, S. F. & W, C. Y. (1968). Longitudinal dispersion of liquid flowing through fixed and fluidized beds. AIChE J. 14, 857–866. C, D. O. (1976). Biomedical Engineering Principles: An Introduction to Fluid, Heat, and Mass Transport Processes, p. 301. New York: Marcel Dekker, Inc. D, E. & B, M. (1977). Dialysate regeneration. In: Artificial Organs (Kenedi, R. M., Courtney, J. M., Gaylor, J. D. S. & Gilchrist, T., eds). London: Macmillan. D, M. & W, E. C. (1964). Enzymes. New York: Academic Press. F, J. H. (1965). Blood and Body Functions. Philadelphia: F. A. Davis Company. F, B. A. (1980). Nonlinear Analysis in Chemical Engineering. New York: McGraw-Hill. G, A., G, T., D P, J. R., G, M. A., M, L. B. MA, M. J. & M, M. H. (1970). A sorbent-based low volume dialysate system: preliminary studies in human subjects. Proc. Eur. Dial. Transpl. Assoc. 7, 63. G, A., B, O. S., G, M. A., M, L. B., G, T., & M, M. H. (1971). Clinical maintenance hemodialysis with a sorbent-based, low-volume dialysate regeneration system. Trans. Am. Soc. Artif. Int. Organs 17, 253. K, W. J. & B, H. T. J. (1944). The artificial kidney: a dialyzer with a great area. Acta. Med. Scand. 117, 121–134. K, B., L, M. & Z, W. (1988). Immobilization of urease for dialysate regeneration system of artificial kidney. Post. Fiz. Med. 23(2), 115–130. L, K. B. (1993). Modelling of urea hydrolysis with microencapsulated urease, PhD Thesis, McGill University, Montreal. L, S. N. & LC, W. C. (1967). Materials and design consideration for a compact artificial kidney. J. Biomed. Mat. Res. 1, 275–284. M, B. B., B, M., S, P. & S, E. (1976). Toxicological problems with the REDY8 system. Scand. J. Urol. Nephrol. Suppl. 30, 23–27. M, A. O. (1972). Mass transfer and biochemical reaction with semipermeable microcapsules. PhD Thesis, Rutgers University, NewBrunswich, New Jersey. M, H. J., L, C. K., C, W. & W, N.-H. L. (1989). Urea hydrolysis by immobilized urease in a fixed-bed reactor: analysis and kinetic parameter estimation. Biotech. Bioengng 34, 951–963. N´, Y. (1990). Artificial kidney research, is it really not necessary? Artif. Organs 14(4), 245. P, F. & C, E. (1976). On acid-base problems in REDY8 dialysis. Scand. J. Urol. Nephrol. Suppl. 30, 28–31. S, U. R., K, C., S, R. & K, W. J. (1983). A portable hemodialysis/hemofiltration system independent of dialysate and infusion system. Artif. Organs 7, 254–256. S , J. (1976). On bacteriological problems in the REDY8 Dialysis system. Scand. J. Urol. Nephrol. Suppl. 30, 19–22. S, R. E., S, R. M., M, P. M., L, M. H. & L, O. (1969). Removal of waste metabolites in uremia by microencapsulated reactants. Trans. Am. Soc. Artif. Int. Organs, 15, 353–359. W, J. M., J, S. C., S, R. L., K, W. J. & R, D. (1977). The role of adsorbents in the wearable artificial kidney. In: Artificial Organs (Kenedi, R. M., Courtney, J. M., Gaylor, J. D. S. & Gilchrist, T., eds), pp. 139–147. London: Macmillan. W, E. J. & G, C. J. (1966). Liquid mass transfer at very low Reynold’s numbers in packed beds. Ind. Engng Chem. Fundam. 5, 9–14.
Blood Urea Clearance with Microencapsulated Urease K. B. L, D. K. B R. J. N
Department of Chemical Engineering, McGill University, 3480 University St., Montreal, Quebec, Canada H3A 2A7
(Received on 4 April 1994, Accepted in revised form on 8 February 1995)
The response of a kidney patient to treatment using a microencapsulated urease artificial kidney (MUAK) system was modelled. The model was used to simulate the patient’s response and reactor performance for an initial blood urea concentration of 10 mM and a MUAK void volume of 0.5. The performance of the reactor was strongly dependent on the enzyme activity. An optimal activity of 10 mM sec−1 was achieved in the analysis. After operation for 4 h at a flowrate of 200 ml min−1 , the reduction in blood urea concentration for reactor dimensions of 2×10 cm, 2×20 cm, 4×10 cm and 4×20 cm were 38%, 52%, 60% and 62%, respectively. The effect of flow rate on the performance of urea removal was studied using the reactor dimensions of 4×10 cm (optimal design). The results for flow rates of 100, 200, 300 and 400 ml min−1 predicted blood urea reductions of 38, 60, 70 and 76%, respectively. Although, the conversion of urea in the reactor decreased from 100% to 96% for the respective flows of 100 to 400 ml min−1 , the high turnover of reactor volume at a higher flow rate was responsible for the improved reduction of the patient’s blood urea level. The model has the ability to predict the performance of the MUAK and the patient’s blood urea level simultaneously, at various operating conditions. 7 1995 Academic Press Limited
Berk, 1944). Approximately half a million patients worldwide are being supported by hemodialysis (Nose´, 1990). Although hemodialysis has been effective for the treatment of kidney failure, there are a number of related problems. The conventional artificial kidney is bulky, heavy, complex and expensive, and difficult to handle, limiting the mobility of the patient (Chang, 1978; Krajewska et al., 1988). Thus, several attempts have been made to simplify the machine and reduce its size (Walker et al., 1977; Denti & Biagini, 1977; Shettigar et al., 1983). Reports by Gordon et al. (1970, 1971) led to the development of the commercially available REDY8 portable absorption dialysis system. However, bacteriological (So nderstrup, 1976), toxicological (Moller et al., 1976) and acid-base (Pederson & Christiansen, 1976) problems in the REDY8 system have necessitated improvements or further research into other alternatives. Investigations into the application of the artificial cell as a basis for the construction of an artificial kidney were initiated by Chang (1964). Urease was microencapsulated in ultra-thin non-toxic polymeric
Introduction The function of the human kidneys is to clear the blood of body wastes, processing up to 1.5 l of blood each minute (Cooney, 1976). Normal renal function includes the removal of the end products of metabolism such as urea, uric acid, creatine, ammonium, sulfates, and phenol; regulation of the body chemistry; regulation of the amount of water in the body to prevent edema and hypertension; and the removal of the substances that are not utilized or metabolized in the body, such as drugs and toxins, thus preventing renal poisoning. During malfunctioning of the kidneys, the patient becomes progressively ill: metabolic waste products, excess electrolytes and water accumulate and the chemical balance is upset and may lead to death within days. Intermittent treatment with a mechanical device such as the artificial kidney reduces the accumulation of waste products, normalizing the blood conditions. The conventional artificial kidney (hemodialyzer) system has been in clinical use since 1943 (Kolff & 0022–5193/95/150295+09 $12.00/0
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7 1995 Academic Press Limited
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membranes which could be used for dialysate regeneration or direct blood urea clearance purposes. The delay in the development of this new concept was due in part to the lack of appropriate theoretical models for the characterization of the microencapsulated system. Additional challenges associated with the direct hemoperfusion of microencapsulated urease include thromboembolism, platelet adhesion and the potential for microcapsule rupture releasing enzyme into the blood stream. One alternative that has been suggested is conventional hemodialysis with dialysate recirculated through a microencapsulated urease/ion exchange resin bed. The present analysis would be applicable by taking into account mass transfer across the membrane dialyser. In the present study, a model was developed to demonstrate the applicability of the microencapsulated urease artificial kidney (MUAK) system with time-varying blood urea composition. The main objective was to study the response of a patient to the operations of the MUAK system and to assess its operability at various enzyme activities. The influence of the reactor design on the patient’s response under various conditions was analysed for optimal design and control purposes.
systems in blood are (Ferguson, 1965): (i) bicarbonate/carbonic acid buffer, HCO− 3 + H2 CO3 especially in plasma; (ii) dihydrogen phosphate/dibasic phosphate 2− buffer, H2 PO− 4 +HPO4 , which is of minor importance in blood; and (iii) protein buffer to some extent in plasma. Among these, the bicarbonate buffer system is the most important for controlling blood pH. Thus, the appropriate ionization reactions for this model can be summarized as follows: + NH+ 4 F H +NH3(aq) ;
K1=5.60×10−10 (M) (2)
H2 CO3 F H++HCO− 3 ;
K2=7.94×10−7 (M) (3)
2− + HCO− 3 F H +CO3 ;
K3=6.31×10−12 (M) (4)
H2 O F H++OH−;
K4=1.0×10−14 (M)
NaHCO3 : Na++HCO− 3 ,
(5) (6)
where the Kj ’s ( j=1,..., 4) denote the ionization equilibrium constants for the respective reactions. Microencapsulated urease was reported to lose some activity with a half-life of one week at a body temperature of 37°C (Chang, 1965). The experimental data was used in the following empirical equation for enzyme deactivation.
Theory
j=1+c1 t+c2 t 2+c3 t 3+c4 t 4+c5 t 5,
The conceptual model is shown in Fig. 1 where a patient is directly connected to an extracorporeal shunt chamber (microencapsulated fixed bed reactor). The urea in the contaminated blood diffuses into the microcapsules and undergoes hydrolysis by urease to ammonia and carbon dioxide as follows:
where j is the activity coefficient which is the ratio of the reaction rate between deactivating urease and the fresh enzyme, t is the time in seconds and c1 to c5 are the empirical constants, with values of 2.97×10−6 , 1.84×10−11 , −7.15×10−17 , 1.33× 10−22 and −9.25×10−29 , respectively. The rate of substrate urea reaction, Rs (Moynihan et al., 1989), is therefore modified to account for urease deactivation as follows:
(NH2 )2 CO+H2 O F 2NH3+CO2
(1)
The blood pH (7.4) is considered to be homeostatically regulated in the body. The three chemical buffer
Rs =
Vm ES j, (S+Km+S 2/Ks )(1+P/Ki )
(7)
(8)
where E, S and P denote the concentration of urease (enzyme), urea (substrate) and ammonium ion (product) respectively, and Ks and Ki are the substrate and product inhibition constants. Vm and Km are the pH-dependent maximum reaction velocity and Michaelis constants, respectively which may be expressed as follows (Dixon & Webb, 1964):
F. 1. A schematic representation of a patient-FBR system.
Vm=Vm,o
1 1+Kes1 /H++H+/Kes2
(9)
Km=Km,o
1+Ke1 /H++H+/Ke2 , 1+Kes1 /H++H+/Kes2
(10)
where Vm,o and Km,o are the pH-independent maximum reaction velocity and Michaelis constants, respectively. The parameters Ke1 and Ke2 are the ionization equilibrium constants of the enzyme, whereas Kes1 and Kes2 are the equilibrium constants of the enzyme–substrate complex. The following assumptions were made in the development of mass conservation equations: (a) Mass transport through and within the semipermeable membrane takes place by passive molecular diffusion. (b) External transport takes place by passive molecular diffusion and convection. (c) The enzyme reaction rate is described by a modified Michaelis–Menten form, which accounts for the substrate and product inhibition, and pH-dependent kinetic constants. (d) No interactions exist among the enzyme, the diffusing solutes and the encapsulating membrane. (e) Enzyme is uniformly distributed in the enzyme phase of the microcapsule. (f) The patient’s blood system is assumed to be a single pool. A single compartment model for the patient (Fig. 1) is considered and a differential mass balance for the body fluid pool, which is assumed to be perfectly mixed, takes the form 1Cs,in QB = (CB,out−CB,in )+G, 1t VB
(11)
where VB represents the volume of body fluids, QB is the volumetric flow rate of blood, CB,out is the urea concentration of blood returning to the patient body, CB,in is the urea concentration of blood leaving the body, and G is the urea production rate by the body metabolism. The subscripts ‘‘in’’ and ‘‘out’’ refer to the reactor inlet (body outlet) and outlet (body inlet) concentrations. The medium in the reactor may consist primarily − of nine species (urea, NH+ 4 , NH3 , H2 CO3 , HCO3 , 2− + − + CO3 , H , OH , Na ). The required nine equations can be reduced to three by combining eqns (2)–(5) for the product species. Therefore, specific balance equations within the microcapsule can be expressed as follows: 1Cs,1 =−9 · Ns,1−Rs 1t
(12)
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1CNH+4,1 =fNH+4 {−9 · (NNH+4,1+NNH3 ,1 )+2Rs } (13) 1t 1CHCO−3,1 =fHCO−3 {−9 · (NH2 CO3 ,1+NHCO−3,1 1t +NCO2− )+Rs } 3 ,1
(14)
where fNH+4= fHCO−3=
H+ H +K1
(15)
+
1 H+ K3 1+ + + K2 H
and N is the molar flux. Rs is the rate of urea removal and subscript 1 denotes the enzyme phase. The concentration of Na+ species would be constant throughout the reactor since it is not involved in any of the reactions. At any given time, the concentrations of H+ and OH− species can be obtained from a balance of charges in the enzyme solution and in the bulk as follows: (CH+,1+CNH+4,1+CNa+,1 ) −(COH−,1+CHCO−3,1+2CCO2− )=0 3 ,1 (CH+,b+CNH+4,b+CNa+,b ) )=0, −(COH−,b+CHCO−3,b+2CCO2− 3 ,b respectively. Fickian diffusion is often used to describe the transport of species within the microcapsule. However, the molecular dynamics of diffusing ions may affect the molar flux of other species. Therefore, the Nernst– Planck equation (Bard et al., 1980) will be adopted as follows:
0
−Ni,1=Di,1 9Ci,1+Ci,1 zi
F9C Rg T
1
(19)
where D, F, Rg and T are the coefficient of diffusion, Faraday constant, molar gas constant, and temperature, respectively. C is the potential of a streaming charge zi , and Ci is the concentration of species i in the solution. The boundary conditions are: r=0; r=r1 ;
Ci,1=Ci,2
1Ci,1 =0 1r
(20)
−Ni,1=ki,O (Ci,b−Ci,r=r 1 ) (21)
where, r1 is the inner radius of curvature of the membrane and ki,O is the combined transfer coefficient of species i for the liquid-phase mass transfer and
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diffusion through the membrane, which is defined by 1 1 1 = + ki,O ki,L ki,M
(22) +
where, ki,L and ki,M are the liquid phase and membrane wall mass transfer coefficients of species i, respectively. The effectiveness factor h, which represents the influence of mass transfer on the overall reaction process, is defined for the present system by
h=
0 1b > 0 10 1
3 1S1 D r1 s,1 1r
2 b
S Ks
1+
(23)
Pb Ki
The overall material balance for the FBR will be presented to account for each species involved in the reaction system. Figure 1 shows the model fixed bed reactor of length L which is packed with microcapsules. The system is fed with blood at a superficial velocity Us and undergoes removal of urea by passage through the encapsulated packing in a reactor of total volume VT . Utilizing the concept of dispersion coefficient Dz to account for axial mixing, a mass balance on urea, − NH+ 4 , HCO3 , in the bulk liquid within the column yields: 1 1 2Cs,b 1 1Cs,b 3aks,O 1Cs,b − (Cs,b−Cs,r 1 ) = − 1t tr Bos 1Z 2 tr 1Z or2 (24) + NH4 ,b
+ NH4 ,b
3akNH+4 ,0 (CNH+4 ,b−CNH+4 ,r 1 ) or2
1 2CNH3 ,b 1 1CNH3 ,b 1 − tr 1Z tr BoNH3 1Z 2
+
3akNH3 ,0 (CNH3 ,b−CNH3 ,r 1 ) or2
6
7
1 2CHCO−3,b 1CHCO−3,b 1 =fHCO−3 − 1t 1Z 2 tr BoHCO3 1 1CHCO−3,b tr 1Z
−
+
1 2CCO2− 1 1 1CCO2− 3 ,b 3 ,b − 2 2− tr BoCO3 1Z tr 1Z
3akCO2− 3 ,0 (CCO2− −CCO2− ) 3 ,r 1 3 ,b or2
−
7
(26)
where, a is the fraction of reactor volume occupied by the microcapsules, t is the physical time for reaction, tr is the residence time, Z is the vertical coordinate direction in the axial length of the bed, and Bo is the Bodenstein number, and Us L Dz,i
(27)
Vc VT
(28)
Lo Us
(29)
Boi= a=
tr=
The boundary conditions for solving eqns (24)–(26) are given by . Z=0,
1 1Ci =Ci,o+−Ci,f Boi 1Z
(30)
1Ci =0 1Z
(31),
where the position Z=0 is chosen to be the bottom inlet port of the bed. The dispersion coefficient Dz is given by the following correlation (Chung & Wen, 1968): NRe omf Dz = (32) 0.48 0.20+0.011N Re rf
−
−
3akH2 CO3 ,0 (CH2 CO3 ,b−CH2 CO3 ,r 1 ) or2
. Z=1,
1C 1CNH+4,b 1 1 1C =fNH+4 − 1t tr BoNH+4 1Z 2 tr 1Z 2
1 2CH2 CO3 ,b 1 1CH2 CO3 ,b 1 − tr BoH2 CO3 1Z 2 tr 1Z
−
Vm ESb
r=r 1
Sb+Km+
6
3akHCO−3,0 (CHCO−3,b−CHCO−3,r1 ) or2
−
(25)
where NRe is the particle Reynolds number (dp Us nf /mf ); dp is the mean microcapsule diameter; Us the superficial velocity; mf , the dialysate viscosity; o, the bed void fraction, and rf is the dialysate density. The method of orthogonal collocation (Finlayson, 1980) was used to carry out the integration for both the intraparticle concentration profiles, Ci,1 (r) and the bulk liquid concentration profiles, Ci,b (Z). The material balance equations were solved using subroutines from
299
the International Mathematical and Statistical Library (IMSL). For steady-state cases, the DNEQNF (double precision nonlinear equations by finite differences) subroutine was used. This routine applies the Levenberg–Marquardt algorithm and a finite difference approximation to the Jacobian. Transient computations were also made using the Gear method from the DIVPAG (double precision initial value problem by Adam-Moulton and Gear) subroutine. Details of the numerical scheme have been discussed elsewhere (Lee, 1993).
Results and Discussion The operation of a microencapsulated urease artificial kidney for the treatment of a patient was simulated. Generally, eqns (11)–(14) and (24)–(26) must be solved simultaneously to obtain information on the intracapsule, bulk reactor and the body urea concentration profiles. Since the human blood contains a bicarbonate buffer system, the intracapsule and bulk charge balances given by eqns (17) and (18), respectively, will be required for a complete solution. Decoupled solutions could only be obtained with the aid of the equilibrium relations given by eqns (2)–(5) and the reaction rate equation (8) which accounts for the deactivation of the enzyme. Due to the complexity of the problem involved, certain assumptions were made to facilitate the computation without loss of details in the trends of the simulation. It was assumed that the urease is coencapsulated with ion exchange resins for ammonia removal and pH control as previously demonstrated (Levine & LaCourse, 1967; Sparks et al., 1969). The partially cleansed blood is continuously recycled to the patient by blood pumps or by the natural arterial to venous pressure difference. Neglecting ammonium ion inhibition and pH kinetic effects reduces eqn (8) to a newly-modified Michelis–Menten model which accounts for urease deactivation as follows: Rs=
Ae S j (S+Km,0+S 2/Ks )
(33)
where Ae=Vm,0 E is the enzyme activity and j is the enzyme deactivation factor which is empirically determined by eqn (7). Secondly, the urea production rate by body metabolism, G [in eqn (11)], was assumed to be zero. The membrane wall mass transfer coefficient ki,M , was obtained from Mogensen (1972) whereas the liquid film coefficient was calculated from the following (Wilson & Geankoplis, 1966): ki,L=1.09
Us1/3Di2/3 od p2/3
(34)
F. 2. Effect of enzyme activity on urea conversion for the various column length to diameter ratios.
Steady-state calculations were initially made across the reactor for better choice of enzyme activity and reactor dimensions. This was followed by transientstate calculations for reactor start-up operations and the body’s response to the treatment. For steady-state calculations, pH and enzyme activities are fixed, and a steady-state urea concentration within the reactor and the microcapsule are assumed for iteration using the method of collocations. The transient-state calculations were made with an initial assumption that the intracapsule and bulk urea concentrations are zero. The body blood pool was chosen to be 50 l with an initial urea concentration of 10 mM. The microcapsule diameter was chosen as 100 mm. An optimal design for a urease microencapsulated fixed bed reactor involves the proper choice of enzyme activity as a first step. The appropriate enzyme activity was selected by calculating the conversion after one pass through the reactor for a flow of 200 ml min−1 and a reactor volume of about 126 ml. In order to study residence time effects for a constant reactor volume, the length to diameter ratio was varied. The steady state conversion for the various length to diameter ratios was computed by assuming various enzyme activities. The results which are presented in Fig. 2 show that for a constant fixed bed reactor volume, the overall conversion is independent of the length to diameter ratio (residence time) but rather dependent on the enzyme activity. The conversion increased with an increase in enzyme activity from 0.1 to 10 mM sec−1 . A transient calculation was made to
300
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monitor the exit urea concentration with time at various enzyme activities as shown in Fig. 3. At activities greater than 5 mM sec−1 , the gain in conversion was only marginal. For the purpose of this study, an activity of 10 mM sec−1 was chosen as optimal on the basis of cost and level of conversion achieved after one pass through the reactor. At lower activities, conversions of less than 90% were achieved whereas at activities greater than 10 mM sec−1 , the cost of the enzyme could be very high with a maximum gain of only 6% in the conversion. The effect of initial blood urea concentration on the conversion (after one pass through the reactor) was calculated using an optimal enzyme activity of 10 mM sec−1 for various reactor height to diameter ratios, as presented in Fig. 4. The higher the substrate concentration, the lower the conversion. Blood urea concentration of 10 mM was used as a practical representation for computing the reactor exit urea concentration with time, for various length to diameter ratios. The results (shown in Fig. 5) reveal that there is only a marginal difference in the steady-state conversion values for reactor length to diameter ratios between 1 and 10. Beyond a ratio of 10, there is no gain in conversion. The proposed artificial kidney with blood recycle (shown in Fig. 1) was studied by modelling the complete process. It was assumed that the treatment process involved 50 l of body fluid which is completely mixed within the patient. The transients in the artificial
F. 3. A transient profile of the exit urea concentration with time at various enzyme activities. Column height (H)=10 cm; Column diameter (D)=4 cm.
F. 4. Effect of blood urea concentration on the conversion using an enzyme activity of 10 mM sec−1 for various reactor height to diameter ratios.
kidney (reactor) are coupled with those of the body (patient). The first stage in the modelling was to determine the required volume of reactor to reduce the body urea concentration to reasonable levels. This was achieved by calculating the blood urea level in the body with time for various reactor volumes. In the calculations, a reactor voidage of 0.5 and an optimal enzyme activity of 10 mM sec−1 were used. The results, which are presented in Fig. 6, show the body’s response (continuous lines)
F. 5. Reactor exit urea concentration profile with time for various reactor length to diameter ratios. (VT=126 ml, tL=18.8 sec).
301
F. 6. The body’s response (continuous line) in comparison to the overall reactor performance (dashed line). The curves labelled 1 to 4 represent reactor volumes of 31.4 (2×10 cm), 62.8 (2×20 cm), 126 (4×10 cm) and 252 ml (4×20 cm), respectively.
F. 8. Effect of flow rate on the patient’s blood urea level (solid line) and the overall urea conversion (dashed line). Curves 1 to 4 represent volumetric flow rates of 100, 200, 300 and 400 ml min−1, respectively.
in comparison to the overall reactor performance (broken). The curves labelled 1–4 represent reactor volumes of 31.4 (2 cm×10 cm), 62.8 (2 cm×20 cm), 126 (4 cm×10 cm) and 252 ml (4 cm×20 cm), respectively. The results show that large reactor volumes, which are normally associated with large quantities of microencapsulated urease, cause substantial reductions in the body urea level. Nevertheless, the performance is little enhanced for reactor volumes exceeding 126 ml (curves labelled 3). Hence, under the conditions used for the calculations, a reactor volume of 126 ml (4 cm×10 cm) would be sufficient for optimal performance. For such a reactor configuration, the body urea level was reduced by about 60%
in 4 h and 80% in 8 h. Figure 7 shows a comparison of the profiles for the reactor’s overall and transient urea conversions at the various reactor volumes considered. The overall conversion in the reactor is defined as the percentage of the initial blood urea level which is hydrolysed, whereas the transient conversion is the percentage of the blood urea entering the reactor which is hydrolysed after one pass. The profiles in Fig. 7 show that the overall urea conversion increases with time—an indication that the body blood urea level is reducing. However, the transient urea conversion is a constant, which is dependent on the volume of reactor or quantity of microencapsulated urease. From the results shown in Figs 7 and 8, a
F. 7. Comparison of the profiles for the reactor’s overall (dashed line) and transient (solid line) urea conversions for the various reactor volumes considered. The definitions of the curves labelled 1–4 are the same as Fig. 6.
F. 9. The urea concentration profiles in a FBR at various times using a flow rate of 200 ml min−1, and a column dimension of 4×10 cm (H/D=2.5).
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. . ET AL .
reactor dimension of 4×10 cm was chosen as the optimum design and further studied. Using the optimum dimension of 4×10 cm, the effect of flow rate was studied by computing the blood urea level at various volumetric flow velocities. In the calculations, the velocity of blood flow in the recycle system was assumed to be constant. The variation with time of the patient’s blood urea level and the overall urea conversion at the various flow velocities are shown in Fig. 8. In this figure, curves 1–4 represent flow velocities of 100–400 ml min−1 respectively. The slower the flow, the higher the conversion (dashed lines) as a consequence of the associated long residence times. It was expected that the high conversions accompanying lower flow velocities would lead to lower blood urea levels in the patient. However, higher flow velocities result in lower blood urea levels with time, as shown in Fig. 8. At high flow rates, the residence time is short and thus the turnover of the reactor volume is large in comparison with that for lower flow velocities. For example, after 4 h, the blood urea level was about 5.5 mM at a conversion of 99% for a flow of 100 ml min−1 in comparison to 2 mM at 94% conversion for a flow of 400 ml min−1 . During this period, the reactor volume turnover is 190 for a flow of 100 ml min−1 in comparison to 762 for 400 ml min−1 flow rate. For a practical flow of 200 ml min−1 , the blood urea level could be reduced from 10 mM sec−1 to 4 mM and 2 mM in 4 and 8 h respectively. The urea concentration profiles in the fixed bed reactor at various times is shown in Fig. 9 for a sample calculation using a flow rate of 200 ml min−1 , and a reactor capacity of 4×10 cm. The profiles show a continuous decrease in the urea concentration along the column. The results have shown the ability of the model to simultaneously predict the reactor performance and patient’s blood urea level under various conditions. In principle, the model could be used for control and monitoring of a patient’s condition (e.g. Figs 7 to 9). - The reactor start-up and transients are very important for control and operational purposes. The transients in the fixed bed reactor were calculated using a flow rate of 200 ml min−1 and a reactor configuration of 4×10 cm. It was assumed that the reactor was initially void of urea. The urea concentration profiles along the reactor were calculated for two enzyme activities as shown in Figs 10(a) (1 mM sec−1 ) and 10(b) (10 mM sec−1 ). The profiles show the dynamic front of the blood at various times. For example in
F. 10. The urea concentration profiles along the column reactor for two enzyme activities: (a) Ae=1 mM sec−1; (b) 10 mM sec−1. Reactor dimensions: 4×10 cm (H/D=2.5).
Fig. 10(a) (1 mM sec−1 urease activity), the blood urea concentration is zero at a dimensionless column height of 0.3 after 4 sec. This is the incipience of blood into the column and therefore the top 70% is void of urea. The column eventually will experience a complete first pass of blood after 20 sec. Steady state was reached
F. 11. The urea concentration profiles along the column reactor for two enzyme activities: (a) Ae=1 mM sec−1; (b) 10 mM sec−1. Reactor dimensions: 2×10 cm (H/D=5).
after 60 sec, which represents three replacements of the reactor volume. Similarly, for an enzyme activity of 10 mM sec−1 [Fig. 10(b)], the first pass is after 20 sec although the urea concentration level within the column is lower than that of a 1 mM sec−1 activity. However, the steady state was reached only after approximately one pass in 24 sec. Similar results [Figs 11(a) and 11(b)] were obtained for a smaller column of dimensions 2×10 cm (half the optimal size). In these figures, the residence time after one pass is about 4.7 sec. Meanwhile, the steady state is reached after four replacements of the reactor for the 1 mM sec−1 [Fig. 11(a)] as compared to about 1.2 reactor volumes for the 10 mM sec−1 [Fig. 11(b)] enzyme activity. Hence, steady-state conditions are achieved quickly at high enzyme activities.
Conclusion A simulation for the prediction of a patient’s response to treatment using a microencapsulated urease artificial kidney system has been made, taking into account the deactivation of the urease. The model has the ability to predict the performance of the artificial kidney and the patient’s blood urea level simultaneously, at various operating conditions. The results showed that an optimal microencapsulated urease activity of 10 mM sec−1 and reactor height to diameter ratio (H/D) in the range of 1 to 10 are recommended for good performance. An optimal reactor configuration of 4×10 cm (H/D=2.5) was achieved in the simulations. There was an increase in the reduction of blood urea from low to high flow rates, although the reactor performance was better at low flow rates. This was due to the high turnover of the reactor volume at high flow.
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