- Email: [email protected]
Importance of the extracorporeal circulation rate in a bioartificial liver
Materials Science and Engineering C 6 Ž1998. 235–243
Importance of the extracorporeal circulation rate in a bioartificial liver H. Iwata ) , Y.G. Park, Y. Ikada Research Institute for Frontier Medical Sciences, Kyoto UniÕersity, 53 Kawahara-cho, Shogoin, Sakyo-ku, Kyoto 606, Japan Received 3 July 1998
Abstract A bioartificial liver ŽBAL. is an extracorporeal medical device which incorporates living hepatocytes in a cartridge. Most efforts have been exerted to develop a BAL cartridge with high metabolic functions. However, little attention has been paid to a extracorporeal circulation rate. In this study, equations by which we can predict the operation conditions of the BAL system for the effective treatment of patients with severe liver failure were derived from pharmacokinetic considerations. Toxins of which intrinsic clearance in the healthy person is more than several hundred millilitersrminute cannot be effectively removed from the blood under the low extracorporeal circulation rate even though the BAL cartridge has high metabolic functions. We do not deny the importance of developing the BAL cartridge with high metabolic functions, but much attention should be also paid to the extracorporeal circulation rate for the effective treatment of the patient with severe liver failure. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Extracorporeal circulation rate; Bioartificial liver; Hepatocytes; Liver failure
1. Introduction A bioartificial liver ŽBAL., a medical device entrapping living hepatocytes in a cartridge, has been expected as one of the most promising approaches to treat patients with severe liver failure. Several research groups w1–3x have started clinical trials of their BAL devices. However, it is still unknown whether BAL devices can effectively support the liver functions of patients. Most efforts have been exerted to develop a BAL cartridge with high metabolic functions. Even though the BAL cartridge contains a large number of hepatocyte and thus actively metabolizes toxins and synthesizes biomolecules, it cannot support the liver functions of the patient with severe hepatic failure under an unsuitable operation. Much attention should be paid on how the BAL system is operated. Pharmacokinetics have been developed for studies on the mechanisms of drug absorption, distribution, and elimination and also of the kinetics of these processes w4x. In a previous study w5x, equations were derived from pharmacokinetic considerations for the quantitative evaluation of BAL metabolic functions. In addition, the drug concentration decay after its loading was analyzed using these equations to quantitatively evaluate the BAL metabolic
capacities and they were expressed to be easily compared with those of the healthy human liver. We believe that such pharmacokinetic considerations can also give a suitable basis to predict the operation conditions of the BAL system for the effective treatment of patients with severe liver failure. This study is aimed to demonstrate the importance of extracorporeal circulation rate.
2. Fundamental equations for detoxification A simple pharmacokinetic model depicted in Fig. 1 is employed in this study. The boxes represent a patient’s body, a natural liver, and a BAL cartridge. Toxin endogenously generated in the patient’s body is carried into the natural liver and the BAL cartridge by the blood at the concentration in the body and its some fraction is metabolized by hepatocytes and then returns to the body at reduced concentrations. Since clearance ŽCL. is defined as a volume of blood from which an entire chemical would be removed in an unit of time w4x, the rate of toxin metabolism is expressed by Removal rate of toxin from blood s w Clearancex = w drug concentration in the inflowx
)
Corresponding author. Tel.: q81-75-7514119; Fax: q81-757514144; E-mail: [email protected]
s CL = Cin
0928-4931r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 8 - 4 9 3 1 Ž 9 8 . 0 0 0 5 6 - 3
Ž 1.
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
236
blood delivers the drug to the liver and the BAL in which it is cleared. The extent of binding of the toxin to plasma proteins and blood cells determines how much of the drug is available in a form that can be cleared. Thus, the clearance value depends on the intrinsic metabolic capacity, the perfusate flow rate, and the binding of drug to plasma proteins and various cells. There are several models which can describe the dependence of the clearance on these factors. In this study, we employ the well-stirred model in which instantaneous and complete mixing occurs within the liver and the BAL, because it is simple and can depict clearly various features of dependencies on these factors without using complicate mathematics. This clearance can be expressed in the well-stirred model by Fig. 1. Schematic representation of the BAL assistance system.
CL s QCL int f Xr Ž Q q CL int f X . Changes of toxin concentrations, C b , C h and Ca in the body, the liver and the BAL, respectively, can be expressed as V b dC brd t s y Ž Q h q Qa . C b q Q h C h q Qa Ca q R V h dC hrdt s yQ h C h q Q h C b y CL h C b Va dCardt s yQa Ca q Qa C b y CL a C b
Ž 2. Ž 3. Ž 4.
where V is the volume of each compartment; Q is the blood circulating rate; CL is the clearance; R is the toxin generation rate in the body and subscripts b, h and a, represent the body, the liver and the BAL cartridge, respectively. When a patient has suffered with hepatic disease for a certain period, he or she is hospitalized. The toxin concentration in the body might be assumed to reach a high plateau level. The toxin concentration can also be expressed by Eqs. Ž2. and Ž3., but Qa should be zero, since the patient is not treated by the BAL. Under the steady state, the right side terms of Eqs. Ž2. and Ž3. become zero. The steady toxin concentrations, C b0 in the body and C h0 in the liver, before the BAL treatment starts can be expressed from these equations as C b0 s RrCL h C h0 s R Ž 1rCL h y 1rQ h .
Ž 5. Ž 6.
When the BAL treatment starts, the toxin is eliminated not only by their own liver, but also by the BAL. Changes of the toxin levels in the three compartments can be calculated using Eqs. Ž2. – Ž4. under the initial conditions expressed by Eqs. Ž5. and Ž6.. After treatment of the patient by the BAL system for a long period of time, the toxin concentration reaches a new plateau level. The right side terms of Eqs. Ž2. – Ž4. become zero in the steady state. The plateau toxin level in the body derived from these equations is given by C b` s Rr Ž CL h q CL a . .
Ž 7.
The clearance value varies with various factors. Blood flow rate influences elimination of toxins because the
Ž 8.
where CL int relates the intrinsic rate of metabolism to the unbound toxin concentration at the enzyme site, and f X is the fraction of unbound toxin in the blood w x x. Eq. Ž8. predicts that the CL increases with the increasing blood flow rate and approaches to a maximum value, that is, the intrinsic clearance ŽCL int .. The liver of the patient and the BAL cartridge have the relative metabolic capacity, h and a, to the liver of the healthy person. Thus, these intrinsic clearances are expressed by hCL int and aCL int , respectively, where CL int is the intrinsic clearance of the healthy liver. The toxin concentration of the patient, C b0 , relative to that of a healthy person, C H , can be expressed by C b0 rC H s Ž 1rQ h q 1rhCL int f X . Ž 1rQ h q 1rCL int f X .
Ž 9. under the assumption that the blood perfusion rate through the liver of the patient is the same as that of the healthy person. The toxin concentration in the patient decreases to a new plateau level during the BAL treatment. Eqs. Ž7. and Ž8. yield it as C b` rC H s Ž 1rQ h q 1rh)CL int f X . Ž 1rQa q1ra)CL int f X . Ž 1rQ h q 1rCL int f X . Ž 1rQ h q1rh)CL int f X . q Ž 1rQa q 1ra)CL int f X . 4 . Ž 10 . The other important parameter to characterize the BAL assistance is the toxin elimination rate. The analytical solution of Eqs. Ž2. – Ž4. gives complicated equations. The toxin concentration decay can be expressed by three exponential terms. Strictly, the elimination rate cannot be characterized by one constant rate. However, the concentration decay is roughly expressed by one exponential term as shown below. The concentration difference between the body and the liver is small when the patient’s liver has a
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
237
Fig. 2. Effect of the BAL assistance on toxin concentration. Relative toxin concentrations to that of the healthy person, CrC H , are plotted against the perfusion time. Ža. Intrinsic clearance in the healthy person for the toxin, CL int s 1000 mlrmin. Žb. Intrinsic clearance in the healthy person for the toxin, CL int s 50 mlrmin. Ratio of intrinsic clearance of the BAL to that of the healthy liver, a s 0.2. Ratio of the intrinsic clearance of the disordered liver to that of the healthy liver, h s 0.1. Volume of the toxin distribution in the body, V b s 10,000 ml; Volume of the toxin distribution in the liver, Vh s 1000 ml. Volume of the toxin distribution in the BAL, Va s 200 ml. Blood perfusion rate through the liver, Q h s 1000 mlrmin. Blood perfusion rate through the BAL, Q h s 200 mlrmin.
small metabolic capacity. Under such circumstances, the differential Eqs. Ž2. – Ž4. can be reduced to
where C2 and C3 are constants, and ay and aq are given below.
Ž V b q Vh . dC brd t s yQa C b q qQa Ca q R y CL h C b , Ž 11 .
aq, ays aCL intV1 q QaV1 q hCL intV2 q QaV2 " Sqrt
Va dCardt s yQa Ca q Qa C b y CL a C b .
Ž 12 .
½
= y4CL int Ž ahCL int q aQa q hQa . V1V2
These differential equations are analytically solved. The toxin concentration in the body can be expressed by:
q Ž aCL intV1 q QaV1 q hCL intV2
C b s C b` q C2 exp Ž yayt . q C3 exp Ž yaqt .
qQaV2 .
Ž 13 .
2 1r2
r Ž 2V1V2 .
Ž 14 .
238
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
where aq is larger than ay. The toxin concentration in the body follows the early rapid and subsequently slower decline characterized by aq and ay, respectively, and then reaches a plateau level, C b` . The initial rapid decline
is mainly determined by dilution of the toxin just after the BAL treatment starts. The later slow decay, that is, the rate limiting step of the toxin level decline, is related to the metabolic elimination of the toxin. The volume of the BAL
Fig. 3. Dependence of the toxin plateau level, C b` rC H , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. Ža. Three-dimensional plot. Intrinsic clearance in the healthy person for the toxin, CL int s 1000 mlrmin. Ratio of the intrinsic clearance of the BAL to that of the healthy liver, a s 0.2. Ratio of the intrinsic clearance of the disordered liver to that of the healthy liver, h s 0.1. Blood perfusion rate through the liver, Q h s 1000 mlrmin. Blood perfusion rate through the BAL, Q h s 200 mlrmin. Žb. Plot of C b` rC H against the extracorporeal flow rate. Intrinsic clearance in the healthy person for the toxin, CL int s 1000 mlrmin. Ratio of the intrinsic clearance of the disordered liver to that of the healthy liver, h s 0.1. Blood perfusion rate through the liver, Q h s 1000 mlrmin.
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
device is much less than the toxin distribution volume in the body in the most of the BAL systems developed. The toxin dilution does not contribute much to the concentration decline process. The effect of the BAL assistance is roughly expressed by an exponential term of the latter process, expŽyay t .. The ay in the exponent gives information about the decline rate, but the elimination half-life, t 1r2 , which is the time for toxin concentration to fall by one-half of the concentration difference between the initial and the final plateau level, is more convenient to imagine
239
the toxin elimination rate. The half-life Ž t 1r2 . for this term is given by: t 1r2 s 0.693ray Ž 15 .
3. Results and discussion Various kinds of toxins which are endogenously generated are eliminated by the healthy liver at different rates. These are accumulated in the body of the patient suffering
Fig. 4. Dependence of the toxin plateau level, C b` rC H , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. All parameters used, except CL int s 50 mlrmin, are same as those in Fig. 3. Ža. Three-dimensional plot. Intrinsic clearance in the healthy person for the toxin. Žb. Plot of C b` rC H against the extracorporeal flow rate.
240
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
from hepatic failure. When the patient is treated by the BAL, the initial high toxin level gradually decreases. The toxin concentration change can be numerically calculated using Eqs. Ž2. – Ž4.. Fig. 2 shows representative calculations of the concentration changes in the body for two toxins of which intrinsic clearances are distinctively different. These figures also contain the toxin concentration changes in the patient’s liver and the BAL, and a stationary level which was reached after the infinite BAL assis-
tance. The toxin of which intrinsic clearance in the healthy person is 1000 mlrmin is rapidly eliminated and approaches to the plateau level during 3 h of BAL assistance. However, the new plateau level is 2.6 times higher than the steady toxin level in the healthy person. After the BAL assistance stops, the toxin level rapidly recovers to the preoperative high level during the next several hours. The toxin, of which intrinsic clearance in the healthy person is small, is slowly eliminated by the BAL assistance and thus
Fig. 5. Dependence of the elimination half-life, t1r 2 , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. All parameters used are same as those in Fig. 3. Ža. Three-dimensional plot. Žb. Plot of t1r 2 against the extracorporeal flow rate.
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
it takes so long to approach the new plateau level as shown in Fig. 2b. The toxin concentration decline is determined by the metabolic capacity of the BAL and the extracorporeal circulation rate. These dependencies can be evaluated from concentration decline curves generated by numerically solving Eqs. Ž2. – Ž4. for various combinations of the metabolic capacities and the circulation rates. However, it is tedious work to draw a number of the decline curves and it is difficult to clearly elucidate these dependencies from
241
them. The effects of the BAL assistance on the toxin level can be mainly characterized by two parameters, the plateau level Ž C b` . which the toxin concentration approaches during long BAL assistance and the half-life Ž t 1r2 . which represents the time for toxin concentration to fall by one-half of the concentration difference between the initial high level and the final plateau level. In the following, the plateau levels and the half-lives under various conditions are calculated using Eqs. Ž10. and Ž15., respectively, to clearly show the effects of the metabolic capacity of the
Fig. 6. Dependence of the elimination half-life, t1r 2 , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. All parameters used, except CL int s 50 mlrmin, are same as those in Fig. 3. Ža. Three-dimensional plot. Žb. Plot of t1r2 against the extracorporeal flow rate.
242
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
BAL and the extracorporeal circulation rate on the toxin concentration decline. The ratio of the plateau level to the toxin level in a healthy person, C b` rC H , was calculated using Eq. Ž10.. Fig. 3a shows three-dimensional plot of the C b` rC H against the extracorporeal flow rate and the ratio of the metabolic capacity of the BAL to that of the healthy liver for a toxin of which intrinsic clearance in the healthy person is 1000 mlrmin. In Fig. 3a, the maximum extracorporeal flow rate is to 200 mlrmin, since most of the BAL systems have been operated at the extracorporeal circulation rate of less than 200 mlrmin to minimize disturbance of the systemic circulation of the patient w1–3x. The C b` rC H does not greatly depend on the metabolic capacity of the BAL until it decreases to approximately one fourth of that of the healthy liver at any extracorporeal flow rate. On the other hand, it highly depends on the flow rate in the flow region, where the BAL is expected to be operated. Its dependence on the flow rate is extracted from Fig. 3a and shown in Fig. 3b for the BALs with different metabolic capacities. The BAL assistance cannot decrease the toxin level in the patient to that in the healthy person, even though the BAL has the same metabolic capacity as the healthy natural liver under such low extracorporeal circulation rates. Furthermore, the plateau level achieved by the BAL with one fifth of metabolic capacity of the healthy natural liver is not so much different with that achieved by the BAL with the same metabolic capacity as the healthy natural liver. The plateau toxin level for the toxin which is rapidly removed by the natural liver is mainly determined by the extracorporeal flow rate, but not the elimination capacity of the BAL. A plasmapheresis circuit is included in some of the BAL systems reported w1x. For this system, the plasma exchange rate which is less than 50 mlrmin should be used instead of the extracorporeal circulation rate to calculate the C b` rC H Žsee Appendix A.. Under such a low flow rate, the toxin cannot be effectively removed even though the BAL with a high metabolic capacity is used. The toxin removal is restricted by the plasma filtration rate in that system. Fig. 4 shows dependence of the C b` rC H on the extracorporeal flow rate and the metabolic capacity of the BAL for a toxin of which intrinsic clearance in the healthy person is 50 mlrmin. The C b` rC H sharply decreases with the increasing flow rate up to 50 mlrmin and then slowly decreases to the plateau level. The C b` rC H does not depend on the extracorporeal flow rate under the conditions, where the BAL is expected to be operated, but depends on the elimination capacity of the BAL as shown in Fig. 4. The C b` rC H of the toxin which is slowly eliminated by the healthy liver is mainly determined by the elimination capacity of the BAL, but not the extracorporeal circulation rate. The other important parameter to characterize the BAL assistance is the elimination rate. Strictly speaking, the toxin concentration does not decrease monoexponentially
during the BAL assistance. However, the toxin concentration decline is roughly expressed by the third term of the right side of Eq. Ž15., since the rate of the toxin concentration decline is mainly determined by the elimination rate by the BAL system as discussed above. The elimination half-life, t 1r2 , was calculated using Eqs. Ž14. and Ž15. was plotted against the extracorporeal flow rate and the ratio of the metabolic capacity of the BAL to that of the healthy liver in Figs. 5 and 6. The dependence of the t 1r2 on these parameters is similar to that of the C b` rC H . For the toxin of which intrinsic clearance in the healthy person is 1000 mlrmin, the t 1r2 highly depends on the extracorporeal flow rate, but not the metabolic capacity of the BAL. On the other hand, the t 1r2 of the toxin which is slowly eliminated by the healthy liver is mainly determined by the metabolic capacity of the BAL, but not the extracorporeal circulation rate. Unfortunately, it is yet unknown what kinds of toxins should be removed by the BAL. Some toxins of which intrinsic clearance in the healthy person is more than several hundred millilitersrminute cannot be effectively removed from the blood under the low extracorporeal circulation rate even though the BAL cartridge has high metabolic functions. We do not deny the importance of developing the BAL cartridge with high metabolic functions, but much attention should be also paid to the extracorporeal circulation rate for the effective treatment of the patient with severe liver failure.
Acknowledgements This work was supported in part by a grant under Research for the Future Program 96I00203 from the Japan Society for the Promotion of Science.
Appendix A When a plasmapheresis cartridge is included in the BAL system Scheme 1, changes of toxin concentrations can be expressed as V b dC brd t s y Ž Q h q Q p . C b q Q h C h q Qa Ca q Ž Q p y Qa . Cp2 q R ,
Ž A-1.
V h dC hrdt s yQ h C h q Q h C b y CL h C b ,
Ž A-2.
Vp dCprdt s yQa Cp1 y Ž Q p y Qa . Cp2 q Q p C b ,
Ž A-3.
Va dCardt s yQa Ca q Qa Cp1 y CL a Cp1
Ž A-4.
where Cp1 and Cp2 are the toxin concentrations in the plasma and the concentrated blood separated by the plasmapheresis respectively. Since the plasmapheresis car-
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
243
plasmapheresis cartridge is included in the BAL system, the same pharmacokinetic equations derived for the BAL system without it can be used to describe the toxin concentration changes. Attention is solely paid to the fact that Qa is not the extracorporeal circulation rate, but is the plasma filtration rate, in the case where the plasmapheresis cartridge is included in the BAL system.
References
Scheme 1. Schematic representation of the bioartificial liver assistance system including a plasmapheresis.
tridge does not eliminate toxin, The dCprd t in Eq. ŽA-3. should be zero. It gives yQa Cp1 q Q p C b s Ž Q p y Qa . Cp2 .
Ž A-5.
Using this relation, Eq. ŽA-1. gives V b dC brd t s y Ž Q h q Qa . C b q Q h C h q Qa Cp1 q R.
Ž A-6. Thus, Eqs. ŽA-1., ŽA-2., ŽA-3. and ŽA-4. are reduced to Eqs. Ž2. – Ž4., if the Cp1 is replaced by C b . Even though the
w1x F.D. Watanabe, C.J. Mullon, W.R. Hewitt, N. Arkadopoulos, E. Kahaku, S. Eguchi, T. Khalili, W. Arnaout, C.R. Shackleton, J. Rozga, B. Solomon, A.A. Demetriou, Clinical experience with a bioartificial liver in the treatment of severe liver failure: a phase I clinical trial, Ann. Surg. 225 Ž5. Ž1997. 484–491. w2x N.L. Sussman, G.T. Gislason, C.A. Conlin, J.H. Kelly, The hepatic extracorporeal liver assist device: initial clinical experience, Artificial Organs 18 Ž5. Ž1994. 390–396. w3x J.C. Gerlach, Organotypical in vitro model for 3-D culture of hepatocytes and hepatic nonparenchymal cells—a bioreactor construction for liver support systems, Presented in The Second International Symposium of Tissue Engineering for Therapeutic Use held at Tokyo, Japan, Oct. 30–31, 1997. w4x M. Rowland, T.N. Tozer, Clinical Pharmacokinetics: Concepts and Application, 3rd edn., Williams & Wilkins, Media, PA, USA, 1995. w5x H. Iwata, T. Sajiki, H. Maeda, Y.-G. Park, B. Zhu, S. Satoh, T. Uesugi, I. Ikai, Y. Yamaoka, Y. Ikada, In vitro evaluation of metabolic functions of a bioartificial liver, ASAIO J. ŽSubmitted..
Importance of the extracorporeal circulation rate in a bioartificial liver H. Iwata ) , Y.G. Park, Y. Ikada Research Institute for Frontier Medical Sciences, Kyoto UniÕersity, 53 Kawahara-cho, Shogoin, Sakyo-ku, Kyoto 606, Japan Received 3 July 1998
Abstract A bioartificial liver ŽBAL. is an extracorporeal medical device which incorporates living hepatocytes in a cartridge. Most efforts have been exerted to develop a BAL cartridge with high metabolic functions. However, little attention has been paid to a extracorporeal circulation rate. In this study, equations by which we can predict the operation conditions of the BAL system for the effective treatment of patients with severe liver failure were derived from pharmacokinetic considerations. Toxins of which intrinsic clearance in the healthy person is more than several hundred millilitersrminute cannot be effectively removed from the blood under the low extracorporeal circulation rate even though the BAL cartridge has high metabolic functions. We do not deny the importance of developing the BAL cartridge with high metabolic functions, but much attention should be also paid to the extracorporeal circulation rate for the effective treatment of the patient with severe liver failure. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Extracorporeal circulation rate; Bioartificial liver; Hepatocytes; Liver failure
1. Introduction A bioartificial liver ŽBAL., a medical device entrapping living hepatocytes in a cartridge, has been expected as one of the most promising approaches to treat patients with severe liver failure. Several research groups w1–3x have started clinical trials of their BAL devices. However, it is still unknown whether BAL devices can effectively support the liver functions of patients. Most efforts have been exerted to develop a BAL cartridge with high metabolic functions. Even though the BAL cartridge contains a large number of hepatocyte and thus actively metabolizes toxins and synthesizes biomolecules, it cannot support the liver functions of the patient with severe hepatic failure under an unsuitable operation. Much attention should be paid on how the BAL system is operated. Pharmacokinetics have been developed for studies on the mechanisms of drug absorption, distribution, and elimination and also of the kinetics of these processes w4x. In a previous study w5x, equations were derived from pharmacokinetic considerations for the quantitative evaluation of BAL metabolic functions. In addition, the drug concentration decay after its loading was analyzed using these equations to quantitatively evaluate the BAL metabolic
capacities and they were expressed to be easily compared with those of the healthy human liver. We believe that such pharmacokinetic considerations can also give a suitable basis to predict the operation conditions of the BAL system for the effective treatment of patients with severe liver failure. This study is aimed to demonstrate the importance of extracorporeal circulation rate.
2. Fundamental equations for detoxification A simple pharmacokinetic model depicted in Fig. 1 is employed in this study. The boxes represent a patient’s body, a natural liver, and a BAL cartridge. Toxin endogenously generated in the patient’s body is carried into the natural liver and the BAL cartridge by the blood at the concentration in the body and its some fraction is metabolized by hepatocytes and then returns to the body at reduced concentrations. Since clearance ŽCL. is defined as a volume of blood from which an entire chemical would be removed in an unit of time w4x, the rate of toxin metabolism is expressed by Removal rate of toxin from blood s w Clearancex = w drug concentration in the inflowx
)
Corresponding author. Tel.: q81-75-7514119; Fax: q81-757514144; E-mail: [email protected]
s CL = Cin
0928-4931r98r$ - see front matter q 1998 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 8 - 4 9 3 1 Ž 9 8 . 0 0 0 5 6 - 3
Ž 1.
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
236
blood delivers the drug to the liver and the BAL in which it is cleared. The extent of binding of the toxin to plasma proteins and blood cells determines how much of the drug is available in a form that can be cleared. Thus, the clearance value depends on the intrinsic metabolic capacity, the perfusate flow rate, and the binding of drug to plasma proteins and various cells. There are several models which can describe the dependence of the clearance on these factors. In this study, we employ the well-stirred model in which instantaneous and complete mixing occurs within the liver and the BAL, because it is simple and can depict clearly various features of dependencies on these factors without using complicate mathematics. This clearance can be expressed in the well-stirred model by Fig. 1. Schematic representation of the BAL assistance system.
CL s QCL int f Xr Ž Q q CL int f X . Changes of toxin concentrations, C b , C h and Ca in the body, the liver and the BAL, respectively, can be expressed as V b dC brd t s y Ž Q h q Qa . C b q Q h C h q Qa Ca q R V h dC hrdt s yQ h C h q Q h C b y CL h C b Va dCardt s yQa Ca q Qa C b y CL a C b
Ž 2. Ž 3. Ž 4.
where V is the volume of each compartment; Q is the blood circulating rate; CL is the clearance; R is the toxin generation rate in the body and subscripts b, h and a, represent the body, the liver and the BAL cartridge, respectively. When a patient has suffered with hepatic disease for a certain period, he or she is hospitalized. The toxin concentration in the body might be assumed to reach a high plateau level. The toxin concentration can also be expressed by Eqs. Ž2. and Ž3., but Qa should be zero, since the patient is not treated by the BAL. Under the steady state, the right side terms of Eqs. Ž2. and Ž3. become zero. The steady toxin concentrations, C b0 in the body and C h0 in the liver, before the BAL treatment starts can be expressed from these equations as C b0 s RrCL h C h0 s R Ž 1rCL h y 1rQ h .
Ž 5. Ž 6.
When the BAL treatment starts, the toxin is eliminated not only by their own liver, but also by the BAL. Changes of the toxin levels in the three compartments can be calculated using Eqs. Ž2. – Ž4. under the initial conditions expressed by Eqs. Ž5. and Ž6.. After treatment of the patient by the BAL system for a long period of time, the toxin concentration reaches a new plateau level. The right side terms of Eqs. Ž2. – Ž4. become zero in the steady state. The plateau toxin level in the body derived from these equations is given by C b` s Rr Ž CL h q CL a . .
Ž 7.
The clearance value varies with various factors. Blood flow rate influences elimination of toxins because the
Ž 8.
where CL int relates the intrinsic rate of metabolism to the unbound toxin concentration at the enzyme site, and f X is the fraction of unbound toxin in the blood w x x. Eq. Ž8. predicts that the CL increases with the increasing blood flow rate and approaches to a maximum value, that is, the intrinsic clearance ŽCL int .. The liver of the patient and the BAL cartridge have the relative metabolic capacity, h and a, to the liver of the healthy person. Thus, these intrinsic clearances are expressed by hCL int and aCL int , respectively, where CL int is the intrinsic clearance of the healthy liver. The toxin concentration of the patient, C b0 , relative to that of a healthy person, C H , can be expressed by C b0 rC H s Ž 1rQ h q 1rhCL int f X . Ž 1rQ h q 1rCL int f X .
Ž 9. under the assumption that the blood perfusion rate through the liver of the patient is the same as that of the healthy person. The toxin concentration in the patient decreases to a new plateau level during the BAL treatment. Eqs. Ž7. and Ž8. yield it as C b` rC H s Ž 1rQ h q 1rh)CL int f X . Ž 1rQa q1ra)CL int f X . Ž 1rQ h q 1rCL int f X . Ž 1rQ h q1rh)CL int f X . q Ž 1rQa q 1ra)CL int f X . 4 . Ž 10 . The other important parameter to characterize the BAL assistance is the toxin elimination rate. The analytical solution of Eqs. Ž2. – Ž4. gives complicated equations. The toxin concentration decay can be expressed by three exponential terms. Strictly, the elimination rate cannot be characterized by one constant rate. However, the concentration decay is roughly expressed by one exponential term as shown below. The concentration difference between the body and the liver is small when the patient’s liver has a
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
237
Fig. 2. Effect of the BAL assistance on toxin concentration. Relative toxin concentrations to that of the healthy person, CrC H , are plotted against the perfusion time. Ža. Intrinsic clearance in the healthy person for the toxin, CL int s 1000 mlrmin. Žb. Intrinsic clearance in the healthy person for the toxin, CL int s 50 mlrmin. Ratio of intrinsic clearance of the BAL to that of the healthy liver, a s 0.2. Ratio of the intrinsic clearance of the disordered liver to that of the healthy liver, h s 0.1. Volume of the toxin distribution in the body, V b s 10,000 ml; Volume of the toxin distribution in the liver, Vh s 1000 ml. Volume of the toxin distribution in the BAL, Va s 200 ml. Blood perfusion rate through the liver, Q h s 1000 mlrmin. Blood perfusion rate through the BAL, Q h s 200 mlrmin.
small metabolic capacity. Under such circumstances, the differential Eqs. Ž2. – Ž4. can be reduced to
where C2 and C3 are constants, and ay and aq are given below.
Ž V b q Vh . dC brd t s yQa C b q qQa Ca q R y CL h C b , Ž 11 .
aq, ays aCL intV1 q QaV1 q hCL intV2 q QaV2 " Sqrt
Va dCardt s yQa Ca q Qa C b y CL a C b .
Ž 12 .
½
= y4CL int Ž ahCL int q aQa q hQa . V1V2
These differential equations are analytically solved. The toxin concentration in the body can be expressed by:
q Ž aCL intV1 q QaV1 q hCL intV2
C b s C b` q C2 exp Ž yayt . q C3 exp Ž yaqt .
qQaV2 .
Ž 13 .
2 1r2
r Ž 2V1V2 .
Ž 14 .
238
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
where aq is larger than ay. The toxin concentration in the body follows the early rapid and subsequently slower decline characterized by aq and ay, respectively, and then reaches a plateau level, C b` . The initial rapid decline
is mainly determined by dilution of the toxin just after the BAL treatment starts. The later slow decay, that is, the rate limiting step of the toxin level decline, is related to the metabolic elimination of the toxin. The volume of the BAL
Fig. 3. Dependence of the toxin plateau level, C b` rC H , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. Ža. Three-dimensional plot. Intrinsic clearance in the healthy person for the toxin, CL int s 1000 mlrmin. Ratio of the intrinsic clearance of the BAL to that of the healthy liver, a s 0.2. Ratio of the intrinsic clearance of the disordered liver to that of the healthy liver, h s 0.1. Blood perfusion rate through the liver, Q h s 1000 mlrmin. Blood perfusion rate through the BAL, Q h s 200 mlrmin. Žb. Plot of C b` rC H against the extracorporeal flow rate. Intrinsic clearance in the healthy person for the toxin, CL int s 1000 mlrmin. Ratio of the intrinsic clearance of the disordered liver to that of the healthy liver, h s 0.1. Blood perfusion rate through the liver, Q h s 1000 mlrmin.
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
device is much less than the toxin distribution volume in the body in the most of the BAL systems developed. The toxin dilution does not contribute much to the concentration decline process. The effect of the BAL assistance is roughly expressed by an exponential term of the latter process, expŽyay t .. The ay in the exponent gives information about the decline rate, but the elimination half-life, t 1r2 , which is the time for toxin concentration to fall by one-half of the concentration difference between the initial and the final plateau level, is more convenient to imagine
239
the toxin elimination rate. The half-life Ž t 1r2 . for this term is given by: t 1r2 s 0.693ray Ž 15 .
3. Results and discussion Various kinds of toxins which are endogenously generated are eliminated by the healthy liver at different rates. These are accumulated in the body of the patient suffering
Fig. 4. Dependence of the toxin plateau level, C b` rC H , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. All parameters used, except CL int s 50 mlrmin, are same as those in Fig. 3. Ža. Three-dimensional plot. Intrinsic clearance in the healthy person for the toxin. Žb. Plot of C b` rC H against the extracorporeal flow rate.
240
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
from hepatic failure. When the patient is treated by the BAL, the initial high toxin level gradually decreases. The toxin concentration change can be numerically calculated using Eqs. Ž2. – Ž4.. Fig. 2 shows representative calculations of the concentration changes in the body for two toxins of which intrinsic clearances are distinctively different. These figures also contain the toxin concentration changes in the patient’s liver and the BAL, and a stationary level which was reached after the infinite BAL assis-
tance. The toxin of which intrinsic clearance in the healthy person is 1000 mlrmin is rapidly eliminated and approaches to the plateau level during 3 h of BAL assistance. However, the new plateau level is 2.6 times higher than the steady toxin level in the healthy person. After the BAL assistance stops, the toxin level rapidly recovers to the preoperative high level during the next several hours. The toxin, of which intrinsic clearance in the healthy person is small, is slowly eliminated by the BAL assistance and thus
Fig. 5. Dependence of the elimination half-life, t1r 2 , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. All parameters used are same as those in Fig. 3. Ža. Three-dimensional plot. Žb. Plot of t1r 2 against the extracorporeal flow rate.
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
it takes so long to approach the new plateau level as shown in Fig. 2b. The toxin concentration decline is determined by the metabolic capacity of the BAL and the extracorporeal circulation rate. These dependencies can be evaluated from concentration decline curves generated by numerically solving Eqs. Ž2. – Ž4. for various combinations of the metabolic capacities and the circulation rates. However, it is tedious work to draw a number of the decline curves and it is difficult to clearly elucidate these dependencies from
241
them. The effects of the BAL assistance on the toxin level can be mainly characterized by two parameters, the plateau level Ž C b` . which the toxin concentration approaches during long BAL assistance and the half-life Ž t 1r2 . which represents the time for toxin concentration to fall by one-half of the concentration difference between the initial high level and the final plateau level. In the following, the plateau levels and the half-lives under various conditions are calculated using Eqs. Ž10. and Ž15., respectively, to clearly show the effects of the metabolic capacity of the
Fig. 6. Dependence of the elimination half-life, t1r 2 , on the extracorporeal flow rate and the relative metabolic capacity of the BAL. All parameters used, except CL int s 50 mlrmin, are same as those in Fig. 3. Ža. Three-dimensional plot. Žb. Plot of t1r2 against the extracorporeal flow rate.
242
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
BAL and the extracorporeal circulation rate on the toxin concentration decline. The ratio of the plateau level to the toxin level in a healthy person, C b` rC H , was calculated using Eq. Ž10.. Fig. 3a shows three-dimensional plot of the C b` rC H against the extracorporeal flow rate and the ratio of the metabolic capacity of the BAL to that of the healthy liver for a toxin of which intrinsic clearance in the healthy person is 1000 mlrmin. In Fig. 3a, the maximum extracorporeal flow rate is to 200 mlrmin, since most of the BAL systems have been operated at the extracorporeal circulation rate of less than 200 mlrmin to minimize disturbance of the systemic circulation of the patient w1–3x. The C b` rC H does not greatly depend on the metabolic capacity of the BAL until it decreases to approximately one fourth of that of the healthy liver at any extracorporeal flow rate. On the other hand, it highly depends on the flow rate in the flow region, where the BAL is expected to be operated. Its dependence on the flow rate is extracted from Fig. 3a and shown in Fig. 3b for the BALs with different metabolic capacities. The BAL assistance cannot decrease the toxin level in the patient to that in the healthy person, even though the BAL has the same metabolic capacity as the healthy natural liver under such low extracorporeal circulation rates. Furthermore, the plateau level achieved by the BAL with one fifth of metabolic capacity of the healthy natural liver is not so much different with that achieved by the BAL with the same metabolic capacity as the healthy natural liver. The plateau toxin level for the toxin which is rapidly removed by the natural liver is mainly determined by the extracorporeal flow rate, but not the elimination capacity of the BAL. A plasmapheresis circuit is included in some of the BAL systems reported w1x. For this system, the plasma exchange rate which is less than 50 mlrmin should be used instead of the extracorporeal circulation rate to calculate the C b` rC H Žsee Appendix A.. Under such a low flow rate, the toxin cannot be effectively removed even though the BAL with a high metabolic capacity is used. The toxin removal is restricted by the plasma filtration rate in that system. Fig. 4 shows dependence of the C b` rC H on the extracorporeal flow rate and the metabolic capacity of the BAL for a toxin of which intrinsic clearance in the healthy person is 50 mlrmin. The C b` rC H sharply decreases with the increasing flow rate up to 50 mlrmin and then slowly decreases to the plateau level. The C b` rC H does not depend on the extracorporeal flow rate under the conditions, where the BAL is expected to be operated, but depends on the elimination capacity of the BAL as shown in Fig. 4. The C b` rC H of the toxin which is slowly eliminated by the healthy liver is mainly determined by the elimination capacity of the BAL, but not the extracorporeal circulation rate. The other important parameter to characterize the BAL assistance is the elimination rate. Strictly speaking, the toxin concentration does not decrease monoexponentially
during the BAL assistance. However, the toxin concentration decline is roughly expressed by the third term of the right side of Eq. Ž15., since the rate of the toxin concentration decline is mainly determined by the elimination rate by the BAL system as discussed above. The elimination half-life, t 1r2 , was calculated using Eqs. Ž14. and Ž15. was plotted against the extracorporeal flow rate and the ratio of the metabolic capacity of the BAL to that of the healthy liver in Figs. 5 and 6. The dependence of the t 1r2 on these parameters is similar to that of the C b` rC H . For the toxin of which intrinsic clearance in the healthy person is 1000 mlrmin, the t 1r2 highly depends on the extracorporeal flow rate, but not the metabolic capacity of the BAL. On the other hand, the t 1r2 of the toxin which is slowly eliminated by the healthy liver is mainly determined by the metabolic capacity of the BAL, but not the extracorporeal circulation rate. Unfortunately, it is yet unknown what kinds of toxins should be removed by the BAL. Some toxins of which intrinsic clearance in the healthy person is more than several hundred millilitersrminute cannot be effectively removed from the blood under the low extracorporeal circulation rate even though the BAL cartridge has high metabolic functions. We do not deny the importance of developing the BAL cartridge with high metabolic functions, but much attention should be also paid to the extracorporeal circulation rate for the effective treatment of the patient with severe liver failure.
Acknowledgements This work was supported in part by a grant under Research for the Future Program 96I00203 from the Japan Society for the Promotion of Science.
Appendix A When a plasmapheresis cartridge is included in the BAL system Scheme 1, changes of toxin concentrations can be expressed as V b dC brd t s y Ž Q h q Q p . C b q Q h C h q Qa Ca q Ž Q p y Qa . Cp2 q R ,
Ž A-1.
V h dC hrdt s yQ h C h q Q h C b y CL h C b ,
Ž A-2.
Vp dCprdt s yQa Cp1 y Ž Q p y Qa . Cp2 q Q p C b ,
Ž A-3.
Va dCardt s yQa Ca q Qa Cp1 y CL a Cp1
Ž A-4.
where Cp1 and Cp2 are the toxin concentrations in the plasma and the concentrated blood separated by the plasmapheresis respectively. Since the plasmapheresis car-
H. Iwata et al.r Materials Science and Engineering C 6 (1998) 235–243
243
plasmapheresis cartridge is included in the BAL system, the same pharmacokinetic equations derived for the BAL system without it can be used to describe the toxin concentration changes. Attention is solely paid to the fact that Qa is not the extracorporeal circulation rate, but is the plasma filtration rate, in the case where the plasmapheresis cartridge is included in the BAL system.
References
Scheme 1. Schematic representation of the bioartificial liver assistance system including a plasmapheresis.
tridge does not eliminate toxin, The dCprd t in Eq. ŽA-3. should be zero. It gives yQa Cp1 q Q p C b s Ž Q p y Qa . Cp2 .
Ž A-5.
Using this relation, Eq. ŽA-1. gives V b dC brd t s y Ž Q h q Qa . C b q Q h C h q Qa Cp1 q R.
Ž A-6. Thus, Eqs. ŽA-1., ŽA-2., ŽA-3. and ŽA-4. are reduced to Eqs. Ž2. – Ž4., if the Cp1 is replaced by C b . Even though the
w1x F.D. Watanabe, C.J. Mullon, W.R. Hewitt, N. Arkadopoulos, E. Kahaku, S. Eguchi, T. Khalili, W. Arnaout, C.R. Shackleton, J. Rozga, B. Solomon, A.A. Demetriou, Clinical experience with a bioartificial liver in the treatment of severe liver failure: a phase I clinical trial, Ann. Surg. 225 Ž5. Ž1997. 484–491. w2x N.L. Sussman, G.T. Gislason, C.A. Conlin, J.H. Kelly, The hepatic extracorporeal liver assist device: initial clinical experience, Artificial Organs 18 Ž5. Ž1994. 390–396. w3x J.C. Gerlach, Organotypical in vitro model for 3-D culture of hepatocytes and hepatic nonparenchymal cells—a bioreactor construction for liver support systems, Presented in The Second International Symposium of Tissue Engineering for Therapeutic Use held at Tokyo, Japan, Oct. 30–31, 1997. w4x M. Rowland, T.N. Tozer, Clinical Pharmacokinetics: Concepts and Application, 3rd edn., Williams & Wilkins, Media, PA, USA, 1995. w5x H. Iwata, T. Sajiki, H. Maeda, Y.-G. Park, B. Zhu, S. Satoh, T. Uesugi, I. Ikai, Y. Yamaoka, Y. Ikada, In vitro evaluation of metabolic functions of a bioartificial liver, ASAIO J. ŽSubmitted..