Pharmacokinetic Considerations in Development of a Bioartificial Liver

Abstract

We consider the pharmacokinetics of bioartificial livers (BALs) prepared using a hollow fibre module with respect to two key functions, detoxification and plasma protein supply, and present the results in a simple form. We then discuss the advantages and disadvantages of BAL therapy in comparison with the non-biological therapies of haemodialysis and plasma exchange.

Nitrogenous and other potentially toxic compounds, such as ammonia, mercaptans, short-chain fatty acids and γ-aminobutyric acid, are produced in the bowels and accumulate in the systemic blood because of impaired elimination by the ailing liver. Adrenal and gonadal steroids, including corticosteroids, estrogens, progestins and androgens, are biosynthesised, and high concentrations of these hormones become harmful. All these endogenously produced toxins require effective metabolism. In haemodialysis, toxins that permeate through the hollow fibre membrane are rapidly removed by the dialysate flow, and their concentrations decrease to almost zero. In a BAL bioreactor, the toxins are slowly metabolised by hepatocytes in the hollow fibres, and decreased to concentrations that are inversely proportional to the number of hepatocytes in the BAL, even after a long-term assist. It is difficult to rationalise the clinical usage of BAL systems containing small amounts of hepatocytes (70–100g) to remove the toxins.

Concentrations of plasma proteins in a patient after long-term BAL treatment are proportional to the number of hepatocytes in the device. BAL reactors prepared using porcine hepatocytes supply porcine proteins, not human proteins, to the recipient. Plasma exchange increases protein concentrations much more effectively than BAL as long as a sufficient amount of plasma is available.

The blood inflow rate to the liver is about 1500 mL/min in a normal adult. On the other hand, blood draw rates to a BAL system are restricted to the range of 100–300 mL/min. Toxins that are rapidly cleared by the liver (for example, the ammonia clearance of the normal human liver is several hundred mL/min) cannot be effectively eliminated from the systemic blood by BAL systems currently under clinical evaluation.

Hepatocytes are the only elements in a BAL reactor that can metabolise toxins and synthesise proteins, and thus BAL performance increases with the increasing number of hepatocytes in the bioreactor. The human liver weighs about 1500g and contains about 80% hepatocytes, i.e. about 1200g of hepatocytes. The blood flow through the liver is about 1500 mL/min in a normal adult. To effectively replace liver functions would require a BAL reactor containing the functional equivalent of several hundreds of grams of human hepatocytes with an extracorporeal perfusion rate of more than 1000 mL/min. At this point, only orthotopic liver transplantation can meet these criteria.

Appendices

6. Appendix I

It might be thought that the differential equation 1 and equation 2, which are used in the text to express the pharmacokinetics of toxins and proteins in a patient, are oversimplified. In the following section, we will discuss the validity of use of the two equations instead of using the more complicated differential equations derived from the more realistic model. The model flow path diagram for the plasma perfusion of a BAL bioreactor is shown in figure 2. The patient’s body is regarded as three compartments consisting of a well-mixed tissue compartment with volume V_t and concentration C_t, a well-mixed blood compartment with volume V_b and concentration C_b, and a well-mixed liver compartment with volume V_l and concentration C_l. Following the assumption employed by Patzer et al.,[20] mass transfer between the tissue and blood compartments is isotropic, and to be characterised by the intercompartment mass transfer parameter k_bt ≊ k_tb. The blood draw rate from the patient is expressed by Q_b.

Fig. 2

Schematic diagram of a plasma perfusion bioartificial liver device (BAL) system. C = concentration; CL = clearance; f = fraction of blood flow removed by plasma separator; k = intercompartment mass transfer parameter; Q = flow; V = volume; the subscripts a, b, l, t and r refer to BAL, blood, liver, tissue and reservoir, respectively.

A plasma separator is included in the BAL system. A fraction, f, of the blood flow is removed and sent to a fourth part, the plasma reservoir with volume V_r. The plasma reservoir volume includes that of the reservoir and the associated perfusion tubing and oxygenator. Plasma from the reservoir passes through the bioreactor where the species concentration is reduced from C_r to C_a, and then returns to the reservoir. Finally, an equivalent volume of plasma to that taken from the patient is recombined with the patient’s blood flow before being returned to the patient at a concentration C_p.

A material balance produces the differential equations (equation 35, equation 36, equation 37, equation 38 and equation 39):

where Q, V, C, R and CL are the circulating rates of blood or plasma, the volume of each of the parts, the toxin concentrations, the toxin production rates and toxin clearance, respectively, and subscripts t, b, l, r and a indicate tissue, blood and liver compartments, and the reservoir and the BAL cartridge, respectively, and f is the extraction fraction of the plasma separator. It is difficult to analytically solve these five simultaneous differential equations under certain initial conditions. We need not directly solve them to appreciate the important points for BAL performance. Some approximations were introduced to simplify these equations. Internal equilibration between the tissue and blood compartments is assumed to be relatively rapid, as indicated by the results in Patzer’s work.[20] The following equation was obtained by addition of equation 35 and equation 36 under the assumption, C_b ≈ C_t (equation 40):

For a healthy person without BAL, Q_b, Q_a and CL_a should be zero. Equation 37, equation 38, equation 39 and equation 40 are reduced to two simple differential equations expressed by (equation 41 and equation 42):

These two equations take the same form as equation 1 and equation 2 in the text.

Next, we deal with a patient with severe liver failure. The liver completely loses its functions, i.e. CL₁ = 0. The liver functions are partially replaced by a blood perfusion BAL. By addition of equation 35, equation 36 and equation 37, these equations are reduced to (equation 43):

Concentration differences between tissue, blood and the liver can be assumed to be small, as discussed above, and concentrations in these compartments can be represented by one parameter, C_b. Equation 43 is reduced to (equation 44):

When the bioreactor is perfused with whole blood, the extraction fraction of a plasma separator, f, should be 1. By taking f = 1 into account, equation 44 becomes (equation 45):

And equation 38 and equation 39 are reduced to (equation 46):

The blood reservoir, which is treated as a well-mixed vessel with a concentration C_r, is used to decouple the bioreactor perfusion blood circulation rate, Q_r, from the patient’s blood draw rate, Q_b. Thus, Q_r can be set much larger than Q_b. Thus, the concentration difference C_r in the reservoir and C_a in the bioreactor can be assumed to be small. Equation 45 is reduced to (equation 47):

where (V_r + V_a) is the volume of the BAL system including the reservoir and the reactor. The two equation 45 and equation 47 are differential equations to express the toxin concentrations of a patient with severe liver failure who was treated with a whole blood perfusion BAL. As is easily seen, the combination of equation 45 and equation 47 takes the same form as the combination of equation 1 and equation 2 in the text.

Finally, a patient treated with a plasma perfusion BAL will be dealt with. Equation 38 and equation 39 are reduced to (equation 48):

As discussed above, Q_r can be set much larger than Q_b, and indeed most BAL systems have been operated at Q_r much larger than Q_b. Thus, the concentration difference between C_r in the reservoir and C_a in the bioreactor can be assumed to be small. Equation 48 becomes (equation 49):

where (V_r + V_a) is the volume of the BAL system. The combination of equation 44 and equation 49 can express toxin elimination dynamics by a plasma perfusion BAL system. Even for a slightly complicated system such as a plasma perfusion BAL system, the pharmacokinetic equations expressing the toxin removal are the same form as equation 1 and equation 2 in the text. However, we should remind researchers of the fact that the flow rate used in the calculation is neither a blood draw rate nor a plasma circulation rate in the BAL system, but the plasma separation rate, fQ_b.

In summary, we conclude that the dynamics of toxin concentrations in a healthy person, a patient treated with a blood perfusion BAL and a patient treated with a plasma perfusion BAL are expressed by pharmacokinetic equations of the same form as equation 1 and equation 2 in the text.

7. Appendix II

An analytical solution to the two simultaneous differential equation 1 and equation 2 was presented in our previous paper.[14] Here, we would like to briefly show the equations in relation to the time course of concentration changes. The initial amount of toxins in the body are assumed to be D. The initial conditions are C₁(0) = D/V₁ and C₂(0) = 0. C₁(t) and C₂(t) are expressed as (equation 50 and equation 51):

Here α and β are given by (equation 52 and equation 53):

Equation 53 can be modified as follows (equation 54):

Equation 54 can be simplified using the Maclaurin expansion by taking Q > CL_a and V₁ >> V₂ into consideration (equation 55):

A simplified form of equation 9 is obtained by substituting equation 55 into β.