www.elsevier.com/locate/procbio

Process Biochemistry 40 (2005) 2763–2770

Control of a continuous bioreactor using model predictive control

S. Ramaswamy a, T.J. Cutright b, H.K. Qammar a,*

a Department of Chemical Engineering, The University of Akron, Akron, OH 44325-3906, USAb Department of Civil Engineering, The University of Akron, Akron, OH 44325-3906, USA

Received 22 April 2004; accepted 9 December 2004

Abstract

Bioreactor control has become very important in recent years due to the difficulty in controlling the highly non-linear behaviour associated

with such systems. Model predictive control (MPC) was used to control a non-linear continuous stirred tank bioreactor to an unstable steady

state, which was the desired set point. The effect of varying predictor horizon, an important MPC controller tuning parameter, was studied. As

the predictor horizon increased, the trajectory moved away from steady state towards a period attractor. The relationship between the changing

manipulated input and the system behaviour for different initial conditions as a result of the non-linearities associated with the bioreactor

dynamics was also analyzed. The results indicated that the controlled bioreactor can exhibit non-intuitive non-linear behaviour and even a

well-designed control scheme may result in very poor performance.

# 2005 Elsevier Ltd. All rights reserved.

Keywords: Model predictive control; Bioreactor; Non-linear; P horizon

1. Introduction

Previously, the control of bioreactors was restricted to the

regulation of variables, such as pH and temperature, for

optimizing the microbial growth [1]. These were the ideal

variables to control since they often have negligible

perturbations. However, variables subject to large fluctua-

tions, such as substrate regulation can be just as important

for growth optimization [2,3]. For instance, too much

substrate can be toxic while too little can force an early

stationary phase or the onset of endogenous decay (i.e.,

death by starvation). The level of substrate required will

depend on the consumption rate (a function of the species

growth rate) and on the desired end product. For example, if

a rhamnolipid biosurfactant produced from Pseudomonas

auerigenosa is desired, the system must be maintained in the

stationary phase. Therefore the key is to be able to select and

control the optimal substrate set point based on the desired

biomass concentration [4–6]. Another approach may need a

specific respiratory quotient to be maintained [7]. Both cases

require the successful use of a controller. For systems where

* Corresponding author. Tel.: +1 330 972 5917; fax: +1 330 972 5856.

E-mail address: hqammar@uakron.edu (H.K. Qammar).

0032-9592/$ – see front matter # 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.procbio.2004.12.019

large biomass production is desired, a controller to enable

continuous operation with a minimal number of washout

cycles (i.e., dilution rates) would be required.

The control of bioreactors based on substrate concentra-

tion has become an interesting problem from both

implementation and controller design points of view. This

is particularly true if the complex microbial interactions

yield significant non-linear behaviour. When this occurs,

conventional control strategies may be ineffective and more

specialized strategies need to be suggested on a case-by-case

basis. Often the reasons why a controller works and the role

of the non-linear dynamics are not fully explained. Previous

studies reported various types of controller designs

including internal model control (IMC) and adaptive

controllers, which can lead to stable control [8–12]. While

these control techniques may be successful for open loop

stable processes or in the vicinity of an unstable operating

point, about which a linearization is applied, they are often

inadequate for highly unstable non-linear bioreactors.

Unstable non-linear bioreactors are commonly encountered

in fed-batch systems or processes that involve a change in

gain [7,13,14]. In this paper, an example is provided showing

how non-linear effects may result in poor control

performance causing unwanted washout or batch operation.

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–27702764

There have been previous investigators of non-linear aspects

of bioreactor dynamics. Ajbar and Gamal [15] studied the

stability and bifurcation mechanisms of a non-linear

bioreactor with cell recycle. From their mathematical

analysis, they reported a range of system parameter values

that caused instability, i.e., washout. These results show the

importance of understanding the types of non-linear

behaviour possible for a given operating range. Tsao and

Wu [12] considered the dynamics for a controlled bioreactor

system. They formulated a multivariable adaptive control

algorithm on a continuous stirred tank bioreactor to control it

to a steady state within a limit cycle. They found that by

imposing bounds on the substrate flow rate and substrate

feed concentration, washout could be prevented. While these

studies emphasize the importance of avoiding washout, they

do not explain how bioreactor non-linearities lead to such

inefficient control performance.

In this paper, we have applied model predictive control

(MPC) to a model of a non-linear continuous fermentor. The

system originally developed by Agrawal et al. [16] has been

selected as a benchmark control problem. MPC has been

implemented on several difficult non-linear systems [17–19]

and is a good candidate for implementing control on a non-

linear bioreactor. The effect of an MPC control parameter,

predictor horizon has been investigated and the range that

yields stable control was determined. The predicator horizon

is used to whatever time in the future the model prediction is

required. (An in-depth review of control theory is not within

the scope of this manuscript. Additional information

pertaining to the basic concepts in model predictive control

is presented and illustrated in Bequette’s control textbook

[20].) The effect of the predictor horizon on the stability

from the underlying non-linear behaviour of the controlled

system has been explained. The MPC controlled system can

exhibit both ‘batch reactor’ mode, and washout and the

operating regions for these have been described. Knowledge

pertaining to the operating regions is critical for successful

economical control. As will be shown in this paper, the

control of a benchmark system, prevention of batch mode

and minimization of washout occurs only in a narrow

operating region.

Fig. 1. Simplified schematic of a continuous bioreactor.

2. Bioreactor control

The bioreactor model used in this paper was originally

studied by Agarwal who performed a bifurcation analysis

and determined regimes of periodic oscillations. Periodic,

sustained oscillations are a common undesirable problem

that occurs during the production of baker’s yeast [7,21].

Brengel and Seider [22] have also applied MPC to control

oscillating systems but used a linearized model. Ungar [23]

selected this as a benchmark control problem because this

system is difficult to control due to non-linearities such as

input–output multiplicities and limit cycles associated with

its behaviour.

Basically a bioreactor is a tank in which several

biological reactions occur simultaneously in a liquid

medium. In the simplest reactor two components are

considered, biomass and substrate. Biomass consists of cells

that consume the substrate. Consider the schematic of a

biochemical reactor in Fig. 1, where X and S are the cell and

substrate concentrations, F is the feed flow rate of substrate

(assuming no biomass in feed) and V the volume of the

reactor.

The continuous time representation of the bioreactor

system is:

dXdt

¼ � F

VX þ mðSÞX (1)

dS F

dt¼

VðSF � SÞ � sðSÞX (2)

where S is the substrate concentration and X is the cell

concentration. The growth model where all of the biomass

cells are ‘‘lumped together’’ in a single (one-hump) model

mðSÞ ¼ kSe�S=K (3)

is used to represent the substrate growth rate as a function of

biomass with K as the growth model coefficient. The specific

substrate consumption rate, s, is related to the yield coeffi-

cient Y in the form

YðSÞ ¼ mðSÞsðSÞ ¼ a þ bS (4)

where a and b are positive constants used to approximate the

increasing portion of the yield curve. Combining Eqs. (1)–

(4), the dimensionless cell and substrate mass balances can

be represented as

dC1

dt¼ �C1 þ Dað1 � C2Þec2=gC1 (5)

dC2 1 þ b c =g

dt¼ �C2 þ Da

1 þ b� C2ð1 � C2Þe 2 C1 (6)

where C1 and C2 are the dimensionless cell mass and

substrate conversion, respectively.

Brengel and Seider transformed Eqs. (5) and (6) to yield

the following system of equations incorporating the

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–2770 2765

manipulated input variable, u which is the dimensionless

feed flow rate.

dX1

dtS

¼�ðC1S þ X1Þð1 þ uÞþ DaSð1� X2� C2SÞeðc2Sþx2Þ=g

� ðC1S þ X1Þ (7)

dX2 ðc2Sþx2Þ=g

dtS

¼�ðC2S þ X2Þð1þ uÞ þDaSð1 �X2 �C2SÞe

� 1 þ b

1 þ b� X2 � C2SðC1S þ X1Þ (8)

where tS represents dimensionless time under steady state

conditions DaS, the Damkohler number at the steady state,

X1 the dimensionless cell mass deviation from steady state

and X2 is the corresponding dimensionless substrate con-

version deviation.

Fig. 2(a) is an open loop bifurcation diagram showing the

steady states and periodic branches over a range of values.

For low values of u, it can be seen that there are two steady

states, the upper state being stable while the lower one is

unstable. At u = 0.0778, the system undergoes a Hopf

bifurcation (i.e., limit cycle formation as a parameter is

varied [24]) resulting in three steady states, an upper repellor

encircled by a periodic attractor, a middle saddle and a lower

unstable state that becomes stable at the bifurcation point

u = 0.3023. An attractor is defined as some form of a long-

term recurrent behaviour that a dissipative dynamic system

will settle to [25]. By this definition, a periodic attractor is an

attractor that has a sin or cosin shape. Conversely, a repellor

is a solution that will go toward an unstable value [20,25].

Fig. 2. Open loop bifurcation diagram of (a) steady states and (b) substrate

conversions of the steady states.

Fig. 2(b) gives the substrate conversion for each of these

states. It was decided to use MPC to control the system to a

full repellor having values X1 = 0.05 and X2 = �0.078 at

u = 0.3846 with both eigenvalues 0.2126 � 2.0835i having

positive real parts. Analysis of the root locus (i.e., plot used

to represent the roots of the characteristic equation)

illustrates that conventional proportional-integral control

cannot be used to stabilize this unstable set point [26].

3. Model predictive control

Non-linear MPC is most naturally formulated in discrete

time and it becomes necessary to discretize the continuous

time differential equations. The discrete time representation

of the non-linear bioreactor model is given as

Xðk þ 1Þ ¼ f ½xðkÞ; uðkÞ; yðkÞ ¼ h½xðkÞ (9)

where x is an n-dimensional vector of state variables, u is an

m-dimensional vector of manipulated input variables, y is a

p-dimensional vector of controlled output variables, and f

and h are functions of x and U. The discretization was

performed using the Runge–Kutta four algorithm for numer-

ical integration [27]. In this case, X1 and X2 are the state

variables, the dimensionless flow rate is the input variable

and the output vector equals the state vector, i.e., x = y.

MPC is based on computing a set of M control moves that

minimize an objective function, based on a desired output

trajectory over a predictor horizon and implementing only

the first control move. This is schematically represented in

Fig. 3. The objective function normally is the sum of the

squares of the residuals between the model predicted outputs

and the set point values over the predictor horizon of P time

steps.

The MPC control algorithm is mathematically repre-

sented [28] as

minuðkjkÞ;uðkþ1jkÞ;...;uðkþM�1jkÞ

J

¼ f½yðk þ PjkÞ þXP�1

j¼0

L½yðk þ jjkÞ; uðk þ jjkÞ;Du

�ðk þ jjkÞ (10)

subject to certain process constraints where u(k + jjk) is the

input u(k + j) calculated from information available at time k,

y(k + jjk) is the output y(k + j) calculated from information

available at time k, Du(k + jjk = u(k + jjk) � u(k + j � 1jk),

M is the control horizon, P is the predictor horizon,f and L and

are non-linear functions of their arguments. It is meaningful to

consider quadratic functions of the form:

L ¼ ½yðk þ jjkÞ � ySðkÞTQ½yðk þ jjkÞ � ySðkÞ

þ ½uðk þ jÞ � uSðkÞTR½uðk þ jjkÞ � uSðkÞ

þ DuTðk þ jjkÞSDuðk þ jjkÞ (11)

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–27702766

Fig. 3. Schematic representation of MPC control algorithm.

f ¼ ½yðk þ PjkÞ � ySðkÞTQ½yðk þ PjkÞ � ySðkÞ (12)

where uS(k) and yS(k) are steady state targets for u and y,

respectively and Q, R, and S are positive definite weighting

matrices. The controller tuning parameters are M, P, Q, R, S

and the sampling period Dt.

Based on a trial and error procedure, the predictor horizon

P was chosen to be 15 and the control horizon M was 1. The

sampling time Dt was 0.01 and the weights were Q = 10,

R = 0 and S = l. The process constraint was a lower bound of

�1 on the manipulated input u. A sequential solution

technique was used to implement the control algorithm

comprising of solution of the ordinary differential equations

as an inner loop to evaluate the objective function with an

optimization code as the outer loop.

Fig. 4(a)–(c) shows the time series response after

implementing MPC on the bioreactor from one initial

condition. It can be seen that the system has been

successfully controlled to the specified set point.

A number of tuning parameters such as control and

predictor horizons, weight matrices and sampling interval

affect controller performance and closed loop stability. In

this discussion, the effect of P on the stability of the closed

loop system was studied. For non-linear MPC, it is generally

known that predictor horizons shorter than a critical value

produce unstable control, typically oscillatory behaviour

[29]. In Fig. 5, we see this effect for predictor horizon values

less than 4. Small values of the predictor horizon greater

than the critical value result in less aggressive control

performance. Fig. 6 shows X1 versus u for one such predictor

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–2770 2767

Fig. 4. Closed loop MPC response for (a) X1; (b) X2; and (c) u.

Fig. 5. Closed loop trajectory of the deviation of the dimensionless cell

mass from steady state, X1, using predictor horizon (P) = 3.

Fig. 6. Deviation of the dimensionless cell mass from steady state, X1, vs.

the manipulated input variable, u when the predictor horizon (P) = 5.

horizon, P. Values of between 10 and 20 gave excellent

controller performance (data not shown) and will not be

further discussed. However, for values of P greater than 20,

the controller performance degraded gradually with increas-

ing predictor horizons resulting in sluggish and oscillatory

responses until the closed loop system became unstable for

P = 149. Fig. 7 shows X2 versus X1 for P = 200. It can be

seen that the large value of the predictor horizon results in

the trajectory moving away from the steady state towards a

periodic attractor. Meadows and Rawlings [29] stated that

the effect of increasing predictor horizon diminishes as the

horizon becomes large, and the advantages of longer

horizons are outweighed by an increase in computational

time. However, we can see that for our system, large

predictor horizons result in loss of closed loop stability. The

loss of stability could be attributed to the fact that since the

operating point has both eigenvalues with positive real parts,

the effect of the repelling manifolds of the operating point

becomes significant when the values of P approach the time

constant of the open loop system. For bioreactor systems,

which normally have time constants that are substantially

larger than other chemical systems, the time constant of the

system becomes important in tuning the predictor horizon.

Increasing horizons result in Hopf-bifurcation-like closed

loop behaviour where a stable point becomes unstable

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–27702768

Fig. 7. Predictor horizon causing trajectories to move toward a periodic

attractor. In this example, P = 200.

Fig. 9. Minimum values of u along closed loop basin of initial conditions in

X1 and X2.

creating stable limit cycles. An optimum value of P was

taken to be 15, as described earlier.

The MPC controller was tested on a basin of initial

conditions in order to analyze the effect of the changing

manipulated input flow rate on the system response. Fig. 8

shows the points in the basin that reach a lower bound on the

input u = �1 during the closed loop simulation. For these

initial conditions at u = �1, the MPC control algorithm

switches off the input to the reactor resulting in increasing

cell mass concentrations and substrate conversions while

u = �1, and the controlled system acts as an intermittent

batch reactor. Figs. 9 and 10 show a basin of feasible initial

conditions against the minimum and maximum values of the

manipulated input u along the closed loop trajectory. In

Fig. 10, it can be seen that the maximum u is very large for

certain initial conditions, the input flow rate increasing

significantly during the closed loop response, resulting in a

certain ‘washout’ condition. It can be inferred from these

figures that the controlled bioreactor is not directly driven to

Fig. 8. Basin of initial conditions that reach u = �l along a closed loop

trajectory.

the required set point. The non-linearities associated with the

system result in inefficient control causing batch or washout

conditions along the controlled system response. This is

confounded by the fact that most bioreactor, although

assumed to be homogeneous actually exhibit large inhomo-

geneities [3,30]. These results are also seen in plots of X1 and

X2 versus u for a number of initial conditions (Fig. 11(a)–(c)).

The results contained in Fig. 11 provide three key

findings. As shown in Fig. 11(a) with initial conditions low

in cell mass, the system acts like a batch reactor with the

controlled input turning off completely initially along the

closed loop trajectory. When the initial condition has high

cell mass and substrate conversion (Fig. 11(b)), the value of

the manipulated input becomes very large along the

trajectory resulting in ‘washout’ (i.e., completely opening

the valve). The washout occurs since the high conversion

rate results in low substrate concentrations that cannot

sustain the large cell mass. As depicted in Fig. 11(c), for

initial conditions that are high in cell mass but low in

Fig. 10. Maximum values of u along closed loop basin of initial conditions

in X1 and X2.

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–2770 2769

Fig. 11. Effect of the manipulated input, u, on the dynamics of a controlled

system with (a) initial conditions low in cell mass; (b) initial conditions of

high cell mass and high substrate conversion; and (c) initial conditions of

high cell mass and low substrate conversion.

Fig. 12. Difference between the maximum and minimum values of u along

the closed loop trajectory for the basin of initial conditions in X1 and X2.

Trough depicts region of initial conditions that will result in stable control.

substrate conversion the value of the manipulated input

initially increases in order to sustain the high cell mass but

then decreases substantially. This considerable decrease is

due to the high concentrations of substrate already present,

that would be enough to sustain large cell concentrations.

Effective controllers requiring minimum input are

particularly advantageous for systems metabolite production

via microbes may be inhibited by catabolite repression or

inhibition from either the substrate or end product [31]. A

plot of difference in the maximum and minimum values of u

versus the initial conditions (Fig. 12) shows a certain trough-

like region where the input manipulation is minimum. The

initial conditions that lie in the trough exhibit minimum

change in input flow rate and hence provide the most

effective control. Thus, this analysis of the closed loop

trajectories clearly explains the relation between manipu-

lated input and the system behaviour for different conditions

and establishes regions for optimum controller performance.

4. Conclusions

In this paper, certain key issues for bioreactor control

have been addressed. It was shown that MPC could

successfully control a non-linear bioreactor to the unstable

operating point that resulted in optimal control of biomass

growth based on substrate concentration. This is a key

finding since very few biological systems are truly

homogenous or strictly first order. The effect of the

controller tuning parameter P on the stability of the

controlled system has been demonstrated. Values of P

between 15 and 20 yielded excellent control. When P > 20,

the controller performance degraded and generated an

oscillatory response. Analysis of the trajectories of the

controlled system generated a narrow operating region

(trough) for efficient control that required minimal

manipulation. It is important to note that even a well-

designed control scheme may result in poor control

performance causing batch or washout modes of operation.

S. Ramaswamy et al. / Process Biochemistry 40 (2005) 2763–27702770

Appendix A. Nomenclature

a, b positive constants in approximated yield equation

C1 d

imensionless cell mass, X/(SFY(SF))

C2 d

imensionless substrate conversion, (SF � S)/SF

Da D

amkohler number

DaS D

amkohler number at steady state

f f

unction of x and u

F f

eed flow rate

h f

unction of x

J o

bjective function

K m

aximum of one-hump growth model

L n

on-linear function

M c

ontrol horizon

P p

redictor horizon

S s

ubstrate concentration

t ti

me

T s

ampling interval

u m

anipulated input variable

V v

olume of bioreactor

x s

tate variable

X c

ell mass

X1 d

imensionless cell mass deviation from steady state

X2 d

imensionless substrate conversion deviation from

steady state

y c

ontrolled output variable

Y y

ield coefficient

Greek letters

b w

eighting coefficient a/(bSF)

f n

on-linear function

k g

rowth model coefficient

m s

pecific growth rate

s s

pecific substrate consumption rate

t d

imensionless time

C w

eighting coefficient (K/SF)

Subscripts

F f

eed

S s

teady state

References

[1] Aguilar R, Gonzalz J, Barron MA, Martinez-Guerra R, Maya-Yescas

R. Robust PI2 controller for continuous bioreactors. Process Biochem

2000;36:1007–13.

[2] Barron MA, Aguilar R. Dynamic behaviour of continuous stirred

bioreactor under control input saturation. J Chem Technol Biotechnol

1998;72:15–8.

[3] Dondo RG, Marques D. Optimal control of a batch bioreactor: a study

on the use of an imperfect model. Process Biochem 2001;37:379–85.

[4] Checkova E, Barton P, Gorak A. Optimal operation processes of

discrete–continuous biochemical processes. Comput Chem Eng

2000;24:1167–73.

[5] Czeczot J, Metzger M, Babary JP, Nihtila M. Filtering in adaptive

control of distributed parameter bioreactors in the presence of noisy

measurements. Simulation Pract Theory 2000;8:39–56.

[6] Wang HH, Krstic M, Bastin G. Optimizing bioreactors by extreme

seeking. Int J Adaptive Control Signal Process 1999;13:651–69.

[7] Liden G. Understanding the bioreactor. Bioprocess Biosyst Eng

2002;24:273–9.

[8] Henson MA, Seborg DE. An internal model control strategy for non-

linear systems. AIChE J 1991;37:1065–81.

[9] Aoyama A, Venkatasubramanian V. Internal model control framework

using neural networks for the modelling and control of a bioreactor.

Eng Appl Artif Intell 1995;8:689–701.

[10] Bastin G, Dochain D. On-line estimation and adaptive control of

bioreactors. Amsterdam: Elsevier; 1990.

[11] Radhakrishnan TK, Sundaram S, Chidambaram M. Non-linear control

of continuous bioreactors. Bioprocess Eng 1999;20:173–8.

[12] Tsao JH, Wu WT. Global control of a continuous stirred tank bior-

eactor. Chem Eng J 1994;56:B69–74.

[13] Mahadevan R, Doyel III FJ. Efficient optimization approaches to non-

linear model predictive control. Int J Robust Non-linear Control

2003;13:309–29.

[14] Rodrigues JAD, Filho RM. Production optimization with operating

constraints for a fed-batch reactor with DMC predictive control. Chem

Eng Sci 1999;54:2745–51.

[15] Ajbar A, Gamal I. Stability and bifurcation of an unstructured model

of a bioreactor with cell recycle. Math Comput Model 1997;25:31–48.

[16] Agrawal P, Lee C, Lim HC, Ramakrishna D. Theoretical investigations

of dynamic behaviour of isothermal continuous stirred tank biological

reactors. Chem Eng Sci 1982;37:453–62.

[17] Dowd JE, Kwok KE, Piert JM. Predictive modeling and loose-loop

control for perfusion bioreactors. Biochem Eng J 2001;9:1–9.

[18] Henson MA. Dynamic modeling and control of yeast cell populations

in continuous biochemical reactors. Comput Chem Eng

2003;27:1185–99.

[19] Sistu PB, Bequette WB. Non-linear predictive control of uncertain

processes: application to a CSTR. AIChE J 1991;37:1711–23.

[20] Bequette BW. Process control: modeling, design and simulation. New

Jersey: Prentice-Hall; 2003.

[21] Zhu GY, Zamamiri A, Henson MA, Hjortso MA. Model predictive

control of continuous yeast bioreactors using cell population balance

models. Chem Eng Sci 2000;55:6155–67.

[22] Brengel DB, Seider WD. Multi-step non-linear predictive controller.

Ind Eng Chem Res 1989;28:1812–22.

[23] Ungar LH. A Bioreactor benchmark for adaptive-network based

process control. In: Miller III WT, Sutton RS, Werbos PJ, editors.

Neural networks for control. Cambridge, MA: MIT Press; 1990. p.

387–402.

[24] Bequette BW. Process dynamics: modeling, analysis and simulation.

New Jersey: Prentice-Hall; 1998.

[25] Thompson JMT, Stewart HB. Non-linear dynamics and chaos. New

York: John Wiley; 1986.

[26] Ramaswamy S. Role of non-linear dynamics in model predictive

control of a non-linear bioreactor. Master’s thesis, The University

of Akron; August 2001.

[27] Kreyszig E. Advanced engineering mathematics, 6th ed. New York:

John Wiley; 1998. p. 1069.

[28] Henson MA. Non-linear model predictive control: current status and

future directions. Comput Chem Eng 1998;23:187–202.

[29] Meadows ES, Rawlings JB. Model predictive control. In: Henson MA,

Seborg DE, editors. Non-linear process control. Prentice-Hall; 1997.

p. 233–310.

[30] Patnaik PR. A simulation study of dynamic neural filtering and control

of a fed-batch bioreactor under non-ideal conditions. Chem Eng J

2001;84:533–41.

[31] Sengupta S, Modak JM. Optimization of fed-batch bioreactor for

immobilized enzyme processes. Chem Eng Sci 2001;56:3315–25.