King M.R., Mody N.A. Numerical and Statistical Methods for Bioengineering: Applications in MATLAB

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¼ 0:3ghyphomicrobium bacteria produced/g methanol consumed,

a ¼ 42:0/hr,

¼ saturated dissolved O

concentration = 0.008 g/l,

¼ 0:08/hr.

The starting mass concentrations of the bacterial species are x

= 0.672 g/l and

= 0.028 g/l, and c

¼ 1  10

5

g/l. At t = 0, the methanol concentration in the

chemostat is increased to 1.6 g/l. Evaluate the transient response of the system to the

methanol shock load over the 72 hour period following the step change in methanol

concentration. Is a steady state obtained after three days?

(Hint: The methanol concentration dips to zero between 16 and 17 hours of

elapsed time. How does this pose a problem for a discrete integration method? Do

you get a meaningful result? What corrective measures can be taken?)

7.5. Binding kinetics for multivalent ligand in solution interacting with cell surface

receptors This problem models the dynamics of the attachment of proteins that

have multiple binding sit es to cell surfaces. When a single ligand molecule binds two

or more cell surface receptors, the bound receptors are said to be “crosslinked” by

the ligand. Viral infection involves attachment of a virus to the cell surface at

multiple points. Once the virus is tightly bound to the cell surface, it injects viral

genes into the host cell, hijacks the cell’s gene replication machinery, and ﬁnally lyses

the cell to release newly packaged viral particles within the cell. Study of the

mechanism of viral docking to cell surface receptors is required to develop inter-

vention strategies or suitable drugs that block the steps leading to viral attachment.

Perelson (1981) developed a kinetic model for mult ivalent ligand binding to cells

(see Box 5.2A). This model makes an assumption (the “equivalent site hypothesis”

assumption) that the forward and reverse crosslinking rate constants that character-

ize the frequency and duration, respectively, of receptors and ligand association are

the same for all ligand binding sites. Note that the two-dimensional crosslinking

forward rate constant is different from the three-dimensional associati on rate con-

stant. The latter applies when a ligand in solution ﬁrst binds to a single receptor on

the cell surface. If L is the time-varying ligand concentration in solution, R is the

density of unbound receptors on the surface, and C

is the number of ligand

molecules bound to a cell by i ligand binding sites, then the set of deterministic

kinetic equations that govern the dynamics of binding is:

¼υk

LR þ k

;

¼ υk

LR  k

 f  1ðÞk

R þ 2k

x

;

¼ f  i þ 1ðÞk

i1

R  f  iðÞk

R  ik

x

þ i þ 1ðÞk

x

iþ1

; 2  i  f  1;

¼ k

f1

R  fk

x

;

R ¼ R



i¼1

;

where k

is the two-dimensional crosslinking forward rate constant and k

−x

is the

two-dimensional cross-linking reverse rate constant. For L, which varies with time,

these f + 2 equations containing f + 2 unknowns constitute an initial-value problem

that can only be solved numerically. This set of equations involves f + 1 ﬁrst-order

differential equations and one algebraic equation.

477

7.9 Problems

The equivalent site hypothesis model for multivalent ligand binding to mono-

valent receptors can be used to model the dynamic binding of von Willebrand factor

(vWF), a multivalent plasma protein, to platelet cells. At abnormally high shear

ﬂows in blood, typical of regions of arterial constriction, von Willebrand fact or

spontaneously binds to platelet cells. This can lead to the formation of platelet

aggregates called thrombi. Platelet thrombi can obstruct blood ﬂow in narrowed

regions of the arteries or microvasculature and cause ischemic organ damage that

have life-threatening consequences. We use this kinetic model to study certain

features of vWF binding to platelets. We adopt the following values for the following

model parameters (Mody and King, 2008):

v = 18,

f =9,

= 3.23 × 10

−8

1/(# molecules/cell) · 1/s

= k

−x

= 5.47/s

= 0.0003172 × 10

−4

1/(# molecules/cell) · 1/s

The initial conditions at time t = 0 are

R = R

= 10 700 #receptors=cell

L=L

= 4380 #unbound molecules=cell

(i = 1,2, ...,9)=0#bound molec ules=cell.

You are asked to do the following.

(a) Develop a numerical scheme based on the fourth-order Runge–Kutta method to

determine the number of ligand molecules that bind to a platelet cell ð

i¼1

after 1 s and 5 s when the cell is exposed to vWF molecules at high shear rates that

permit binding reactions. Use a time step of Δt =0.01s. Use sum(ﬂoor(Ci))

after calculating C

to calculate the integral number of bound ligand molecules.

(b) Mo dify the program you have written to determine the time taken for the system

to reach thermodynamic equilibrium. The criterion for equilibrium is |R(t + Δt)

− R(t)| < 0.01Δt. Also determine the number of ligand molecules bound to the

cell at equilibrium.

you obtained in Box 5.2B.

(d) In von Willebrand disease, the dissociation rate decreases six-fold due to a muta-

tion in the ligand binding site. If k

and k

−x

are reduced by a factor of 6, determine

(i) the number of bound receptors (R

− R) after 1 s and 5 s;

(ii) the time required for equilibrium to be established. Also determine the

number of ligand molecules bound to the cell at equilibrium.

7.6. Settling of a microparticle in air Determine the time it takes for a microsphere of

radius a =30μm and density 1.2 g/cm

to attain 99% of its terminal Stokes velocity

when falling from rest in air at T =20 ˚C; assume ρ

air

= 1.206 × 10

−3

g/cm

(no

humidity) and μ

air

= 0.0175 cP. The governing equations (Equations (7.27) and

(7.28)) are presented in Example 7.2.

References

Bailey, J. E. and Ollis, D. F. (1986) Biochemical Engineering Fundamentals (New York: The

Mc-Graw Hill Companies).

Barabasi, A. L. (2003) Linked: How Everything Is Connected to Everything Else. What It

Means for Business, Science and Everyday Life (New York: Penguin Group (USA)).

478

Numerical integration of ODEs

Burden, R. L. and Faires, J. D. (2005) Numerical Analysis. (Belmont, CA: Thomson

Brooks/Cole).

Davis, T. H. and Thomson, K. T. (2000) Linear Algebra and Linear Operators in Engineering

(San Diego, CA: Academic Press).

Grasselli, M. and Pelinovsky, D. (2008) Numerical Mathematics (Sudbury, MA: Jones and

Bartlett Publishers).

Haase, A. T., Henry, K., Zupancic, M. et al. (1996) Quantitative Image Analysis of HIV-1

Infection in Lymphoid Tissue. Science, 274, 985–9.

Ko, M. K. W., Sze, N. D., and Prather, M. J. (1994) Better Protection of the Ozone-Layer.

Nature, 367, 505–8.

Lauffenburger, D. A. and Linderman, J. J. (1993) Receptors: Models for Binding, Trafﬁcking

and Signaling (New York: Oxford University Press).

Mody, N. A. and King, M. R. (2008) Platelet Adhesive Dynamics. Part II: High Shear-

Induced Transient Aggregation Via GPIbalpha-vwf-GPIbalpha Bridging. Biophys. J.,

95, 2556–74.

Perelson, A. S. (1981) Receptor Clustering on a Cell-Surface. 3. Theory of Receptor Cross-

Linking by Multivalent Ligands – Description by Ligand States. Math. Biosci., 53, 1–39.

Perelson, A. S., Neumann, A. U., Markowitz, M., Leonard, J. M., and Ho, D. D. (1996) Hiv-1

Dynamics in Vivo: Virion Clearance Rate, Infected Cell Life-Span, and Viral Generation

Time. Science, 271, 1582–6.

Pozrikidis, C. (2008) Numerical Computation in Science and Engineering (New York: Oxford

University Press).

Siepmann, J., Elkharraz, K., Siepmann, F., and Klose, D. (2005) How Autocatalysis

Accelerates Drug Release from PLGA-Based Microparticles: A Quantitative

Treatment. Biomacromolecules, 6, 2312–19.

Sugano, K. (2008) Theoretical Comparison of Hydrodynamic Diffusion Layer Models

Used for Dissolution Simulation in Drug Discovery and Development. Int. J. Pharm.,

363, 73–7.

Tsuchiya, H. M., Drake, J. F., Jost, J. L., and Fredrickson, A. G. (1972) Predator-Prey

Interactions of Dictyostelium Discoideum and Escherichia Coli in Continuous Culture.

J. Bacteriol., 110, 1147–53.

Vauthier, C., Schmidt, C., and Couvreur, P. (1999) Measurement of the Density of Polymeric

Nanoparticulate Drug Carriers by Isopycnic Centrifugation. J. Nanoparticle Res., 1,

411–18.

Wilkinson, T. G., Topiwala, H. H., and Hamer, G. (1974) Interactions in a Mixed Bacterial

Population Growing on Methane in Continuous Culture. Biotechnol. Bioeng., 16, 41–59.

479

References

8 Nonlinear model regression

and optimization

8.1 Introduction

Mathematical modeling is used to analyze and quantify physical, biological, and

technological processes. You may ask, “What is modeling and what purpose does a

model serve in scientiﬁc research or in industrial design?” Experimental investigation

generates a set of data that describes the nature of a process. The quantitative trend

in the data, which describes system behavior as a functio n of one or more variables,

can be ﬁtted to a mathemati cal function that is believed to represent adequately the

relationship between the observations and the independent variables. Fitting a

model to data involves determining the “best-ﬁt” values of the parameters of the

model.

A mathematical model can be of two types.

(1) A model form ulated from physical laws is called a mechanistic model. A model of this

type, if formulated correctly, provides insight into the nature of the process. The

model parameters can represent physical, chemical, biological, or economic proper-

ties of the process.

(2) An empirical model, on the other hand, is not derived from natural laws. It is

constructed to match the shape or form of the data, but it does not explain the

origin of the trend. Therefore, the parameters of an empirical model do not usually

correspond to physical properties of the system.

Mechanistic models of scientiﬁc processes generate new insights into the mecha-

nisms by which the processes occur and proceed in nature or in a man-made system.

A mathe matical model is almost always an incomplete or imperfect description of

the system. If we were to descri be a process completely, we would need to incorpo-

rate a prohibitively large number of parameters, which would make mathematical

modeling exceedingly difﬁcult. Often, only a few parameters inﬂuence the system in

an important way, and we only need to consider those. The model analysis and

development phase is accompanied by a systematic detailing of the underlying

assumptions used to formulate the model.

Models serve as predict ive tools that allow us to make critical decisions in our

work. The parameters of the mechanistic model quantify the behavior of the system.

For example, a pharmacodyn amic model of the dose-response of a drug such as

E ¼ E

max

C=C

ðÞ

1 þ C=C

ðÞ

quantiﬁes the potency of a drug, since the model parameter C

is the drug concen-

tration in plasma that produces 50% of its maximum pharmacological effect. Here E

is the drug effect and γ is the shape factor. By quantifying and compari ng the

potency, efﬁcacy, and lethal doses of a variety of drugs, the drug research er or

manufacturer can choose the optimal drug to consider for further development.

Biochemical kinetic models that characterize the reactive properties of enzymes with

regard to substrate conversion, enzyme inhibition, temperature, and pH are needed

to proceed with the industrial design of a biocatalytic process. It is very important

that the mathematical model serve as an adequate representation of the system.

Careful attention should be paid to the model development phase of the problem.

Performing nonlinear regression to ﬁt a “poor” model to the data will likely produce

unrealistic values of the system parameters, or wide conﬁdence intervals that convey

large uncertainty in the parameter estimates.

In Chapters 2 and 3, you were introduced to the method of linear least-squares

regression. The “best-ﬁt” values of a linear model’s parameters were obtained by

minimizing the sum of the squares of the regression error. The linear least-s quares

method produces the best values of the parame ters when the following assumptions

regarding the data are valid.

(1) The values of the independent variable x are known exactly. In other words, the

error in x is much less compared to the variability in the dependent variable y.

(2) The scatter of the y points about the function curve follows a Gaussian distribution.

(3) The standard deviation that characterizes the variation in y is the same for all x

values. In other words, the data are homoscedastic.

(4) All y points are independent and are randomly distributed about the curve.

The goal of least-squares regression is to minimize the sum of the squares of the error

between the experimental observations y and the corresponding model predictions

We seek to minimize the quantity

SSE ¼

i¼1



ðÞ

;

where m is the number of independent observations. The expression above is called

an objective function. To ﬁnd the parametric values that minimize an objective

function, we take the derivative of the function and equate it to zero. In

Chapter 2, it was shown that, for an objective function that is linear in the

unknowns, this procedure produces a matrix equation called the normal equations,

Ac ¼ A

y, the solution of which yields the best-ﬁt model parameters c.

Most mathematical models in the biological and biomedical sciences are nonlinear

with respect to one or more of the model parameters. To obtain the best-ﬁt values for

the constants in a nonlinear model, we make use of nonlinear regression methods.

When the four assumptions stated above for the validity of the least-squares method

apply – when variability of data is exclusively observed in the y direction, the

observations are distributed normally about the curve in the y direction, all obser-

vations are independent, and the variance of the data is uniform along the curve – we

In Chapter 2 it was shown that some forms of non-linear models can be linearized using suitable

transformations. However, this may not always be a good route to pursue when the goal is to obtain

a good estimate of the model parameters. The new x values of the transformed data may be a function of

the dependent variable and therefore may exhibit uncertainty or variability in their values. Moreover,

the transformation can change the nature of scatter about the curve converting a normal distribution to

a non-normal one. As a result, the assumptions for least-squares regression may no longer be valid and

the values of the parameters obtained using the linearized model may be inaccurate. Exercise caution

when using non-linear transformations to convert a non-linear model into a linear one for model

regression. Non-linear transformations are usually recommended when the goal is to remove hetero-

scedasticity in the data.

481

8.1 Introduction

perform nonlinear least-squares regression to compute the optimal values of the

constants.

In some cases, the variance of y is found to increase with y. When the variances are

unequal, y values with wide scatter inﬂuence the shape of the curve more than y

values with narrow scatter, and inaccurate values of the model parameters will

result. To make the data suitable for least-squares regression, a weighting scheme

must be applied to equalize the scatter. If the standard deviation is proportional to

the magnitude of y, then each square y



ðÞ

is weighted by 1=y

This is called the relative weighting scheme. In this scheme, the goal of least-

squares regression is to minimize the relative squared error. Therefore, the obj ective

function to be minimized is the sum of the squares of the relative error or

SSE ¼

i¼1



ðÞ

How does one calculate the minimum of an objective function? For a nonlinear

problem, this almost always requires an iterative approach using a very useful

mathematical tool called optimization. In industrial design, manufacturing, and

operations research, optimization is a valuable tool used to optimize industrial

processes such as daily production quantities, inventory stock, equipment sizing,

and energy consumption. An objective function fðxÞ that calculates some quantity to

be maximized, such as proﬁt, or to be minimized, such as cost, is formulated in terms

of the process variables. The objective function is the performance criterion that is to

be either minimized or maximized. The optimization problem involves ﬁnding the

set of values of the process variables that yields the best possible value of fðxÞ. In this

chapter, we will focus on the minimization problem since this is of primary concern

when performing nonlinear least-squares regression. If fxðÞis to be maxi mized, then

the problem is easily converted to the minimization of fðxÞ.

Sometimes, limitations or constraints on the variables’ values exist. Some physical

variables such as mass, age, or kW of energy consumed, cannot become negative. In

industrial production, availability of raw material may be limited, or demand for a

certain product in the market may be capped at a certain value, and therefore

production must be constrained. A variety of constraints may apply that limit the

feasible space within which the solution is sought. An optimization problem that is

not subject to any constr aints is an unconstrained opti mization problem. When

constraints narrow the region within which the minimum of the objective function

must be searched, the problem is a constrained optimization problem.

In mathematical terms, we deﬁne the optimization problem as the minimization

of the objective function(s) fðxÞ subject to the equality constraint(s) hxðÞ¼0, and

the inequality constraint(s) gðxÞ0, where x is the variable vector to be optimized,

f is the objective function, and g and h are each a set of functions.

Consider an optimization problem in two varia bles x

and x

. Suppose the

objective function is a quadratic function of x

and x

and is subject to one equality

constraint,

; x

ðÞ: px

þ qx

¼ b;

and three inequality constraints, of which two are

; x

ðÞ: x

 0; g

; x

ðÞ: x

 0:

In a two-variable optimization problem, the inequality constraints together deﬁne

an area within which the feasible solution lies. On the other hand, an equality

482

Nonlinear model regression and optimization

constraint deﬁnes a line on which the feasible solution lies. In this problem, if two

equality constraints had been deﬁned , then the solution of the problem would be

given by the intersection of the two equality curves, and an optim ization of the

objective function would be unnecessary since two equality constraints reduce the

feasible region to a point. Figure 8.1 illustrates the feasible region of the solution.

The feasible region is the variable space within which the optimization solution is

searched, since this space satisﬁes all the imposed inequality and equality con-

straints. Superimposed on the plot are the contour lines of the objective function

in two variables. Note that the minimum of the objective function lies outside the

feasible region. The minimum of the function is not a feasible solution according to

the constraints of the problem.

Optimization methods can be used to solve a variety of problems as follows.

(1) The constrained minimization problem, e.g. linea r programming and nonlinear

programming problems.

(2) Nonlinear regression. The performance criterion of a regression problem is the

minimization of the error or discrepancy between the data and the model prediction.

(3) Systems of nonlinear equations fxðÞ¼0, where x is a vector of the variables, and the

objective function is the norm of the fxðÞ:

Optimization techni ques come in handy for solving a wide variety of nonlinear

problems.

Optimization uses a systematic algorithm to select the best solution from a variety

of possible solutions. Different combinations of variables (or parameters) will yield

different values of the objective criterion, but usually only one set of values will

satisfy the condition of the minimum value of the objective function. Sometimes, a

nonlinear function has more than one minimum (or maximum). A local minimum x*

is deﬁned as a point in the function space where the function values at all points x in

Figure 8.1

Two-dimensional constrained optimization problem. The constrained minimum is the optimal solution

f (x

, x

)

f (x

, x

)

f (x

, x

)

≥ 0

Inequality constraints

Equality constraint

Feasible region

Unconstrained minimum lies

outside feasible re

ion

Contour lines of

objective function

Constrained

minimum

483

8.1 Introduction

the vicinity of x* obey the criterion f xðÞfðx



Þ. A local minimum may correspond

to a subregion of the variable space, but may not be the smallest possible value of the

function when the entire variable space is considered. A minimum point at which the

function value is smallest in the domain space of the function is called a global

minimum. While an optimization method can locate a minimum, it is usually difﬁcult

(for multivariable problems) to ascert ain if the minimum found is the global

minimum.

A local minimum of a function in one variable is deﬁned in calculus as the point at

which the ﬁrst derivative f

ðxÞ of the function equals zero and the second derivative

ðxÞ of the function is positive. A global minimum of a single variable function is

deﬁned as value of x at which the value of the function is the least in the deﬁned

interval x 2½a; b. A global minimum may either be located at a local minimum, if

present within the interval, or at an endpoint of a closed interval, in which case the

endpoint may not be an extreme point. In Figure 8.3, the global minimum over the

interval shown is at x =0;f

ðx ¼0Þ may not necessa rily equal zero at such points.

The goal of an optimization problem is to ﬁnd the global minimum, which is the

best solution possible. For any linear optimization problem, the solution, if it exists,

is unique. Thus, a linear programming problem has only one minimum, which

corresponds to the global minimum. However, a nonlinear function can have

more than one minimum. Such functions that have multiple minima (and/or max-

ima) are called multimodal functions. In contrast, a function that has only one

extreme point is called a unimodal function. A function fðxÞ with a minimum at x*

is said to be unimodal if

f xðÞf x



ðÞ x



;

f xðÞf x



ðÞ x



;

where x is any point in the varia ble space. For a function of one or two variables,

unimodality can usually be detected by plotting the function. For functions of three

or more variables, assessing the modali ty of a function is more difﬁcult.

Our discussion of optimization techniques in this chapter applies in most cases to

objective functions that are continuous and smooth. Discontinuities in the function

or in its derivatives near the optimal point can prevent certain optimization techn i-

ques from converging to the solution. Optimization problems that require a search

for the optimum within a discretized function space, arising due to the discrete

nature of one or more independent variables, is a more challenging topic beyond

the scope of this book. Unless mentioned otherwise, it is assumed that the objective

function has continuous ﬁrst and second derivatives.

Box 8.1A Pharmacokinetic and toxicity studies of AZT

The thymidine nucleoside analog AZT is a drug used in conjunction with several other treatment

methods to combat HIV. The importance of AZT is increased by the fact that it is found to reduce HIV

transmission to the fetus during pregnancy. AZT crosses the placenta into the fetal compartment

easily since its lipophilic properties support rapid diffusion of the molecule. However, AZT is found to

be toxic to bone marrow, and may possibly be toxic to the placenta as well as to the developing

fetus. In vitro studies have shown that AZT inhibits DNA synthesis and cell proliferation. Boal et al.

(1997) conducted studies of the transport of nucleoside analog AZT across the dually perfused

(perfused by both maternal and fetal perfusate) human term placental lobule. The purpose of this

study was to characterize pharmacokinetic parameters that describe the transport and toxicity

484

Nonlinear model regression and optimization

properties of AZT in placental tissues. AZT was found to transfer rapidly from the maternal perfusate

(compartment) to the fetal perfusate (compartment). The concentration of AZT in the fetal perfusate

never rose beyond the maternal levels of AZT, indicating that drug accumulation in the fetal

compartment did not occur. Table 8.1 lists the concentration of AZT in the maternal and fetal

perfusates with time.

In Chapter 2, Boxes 2.3A, B gave you an introduction to the field of pharmacokinetics and presented

a mathematical model of the first-order elimination process of a drug from a single well-mixed

compartment. In this example, we model the maternal compartment as a first-order drug absorption

and elimination process. Figure 8.2 is a block diagram describing the transpor t of drug into and out of a

single well-mixed compartment that follows first-order drug absorption and elimination. The following

description of the model follows Fournier’s(2007 ) pharmacokinetic analysis. Let A

be the total mass of

drug available for absorption by the compartment. Let A be the mass of drug waiting to be absorbed at

any time t.Ifk

is the rate of absorption, then the depletion of unabsorbed drug is given by the following

mass balance:

¼k

The solution of this first-order differential equation is A ¼ A

k

The drug is distributed in the compartment over a volume V, and C is the concentration of drug in the

compartment at time t. A stream with flowrate Q leaves the compartment and enters the elimination site

where the entire drug in the stream is removed, after which the same stream, devoid of drug, returns to

the compartment at the same flowrate. Note that this perfect elimination site is more of a mathematical

abstraction used to define the elimination rate in terms of a flowrate Q. The concentration of drug in the

compartment is governed by the first-order differential equation

¼ k

A  QC:

Table 8.1. AZT concentration in maternal and fetal perfusates following exposure

of the maternal compartment to AZT at 3.8 mM

Maternal Fetal

AZT (mM) Time (min) AZT (mM) Time (min)

0000

1.4 1 0.1 1

4.1 3 0.4 5

4.5 5 0.8 10

3.5 10 1.1 20

3.0 20 1.2 40

2.75 40 1.4 50

2.65 50 1.35 60

2.4 60 1.6 90

2.2 90 1.7 120

2.15 120 1.9 150

2.1 150 2.0 180

2.15 180 1.95 210

1.8 210 2.2 240

2.0 240

485

8.1 Introduction

The selection of the type of optimization algorithm to use depends on the nature

of the objective function, the nature of the constraint functions, and the number of

variables to be optimized. A variety of optimization techniques have been developed

to solve the wide spectrum of nonlinear problems encountered in scientiﬁc research,

The volumetric clearance rate from the compartment per unit distribution volume is set equal to an

elimination constant

The differential equation is rewritten as

¼ k

k

 k

VC:

Solving the differential equation, we obtain the concentration profile of the drug in the compartment as a

function of time:

CtðÞ¼

 k



k

 e

k



If I

¼ A

=V, then

CtðÞ¼I

 k



k

 e

k



You are asked to model the maternal and fetal compartments separately. The maternal compartment is to

be modeled as a first-order drug absorption and elimination problem. You will need to perform a nonlinear

regression to solve for k

, k

,andI

. Check the fit by plotting the data along with your best-fit model.

The fetal compartment is to be modeled as a first-order absorption problem with no elimination

= 0). While there is no exponential decay of the drug in the maternal perfusate (the source of drug

for the fetal compartment), the first-order absorption model will mimic the reduced absorption of drug

into the fetal compartment at long times when diffusion of drug across the compartmental barrier stops

due to equalization of concentrations in the two compartments:

ðÞ

¼ I

1  e

k



Find suitable values of the model parameters I

and k

for the fetal compartment.

Figure 8.2

Block diagram of the mass balance of dr ug over a single compartment in which drug absorption and elimination

follow a first-order process

Distribution volume V

Drug concentration in

compartment C

Compartment

Elimination site

Q, C

Q, C = 0

Elimination rate = Q

486

Nonlinear model regression and optimization