Engineers (Chemical Industries) by Peter Englezos [Cap 7]

8/22/2019 Engineers (Chemical Industries) by Peter Englezos [Cap 7]

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Shortcut Estimation Methods forOrdinary Differential Equation (ODE)Models

W h e n e v e r the w h o l e state vector is measured, i .e . , w h e n y ( t j )=x ( t j ) , i = l , . . , N ,we can emp loy a p p r ox i ma t i ons of the t ime-der iva t ives of the state variables orm a k e use of su i tab le integrals and thus reduce the parameter est imat ion problemfrom a n O D E sys tem to one for an algebraic e q ua t io n sys t em. W e shall presenttw o approaches, one a p p r ox i ma t i ng t ime der ivat ives of the state variables (thederivative approach) a n d o th e r a p p r o x i m a t i n g su i t ab le in tegra ls of the state var i -ab les ( t h e integral approach). In addi t ion , in th i s chapter we p resen t th e m e t h o d -o l o gy fo r e s t i m a t i n g average k i n e t i c rates (e.g., spec i f ic g rowth rates, spec i f ic up -take rates, or specif ic secret ion rates) in b io log ic a l sys tems o pera t ing in the batch,fed-batch, cont inuous or per fus ion m o d e . These est imates are rout ine ly used byanalysts to co mpare the product iv i ty or growth character is t ics a m o n g differentmicroorganism po pu l a t io ns o r cell l ines.

7.1 O DE M O D E L S W I T H L I N E A R D E P E N D E N C E O N T H EP A R A M E T E R S

L et us cons ider the special class of problems where a l l state variables aremea su r ed and the parameters enter in a l inear fashion into the g o v e r n i n g differen-tial equat ions . As usual , w e assume that x is the n-dimensional vector of state vari-ables and k is the p-dimensional vector of u n k n o w n parameters . The structure o fthe OD E mo del i s o f the form1 1 5

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116 Chapter 7

dx]d x 2

dtd x n~dT

(7.1)

w h e r e ( P i j ( x ) , i = ! , . . . , ; j = l , . . . , p a r e k n o w n f u n c t i o n s of the state v a r i a b l e s o n l y .Q u i t e often these f u n c t i o n s are su i t ab le p roduct s o f the state var iab les espec ia l l ywhe n the m o d e l is that of a h o m o g e n e o u s react ing sys tem. In a mor e c omp a c tform Equa t ion 7 .1 can be wr i t t en as

dt (7.2)

w h e r e th e nxp dimensional matr ix O(x) has as e lements the f u n c t i o n s c p y ( x ) ,i = l , . . . , ; j = l , . . . , / ? . It i s also a s s u m e d t h a t th e in i t ia l c o n d i t i o n x ( t0 ) = x 0 i s k n o w np re c i s e l y .

7 .1 .1 Derivative approachIn th i s case w e a p p r o x i m a t e th e t ime der ivat ives on the l e f t hand side ofEquat ion 7.1 n um e r i c a l l y . In order t o m in im iz e t h e potent ial num e r ica l errors inth e e v a l u a t i o n of the der ivat ives i t i s s t rong ly r e c o m m e n d e d to s m o o t h th e data

first by p o l y n o m i a l fi t t ing. The order of the p o l y n o m i a l should be the lowest p os -sib le tha t fits th e m e as ur e m e nts satisfactorily.The t ime-der iva t ives can be est imated ana ly t ica l ly from th e s m o o the d dataasdx(t)1 (7.3)

w h e r e x ( t ) i s the fi t ted p o l y n o m i a l . I f for e x a m p l e a second o r th i rd o r de r p o l y -n o m i a l has been used , x ( t ) w i l l b e given respec t ively by

or X( t ) =

(7.4)

(7.5)

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Shortcut Estimation Methods for ODE Models 11 7

The best and easiest way to s m o o t h the data and avoid m i s u s e of the p o l y -n o m i a l c u r v e f i t t ing i s b y e m p l o y i n g smooth cubic splines. I M S L p r o v i d e s t w oroutines for th is p u r p ose : CS S CV and C S S M H . The latter is more versat i le as itgives th e option t o t h e u s e r t o a p p l y d i f f e r e n t l e v e l s o f s m o o t h i n g b y c o n t r o l l i n g as ingle parameter . F ur the r m o r e , I M S L rou t ines C S V A L a nd CSDER c a n be u sedonce the coeffic ients of the cubic spines have b een c omp u t ed by CSSM H t o cal-culate the smoot hed values of the state var iables and their der ivat ives respec t ively .Having the smoothed va lues of the state var iab les at each sa m p l in g p o in tand the der iva t ives , r j ; , w e ha ve essent ia l ly t r ansformed the problem to a "usual"linear regress ion problem. The p a r a met e r vector is obtained by m i n i m i z i n g thef o l l o w i n g L S object ive funct ion

( 7 . 6 )

w h e r e i j i s the smoothed v a lu e of the m e as ur e d state var iables at t= t ;. A n y goodl inear regression package can be used to solve th i s pr o b l e m .H o w e v e r , an im po r tan t ques t ion that needs to be answered is " w ha t const i -tutes a satisfactory p o l y n o m i a l f i t?" A n answer c a n come f rom th e f o l l o w i n g sim-p le reasoning. T h e purpose o f th e p o l y n o m i a l fi t i s to s m o o t h th e data, n a m e l y , tor e mo ve only the m e a s u r e m e n t error (noise) from the data. I f the mathemat ical( O D E ) m od el u n d er considerat ion is indeed the true m o de l (o r s im p ly an adequateo n e ) then the calcula ted values of the output vector based on t he ODE m o d e lshould cor respond to the error-free mea su r emen t s . Obvious ly , these mod el -calculated values sh ou ld ideal ly be the sa me as the smoothed data a ssu mi ng thatthe correct a m o u n t o f data-f i l ter ing has taken place .

Therefore, i n o u r v i e w p o l y n o m i a l f i t t ing fo r data s m o o t h i n g s h o u l d b e a ni terat ive process . Firs t we start wi th the visua l ly best choice of p o l y n o m i a l order orsmoot h cub ic spl ines . Su b sequ en t l y , w e est imate the u n k n o w n parameters in theO D E m o d e l an d generate model ca lcu la ted va lues of the ou tp u t vector. Plot t ing o fthe raw data, the smoothed data and the O D E m o de l calculated values in the samegraph enables v i s ua l inspect ion. If the smoothed data are reasonably close to them o d e l calculated v a lue s and the res iduals appear to be normal , the p o l y n o m i a lfi t t ing w as done correct ly . If not, w e should go back and redo t he po l y no m ia l fit-t ing . In addi t ion , w e should m a k e sure tha t th e ODE m o d e l is adequate . If it is not,th e abo ve procedure fails. In this case, the data s m o o th ing s h o u ld be based s i m plyon the r equ i rem en t tha t th e differences be t w een the raw data and the smoothedvalues are n o r m a l l y dis t r ibu ted w i t h zero mea n .Fina l ly , the user should a l w a y s be aware of the danger in get t ing n u m er i c a lest imates of the der ivat ives from the data. Dif fe ren t smooth ing cubic sp l ines o rp o l y n o m i a l s can resu l t in s imi la r v a lu es for the state var iab les and at the same t im ehave wide ly different est imates of the der iva t ives . Th i s problem can be contro l led



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b y th e p r e v i o u s l y m e n t i o n e d i t e ra t ive pr o c e dur e i f we pay a t t en t ion no t o n l y t o thevalues of the state var iab les bu t to the i r der iva t ives as w e l l .7.1.2 Integral Approach

In this case ins tead of approx imat ing the t i m e der iva t ives on the l e f t h a n dside of Equ a t ion 7.1, w e integrate both sides with respect to t i m e . N a m e l y , inte-g ra ti on b e tw een to and t t of Equa t ion 7.1 y i e l d s ,

t o t o

t ot j

t 0t j

t o t0w h i c h u p o n e x p a n s i o n of the in tegra l s y i e l d s

(7.7)

,x,( t i ) -x , ( t0 ) = kl J < p u ( x ) d t + k2 j ( p 1 2 ( x ) d t + ... + kp |c p l p ( x ) d tt o t0 t0t i t i t i

X 2 ( t i ) - X 2 ( t o ) = ki J < P 2 i ( x ) d t + k2 J c p 2 2 ( x ) d t + .. . + kp J ( p 2 p ( x ) d t

xn (

ti ) -

xn (

to ) =

kl t o

J ( p n 2 ( x ) d t + ... + kp j ( p n p ( x ) d tt 0 t 0

(7.8)

N ot in g th a t t h e in i t ia l c on d i t i on s a re k n own ( x ( t0) = x 0) , th e above equ a t ion scan be rewri t ten asX l ( t i ) - x 1 ( t 0 ) = k 1

x 2 ( t i ) - x 2 ( t o ) = k 1

x n ( t j ) - x n ( t 0 ) = k 1( 7 . 9 )



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o r m o r e c o m p a c t l y asx ( t ) = x n + 1*(t ) k ( 7 1 0 )

w h e r et j

^ j r O i ) = J9jrWdt ; j = l , . . . , & r = l , . . . , p (7.11)t o

T h e a b o v e in tegra ls can be ca lcula ted s ince we h a v e m e a s ur e m e n t s o f a l l t h estate var iables as a funct ion of t im e . In par t icu lar , to obtain a good es t imate ofthese in tegra ls it is s trongly r e c o m m e n d e d to smooth the data first using a po l y -n o m i a l fit of a sui table order. Therefore, the integrals * - M [ ) sh ou ld be calculated asM

r ( t j ) = J < p j r ( i ) d t ; j = l , . . . , / i & r = l , . . . , p (7.12)

w h e r e x ( t ) i s the fi t ted p o l y n o m i a l . T h e s a m e g u i d e l i n e s for the se lec t ion of thesmoot h i ng p o l y n o m i a l s a p p l y .

H a v i n g the s m o o t h e d v a lu es of the state variables at each s am pl ing poin t ,X j and the integrals, * F ( t j ) , w e ha ve essent ia l ly t ransformed the problem to a"usual" linear regression p r o b l e m . T he parameter vec tor is obtained b y m i n i m i z -ing the fo l l o w i n g LS ob jec t ive func t ion

N

i = lThe above l inea r regression problem can be readi ly solved using any stan-dard l inear regression package.

7.2 G E N E R A L I Z A T I O N T O O D E M O D E L S W I T H N O N L I N E A RD E P EN D E N C E O N T H E P A R A M E T E R S

The der ivat ive approach descr ibed previous ly can be read i ly ex tended toO D E m o d e l s where the u n k n o w n parameters enter in a no nl ine ar fashion. Theexa c t same pr o ce dur e to obta in good es t imates of the t i m e der ivat ives of the statevar iables at each s am pl ing po int , t j , can be fo l l o w e d . T hus the go ve rn ing O D ECopyright 2001 by Taylor & Francis Group, LLC


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120 Chapter 7

(t),k) (7.14)

can be writ ten at each sa m p l in g p o i n t asT l, = f (x ( t , ) ,k) (7.15)

w her e aga in

H a v in g the smoothed values of the state var iab les at each sa m p l in g po i n tand hav ing est imated analy t ica l ly the t ime der ivat ives , T | J , w e have t ransformed thep rob lem to a "usua l" n o n l i n e a r regression problem for a lgebraic models . The pa-r a m e t e r vector is o b t a i n e d b y m i n i m i z i n g th e f o l l o w i n g L S o b j e c t iv e func t i onNSL sO O = Zhi - f t f j . l o r ' Q j h j - f ( x j , k ) ] ( 7 . 1 6 )

w h e r e x ; i s the s m o o t h e d v a lue of the m e a s ur e d state var iab les a t t= t j .T he above parameter es t imat ion problem can now be so lved w i t h any esti-m a t ion met hod fo r algebraic m o de l s . Again, our preference is to use the Gaus s -N e w t o n method as descr ibed in Chapter 4.Th e on ly drawback in us i n g th is m e t h o d is t h a t any n um e r i c a l errors in t ro-duced in the es t imat ion of the t ime der iva t ives of the state var iables have a directeffect on the est imated parameter va lues . Fur thermore , b y th i s approach we c a nn o t readi ly calculate conf idence in te rva ls for the u n k n o w n parameters . Th ism e t h o d i s th e s ta n da rd p r o c e du r e u s e d b y t h e Ge n e r a l A lge b r a i c Mo d e l i n g S ys t e m( G A M S ) for the es t imat ion of parameters i n ODE m od e l s wh en a l l state var iab les

are ob se rv ed .

7.3 E S T I M A T I O N O F A P P A R E N T R A T E S IN B I O L O G I C A L S Y S TE M SIn b ioc h em ic a l e ng ine e r ing we are often faced w i t h the problem of es t imat -in g average apparent growth or uptake/secret ion rates. Such est imates are part icu-larly useful w h e n w e co mpare the produc t iv i t y o f a cul ture unde r different operat-ing cond i t ions or m o d e s of operat ion. Such co m puta t io ns are rout ine ly d one b y -

analys ts wel l before any at tempt is ma d e to est imate "true" kine t i cs parametersl ike those appear ing in the M o n o d growth model for e x a m p l e .

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In t h i s sect ion w e sha l l use the s tandard n o t a t i o n e m p l o y e d b y b i o c h e m i c a len g in eer s an d i n d u s t r i a l mi c r o b i o l o g i s t s i n p r e s e n t i n g t h e ma t e r i a l . T h u s i f wedenote by X v the viable cell {cells/L) or b i oma ss (mg/L) concentra t ion, S the l i m -i t ing substrate concentrat ion (mmol/L) and P the product concentrat ion (mmol/L)in the bioreac tor , th e d y n a m i c c o m p o n e n t m a ss balances y i e ld th e fo l l o w i n g O D E sfo r each mode of opera t ion:Batch Experiments -=u X v (7.17)dt

=- q s Xv (7.18)

(7.19)

w h e r e \\ is the a p p a ren t spec i f ic gr o wth rate (1/h), q p is the a p p a ren t spec i f ic secre-t i o n / p r o d u c t i o n rate, (mmoU(mg-h-L)) an d q s i s the a p p a ren t spec i f ic up t a ke rate(mmol/ (mg-h-L)) .

Fed-Batch ExperimentsIn th i s case there i s a cont inuous addi t ion o f nut r ien ts to the cu l tu re by thefeed stream. There is no eff luent stream. The g over n i ng equat ions a r e :(7.20)dt

=- q s Xv +D(S f - S ) (7.21)

=q p Xv - D P (7.22)

w h e r e D is the d i l u t i o n factor ( 1 / h ) d e f i n e d as the feed f lowra te o v e r v o l u m e ( F / V )and Sf {mmol/L) is the substrate concentrat ion in the feed. For fed-batch cu l tu resthe vo lume V is increas ing c o n t i n uo us l y un t i l the end of the cul ture period ac-cording to the feed ing rate as fo l l o w s = F = D V (7.23)dt



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122 Chapt er?

Continuous (Chemostat) ExperimentsIn t h i s case, t h e re is a c o n t i n u o u s s u p p l y o f nut r ien t s and a c o n t i n u o u s w i t h -drawal o f the cul ture bro th i n c l u d i n g t h e s u b me r g e d free cel ls . T h e go v e r n i n ge q ua t io ns for co n t inuo us cul tures are the same as the ones for fed-batch cultures

(Equ a t ion s 7.20-7.22). Th e only difference is that feed f lowra te i s normal ly equalto the eff luent f lowra te (F j n= F o u t= F ) and hence the v o l u m e , V , s tays cons tantt h rou g h ou t t h e culture.

Perfusion ExperimentsPerfus ion cu l tu res of s ubm e r ge d free cel ls are essent ial ly c o n t i n uo us cul-tures with a cel l re ten t ion device so tha t no cel ls e x i t in the eff luent s t ream. Th eg o v e r n i n g O D E s are

(7.24)

- S ) (7.25)

d Pdt = qpXv -DP (7.26)

Formulat ion of th e problemHaving m e as ur e m e nts o f X v , S and P over t im e , d e te rm in e the apparent spe-cific growth ra te (u ) , th e apparent specif ic uptake rate (q s) and the apparent spe-cif ic secretion rate (q p) a t any par t icu lar p o in t dur ing the cu l tu re or obtain an aver-a g e v a l u e o v e r a u s e r - s p e c i fi e d t i m e p e r i o d .

7 . 3 . 1 Derivative Ap proachT his approach is based on the rear rangement of the governing O D E s to y i e l dex press ions of the different specif ic rates, nam e l y ,

batch & perfusionXv(t)Xv(t) dt(7.27)

D + - fedbatch & continuousX v ( t ) d t



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Shortcut Estimation Methods for ODE Models 123

w h i c h can be r e d u c e d to~ d f a { X Y ( t ) }

dt batch & perfusion

D + --- fedbatch & continuousdtThe specif ic up take and secret ion rates are g iven by

- 1 dS(t)

(7.28)

qs = < 0.4

0.30.20.10.0

Rg"" DRawData^ft X S / N = 0.01

. , . . > - Xl^ A S /N =0.1 rff l +s/N=1 J l! oRawData | ^

0s/N -0.01 I X2X s/N = 0.1 |-s/N -1 J

gj g* 8

0.005 001000150.02CtFigure 7.1: Smoothed data for variables xt and x2 using a smooth cubic splineapproximation (s /N-0.01, 0.1 and 1).

O n c e w e h a v e th e s m o o t h e d v a l u e s of the sta te var iables , we c a n p roc eedand c o m p u t e q > n , c p i 2 , ( p 2 i and c p 2 2 . A ll these co m pute d quan t i t i es ( n , , r ) 2 , ( p n , ( p l 2 ,( p 2 i and c p 2 2 ) cons t i tu te the data set for the l inear least squares regress ion. In Figure7.1 the or ig ina l data and the i r smoot hed values are shown for 3 different values ofthe smo o th ing parameter "s" requ ired by C SSMH. An one percent ( 1 % ) standard



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132 Chapter 7

error i n t h e m e a s u r e m e n t s h a s b een a ssu m ed and used as w e i g h t s in data s m o o t h -ing b y C S S M H . I M S L r e c o m m e n d s th i s s m o o t h i n g parameter to be in the range[N- (2N) '5 , N + ( 2 N ) 0 5 ]. F o r t h i s pr o b le m t h i s m e a n s 0 . 5

Engineers (Chemical Industries) by Peter Englezos [Cap 7]

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Transcript of Engineers (Chemical Industries) by Peter Englezos [Cap 7]