Distributed Modeling and Parameter

Estimation of Influenza Virus Replication

During Vaccine Production

R. Durr ∗,⋆ T. Muller ∗∗ B. Isken ∗ J. Schulze-Horsel ∗

U. Reichl ∗,∗∗ A. Kienle ∗,∗∗

∗ Max Planck Institute for Dynamics of Complex Technical Systems,Magdeburg, Germany

∗∗ Otto-von-Guericke University, Magdeburg, Germany

Abstract: This contribution is concerned with population balance modeling of influenza virusreplication in mammalian cell cultures. The cells are heterogeneous with respect to intracellularcompounds like viral proteins. The amount of viral NP protein can be measured directly bymeans of flow cytometry. The corresponding degree of fluorescence is introduced as internalcoordinate for a distributed deterministic modeling approach. The resulting model includeskinetic processes like infection, virus replication and release, apoptosis and cell death. It consistsof three partial differential equations describing the distribution dynamics which are coupled totwo differential equations that characterize the concentration of active and inactive virions inthe medium. Kinetic parameters are determined from experimental data. The parameters areassumed to depend on the internal coordinate. The emerging infinite dimensional parameterestimation problem is translated to a finite dimension using a hermite spline representation ofthe distributed parameters. Hence the resulting inverse problem can be solved in a weightednonlinear least squares framework. Spline approaches of different complexity are discussed andthe estimation results are compared.

Keywords: Influenza vaccine production, population balance modeling, partial differentialequations, inverse problems .

1. INTRODUCTION

In influenza vaccine production the use of permanentmammalian cell lines becomes more and more important.Besides sophisticated cell culture technologies and down-stream processing methods, mathematical modeling playsa crucial role in improving production efficiency. Mostnotably for analysis, experimental design and optimizationof the process, the benefit of combining extensive exper-iments with mathematical modeling approaches becomesapparent.

In our previous work, infection of equine influenza Avirus in Madin-Darby-Canine-Kidney (MDCK) cell cul-tures was investigated in either deterministic (Muller et al.,2008) or stochastic modeling frameworks (Sidorenko et al.,2008a,b). More recently, focus was on population balancemodeling of human influenza replication in MDCK cellcultures (Muller et al., 2011). Distribution dynamics aremeasured by means of fluorescence intensity, which is pro-portional to the intracellular amount of viral protein NP.Interesting new phenomena like transient multimodalityand reversal of propagation direction could be observed.

Our contribution refers to the lab-scale influenza virusproduction as described by Schulze-Horsel et al. (2009).

⋆ Correspondence to: R. Durr, MPI for Dynamics of Com-plex Technical Systems, 39106 Magdeburg, Germany (e-mail:robert.duerr@ovgu.de)

Adherent MDCK cells are cultivated on microcarriers insmall scale bioreactors as depicted in Figure 1. At thebeginning of the process the cells are inoculated withhuman influenza A virus seed. The uninfected cells becomeinfected and start synthesizing viral components. Com-pletely assembled virions are released into the medium andcan infect uninfected cells. With progress of infection cellsbecome apoptotic. Apoptosis is programmed cell deathwhich can be activated by a large variety of external andinternal stimuli, particularly by viral infection. It invari-ably leads to cell lysis and has major influence on theprocess productivity.

To adapt the model to the data an inverse problem has tobe solved. As the kinetic parameters are not only constants

Fig. 1. Scheme of the process: Adherent MDCK cells grow-ing on microcarriers and cells showing fluorescencedue to intracellular accumulation of viral proteins

but depend on the internal characteristics the inverseproblem is set in an infinite dimension. In contrast to least-squares based parameter estimation in a finite dimension,which is extensively studied in literature (Walter andPronzato, 1997), a relatively low number of publicationsdeals with estimation techniques for infinite dimensionalproblems (Banks et al., 2011; Luzyanina et al., 2009).

2. MODEL FORMULATION

Heterogeneity of the cell population with respect to inter-nal compounds like viral proteins gives rise to a distributedmodeling approach with population balances. A degree offluorescence ϕ ∈ [ϕmin, ϕmax], which is proportional tothe intracellular amount of viral protein NP is introducedas an internal coordinate. The fluorescence level of eachcell is interpreted as degree of infection and can be mea-sured by means of flow cytometry. The presented modelcomprises a system of three partial differential equationsfor description of the distribution dynamics. In addition,they are coupled to two ordinary differential equationsthat characterize virus particles in the medium. It is anextension of the model given by Muller et al. (2011).

On contact with active virions the uninfected cells Uc

become infected with the rate constant kvi. Due to mediumexchange, limited space on microcarriers and fast progres-sion of infection, growth and death of the uninfected cellsare neglected.

∂Uc(ϕ, t)

∂t= −kvi Uc(ϕ, t)Vac(t) (1)

Infected cells Ic originate from uninfected cells and startreplicating and releasing virions. The first kinetic processincreases and the latter decreases the degree of fluores-cence. Both are summed up in a net rate coefficient knet(ϕ)that may depend on the fluorescence level and which isstrictly positive in ϕ ∈ [ϕmin, ϕmax]. A time delay τ isconsidered to summarize intracellular processes that causea lag between infection and virus replication and release.In the course of the infection more and more cells becomeapoptotic. This is accounted for by the rate of apoptosiskapo(ϕ) which is also considered to depend nonlinearly onthe degree of fluorescence.

∂Ic(ϕ, t)

∂t= −

∂ {knet(ϕ) Ic(ϕ, t)}

∂ϕ− kapo(ϕ) Ic(ϕ, t)

+ kvi Uc(ϕ, t − τ)Vac(t − τ) (2)

After the infected cells have become apoptotic their repli-cation ability is altered significantly. Its fluorescence de-gree is assumed to be generally decreasing which resultsin a negative net rate knet,apo(ϕ) in ϕ ∈ [ϕmin, ϕmax]. Inaddition, the apoptotic cells drop out of the process withlysis rate kcd(ϕ).

∂Ac(ϕ, t)

∂t= −

∂ {knet,apo(ϕ)Ac(ϕ, t)}

∂ϕ

+ kapo(ϕ) Ic(ϕ, t) − kcd(ϕ)Ac(ϕ, t) (3)

Fully assembled virions are released to the medium byinfected as well as apoptotic cells with the release ratekrel. Due to errors in the virus replication processes,only a small fraction of released virus particles is able to

infect uninfected cells. This fact is accounted for by theconstant parameter Peff . The so called active virions Vac

can become inactive with the constant degradation ratekdeg.

dVac(t)

dt= − kvi Vac(t)

∫

ϕ

Uc(ϕ, t) dϕ − kdeg Vac(t)

+ Peff krel

∫

ϕ

(Ic(ϕ, t) + Ac(ϕ, t)) dϕ (4)

Inactive virions Vinac accumulate in the medium and theratio of inactive virions to the overall amount of releasedvirions is given by (1 − Peff ).

dVinac(t)

dt= (1 − Peff ) krel

∫

ϕ

(Ic(ϕ, t) + Ac(ϕ, t)) dϕ

+ kdeg Vac(t) (5)

3. NUMERICAL SOLUTION

For simulation, the model was discretized with a finitevolume method with 128 logarithmically distributed con-trol volumes with respect to the internal coordinate. Thedelay was approximated with a transport system with 50control volumes. The overall system to be solved consistsof the discretized partial differential equations (1)-(3), thetransport system and the equations (4) and (5). The re-sulting large scale system of ordinary differential equationsis solved with the MATLAB solver ode15s. Computationscould be speeded up significantly by providing analyticalpatterns of the Jacobian with the solver option ”JPat-tern”.

4. ESTIMATION OF KINETIC PARAMETERS

4.1 Preprocessing of flow cytometric measurements

The flow cytometric data comprises the numbers of cellswith a specific fluorescence intensity on a logarithmicfluorescence grid with 1024 compartments. As the modelis solved numerically on an logarithmic grid with 128 finitevolumes, the measurement records are transformed to thesame grid by simply averaging the cell concentrationsover eight channels. For comparability, the flow cytometricmeasurements have to be converted into number densityfunctions as described in Muller et al. (2011). For thispurpose the average number of cells in each compartmentZi(tk) is normalized with the overall cell number of theflow cytometric sample. In addition, the number densityfunction is scaled by the overall concentration of cells inthe reactor C(tk) which can be measured independently:

ci(tk) =Zi(tk)∑i

Zi(tk)C(tk) , i = 1, . . . , 128 (6)

4.2 Translation of the inverse problem to a finite dimension

To adapt the presented model to experimental data aninverse problem has to be solved. This problem is set inan infinite dimension as the parameters characterizing the

kinetic processes of apoptosis, cell death, and virus repli-cation/release depend on the internal coordinate ϕ. Twogeneral solution approaches are described in literature. Inthe first one, the functional dependency of the parameteron the internal coordinate is assumed to be known and canbe described by an analytic funtion with a few parameters,e.g. a gaussian distribution characterized by mean andvariance (Sherer et al., 2008). If the shape of the function isnot known a priori, it can be parametrized with a suitableapproximation, e.g. piecewise constant, piecewise linearor spline approximation. The function is represented bya suitable number of nodes na. In this contribution, wefollow the approach suggested by Luzyanina et al. (2009)where the unknown functional dependencies are approxi-mated by piecewise cubic hermite - splines:

a(ϕ) =

na∑

j=1

ajφj(ϕ) (7)

where φj is a piecewise cubic polynomial defined on themesh Φ = [ϕ0, ϕ1, . . . , ϕna

]. This approximation yieldsa function which is continuous up to its first derivative.Two types of parametrization may be considered. Thefirst one assumes the nodes to be equidistantly distributedand only the values of the nodes are subject to theoptimization. The second one considers node location andnode value as parameters for the estimation procedure.Due to complexity issues, this contribution only deals withthe first approach.

4.3 Overall Parameter Estimation Setup

The overall vector of unknown parameters is given by

p = [ kvi, krel, kdeg, Peff ,

knet(ϕ1), . . . , knet(ϕns),

kapo(ϕ1), . . . , kapo(ϕns),

knet,apo(ϕ1), . . . , knet,apo(ϕns),

kcd(ϕ1), . . . , kcd(ϕns) ] (8)

In addition to flow cytometric distribution data, the con-

centration of active virions Vac and the total concentrationof virus particles V are recorded yielding the measurementvector

y(tk) =

c1(tk)...

c128(tk)

Vac(tk)

Vinac(tk)

=

c1(tk)...

c128(tk)

Vac(tk)

V (tk) − Vac(tk)

. (9)

The parameters can now be estimated in a generalizedleast squares sense

minp

nt∑

k=1

e(tk)T W(tk)e(tk) (10)

with

e(tk) = [y(tk) − y(tk,p)] . (11)

The weighting matrix W(tk) is chosen as

W(tk) = diag[max y−2

]. (12)

The reference values

y(tk) =

c1(tk)...

c128(tk)Vac(tk)

Vinac(tk)

(13)

with

ci = Uc,i + Ic,i + Ac,i , i = 1, . . . , 128 (14)

are obtained from the simulation of the model equationsas described previously.

The emerging nonlinear least squares problem is solvedwith the gradient based optimization routine SNOPT (Gillet al., 2002) which can be called from MATLAB throughan interface. SNOPT is a commercial software but a freestudent version is available. As this method can trap inlocal minima for non-convex objective functions, severalruns with different initial guesses were always performed.

Due to the complexity in the parameter estimation pro-cedure, in particular the high sensitivity of the delay onthe cost function, a two step procedure was implemented.At first, τ was kept fixed and the other parameters wereestimated. In a second step, all parameters exclusive of τwere set to the estimation results from the first step andτ was estimated.

Initial estimates for the unknown parameters without τare based on the ones given in Muller et al. (2011). Aninitial value for the delay of τ0 = 4 h is chosen.

5. RESULTS

5.1 Linear Parameter Dependency

Table 1. Initial estimates and estimation re-sults for constant model parameters

Parameter kvi krel kdeg Peff τ

Unit ml · h−1 h−1 h−1− h

Initial 5 · 10−8 600 0.1 0.02 4Linear 23.7616 · 10−8 598.63 0.1211 0.0308 3.99523-nodes 30.1773 · 10−8 587.57 0.1124 0.0279 3.9976

In a first step, the functional shape is approximated bya hermite spline with two nodes located at ϕmin andϕmax. Thereby, the unknown distributed parameters areassumed to depend linearly on the degree of fluorescence.In Figure 2 - 3 it can be seen that the model can befitted adequately to the data. The value of the leastsquares cost function (10) is 4.5752. The resulting constantparameters are summarized in Table 1. In Figure 4, thefunctional dependencies of the distributed parameters aredepicted. It can be seen that knet increases with risingdegree of fluorescence which is contrary to the assumptionspresented in our previous work. The apoptosis rate is alsoincreasing. One possible interpretation is that cells aremore likely to become apoptotic if their degree of infectionincreases. To interpret the estimation results for the netrate knet,apo one has to keep in mind that the fluorescencedegree of the apoptotic cells is generally decreasing.

100

101

102

103

0

1

2

3

4

5x 10

6 t=0.25 h

100

101

102

103

0

1

2

3

4

x 106 t=14 h

100

101

102

103

0

0.5

1

1.5

2

x 106 t=18 h

100

101

102

103

0

0.5

1

1.5

2

x 106 t=34 h

100

101

102

103

0

2

4

6

8

10

12x 10

5 t=54 h

100

101

102

103

0

1

2

3

4

x 105 t=90 h

ϕϕ

ϕϕ

ϕϕ

c[ (l

og(F

U)·m

l)−

1

]

c[ (l

og(F

U)·m

l)−

1

]

c[ (l

og(F

U)·m

l)−

1

]

c[ (l

og(F

U)·m

l)−

1

]

c[ (l

og(F

U)·m

l)−

1

]

c[ (l

og(F

U)·m

l)−

1

]

Fig. 2. Flow cytometric measurements (dotted) and model predictions for linear (solid) and nonlinear approach (dashed)

0 20 40 60 80 10010

2

103

104

105

106

107

108

0 20 40 60 80 100

106

107

108

109

1010

tt

Vac

[ ml−

1]

Vin

ac

[ ml−

1]

Fig. 3. Active and inactive virions: measurements (dotted) and model predictions for linear (solid) and nonlinearapproach (dashed)

1000 2000 30000

1

2

x 104

1000 2000 30000

20

40

60

1000 2000 3000−50

−40

−30

−20

−10

0

1000 2000 30000

0.02

0.04

0.06

ϕ

ϕ

ϕ

ϕ

knet(

ϕ)

kapo(ϕ

)k

net,

apo(ϕ

)k

cd(ϕ

)

Fig. 4. Estimation results for fluorescence dependent pa-rameters, linear approach (solid), nonlinear approach(dashed); corresponding spline nodes are displayed bydots and squares

It can be seen in Figure 4 that the absolut value of knet,apo

is decreasing into the direction of lower fluorescence levels.Hence the loss of fluorescence intensity is slowed down themore the apoptotic cells move to lower fluorescences. Theestimated cell lysis rate kcd implies that apoptotic cells inlower fluorescence regions are more probable to lyse. Incombination with the overall negative net rate knet,apo thefollowing interpretation comes to mind: the longer a cell isapoptotic the more probable it is to lyse.

5.2 Nonlinear Parameter Dependency

When the distributed parameters are approximated byhermite splines with three nodes, a nonlinear functionaldependency is assumed. As a result, the fit in the fluo-rescence distribution can be improved significantly par-ticularly for later sample points (see Fig. 2). The fit ofthe active virions is improved in later time samples, too.Comparing the least squares cost function (10) shows thatthe value drops to 3.8614. The estimation results for knet

and kapo show the same trend as in the case of linearparameter dependencies though the slopes are larger. Thecell death rate is large for apoptotic cells with a low degreeof fluorescence and decreases with increasing fluorescenceintensities. For ϕ → ϕmax the rate is increasing whichcan be explained in combination with the results for thenet rate of the apoptotic cells. The rate has a parabolicshape. This means that apoptotic cells with a low or a veryhigh intracellular amount of NP protein do only move veryslowly into regions of lower ϕ. One possible explanationmay be that apoptotic cells within regions of very highfluorescence intensity tend to die instead of staying apop-totic and moving to regions of lower fluorescence levels.

For more complex parameter functions with more thanthree nodes the parameter estimation problem became illposed resulting in multiple local minima.

6. CONCLUSION

A deterministic population balance model for influenzavirus replication in MDCK cell cultures has been pre-sented. The model comprises a system of hyperbolic par-tial differential equations using a fluorescence degree asinternal coordinate and ordinary differential equations.Unknown kinetic parameters were estimated from exper-imental data. Therefor, hermite spline approximations ofdifferent complexity for the distributed parameters wereconsidered to translate the inverse problem to a finitedimension. In a first step, a linear dependency on thedegree of fluorescence was assumed. This allows to repro-duce the virus dynamics and the distribution dynamicswith a reasonable agreement to the data. The fit couldbe improved further using nonlinear functional dependen-cies of the parameters on the internal coordinate. Themodel gives insight into the general mechanisms duringvaccine production. Future efforts will be made to includestructured information of intracellular processes like thevirus replication mechanism or apoptosis induction intothe model formulation.

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