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Model Reduction for Biochemical Systems:

Computational Methods

Tom Snowden

T Snowden Model Reduction 1 / 42

Computational Reduction

For high dimensional, complex models many of the analyticalapproaches to model reduction (discussed in the previous presentation)will be di�cult to apply, as they often depend upon the researcherpossessing high degree of model intuition.

Instead it is common to seek computational algorithms for theapplication of model reduction in such settings.

In this presentaton we discuss a range of such methods anddemonstrate computational reduction via application to an example.


Presentation outline

How I de�ne model reduction

Review of existing methods

An example

Linking with pharmacokinetics

Conclusions


Chemical reaction network theory

Biochemical reaction networks are typically de�ned via systems ofinteracting chemical equations. Such networks can be expressed via threesets of information:

An n dimensional set S representingthe species in the network.

A p dimensional set C representingthe `complexes' in the network.

An m dimensional set R ⊂ C × Crepresenting the reations in thenetwork.

Example:

2A→ D

A + B � C → D + B

S = {A,B,C ,D} ,C = {2A,C ,D + B,A + B,D} ,R = {(2A,D), (A + B,C),

(C ,A + B), (C ,D + B)} .


Stoichiometric representation

Example:

2Ak1→ D

A + Bk2

�k3

Ck4→ D + B

N =

−2 −1 00 −1 10 1 −11 0 1

v =

k1x21 (t)

k2x1(t)x2(t)− k3x3(t)k4x3(t)

It is common to describe thedynamics of such networks enmasse via the Law of MassAction.

One common representation isvia the product of astoichiometry matrix N and avector of reaction rates v(x ,p),such that

x = Nv(x ,p)

where x gives the time-varyingmolecular concentration of eachof the species and p is a set ofparameters.


Control theoretic representation

However, it is also common for certain applications to seek to representsuch models in a control theoretic state-space representation, such that

x(t) = f (x(t),p) + g(x(t),p)u(t),

y(t) = h(x(t),p),

with:

u(t) ∈ Rl representing inputs which can be interpreted in some way ascontrolling the system.

y(t) ∈ Rv representing combinations of the species that can beconsidered outputs.

Within the context of QSP, the inputs may represent the dose of a drugwhilst the ouputs might represent the concentrations of species associatedwith some clinical response.


De�nition of model reduction

ε = ‖y(t)− y(t)‖

Hence, I de�ne a method of model reduction to be any method designed togive a system capable of satisfactorily reproducing the input-output

behaviour of the original model (under some given metric of error)whilst producing a reduction in the number of species S, reactions R,or complexes C.


Reducing systems biology models

Common disadvantages

1 Sti�ness:

K = λmax(Jf (x))λmin(Jf (x)) � 1

Presents issues for numericalmethods.

2 Nonlinearity: f (ax) 6= af (x)Presents issues for analyticalmethods.

3 Conservation relations:

∃Γ ∈ Rα×n : Γx(t) = xT , ∀tMust be handled carefully toavoid violation.

Common advantages

1 Asymptotic Stability:

limt→∞ ‖x(t)− x∗‖ = 0Enables a lot of theory.

2 Conservation relations:

xc = xT − Γcx i

Can be exploited to reducesystem for `free'.

Di�culty also arises from the wide range of aims associated with modelling in the �eld

of systems biology. The best available reduced model necessarily depends upon what it

will be used for.





An example


Conclusions


Literature Review Introduction

The review limited itself to methods addressing deterministic

systems of ODEs and which had seen application to models of

biochemical reaction networks. Emphasis was placed on methods

with published use since 2000.

This section begins by reviewing computational approaches for theapplication of conservation analysis.

It then moves on to reviewing model reduction methods, these are dividedinto 4 categories:

1 Time-scale exploitation methods;

2 Optimisation approaches and sensitivity analysis;

3 Lumping; and

4 Singular value decomposition (SVD) based methods.


Conservation relations

α conservation relations imply that ∃Γ ∈ Rα×n : Γx(t) = xT , ∀t.

The conservation relations correspond to linear dependencies in therows of the stoichiometry matrix N.

It is possible to show1 that Γ = Null(NT ).

A numerically stable method for obtaining this null-space for largesystems is to employ QR factorisation via Householder re�ections 2.

1Reder, J. Theor. Biol., 1988.2Vallabhajosyula et al., Bioinformatics, 2006.


Time-Scale Exploitation Methods I

This refers to any method thatexploits the often largedi�erences in reaction rates thatcan occur within a biochemicalsystem.

Typically such methods partitionthe system into fast and slowcomponents - after some initialtransient period those fastportions are assumed to be inequilibrium with respect to theremainder of the network.

Such methods include singularperturbation approaches, ILDM,and CSP.

X1

X2

X3

X4

X5

S

L

O

W

F

A

S

T

Figure: An example of model reductionvia time-scale analysis


Time-Scale Exploitation Methods II

Singular Perturbation

If a system of ODEs can beexpressed in the form

x(t) = f (x , z , t) ,

δz(t) = g (x , z , t) ,

then as δ → 0 this system can beapproximated by

x(t) = f (x , z , t) ,

z(t) = φ (x , t) ,

with φ (x , t) a root of theequations g (x , z , t) = 0.

Species Partitioning(xs

δx f

)=

(Ns

Nf

)v (xs , x f ,p)

Reaction Partitioning

x = (Ns Nf )

(v s (x ,p)

δ−1v f (x ,p)

).

x can then be decomposed intofast and slow contributions as asum, such that x = [x ]

s+ [x ]

f.

Hence

[x(t)]s

= Nsv s (x(t),p) ,

0 = Nf v f (x(t),p) .


Time-Scale Exploitation Methods III

PROS:

Species can maintain biologicalmeaning.

A large number of such methodsexist in the literature.

These methods are typicallyvalid in the reduction ofnonlinear systems.

CONS:

A system may not have a largeenough time-scale seperation tojustify reduction.

What happens during the initialtransient period may be ofinterest.

If a slow/fast partitioning is notknown a priori approaches fordetermining the mostappropriate one can becomputationally expensive.


Optimisation and Sensitivity Analysis Methods I

Reduction can be expressed as anoptimisation problem - i.e. obtain thelowest possible dimensional model(either in terms of species, reactions orcomplexes) for which a metric of error εremains within an acceptable bound,such that ε < εc .

Hence it is common to either:

1 Seek to measure how `sensitive' theconstraint variable ε is toperturbations and use this to guide areduction. Or;

2 Employ an iterative optimisationprocedure.

X1

X2

X3

X4

X5

X1

X2

X3

C

O

N

S

T

A

N

T

Figure: An example of modelreduction via optimisation


Optimisation and Sensitivity Analysis Methods II

A typical optimisationproceedure might involve`switching o�' ofreactions or species.

For example, kineticparameters can be givenswitch variables,

It is then an integerprogramming problemwith these switches todetermine a minimalreduced modelconstrained by an errorbound 3.

3Maurya et al., IET Syst Biol.,

2009.T Snowden Model Reduction 16 / 42

Optimisation and Sensitivity Analysis Methods III

PROS:

Species can maintain theirbiological meaning.

The application of such methodscan be highly algorithmic andcomputationally e�cient (e.g.heuristic approaches such asgenetic algorithms).

Common procedures areimplemented well in a number ofsoftware packages.

CONS:

For very large systemsperforming a su�ceint searchthrough the range of candidatesolutions may be highlycomputationally expensive.

Similarly, for sensitivity analysisconvincingly searching the entireparameter space may beimpossible.


Lumping Based Methods I

Lumping is a classi�cationthat encompasses a range ofmethods.

In particular it pertains toany method that constructsa reduced system withstate-variables correspondingto subsets of the originalspecies.

These new states are referredto as `lumped' variables.

X1

X2

X3

X4

X5

Y1

Y2

Y3

X1

X2

X3

X4

X5

Y1

Y2

Y3

(a) (b)

Figure: (a) Proper lumping - each of theoriginal species corresponds to, at most, oneof the lumped states. (b) Improper lumping- each of the original states can correspondto one or more of the lumped states.


Lumping Based Methods II

Applying a lumping:

A set of species can be reduced via

some proper, linear lumping4

L ∈ {0, 1}r×n giving a reduced set of

species x ∈ Rr where x = Lx .

Via the Galerkin projection we can

obtain a reduced dynamical system of

the form:

˙x = Lf (Lx , p) + Lg(Lx , p)u

y = h(Lx , p).

Here L represents a generalised inverse

of L such that LL = Ir .

4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42

Lumping Based Methods III

PROS:

Lumping is a common methodin the reduction of chemicalkinetics - quite a large range ofliterature exists.

Algorithmic approaches that canbe implemented computationallyexist.

Lumped variables can be chosento be biological meaningful suchthat the reduced model maintinssome degree of biologicalintuitiveness.

CONS:

Many of the procedures in theliterature are highlycomputationally expensive forlarge systems.

Most methods in the literaturepertain to linear, proper lumping- better reduction is likely to beachieved by nonlinear and/orimproper lumping techniques,but this may lead to loss ofbiological meaning.


Singular Value Decomposition Based Approaches I

These methods are based upon thesignular value decomposition (SVD).

Crucially, via Eckart-Young-Mirskytheorem5 the SVD provides a way toapproximate a matrix via one oflower rank.

The most commonly applied suchmethod is balanced truncation.

X1

X2

X3

X4

X5

Z1

Z2

Z3

u

u

y

y

~

Figure: Balanced truncation reducesa model whilst seeking to preservethe input-output relationship

5Eckart& Young, Psychometrika, 1936.T Snowden Model Reduction 21 / 42

Singular Value Decomposition Based Approaches II

Linear balanced truncation istypically applied to linearsystems of the form

x = Ax + Bu,

y = C x .

It requires the computation oftwo matrices P and Q:

1 The controllability GramianP provides information onhow the state-variables xrespond to perturbations ininputs u.

2 The observability Gramian Qprovides information on howthe outputs y respond toperturbations in thestate-variables x .

Balanced truncation done quick

1 Perform Cholesky factorisation ofboth gramians

P = LTL, Q = R

TR.

2 Take SVD of matrix LRT to obtain

LRT = (U1 U2)

(Σ1 00 Σ2

)(VT

1VT

2

)Where U1 is an n × r matrix, Σ1 is

an r × r diagonal matrix and VT

1 is a

r × n matrix.

3 Set

T1 = Σ− 1

2

1 VT

1 R, S1 = LTU1Σ

− 1

2

1 .

4 Finally

˙x = T1AS1x + T1Bu,

y = CS1x .


Singular Value Decomposition Based Approaches II

PROS:

Control theoretic description �tsneatly with the idea of systemspharmacology (i.e. the drugcontrolling subcellularprocesses).

They are highly algorithmicmethods - can potentially beautomated in a straightforwardmanner.

An a priori error bound can beobtained.

CONS:

Transformed/reduced states nolonger have biological meaning -only inputs and outputs preservetheir meaning.

Standard approach only existsfor linear models - butgeneralisations for nonlinearsystems do exist.

For large systems, empiricalbalanced truncation can behighly computational expensive.


Miscellaneous Methods

A number of other methods, with a limited publication record, do existincluding:

Motif replacment methods;

Methods for reduction of combinatorial complexity;

Complex reduction; and

Publications addressing general reduction heuristics.


Conclusions of Literature Review I

This literature review enabled several speci�c conclusions:

There is no `one-size-�ts-all' method of model reduction.

Whilst many of these methods can be highly automated, the onus ison the modeller to choose the correct tool for the task.

Consider what the reduced model will be used for to judge whichmethod is most appropriate.


Conclusions of Literature Review II





An example


Conclusions


Aims

X1

X2

X3

X4

X5

Y1

Y2

Y3

Figure: An example of aproper lumping

This section introduces acomputational modelreduction algorithm developedduring my PhD.

Three existing methods arebrought together in thisapproach:

Conservation analysis.

Proper lumping.

Empirical balancedtruncation.

X1

X2

X3

X4

X5

Z1

Z2

Z3

u

u

y

y

~

Figure: Schematic outline

of Balanced Truncation -

the method focuses on

preserving the

input-output relationship

of the system.


Combined model reduction algorithm

Given this context, we have

developed the an algorithm for

model reduction which combines

previously existing methods in a

novel way. The following

schematic outlines its operation:


Combined method justi�cation

The core justi�cation of the combined reduction algorithm is the

use of proper lumping as a preconditioner for the application of

empirical balanced truncation.

Empirical balanced truncation (EBT) should, in theory, produce moreaccurate reduced networks than proper lumping.

In practice, EBT often fails for highly sti� systems.

Proper lumping, however, will tend to sum together thosestate-variables that interact on faster timescales than their neighbours.

Hence the reduced model will often contain a smaller range oftimescales and be less sti� with each additional dimension eliminated.


ERK Activation Model

Figure: Block schematic overview of EGF

and NGF dependent ERK signalling

network7. Model consists of 150 reactions

and 99 species. There are 23 conservation

relations in this system enabling the model

to be reduced to 76 states.

c1.SOS

c1.pSOS

c1.SOS_Grb2

c1.Grb2

c1.Dok

c1.pDok

c1.Crk

c1.FRS2

c1.Shc

c1.pSOS_Grb2

c1.Rap1_GDP

c1.MEK

c1.MKP3

c1.pShc_dpEGFR

c1.dpEGFR_c_Cbl

c1.B_Raf_Rap1_GTP

c1.pShc_dpEGFR_c_Cbl

c1.pFRS2_dpEGFR_c_Cbl

c1.Shc_dpEGFR

c1.c_Cbl

c1.RasGAP c1.c_Raf

c1.B_Raf

c1.ERK

c1.PP2A

c1.Ras_GDP

c1.Rap1GAP

c1.C3G

c1.pShc

c1.pFRS2_dpEGFR

c1.pTrkA_endo

c1.MEK_ERK

c1.pMEK_ERK

c1.FRS2_dpEGFR_c_Cbl_ubiq

c1.Crk_C3G_pFRS2_dpEGFR_c_Cbl

c1.pShc_dpEGFR_c_Cbl_ubiq

c1.Crk_C3G_pFRS2_dpEGFR

c1.Grb2_SOS_pShc_dpEGFR_c_Cbl_ubiq

c1.Grb2_SOS_pShc_dpEGFR_c_Cbl

c1.Shc_dpEGFR_c_Cbl_ubiq

c1.dpEGFR_c_Cbl_ubiq

c1.proteasome

c1.Grb2_SOS_pShc

c1.Shc_dpEGFR_c_Cbl

c1.Grb2_SOS_pShc_dpEGFR

c1.pFRS2

c1.FRS2_dpEGFR

c1.pDok_RasGAP

c1.pMEK

c1.FRS2_dpEGFR_c_Cbl

c1.pFRS2_dpEGFR_c_Cbl_ubiq

c1.Ras_GTP

c1.Crk_C3G_pFRS2_dpEGFR_c_Cbl_ubiq

c1.c_Raf_Ras_GTP

c1.B_Raf_Ras_GTP

c1.ppMEK

c1.ppERK

c1.Crk_C3G

c1.Rap1_GTP

c1.ppMEK_ERK

c1.dppERK

c1.Shc_pTrkA

c1.Shc_pTrkA_endo

c1.pShc_pTrkA

c1.pFRS2_pTrkA

c1.FRS2_pTrkA

c1.pShc_pTrkA_endo

c1.FRS2_pTrkA_endo

c1.pFRS2_pTrkA_endoc1.Crk_C3G_pFRS2_pTrkA_endo

c1.Grb2_SOS_pShc_pTrkA

c1.Crk_C3G_pFRS2_pTrkA

c1.Grb2_SOS_pShc_pTrkA_endo

c1.c_Raf_Ras_GTP_MEK

c1.c_Raf_Ras_GTP_pMEK

c1.c_Raf_Ras_GTP_MEK_ERKc1.c_Raf_Ras_GTP_pMEK_ERK

c1.B_Raf_Ras_GTP_MEK

c1.B_Raf_Ras_GTP_pMEK

c1.B_Raf_Ras_GTP_MEK_ERK

c1.B_Raf_Ras_GTP_pMEK_ERK

c1.B_Raf_Rap1_GTP_MEK

c1.B_Raf_Rap1_GTP_pMEK

c1.B_Raf_Rap1_GTP_MEK_ERK

c1.B_Raf_Rap1_GTP_pMEK_ERK

c1.ppERK_MKP3

c1.dppERK_MKP3

c1.pro_TrkA

c1.pro_EGFR

c1.degradation

compartment.EGFR

compartment.L_EGFR

compartment.L_EGFR_dimer

compartment.L_dpEGFR

compartment.NGFRcompartment.pTrkA

compartment.L_NGFR

compartment.NGF

compartment.EGFform_EGFreceptor

EGFbinding

dimerization

binding_SOS_Grb2

binding_pSOS_Grb2

EGFRphosphorylation

binding_cCbI_dpEGFR

binding_pShc_LdpEGFR

pDOKdephosphorylation

binding_cCbl_pShc_dpEGFR

SOSdephosphorylation

pSOS_Grb2_dephosphorylation

binding_Shc_LdpEGFR

Shc_dpEGFR_phosphorylation

dpEGFR_c_Cbl_ubiquitination

dpEGFR_cCbl_degrad

binding_cCbl_Shc_dpEGFR

Shc_dpEGFR_c_CBl_Ubiquitination

Shc_dpEGFR_c_Cbl_ubiq_DegradationpShc_dpEGFR_c_Cbl_ubiquitination

pShc_dpEGFR_c_Cbl_ubiq_degradation

Shc_dpEGFR_c_Cblphosphorylation

binding_Grb2_SOS_pShc

binding_Grb2_SOS_pShc_dpEGFRbinding_Grb2_SOS_pShc_dpEGFR_1

binding_c_Cbl_Grb2_SOS_pShc_dpEGFR

binding_Grb2_SOS_pShc_to_dpEGFR_c_Cbl

Grb2_SOS_pShc_dpEGFR_c_Cbl_ubiquitination

Grb2_SOS_pShc_dpEGFR_c_Cbl_ubiq_degradation

Grb2_SOS_pShc_Dissociation

Unnamed Reaction

pShc_dephosphorylation

pFRS2_dephosphorylation

binding_Crk_to_C3G

binding_L_dpEGFR_to_FRS2

binding_pFRS2_to_L_dpEGFR

FRS2_dpEGFRphsphorylation

binding_Crk_C3G_to_pFRS2_pRTK

binding_c_Cbl_to_FRS2_dpEGFR

binding_c_Cbl_to_pFRS2_dpEGFRpFRS2_dpEGFR_c_Cbl_ubiquitiation

FRS2_dpEGFR_c_Cbl_ubiquitination

FRS2_dpEGFR_c_Cbl_phosphorylation

binding_Crk_C3G_to_pFRS2_pFRS2_dpEGFR_c_Cbl

Crk_C3G_pFRS2_dpEGFR_c_Cbl_ubiquitination

FRS2_dpEGFR_c_Cbl_ubiq_dissociation

pFRS2_dpEGFR_c_Cbl_ubiq_dissociation

binding_RasGAP_to_pDOK

SOS_Grb2_phosphorylationSOS_phosphorylation

binding_c_Raf_to_Ras_GTP

binding_B_Raf_to_Rap1_GTP

binding_B_Raf_to_Ras_GTP

ppMEK_dephosphorylation

pMEK_dephosphorylation

ppMEK_ERK

pMEK_ERK_dephosphorylation

ppERK_dimerization

Ras_GTP_dephosphorylation

Rap1_GTP_dephosphorylation

Rap1_GTP_phosphorylation

Ras_GDP_phosphorylation

binding_NGF_to_NGFRTrkA_phosphorylation

pTrkA_intermalization

pTrkA_endo_degradation

pTrkA_degradation

binding_Shc_to_pTrkA

binding_pShc_to_pTrkA

binding_FRS2_to_pTrkA

binding_pFRS2_to_pTrkA

binding_Shc_to_pTrkA_endo

binding_pShc_to_pTrkA_endo

Shc_pTrkA_endo_phosphorylationShc_pTrkA_phosphorylation

pFRS2_pTrkA_phosphorylation

binding_FRS2_to_pTrkA_endo

binding_pFRS2_to_pTrkA_endo

FRS2_pTrkA_endo_phosphorylation

FRS2_pTrkA_degradation

pFRS2_pTrkA_degradation

Shc_pTrkA_degradation

pShc_pTrkA_degradation

FRS2_pTrkA_endo_degradation

Shc_pTrkA_endo_degradation

pShc_pTrkA_endo_degradation binding_Grb2_SOS_to_pShc_pTrkA_endo

binding_Grb2_SOS_to_pShc_pTrkAGrb2_SOS_pShc_pTrkA_ubiquitination

Crk_C3G_pFRS2_pTrkA_ubiquitinationpFRS2_pTrkA_ubiquitination

FRS2_pTrkA_ubiquitination

pShc_pTrkA_ubiquitination

Shc_pTrkA_ubiquitination

binding_Crk_C3G_to_pFRS2_pTrkA

binding_Crk_C3G_to_pFRS2_pTrkA_endo

binding_Grb2_SOS_pShc_to_pTrkA

binding_Grb2_SOS_pShc_to_pTrkA_endo

Crk_C3G_pFRS2_pTrkA_degradation

Crk_C3G_pFRS2_pTrkA_endo_degradation

Grb2_SOS_pShc_pTrkA_degradation

Grb2_SOS_pShc_pTrkA_endo_degradation

pFRS2_pTrkA_endo_degradation

form_NGFreceptor

binding_Shc_to_dpEGFR_c_Cbl

binding_pShc_to_dpEGFR_c_Cbl

binding_SOS_Grb2_to_pShc_dpEGFR_c_Cbl

binding_c_Cbl_to_Crk_C3G_pFRS2_dpEGFR

binding_FRS2_to_dpEGFR_c_Cbl

binding_pFRS2_to_dpEGFR_c_Cbl

Ras_GTP_dephosphorylation_1

RAP1_GTP_dephosphorylation

Dok_phosphorylation

Grb1_SOS_pShc_dissociation

binding_MEK_to_ERK

binding_ERK_to_pMEK

binding_ERK_to_ppMEKppMEK_ERK_dissociationc_Raf_Ras_GTP_dissociationB_Raf_Ras_GTP_dissociation

B_Raf_Rap1_GTP_dissociation

Unnamed 1

Unnamed 2

Unnamed 3Unnamed 4

Unnamed 5

Unnamed 6

Unnamed 7

Unnamed 8

Unnamed 9

Unnamed 10

Unnamed 11

Unnamed 12

Unnamed 13

Unnamed 14

Unnamed 15Unnamed 16

Unnamed 17

Unnamed 18

Unnamed 19

Unnamed 20

Unnamed 21

Unnamed 22

Unnamed 23

Unnamed 24

Unnamed 25

Unnamed 26

Unnamed 27

Unnamed 28

Unnamed 29

Figure: Full ERK activation pathway model

in petri-net form

6Sasagawa et al., Nat. Cell Biol., 2005.7Sasagawa et al., Nat. Cell Biol., 2005.T Snowden Model Reduction 31 / 42

ERK Activation Reduction Results I

Results for thereduction of the 99dimensionalErk-activation model.`#' implies Matlabcould not simulatethis reduction usingode15s due tonumerical error. `-'implies the error atthis point was equalto the lumping error.


ERK Activation Reduction Results II

Figure: Simulated results for the output of the original 99-dimensional ERK activation modelvs the reduced 8 dimensional model. This plot emphasises the fact that the reduced model isdesigned to remain valid for any reasonable change in input. The system starts by being a�ectedby an agonist that increases the rate of EGF binding by 25% for over an hour (4000 seconds), atthis point the input �ips to an antagonist decreasing the rate of EGF binding by 25% and runsfor the same time period (an additional 4000 seconds). At any given time point the errorbetween the original and reduced model exceeds no more than 5%.





An example


Conclusions


Reducing PBPK models I

In this section we explore the application of model reduction methodsto models of pharmacokinetics.

Pharmacokinetic models are typically linear which enables moreaccurate reduction as compared with, typically nonlinear, models ofbiochemical reaction networks.

A brief study of applying model reduction methods to physilogicallybased pharmacokinetic models was underaken.

The PBPK system we chose to employ was a deterministic, linear,16-dimensional, compartmental model.


Reducing PBPK models II

Analysis was made of bothlumping and standardbalanced truncation as ameans for the reduction thissystem. Balanced truncationwas found to give the bestresults


Linking I

Questions include:

Should spatial inhomogeneityin di�usion be explicitlyaccounted for?

What is the cumulativee�ect of the cellularresponse?

Should di�erent cell types(e.g. diseased and healthy)and their di�erences in druga�nity be accounted for?


Linking II

We made the simplifyingassumption that the tissuee�ects were accounted for bythe PBPK model and thatthe cells/receptors werehomogeneously distributed inthe relevant tissuecompartment.

Hence they are partiallydecoupled and can bereduced separately as in theschematic given on the right.


Linking Results: ERK activation

Figure: Linking the 10 dimensional reduced version of the ERK activation modelobtained under the combined model reduction algorithm with a 3 dimensionalreduced version of the PBPK model obtained via balanced truncation yields theresults above. In comparison to a linked version of the original model, the reducedversion maintains a 3% error bound.





An example


Conclusions


Conclusions

We have hopefully demonstrated that model reduction methods canproduce signi�cant simpli�cations in a system whilst retaining a highdegree of accuracy.

The literature review shows that a wide range of such methods currentlyexist.

The aims of such reduction might include seeking to speed up simulationtime, obtaining a model of an appropriate scope relative to the avilabledata, or trying to analyse which components of a model are mostresponsible for driving the dynamical behaviour of interest.

Crucially, the optimal reduction method is deeply dependent upon yourresearch question!


Thank you for listening.

Acknowledgments

Thank you to P�zer and EPSRC for their �nancial support throughoutthe PhD.

Thank you to Marcus Tindall and Piet van der Graaf for theirsupervision throughout the project.


APPENDIX


Petrov-Galerkin Projection

System trajectories can often be well approximated in a lower dimensional

subspace S : dim (S) = r .

Select a test basis B ∈ Rn×r of S, such that x(t) ≈ B x(t) with x(t) ∈ Rr

represents our reduced state vector.

Hence, B ˙x(t) = f (B x(t), p, u(t)) + r(t) where r(t) represents the residual

incurred via our approximation.

Constrain the residual to be orthogonal to a subspace C with an associated test

basis C ∈ Rn×r such that CT r(t) ≈ 0.

Therefore we left multiply by CT to obtain CTB ˙x(t) = C

T f (B x(t), p, u(t))

Assuming CTB is non-singular we can obtain

˙x(t) =(CTB

)−1

CTf (B x(t), p, u(t))

y = g (B x(t), p)

If B = C this is a special case known as a Galerkin projection.


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