Inverse Problem for Dynamical Systems with Uncertain and Scarce Data DAVID SWIGON UNIVERSITY OF PITTSBURGH

Inverse Problem for Dynamical Systems with Uncertain and Scarce Data DAVID SWIGON UNIVERSITY OF PITTSBURGH

Immune response to IAV infection: Experimental design Intranasal infection of Balb/c mice by Influenza virus A/PR/8/34 (H1N1) 5 pfu (lethal) or 5 pfu (sub-lethal) in PBS Flow Cytometry B-cells: CD19 (B-cell marker), CD69 (early activation marker), CD138 (plasma cells) T-cells: CD3, CD4 and CD8, CD69 Innate Cells: Granulocytes (Gr-1 (Ly-6G)high, CD11b high), Lung Macrophages (Gr-1 med, CD11b+, CD11c-, F4/8-), Common DC (cdc) (CD11b+, CD11c+), Plasmacytoid DC (pdcs) (CD11c+, CD11b-, Gr-1+, B22+, F4/8-) Luminex Analysis of cytokine profiles (IL-1a, IL-1b, IFN-γ, TNF-α, IL-1, IL-12, IL-6, IL-9, IL-13, KC, MIP- 1b, RANTES, MCP-1) Data measured on days, 1, 2, 3, 5, 7, (9, 11) Each data point is measured from a different mouse (>1 mice used in total). [Toapanta & Ross, Resp. Res., 29]

Antibodies Chemokines Effector cells (CD8) B-cells Type I IFN Virus NK cells Plasma cells Neutrophils Macrophages IL-12 Type II IFN IL-1 TNF-α Th cells (CD4) 5 pfu/ml 5 pfu/ml Why are the mice dying?

Immunology [Tamura & Kurata, JJID, 24]

Model Neutr tissue NO Healthy Infected IFNα Neutr blood Virus Ab B-cell IL-1 TNFα Σ 2 Dead CD8 P-cell IL-12 Mac Σ 1 CD4 NK IFNγ Chemo Inflammatory response Immune response [Price, Mochan, DS, Ermentrout, Lukens, Toapanta, Ross & Clermont, JTB, 215]

2 variables, 97 parameters Σ= + + 21 at 11 ad 12 a 22 + V hc bmcc M ' = µ ( ) hc hc M M bm a + C mc bm t Σ T' = µ lt gl 1 + g2 kl 1 + k2 Σ+ ( Σ+ )( ) L+ d L+ d V ' = g gvv (1 R) I g HV g AV µ V 1+ av hx gixix I ' = ghvvh b (1 ) hx hx ikrik bierie µ i R I a + X F ' = b (1 R) I + b P g IF µ F R = F / ( a vi vh va v v ix fi fp fi f rf + F) hk bc kc K ' = bk + g hk hk kirik µ kk a + C kc bgow + ggo bgkw + ggk G' = O+ K µ gg a + W a + W go 2 2 av 2 1 bm l Σ L' = µ l ( Lblh (1 R) H) gl 1 + g2 Σ+ L+ d bm c Σ C ' = µ cc gl 1 + g2 Σ+ L+ d gk ˆ ˆ bt nt rncnc N ' = µ ˆ Nˆ N a + a L+ T a + C r ˆ ncnc N ' = µ nn a + C bxnn X ' = gxhhx gxiix µ x X a + N g H ' = b (1 R) H D ( H θ ) g VH a hx ghxhx DH ' = b (1 ) ( ) hx hx h R H DH H θ a + X nt nl nc nc xn hx h hv hx hx hx H = H + I + DH + DI P h HX + X b bp max b x h x g V a G = P( + g D )( g + ) µ Pb ) pv p1 ' pd I p p( p apv + V a p2 + G b E ' = h a ho bopp O' = µ ho ho oo a + P bwoo W ' = bw + Pµ ww a + O B' = b ep op ep e P h e + P wo h e b RIE µ E + bwp( B B) µ B A' = b + b B g AV µ A a ab av a [Price, Mochan, DS, Ermentrout, Lukens, Toapanta, Ross & Clermont, JTB, 215] ei Key Ʃ Inflammatory Signal D Damage (dead cells) M Macrophages T Pro-inflammatory (TNFα) L Anti-inflammatory (IL1) C Chemokines Ñ Neutrophils (blood) N Neutrophils (tissue) X Toxin (NO) H Healthy Cells DH Cells killed by toxins V Influenza A Virus I Infected Cells F Type I Interferon (IFNα,β) K Natural Killer Cells G Type II Interferon (IFNγ) R Resistance A Antibodies P Antigen-presenting Cells (cdc) E CTLs (CD8) O T Helper Cells (CD4) W IL-12 B B-Cells A Antibodies (IgM) e DATA

Trajectory quantification (sublethal dosage) Sample size: 1 7 Running time: 1-2 weeks

Trajectory quantification (lethal dosage)

Survival quantification

Biological systems Model complexity Large number of variables and parameters Connectivity of interaction graph Data scarcity Insufficient number of variables observed Insufficient time resolution Quality Data inhomogeneity Multiple subjects combined in one data set Inadequate prior characterization Inclusion of dynamical expectations Noninformative priors Quantity Homogeneity

New (old?) directions Inverse problem for dynamical systems Identifiability from a single trajectory Existence and uniqueness of inverse Maximal permissible uncertainty Parameter estimation for ensemble models Transformation of measure Accurate prediction from scarce data Prediction uncertainty Noninformative and behavioral priors Parameter inference under L1 norm

Problems Initial value problem for ODEs Solution xx = ff xx, xx = bb R nn xx tt; bb CC 1 R nn FORWARD PROBLEM ff. xx(. ) INVERSE PROBLEM Existence & uniqueness Robustness Identifiability (uniqueness over a set of problems) Uncertainty of inverse and predictions for noisy data Answers depend on how much information is known about xx. and ff.

Identifiability from a single trajectory System xx = ff(xx; AA) is identifiable from trajectory with xx = bb iff there exists no BB R nn mm with AA BB such that xx ; AA, bb = xx( ; BB, bb). Not every trajectory can identify a system Some systems cannot be identified from any single trajectory

Linear-in-parameter systems Initial value problem for ODEs Solution xx = AAAA xx, xx = bb R nn xx tt; AA, bb CC 1 R nn AA R nn mm gg(xx) CC kk (R mm, R nn ) is assumed to be known (AA, bb) FORWARD PROBLEM INVERSE PROBLEM xx(. ; AA, bb)

= y xy x y x 3 1 1 2 b = 1 1 Solution obeys xx tt = yy(tt) and hence it is confined to a proper subspace. System with the same trajectory: ) ( ) ( y x y x = + = y xy x y x β β α α 3 1 1 2 Theorem (identifiability): For the linear-in-parameters model there exists no BB R nn mm with AA BB such that xx ; AA, bb = xx( ; BB, bb) iff gg(xx tt; AA, bb ) is not confined to a proper subspace of R mm. Example [Stanhope, Rubin, & DS, SIADS, 214]

Linear systems xx = AAAA, xx = bb R nn Theorem (identifiability): There exists no BB R nn nn with AA BB such that xx( ; AA, bb) xx( ; BB, bb) iff xx(tt; AA, bb) is not confined to a proper subspace of R nn. Theorem (non-confinement): xx(tt; AA, bb) is not confined to a proper subspace of R nn iff (a) bb is not confined to a proper AA-invariant subspace of R nn. (b) there is no left eigenvector of AA orthogonal to bb. Theorem (potential for identifiability): There exists bb R nn such that xx(tt; AA, bb) is not confined to a proper subspace of R nn if and only if AA has only one Jordan block for each of its eigenvalues. [Stanhope, Rubin, & DS, SIADS, 214]

= 2 2 4 2 3 1 A =.3.2 1 1.2 A = 1 1 1 1 A Examples No single trajectory will identify this system

Identification from discrete data on a single trajectory Discrete data: dd = xx, xx 1,, xx nn, xx ii = xx(tt ii ; AA, bb) One can partition the data space into domains in which the system is identifiable from data the system is robustly identifiable (on an open neighborhood) the system is identifiable to within countable number of alternatives the system has a continuous family of compatible parameters there is no parameter set corresponding to the data the system has a specific dynamical behavior (stable, saddle, )

Linear models Discrete data (uniformly spaced): dd = (xx, xx 1,, xx nn ) Solution map: xx kk = ee AAAA bb 1 Step 1: Get Φ = XX 1 XX XX kk = [xx kk xx kk+1 xx nn+kk1 ] Step 2: Solve ee AA = Φ Conditions for existence and uniqueness of real matrix logarithm were characterized by Culver.

Robust identification Theorem: There exists an open set UU R nn nn containing Φ such that for any Ψ UU the equation Ψ = ee AA aa ssssssssssssssss has aa uuuuuuuuuuuu ssssssssssssssss nnnn ssssssssssssssss AA R nn nn iff (a) {xx, xx 1,, xx nn1 } are linearly independent and only positive real or complex eigenvalues. (b) Φ has n distinct positive real eigenvalues. at least one negative eigenvalue of odd multiplicity. [Stanhope, Rubin, & DS, JUQ, 217]

Region of unique inverses Let Eigenvalues of Φ are determined by Conditions on y that guarantee positive real eigenvalues have been worked out for arbitrary dimension [Gantmacher; Yang] 1 1 1 ˆ X X X X = Φ Φ = Φ = y n y y y 1 1 ˆ 3 2 1 x n X y 1 = Example in 2D (regions of data which give unique inverse)

Regions of special properties T = tr Φ, D = det Φ xx xx 1

Uncertainty in identification from discrete trajectory Uncertain data: dd = ( xx, xx 1,, xx nn ) Bounds on uncertainty: CC dd, εε = { dd: max ii,jj Δxx ii,jj < εε} Neighborhood CC dd, εε is called permissible for property P iff P is shared by all systems corresponding to the data dd CC dd, εε. Value εε PP > is maximal permissible uncertainty for property P iff CC dd, εε is permissible for P for all < ε < εε PP, and CC dd, εε is not permissible for P for any εε > εε PP P can be Existence of inverse (unique or nonunique) Unique inverse Unique stable inverse

Inverse problem solution can be very sensitive to small changes in data. xx 5 =.4 xx 5 =.12 xx 5 =.13

Theorem (lower bound on uniqueness uncertainty): Let d be such that Φ has n distinct positive eigenvalues λλ 1, λλ 2,, λλ nn. Let mm 1 = 1 min λλ 2 ii λλ jj, mm 1 = min λλ ii, and ii<jj ii δδ UU = min{mm 1, mm 2 }. If < εε εε UU where δδ UU εε UU = nn(δδ UU + 1 + Λ ) SS 1 XX 1 SS with Φ = SSΛSS 1, then for any dd CC(dd, εε), Φ has n distinct positive eigenvalues. Proof uses a bound for perturbation of eigenvalues of Φ [Bauer and Fike]: λλ ii λλ ii SSSSSS 1 and the following estimate for EE = Φ Φ: SSSSSS 1 SS1 XX (1 + Λ ) XX 1 SS 1 SS 1 XX XX 1 SS [Stanhope, Rubin, & DS, JUQ, 217]

Theorem (upper bound on uniqueness uncertainty): Let d be such that Φ has n distinct positive eigenvalues λλ 1, λλ 2,, λλ nn, and let y be the last column vector of the companion matrix form Φ of Φ. The smallest neighborhood CC(dd, εε) that contains data dd for which companion matrix Φ has last column yy has εε = εε UU where εε UU = XX (yy yy) yy 1 + 1 Proof: For any perturbed data dd with companion matrix Φ, XX yy = xx nn XX yy + XX yy yy = xx nn εε UU is the solution of the linear programming problem of minimizing εε such that ww ii = nn1 jj= Δxx jj,ii yy jj+1 Δxx nn,ii, ii = 1,, nn where ww = XX yy yy. εε Δxx ii,jj εε, ii = 1,, nn, jj =,, nn A tight upper bound on εε UU is found by optimizing εε UU over all yy for which Φ does not have n distinctive positive eigenvalues. [Stanhope, Rubin, & DS, JUQ, 217]

Example: Comparisons of uncertainty estimates by different methods. d 1 6.65,, x 2 4.44 = 2

= 4.3 3.6, 4.44 6.65, 2 1 d Example: Different properties for the same data set give different uncertainties. = 4.3 3.1, 4.44 6.65, 2 1 ~ d

= 5.6 5, 4.44 6.65, 2 1 d Example: Different properties for the same data set give different uncertainties. = 5.6 5.4, 4.44 6.65, 2 1 ~ d

Ensemble model Initial value problem Solution map (discrete data) xx = ff xx, aa, xx = bb R nn yy = FF aa invertible on Ω Parameter distribution: ρρ(aa) Data distribution: ηη yy aa Ω yy FF(Ω) ρρ aa = ηη FF aa JJ(aa) JJ aa = det DD aa FF aa Inverse problem: Reconstruct ρρ(aa) from the knowledge of ηη yy and FF aa.

Parameter inference Common technique for inference of ρρ(aa): 1. Approximate ηη yy with multivariate Gaussian ηη yy obtained from data sample YY = yy 1, yy 2, yy NN 2. Estimate ρρ(aa) as a Bayesian posterior σσ aa YY with likelihood LL aa YY = ηη(ff aa ) and some noninformative prior ππ(aa) Compare: σσ aa YY ηη(ff aa )ππ(aa) (Bayes formula) ρρ aa = ηη FF aa JJ(aa) (Ensemble model) Observation: Asymptotically accurate estimate of ρρ aa can be obtained using Bayes formula only if The data distribution ηη yy is accurately represented Jacobian prior is used: ππ aa = JJ(aa) [DS, Stanhope, Rubin, 217]

Examples Method 1 Method 2 Prior choices Uniform, Jeffreys (1 aa), Jacobian JJ aa [DS, Stanhope, Rubin, 217]

Method 1, linear dynamical system, uniform data distribution U ( x ij ε, x + ε ) ij x = Λx d 1 6.253 5 =,, 2 1.5624 2.3 ε =.436.524 Λ =.4839.7491.9519 a = ( λ11, λ12, λ21, λ22) MM Jeff MM Jac AA MM Unif

1 5 4 d =,, 2 2.3 2.2 ε U =. 134 1 Λ = 1 1.5 2 MM Jac MM Jeff AA MM Unif Jeffreys works well 5 1 3 d =,, ε U =1. 272 15 1 15 1.1347 Λ = 1.4213.2187.3694 MM Jeff Uniform works well

Method 2, linear dynamical system, Gaussian parameter distribution, KDE used for smoothing the data distribution. 1 Λ = 1 1.5 2 σ =.25 xx 11 xx 12 xx 21 xx 22 MM Jac MM Jeff AA MM Unif

Method 1, influenza dynamics, uniform data distribution U (( 1σ ) x ij,(1 + σ ) xij ) V = ri cv H = βhv I = βhv δi σ =.1 t = {1, 2} AA MM Jac 8 a = ( V, H, β, r, c, δ ) = (.93, 4 1, 2.7 1 5,.12, 3., 4.) MM Jeff MM Unif Uniform works well

Method 1, influenza dynamics, uniform data distribution U 1σ ) x ij,(1 + σ ) x ) V = ri cv H = βhv I = βhv δi (( ij σ =.1 t = {2, 3} AA MM Jac 8 a = ( V, H, β, r, c, δ ) = (.93, 4 1, 2.7 1 5,.12, 3., 4.) MM Jeff MM Unif Jacobian works well

Method 2, influenza dynamics, Gaussian parameter distribution, KDE used for smoothing the data distribution. V = ri cv H = βhv I = βhv δi t = {1, 2} AA 8 a = ( V, H, β, r, c, δ ) = (.93, 4 1, 2.7 1 σ = (.1,.5,.1,.5,.2,.1) 5,.12, 3., 4.) MM Jac MM Jeff MM Unif

V = ri cv H = βhv I = βhv δi t = {2, 3} AA 8 a = ( V, H, β, r, c, δ ) = (.93, 4 1, 2.7 1 σ = (.1,.2,.1,.1,.2,.1) 5,.12, 3., 4.) MM Jac MM Jeff MM Unif

Computational aspect Accurate representation of ηη yy requires large amount of data. KDE method can be used but introduces errors. Jacobian is expensive to compute, but still less expensive than direct inverse of FF aa. Methods Exact formula (available for linear models) Numerical integration of the first variational equation (sensitivity coefficients) Numerical differentiation Broyden s method (rank-1 update) [DS, Stanhope, Rubin, 217]

Summary Challenges with poorly supported models require rethinking of the model parametrization and inference procedures Inverse problem techniques should be explored and developed, bound estimates complement probabilistic approaches Parameter estimation of ensemble models requires the computation of Jacobian Priors should take into account dynamical features of modeled systems

Credits Inverse problems Jon Rubin Shelby Stanhope Influenza modeling Gilles Clermont Bard Ermentrout Baris Hancioglu Ian Price Ericka Mochan Funding NIGMS NSF-RTG