Sensitivity Analysis of a Cancer Model with Drug Treatment

Transcription

1 Sensitivity Analysis of a Cancer Model with Drug reatment John A. Burns & Golnar Newbury Interdisciplinary Center for Applied Mathematics Virginia Polytechnic Institute and State University Blacksburg, Virginia Atlantic Coast Symposium on the Mathematical Sciences in Biology and Biomedicine April 24-26,28 North Carolina State University Raleigh, NC Atlantic Coast Symposium MSBB

2 Observations Math-Bio Modelers urbulence Modelers REPLACE IGNORANCE BY FICION d () t = a( t τ) a () t d () t c () t I() t u ( t) ( t) dt Q () 5 I 6 Q 4 Q 5 Q Q di () t (2) = 2 a4m() t a5t I( τ) a6t Q() ctit I() () d2t I() at I( τ) dt dm () t (3) = ai( t τ) d3m() t a4m() t c3m() t I() t u2( t) M( t) dt n di() t ρ I()[ t Q() t + I() t + M ()] t (4) = k + c n dt α + [ ( t) + ( t) + ( t)] Q I M 4 M 6 Q 3 2 I() t () t c I() t () t c I() t () t d I() t u () t I() t I Who are going to believe, me or your own eyes? Juanita Hutchins first husband - Baja Oklahoma CAN BE A VERY GOOD HING Atlantic Coast Symposium MSBB 2

3 People and Problem Adam Childers, Golnar Newbury (Virginia ech) John Singler (Oregon State) Ed Allen, David Gilliam (exas ech University) Lisa Davis (Montana State University) GIVEN A PARAMEERIZED MODEL AND A COMPUED VALUE BASED ON HIS MODEL, WHEN CAN WE RUS HE RESULS? -- CAN WE BE SURE WE HAVE CONVERGED O HE CORREC VALUE OF INERES? ARE HERE PARAMEERS (SMALL, UN-MODELED) HA PRODUCE FALSE OR UNEXPECED RESULS? -- CAN WE FIND INDICAORS / PRECURSORS O HESE PROBLEMS? -- WHEN IS A COMPUAIONAL PROBLEM DUE O AN ALGORIHM OR EVEN MACHINE PRECISION AND HOW CAN WE ELL? Atlantic Coast Symposium MSBB 3

4 Some Fundamental Issues GIVEN A PARAMEERIZED MODEL AND A COMPUED VALUE BASED ON HIS MODEL, WHEN CAN WE RUS HE RESULS? -- CAN WE BE SURE WE HAVE CONVERGED O HE CORREC VALUE OF INERES? ARE HERE PARAMEERS (SMALL, UN-MODELED) HA PRODUCE FALSE OR UNEXPECED RESULS? -- CAN WE FIND INDICAORS / PRECURSORS O HESE PROBLEMS? -- WHEN IS A COMPUAIONAL PROBLEM DUE O AN ALGORIHM OR EVEN MACHINE PRECISION AND HOW CAN WE ELL? SENSIIVIIES HELP Atlantic Coast Symposium MSBB 4

5 Cancer Models BREAS CANCER CELLS Cancer cell being attacked by the immune system Atlantic Coast Symposium MSBB 5

6 Background In the U.S. 4% chance for the average person to develop cancer Breast cancer is the 2nd most common cancer among American women Risk factors: incidence in family, oral contraceptives, obesity Normal cells have many checkpoints During checkpoints reproduction is stopped if abnormality is detected Cancer cells don t have these checkpoint. Unmanageable proliferation leads to loss of genetic information Recent cancer cells more mutated than older cancer cells. Atlantic Coast Symposium MSBB 6

7 he Cancer Cell Cycle 4 stages to cell cycle: G (presynthetic) S (synthetic) G2 (postsynthetic) mitosis G (quiescent) INERFACE SAGE Immune cells cytotoxic -cells flow increases to area of tumor cells Paclitaxel is a common drug used for Breast, Ovarian, Head and Neck Cancer - attack tumor cells during a cell cycle Atlantic Coast Symposium MSBB 7

8 he Cancer Cell Cycle G: S: - parent cells grows, organelles are reproduced - longest phase; can last up to 48 hours G2: - DNA is replicated; lasts 8-2 hours - prepares for cell division; lasts up to 4 hours Mitosis: - DNA is distributed evenly among daughter cells; lasts up to 3 minutes Atlantic Coast Symposium MSBB 8

9 Cell Population Dynamics t () I () M t () Q t I() t ct () -- Population of cells in Interface stage -- Population of cells in Mitosis stage -- Population of cells in Quiescent stage -- Population of Immune cells (cytotoxic -cells) -- Concentration of the drug Paclitaxel M. Villasana and G. Ochoa, Heuristic Design of Cancer Chemotherapies, IEEE ransactions of Evolutionary Computation, 8 (24), R. Yafia, Dynamics Analysis and Limit Cycle in a Delayed Model for umor Growth with Quiescence, Nonlinear Analysis, Modeling and Control, (26), 95. G. Newbury, A Numerical Study of a Delay Differential Equation Model for Breast Cancer, MS hesis, Department of Mathematics, Virginia ech, Blacksburg, VA, August, 27. Atlantic Coast Symposium MSBB 9

10 Models Model (Drugs but no quiescent cells) M. Villasana, Delay Differential Equation Model for umor Growth., Ph.D. Dissertation, Claremont University, 2. M. Villasana and A. Radunskaya, A Delay Differential Equation of the Model for umor Growth, Journal of Mathematical Biology, 47 (23), M. Villasana and G. Ochoa, Heuristic Design of Cancer Chemotherapies, IEEE ransactions of Evolutionary Computation, 8 (24), Model 2 (Quiescent cells but no drugs) R. Yafia, Dynamics Analysis and Limit Cycle in a Delayed Model for umor Growth with Quiescence, Nonlinear Analysis, Modeling and Control, (26), 95-- Atlantic Coast Symposium MSBB

11 Model (Villasana) () (2) (3) di () t dt dm () t dt = 2 a() t ctit () () dt () at ( τ) 4 M I 2 I I = a t τ d t a t c t I t k e t kwt 2 () I( ) 3 M() 4 M() 3 M() () ( ) M() n di() t ρ I()[ t I() t + M ()] t = k + n dt α + [ ( t) + ( t)] I M k4w( t) ( ) ( ) ( ) ( ) ( ) 4 M 3 c I() t () t c I t t d I t k e I t 2 I (4) (5) dw () t = λwt () + ct (), w() = dt dw2 () t = λ2wt 2() + ct (), w2() = dt wt () = rw() t + rw() t 2 2 Atlantic Coast Symposium MSBB

12 Model 2 (Yafia) Pt () = [ () t + ()] t M I Nt () = [ () t + () t + ()] t M I Q () (2) dp() t dt d () t Q dt = bp( t τ) r ( N()) t P() t + r ( N()) t () t P Q Q = r( Nt ()) Pt () r( Nt ()) () t μ P q Q Q Q r ( ) P N rq ( N) -- NON-DECREASING -- NON-INCREASING Atlantic Coast Symposium MSBB 2

13 Model 3 ( Newbury) () (2) (3) (4) d () t Q dt di () t dt dm () t dt = a( t τ) a () t d () t c () t I() t u ( t) ( t) 5 I 6 Q 4 Q 5 Q Q = 2 a() t at ( τ) at () ctit () () dt () at ( τ) 4 M 5 I 6 Q I 2 I I = a( t τ) d () t a () t c () t I() t u ( t) ( t) I 3 M 4 M 3 M 2 M n di() t ρ I()[ t Q() t + I() t + M ()] t = k + c n dt α + [ ( t) + ( t) + ( t)] Q I M 4 M 6 Q 3 2 I() t () t c I() t () t c I() t () t d I() t u () t I() t I u () t = gwtc ((),() t ) i i Atlantic Coast Symposium MSBB 3

14 Delay Equation Model (5) (6) dw () t = λ wt () + ct (), w() = dt dw2 () t = λ2wt 2() + ct (), w2() = dt wt () = r w() t + r w () t 2 2 o compare with existing models. ut k6wt () () k ( e ) k2wt () k4wt () = ut () = k ( e ) ut () = k ( e ) 5 2 We also investigated the ODE model: τ = and a PDE model Atlantic Coast Symposium MSBB 4 3 3

15 ypical Parameters and Inputs [ (), (), (), I()] = [.7,.8,.6, θ] θ =.2 Q I M lim It ( ).2 t + NORMAL LEVEL [ (), (), (), I()] = [.47,.68,.26, θ] θ.9 Q I M concentration c(t) PARAMEERS c(t) -- Pulsed concentration of drug Atlantic Coast Symposium MSBB 5

16 Model 3: Response - Healthy Patient [ (), (), (), I()] = [.7,.8,.6, θ] θ =.2 Q I M θ = % more immune cells ALL Cells With Cytotoxic Cells: NO Drugs.4 population y()= Q y(2)= I y(3)=m y(4)= C cells years years Atlantic Coast Symposium MSBB 6

17 Input 2 for Healthy Patient concentration c(t) Atlantic Coast Symposium MSBB 7

18 Model 3: Response - Healthy Patient [ (), (), (), I ()] = [.7,.8,.6,.5] Q I M.6 ALL Cells With Cytotoxic Cells: WIH Drugs.4 population y()= Q y(2)= I y(3)=m y(4)= C cells years years Atlantic Coast Symposium MSBB 8

19 Response - Unhealthy Patient [ (), (), (), I()] = [.7,.8,.6, θ] θ =.2 Q I M θ = % less immune cells.2 ALL Cells With Cytotoxic Cells: NO Drugs..8 population.6.4 y()= Q y(2)= I y(3)=m y(4)= C cells.2 SHOR IME SIMULAION Atlantic Coast Symposium MSBB year 9

20 θ =.9 Response - Unhealthy Patient [ (), (), (), I()] = [.7,.8,.6, θ] θ =.2 Q I M -- 25% less immune cells.25 ALL Cells With Cytotoxic Cells: NO Drugs SE POINS?.2 population.5..5 LONG IME SIMULAION y()= Q y(2)= I y(3)=m y(4)= C cells years Atlantic Coast Symposium MSBB 2

21 ypical Long ime Simulation [ (), (), (), I()] = [.7,.8,.6, θ] θ =.2 Q I M θ = % less immune cells.4 ALL Cells With Cytotoxic Cells: WIH Drugs.2 population y()= Q y(2)= I y(3)=m y(4)= C cells? IS HE DRUG WORKING?.2? SHOULD YOU BELIEVE YOUR SIMULAION? years Atlantic Coast Symposium MSBB 2

22 Some Fundamental Issues GIVEN A PARAMEERIZED MODEL AND A COMPUED VALUE BASED ON HIS MODEL, WHEN CAN WE RUS HE RESULS? -- CAN WE BE SURE WE HAVE CONVERGED O HE CORREC VALUE OF INERES? ARE HERE PARAMEERS (SMALL, UN-MODELED) HA PRODUCE FALSE OR UNEXPECED RESULS? -- CAN WE FIND INDICAORS / PRECURSORS O HESE PROBLEMS? -- WHEN IS A COMPUAIONAL PROBLEM DUE O AN ALGORIHM OR EVEN MACHINE PRECISION AND HOW CAN WE ELL? SENSIIVIIES HELP Atlantic Coast Symposium MSBB 22

23 ypical Parameters and Inputs lim It ( ).2 t + NORMAL LEVEL [ (), (), (), I()] = [.47,.68,.26, θ] θ.9 Q I M Concentration ime History Concentration c PARAMEERS c(t) -- Pulsed concentration of drug Atlantic Coast Symposium MSBB 23

24 [ (), (), (), I()] = [.47,.68,.26, θ] θ =.9 Q I M θ =.9 θ = ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells.2.2 population.8 population ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells.2.2 population.8 population Atlantic Coast Symposium MSBB 24

25 [ (), (), (), I()] = [.47,.68,.26, θ] θ =.9 Q I M θ =.9 θ = ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells.2.2 population.8 population ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells.2 3 population.8 population Atlantic Coast Symposium MSBB 25

26 [ (), (), (), I()] = [.47,.68,.26, θ] θ = Q I M θ =.9 θ = ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells 4.5 x 4 ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells 4 population.8 population CAN WE BE SURE WE HAVE CONVERGED O HE CORREC VALUE OF INERES? ARE HERE PARAMEERS (SMALL, UN-MODELED) HA PRODUCE FALSE OR UNEXPECED RESULS?? ANY CLUES? Atlantic Coast Symposium MSBB 26

27 Sensitivities [ (), (), (), I()] = [.47,.68,.26, θ] θ =.9 Q I M CRIICAL PARAMEER SENSIIVIIES SQ() t Q(, t θ ) θ = θ θ SI() t I(, t θ ) θ = θ θ SM() t (, t ) θ θ M θ = θ = θ SI() t I(, t ) θ θ θ Atlantic Coast Symposium MSBB 27

28 [ (), (), (), I()] = [.47,.68,.26, θ] θ = Q I M θ =.9 θ = ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells 4.5 x 4 ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells population.8 population SENSIIVIIES x 7 SE - Interactions: WIH Drugs x 9 SE - Interactions: WIH Drugs 7 9 Sensitivity WR C() yse()= Q S E yse(2) = I S E Sensitivity WR C() yse()= Q S E yse(2) = I S E yse(3)= M S E yse(4)= C S E - cells yse(3)= M S E -2 yse(4)= C S E - cells Atlantic Coast Symposium MSBB 28

29 [ (), (), (), I()] = [.47,.68,.26, θ] θ = Q I M θ =.9 θ = ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells 4.5 x 4 ALL Cells With Cytotoxic -Cells: WIH Drugs y()= Q y(2)= I y(3)=m y(4)= C-cells population.8 population x 6 7 SENSIIVIIES x s(t) 7 6 s(t) Atlantic Coast Symposium MSBB 29

30 .5 x 7 SE - Interactions: WIH Drugs [ (), (), (), I()] = [.47,.68,.26, θ] θ =.9 Q I M θ =.9 θ = x 9 SE - Interactions: WIH Drugs Sensitivity WR C() yse()= Q S E yse(2) = I S E yse(3)= M S E yse(4)= C S E - cells Sensitivity WR C() yse()= Q S E yse(2) = I S E yse(3)= M S E yse(4)= C S E - cells x 7 SE - Interactions: WIH Drugs x 9 SE - Interactions: WIH Drugs Sensitivity WR C() yse()= Q S E yse(2) = I S E Sensitivity WR C() yse()= Q S E yse(2) = I S E yse(3)= M S E yse(4)= C S E - cells yse(3)= M S E -2 yse(4)= C S E - cells Atlantic Coast Symposium MSBB 3

31 Some Fundamental Issues GIVEN A PARAMEERIZED MODEL AND A COMPUED VALUE BASED ON HIS MODEL, WHEN CAN WE RUS HE RESULS? -- CAN WE BE SURE WE HAVE CONVERGED O HE CORREC VALUE OF INERES? ARE HERE PARAMEERS (SMALL, UN-MODELED) HA PRODUCE FALSE OR UNEXPECED RESULS? -- CAN WE FIND INDICAORS / PRECURSORS O HESE PROBLEMS? -- WHEN IS A COMPUAIONAL PROBLEM DUE O AN ALGORIHM OR EVEN MACHINE PRECISION AND HOW CAN WE ELL? SENSIIVIIES HELP Atlantic Coast Symposium MSBB 3

32 Sensitivity as a Predictor [ (), (), (), I()] = [.7,.8,.6, θ] θ =.2 Q I M θ = % less immune cells.4 ALL Cells With Cytotoxic Cells: WIH Drugs.2 population y()= Q y(2)= I y(3)=m y(4)= C cells? IS HE DRUG WORKING? years Atlantic Coast Symposium MSBB 32

33 Sensitivity [ (), (), (), I ()] = [.7,.8,.6,.9] Q I M Sensitivity WR C() SE Interactions: WIH Drugs yse()= Q S E yse(2) = I S E yse(3)= M S E yse(4)= C E cells S YES years Atlantic Coast Symposium MSBB 33

34 Feedback Control of Flows Sensitivity with respect to wall disturbance α is a precursor to transition. information on when to turn on feedback control HOW DO WE KNOW WHEN O URN ON FEEDBACK CONROL? SENSIIVIIES ARE IS PRECURSOR O FLOW RANSIION - op figure: norm of sensitivity - Bottom figure: norm of solution to disturbed problem Atlantic Coast Symposium MSBB 34

35 Parabolic PDEs wtx wtx f wtx wtx t t 2 (, ) = μ 2 (, ) + ( (, ), (, ), λ), >, x FigtzHugh-Nagumo, Hodgkin-Huxley and x wtx wtx wtx wtx t t 2 3 (, )) = 2 (, ) + λ( (, ) [ (, )]), >, μ x wt (,) =, wtπ (, ) = t Chaffee-Infante Equation w (, x ) = w ( x ) wtx wtx wtxwtx t 2 (, ) = μ 2 (, ) (, ) (, ), >, x FEEDBACK MECHANISM Burgers Equation x x w( t,) =, wt (,) = Atlantic Coast Symposium MSBB 35

36 Chaffee-Infante Equation 2 2 If μ = and n < λ ( n+ ), then there are 2n+ fixed points ± ± wˆ ( x) and wˆ ( x), i=,2,..., n. he functions wˆ ( x) have exactly i wt (,) =, wtπ (, ) = zeros in (, π). he global attractor is the union of ndimensional man ifolds. i 2 wtx (,) = wtx (,) + λ((,)[(,)]), wtx wtx t>, t μ x 2 Z = L 2 (,) i 3 w (, x ) = w ( x ) ( t) : IS A DISSIPAIVE DYNAMICAL SYSEM S Z Z Atlantic Coast Symposium MSBB 36

37 Chaffee-Infante Equation 2 wtx (,) = wtx (,) + λ((,)[(,)]), wtx wtx t>, t μ x 2 wt (,) =, wtπ (, ) = μ = 3 w (, x ) = w ( x ) λ = 4. w(, x) = ϕ ( x) =.5*sin( x) w(, x) = ϕ ( x) =.5*sin(3* x) 3 SIMULAIONS: pdepe (MALAB) RUN FOR t < 9 AND FOR t < 8 Atlantic Coast Symposium MSBB 37

38 Chaffee-Infante Equation LONG IME SIMULAIONS w(, x) =.5*sin( x) w(, x) =.5*sin(3 x) t 4 w ( x ) = w + ( x) w ( x) t 8? IS HERE ANYWAY O PREDIC HE RANSIION? Atlantic Coast Symposium MSBB 38

39 Chaffee-Infante Equation wˆ () x + wˆ () x 2 ŵ + 2 () x wˆ () x = wˆ () x Atlantic Coast Symposium MSBB 39

40 Chaffee-Infante Equation w(, x) =.5*sin(3 x) w ( x ) = w ( x) t 4 t 8 ONE MIGH BE RAHER CERAIN HA HE SOLUION HAS CONVERGED SMALL PARAMEER CHANGES CAN PRODUCE LARGE DIFFERENCES --- vary α by ? WHA IS α? w + ( x) = Atlantic Coast Symposium MSBB 4

41 t Chaffee-Infante Equation wtx wtx wtx wtx t 2 3 (, ) = μ 2 (, ) + λ( (, ) [ (, )] ), >, x wt (,) =, α, wtπ (, ) = α w (, x ) = w ( x ) μ = λ = 4. HOW CAN WE DEAL WIH HE COMPUAIONAL UNCERAINY HA ARISES IN SIMULAIONS WIH UNCERAIN (OR UN-MODELED) PARAMEERS SENSIIVIY WIH RESPEC O BOUNDARY DISURBANCE wt (, x, α ) stx (, ) α= wα(, tx, α ) α = = s tx s tx wtx stx 2 t(, ) = μ xx(, ) + λ( 3 (,, )) (, ), st (,) =, st (, π ) =, s(, x) =, < x< π. SHOW MOVIES Atlantic Coast Symposium MSBB 4

42 Movies? DO YOU BELIEVE YOUR OWN EYES?? SHOULD YOU BELIEVE YOUR OWN EYES? meshci8 sen meshci8_3_sen SOLUIONS and SENSIIVIIES ON t 8 plotci8 sen plotci8_3_sen SOLUIONS and SENSIIVIIES ON t 8 Atlantic Coast Symposium MSBB 42

43 Chaffee-Infante Equation CONINUOUS SENSIIVIY EQUAION 5 L 2 Energy of the Solution -9 5 L 2 Energy of the Solution x x 4 L 2 Energy of the Sensitivity L 2 Energy of the Sensitivity SENSIIVIY BASED MEHOD PROVIDES INSIGH INO LONG IME SIMULAIONS & COMPUAIONAL UNCERAINY LYAPUNOV EXPONENS YPE MEHODS CAN PROVIDE RIGOROUS ESIMAES OHER APPLICAIONS Atlantic Coast Symposium MSBB 43

44 Some Fundamental Issues GIVEN A PARAMEERIZED MODEL AND A COMPUED VALUE BASED ON HIS MODEL, WHEN CAN WE RUS HE RESULS? -- CAN WE BE SURE WE HAVE CONVERGED O HE CORREC VALUE OF INERES? ARE HERE PARAMEERS (SMALL, UN-MODELED) HA PRODUCE FALSE OR UNEXPECED RESULS? -- CAN WE FIND INDICAORS / PRECURSORS O HESE PROBLEMS? -- WHEN IS A COMPUAIONAL PROBLEM DUE O AN ALGORIHM OR EVEN MACHINE PRECISION AND HOW CAN WE ELL? SENSIIVIIES HELP Atlantic Coast Symposium MSBB 44

45 Algorithm: Model (Villasana) 6 5 v()=tumor cells during interphase v(2)=tumor cells during mitosis v(3)= cytotoxic cells 4 population 3 2 Matlab ode45 Use a Stiff Solver ODE23S NEGAIVE CELL COUN!! Atlantic Coast Symposium MSBB 45

46 Machine Precision: Simple Example < α << α, zt ( ) + α z(t) = f ( z( t), α) = z( ) =, zt ( ) > + α z(t, α ) + αt, t = + α + ( t ), t Atlantic Coast Symposium MSBB 46

47 Simple Example FORWARD EULER m m m z + = z +Δtf( z, α) z = IF Δ tα < eps z z z tf z t m + = m =... = = m +Δ (, α) = +Δ α = PROBLEM IS FINIE PRECISION ARIHMEIC MESH REFINEMEN MAKES HE PROBLEM WORSE Atlantic Coast Symposium MSBB 47

48 Some Comments COMPUING WIHOU HEORY MAY BE DANGEROUS BU HEORY WIHOU COMPUING IS DANGEROUS WAN O KNOW WHEN HE COMPUED ANSWER IS CLOSE O HE ANSWER PREDICED BY HE MODEL (VERIFICAION) WAN O KNOW HOW UNCERAINY IMPACS HE SOLUION AND IS SENSIIVIY AND NEED HELPFUL WARNINGS OF NUMERICAL PROBLEMS Atlantic Coast Symposium MSBB 48

49 Closing Comments WIHOU SOME HEOREICAL ANALYSIS ONE MIGH MISINERPRE HE NUMERICAL RESULS FOR HE DYNAMICAL SYSEMS WIHOU COMPUING ONE CAN MISS AN IMPORAN HEOREICAL SRUCURE IN HE SYSEM SENSIIVIIES CAN OFFER INSIGH INO NUMERICAL PROBLEMS AND COMPUAIONAL UNCERAINY Atlantic Coast Symposium MSBB 49

50 HANK YOU Atlantic Coast Symposium MSBB 5