Controlling Systemic Inflammation Using Nonlinear Model Predictive Control with State Estimation Gregory Zitelli, Judy Day University of Tennessee, Knoxville July 2013 With generous support from the NSF, Award 1122462 Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Motivation ◮ We’re going to be interested in the acute inflammatory response to biological stress in the form of a bacterial pathogen. ◮ The inflammatory response aims to remove the presence of the pathogen. ◮ However, an excessive response may lead to collateral tissue damage, organ failure, or worse. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Motivation ◮ To recover from a high-inflammation state, it is necessary for the inflammatory mechanisms to downregulate itself through the use of some anti-inflammatory mediator. ◮ We will consider a mathematical formulation of these ideas in terms of a highly coupled, nonlinear ODE model, as well as control applied to both the pro and anti-inflammatory influences within the system. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Interaction Diagram Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Nonlinear Model ◮ P represents the bacterial pathogen population. ◮ N ∗ represents the concentration of pro-inflammatory mediators, such as activated phagocytes and their produced cytokines. ◮ D acts as a marker of tissue damage. ◮ C A represents the concentration of anti-inflammatory mediators. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Nonlinear Model � � dP 1 − P k pm s m P µ m + k mp P − k pn f ( N ∗ ) P dt = k pg P − P ∞ dN ∗ s nr R ( P, N ∗ , D ) µ nr + R ( P, N ∗ , D ) − µ n N ∗ = dt f ( N ∗ ) 6 dD dt = k dn dn + f ( N ∗ ) 6 − µ d D x 6 f ( N ∗ + k cnd D ) dC A = s c + k cn 1 + f ( N ∗ + k cnd D ) − µ c C A dt Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Nonlinear Model The system ( P, N ∗ , D, C A ) has three fixed points corresponding to the three biologically relevant scenarios: ◮ P = N = D = 0 and C A = s c µ c , where the patient is healthy. ◮ All states elevated, where the patient is septic. ◮ P = 0 , and N ∗ , D, C A > 0 , where the patient is aseptic. For values of k pg (the pathogen growth rate) in the interval (0 . 5137 , 1 . 755) , all three states are stable. (Reynolds et al 2006) Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
??? So what does this model actually look like? Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Septic Simulation 10 1.5 8 N* (state (−)) 1 P (state (−)) 6 4 0.5 2 0 0 0 50 100 150 200 0 50 100 150 200 Time Time 20 0.6 0.5 15 C a (state (−)) D (state (−)) 0.4 10 0.3 5 0.2 0 0.1 0 50 100 150 200 0 50 100 150 200 Time Time Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Aseptic Simulation 6 1 5 N* (state (−),est(−−)) 0.8 P (state (−),est(−−)) 4 0.6 3 0.4 2 0.2 1 0 0 0 50 100 150 200 0 50 100 150 200 Time Time 20 0.5 C a (state (−),est(−−)) D (state (−),est(−−)) 15 0.4 10 0.3 5 0.2 0 0.1 0 50 100 150 200 0 50 100 150 200 Time Time Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Healthy Simulation 0.8 0.2 0.6 0.15 N* (state (−)) P (state (−)) 0.4 0.1 0.2 0.05 0 0 0 50 100 150 200 0 50 100 150 200 Time Time 1.5 0.6 0.5 C a (state (−)) D (state (−)) 1 0.4 0.3 0.5 0.2 0 0.1 0 50 100 150 200 0 50 100 150 200 Time Time Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Goals ◮ We would like to apply some kind of control to this model, to direct septic and aseptic patients towards the healthy fixed state. ◮ This is accomplished using Nonlinear Model Predictive Control, or NMPC. ◮ The NMPC is applied to the pro and anti-inflammatory mediators, N ∗ and C A . ◮ We only apply positive control, and there are restrictions on how much can be introduced within certain time frames. ◮ We assume that the level of pro and anti-inflammatory mediators, N ∗ and C A , are measurable with Gaussian noise. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Overview of NMPC Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Overview of NMPC ◮ Our reference trajectory is zero levels of P and D , and minimal amounts of control. AI ( t ) ,PI ( t ) � Γ D D � 2 2 + � Γ P P � 2 2 + � Γ AI AI � 2 2 + � Γ PI PI � 2 J = min 2 Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Note on Observations ◮ We take noisy measurements of N ∗ and C A at each time interval which update our predictive model. ◮ The levels of P and D are not measured directly, and are instead estimated from the predictive model. ◮ Being unable to measure P often leads the NMPC to be too aggressive. Many virtual patients were unnecessarily harmed under this scheme. ◮ Since there are biologically relevant scenarios for loose measurements of P (indicators like body temperature or blood pressure give an idea whether or not the infection persists), a pathogen update is done every four timesteps. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Pathogen Update ◮ If the predictive model reads low ( < 0 . 05 ) but the pathogen levels are high ( P > 0 . 5 ) then the level in the predictive model is reset to 0 . 5 . ◮ If the predictive model reads high ( > 0 . 5 ) but the pathogen levels are low ( P < 0 . 05 ) then the level in the predictive model is reset to 0 . Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Virtual Patient Pool ◮ 1,000 patients are randomly generated with different parameters, including unique values of the pathogen growth rate k pg . ◮ 620 acquire elevated P values to suggest treatment. ◮ Of the 620, 251 (40%) will return to a healthy state on their own. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Initial Results (Patient Snapshot) 1.5 0.8 N* (state (−),est(−−)) P (state (−),est(−−)) 0.6 1 0.4 0.5 0.2 0 0 0 50 100 150 200 0 50 100 150 200 Time Time 4 0.8 C a (state (−),est(−−)) D (state (−),est(−−)) 3 0.6 2 0.4 1 0.2 0 0 0 50 100 150 200 0 50 100 150 200 Time Time Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Initial Results (Day et al 2010) Mismatch Mismatch Mismatch Therapy Type: Placebo k pg = 0 . 52 k pg = 0 . 6 k pg = 0 . 8 Percentage Healthy: 40% (251) 60% (369) 82% (510) 83% (513) Percentage Aseptic: 37% (228) 19% (120) 8% (49) 17% (107) Percentage Septic: 23% (141) 21% (131) 10% (61) 0% (0) Percentage Harmed: na 0% (0/251) 1% (2/251) 6% (16/251) Percentage Rescued: na 32% (118/369) 71% (261/369) 75% (278/369) Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Particle Filter ◮ One way to go about improving these results is to impliment robust state estimation for the unobserved variables P and D . ◮ This is accomplished using a particle filter, which tracks its accuracy by comparing its predictions of N ∗ and C A to the observed values. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Particle Filter ◮ The particle filter is initialized with 1,000 particles randomized near our measurements for N and C A and initial predictions for P and D . ◮ Each particle p i = ( P i , N ∗ i , C A,i , D i ) is simulated for one time step. ◮ At the next time step, the particles p i are assigned weights q i depending on how close N ∗ i , C A,i are to the new measurements for N ∗ and C A . Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Particle Filter ◮ At the end of the time step, each of the 1,000 slots holding a particle is randomly assigned a new particle p i according to their weights q i . ◮ This causes bad particles with low weights q i to die off, while good particle with high weights replicate. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Particle Filter Results Original Original Original Therapy Type: k pg = 0 . 52 k pg = 0 . 6 k pg = 0 . 8 Percentage Healthy: 60% (369) 80% (497) 84% (519) Percentage Aseptic: 19% (121) 10% (60) 16% (101) Percentage Septic: 21% (130) 10% (63) 0% (0) Particles Particles Particles Therapy Type: k pg = 0 . 52 k pg = 0 . 6 k pg = 0 . 8 Percentage Healthy: 60% (369) 80% (493) 82% (511) Percentage Aseptic: 19% (120) 10% (66) 18% (109) Percentage Septic: 21% (131) 10% (61) 0% (0) Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
Particle Filter Results ◮ The particle filter does slightly worse than our original mismatched predictive model. ◮ On the other hand, the particle filter is self correcting, so it needs no pathogen update to correct a misled pathogen prediction. Controlling Systemic Inflammation Using NMPC University of Tennessee, Knoxville
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