Introduction Model Results Discussion Analyzing the Dynamics of an Inflammatory Response to a Bacterial Infection in Rats Allison Torsey 1 Amy Carpenter 2 Dr. Julia Arciero 3 1 Department of Mathematics, SUNY Buffalo State 2 Department of Natural Sciences and Mathematics, Lee University 3 Department of Mathematical Sciences, IUPUI January 26th, 2019 Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Background Sepsis ◮ Sepsis is a life threatening condition that results from an overwhelming inflammatory response to a bacterial infection Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Background Inflammation ◮ Inflammation is the body’s response to an infection ◮ Too much inflammation can cause damage to healthy tissue ◮ Bacterial infections that often cause sepsis: ◮ Staphylococcus aureus ( staph ) ◮ Escherichia coli (E. coli) ◮ some types of Streptococcus Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Background Virulence ◮ The strength of the pathogen ◮ Low virulent strains - large quantity in blood does not cause significant damage ◮ High virulent strains - small quantity will cause significant damage ◮ We interpret virulence as a pathogen’s ability to cause more inflammation ◮ Bacteria are constantly mutating, so the virulence may vary over time Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Observations Experimental Observations ◮ Rats were injected with a fibrin clot containing four different levels of bacteria (E. coli) ◮ For very high levels of bacteria, the clot was saturated and the bacteria was released immediately ◮ The bacteria levels in the blood were measured over time ◮ Once the bacteria levels in the blood reached a certain level, the rats were unable to recover ◮ Used the data to parameterize sepsis model and predict health or disease outcome Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Observations Data Set: Time Dynamics Dose administered to rats: Amount (x10 6 /cc) Bsource 1 128 2 248 3 505 4 1940 Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Observations Expected Outcomes ◮ Septic death: bacteria remains in the blood ◮ Aseptic death: bacteria is eliminated but damage remains elevated ◮ Health: both bacteria and damage are eliminated Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Objectives Objective Our goal is to use a mathematical model to predict the survivability range in rats for an infection while varying the initial dose, growth rate, or virulence of the bacteria Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Conceptualization Model Schematic Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Model Bacteria: dB dt = D ( t ) + k 1 B (1 − B µ l + k 3 B − k 5 BM k 2 s l B ) − (1) 1 + k A A B ∞ Pro-inflammatory Response: ν 1 ( k M M + k B c 1 B + k ǫ ǫ ) dM dt = ( ν 2 + k M M + k B c 1 B + k ǫ ǫ )(1 + k A A ) − µ M M (2) Anti-inflammatory Response: dA a 1 ( M + k 4 ǫ ) dt = s A + (1 + M + k 4 ǫ )(1 + k A A ) − µ A A (3) Damage Markers: τ + [ f M − T ] + d ǫ dt = − ǫ (4) 1 + k A A Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Model: Bacteria dB + k 1 B (1 − B k 2 s l B k 5 BM dt = D ( t ) ) − − B ∞ µ l + k 3 B 1 + k A A ���� � �� � � �� � � �� � Dosing function growth immune response local immunity Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Dosing Function D ( t ) = k D B source e − k D t ◮ The dosing function simulates how bacteria is released from the fibrin clot ◮ No initial bacteria in the blood, B (0) = 0 ◮ Constant rate of decay, k D Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Model: Bacteria dB + k 1 B (1 − B k 2 s l B k 5 BM dt = D ( t ) ) − − B ∞ µ l + k 3 B 1 + k A A ���� � �� � � �� � � �� � Dosing function growth immune response local immunity Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Model: Pro-inflammatory Response dM ν 1 ( k M M + k B c 1 B + k ǫ ǫ ) dt = µ M M − ( ν 2 + k M M + k B c 1 B + k ǫ ǫ )(1 + k A A ) � �� � � �� � natural decay inflammation activation Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Model: Anti-inflammatory Response dA a 1 ( M + k 4 ǫ ) dt = + s A µ A A − (1 + M + k 4 ǫ )(1 + k A A ) ���� ���� source term � �� � natural decay anti-inflammation activation Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion System of ODEs Model: Damage d ǫ +[ f M − T ] + dt = − ǫ 1 + k A A τ ���� � �� � repair damage from pro-inflammatory response Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Parameters Model Parameters Description Value/ unit Reference k 1 pathogen growth rate varied 145 x 10 6 / cc maximum carrying capacity optimized B ∞ rate at which the non-specific local response eliminates pathogen .6/l-units/h Reynolds (2006) k 2 source of non-specific local response .005/l-units/h Reynolds (2006) s l µ l decay of non-specific local response .002/h Reynolds (2006) rate at which the non-specific local response is exhausted by pathogen .01 B-units Reynolds (2006) k 3 rate at which activated inflammatory response consumes pathogen 1.6/ M-units /h optimized k 5 inhibition rate of the anti-inflammatory response 2.6/ A-units optimized k A ν 1 source of pro-inflammatory response .08 M-units/h Reynolds (2006) ν 2 decay of pro-inflammatory response .12/h Reynolds(2006) k M activation of resting inflammatory response by activated .01/M-units/h Reynolds (2006) inflammatory response Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Parameters Model Parameters Cont. Description Value/unit Reference k B activation of resting inflammatory response by pathogen .1 /B-units/h Reynolds (2006) c 1 virulence of pathogen varied k ǫ activation of resting inflammatory response by damage .02/ ǫ -units h Reynolds (2006) µ M decay of pro-inflammatory response .12/h Reynolds (2006) s A source of anti-inflammatory response .0125 A-units/h Reynolds (2006) a 1 maximum production rate of anti-inflammatory response .04 A-units/h Reynolds (2006) k 4 relative effectiveness of pro-inflammatory response and 48 M-units/ ǫ -units Reynolds (2006) damage inducing the production of the anti-inflammatory response µ A decay of the anti-inflammatory response .1/h Reynolds (2006) τ rate of recovery from damage estimated f maximum rate of damage produced by the pro-inflammatory 15 /M-units h optimized response T threshold for damage estimated Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Fitting Model Parameters Parameter Estimation ◮ Using least squares optimization, the following parameters are fit to the data set ◮ k A ◮ k 5 ◮ f ◮ B ∞ ◮ k D � B source , 4 B source , 1 ( B actual − B model ) 2 min [ ] � B source , 4 B source , 1 B 2 actual Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Fitting Model Parameters Fitting the Model to the Data + Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Model Analysis Steady States ( B ∗ , M ∗ , A ∗ , ǫ ∗ ) ◮ Health: (0,0, A 1 ,0) ◮ Aseptic Death: (0, M 2 , A 2 , ǫ 2 ) ◮ Septic Death: ( B 3 , M 3 , A 3 , ǫ 3 ) Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Model Analysis Time Dynamics ◮ Sepsis: B source = 2 ◮ Asepsis: B source = 1 . 3 ◮ Health: B source = 1 ◮ k 1 = . 5 Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Pathogen Growth Rate Bifurcations Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Pathogen Growth Rate B source vs. Pathogen Growth Rate Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Virulence Bifurcations - Virulence ( k 1 = 0 . 5) Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Virulence Bsource vs. Virulence ( k 1 = 0 . 5) Torsey, Carpenter, Arciero NCUWM Presentation
Introduction Model Results Discussion Conclusions Conclusions ◮ This model highlights the balance between the pro-inflammatory response and the damage caused to the healthy tissue ◮ Parameters ranges are predicted that yield outcomes: health, aseptic death, and septic death ◮ May be useful for determining optimal treatment strategies (e.g. timing and amount of antibiotics or anti-inflammatory medication) Torsey, Carpenter, Arciero NCUWM Presentation
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