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Parameter Estimation and Exploration in a Model of Immune Cell Interactions During Inflammation
- 1. ParameterEstimationandExplorationinaModelof ImmuneCellInteractionsDuringInflammation Marcella Torres1, Jing Wang3, Paul J Yannie3, Rebecca A Segal2, Angela M Reynolds2 and Shobha Ghosh3 Department of Mathematics & Computer Science, University of Richmond1, Department of Mathematics and Applied Mathematics2 and Internal Medicine3, Virginia Commonwealth University, Richmond, VA
- 4. Injury or Infection Neutrophils are first cells to arrive Neutrophil
- 6. Monocytes differentiate to M1 or M2 Monocyte M1 M2
- 7. Monocytes differentiate to M1 or M2 Monocyte M1 M2 Destroys tissue Produces pro- inflammatory cytokines Builds tissue Produces anti- inflammatory cytokines
- 8. M1M1M1 M1M1 M1 M1 Macrophage phenotype imbalance → Chronic wounds or disease M2
- 9. How do macrophages transition from contributing to inflammation to resolving it? • Can switch from M1 to M2 • Driven by phagocytosis of apoptotic cells • Driven by cytokines
- 10. Mathematical Model of Immune Cell Response KEY B Broth P Pathogen N Neutrophils AN Apoptotic Neutrophils M1 Classically Activated Macrophages M2 Alternatively Activated Macrophages Arrows: up-regulation Bars: destruction or inhibition Gray: transition kpm
- 11. Inflammatory Stimulus logistic broth-dependent growth removal by N removal by M1 removal by M2 1 2 consumption by P 1 ( 1, ) ( 2, )( )pg pn pm i pm i b dP P k P k PN k Pf M N k Pf M N dt P B dB k BP dt 6 4 4 7 4 4 8 6 7 8 6 4 4 7 4 48 6 4 4 7 4 4 8 6 7 8
- 12. Neutrophils } } } activation rate removal by M1 removal by M2apoptosis apoptotis of N 1 2 secremoval by N ( , ) ( 1, ) ( 2, ) ( , ) nr N an an anm i anm i nr N ann an s R P ANdN dAN k N k N k ANf M N k ANf M N dt R P AN dt k N AN 6 4 4 7 4 4 8 6 4 44 7 4 4 48 6 4 44 7 4 4 48 } } } ondary necrosis removal by M1 removal by M2 secondary necrosisapoptotis of N removal by N 1 2 activactivation by P ( 1, ) ( 2, )an anm i anm i ann an N np nan an dAN k N k ANf M N k ANf M N k N AN dt R k P k AN 6 7 8 6 4 44 7 4 4 48 6 4 44 7 4 4 48 6 7 8 ation by necrotic AN 2 ( , ) 1 ( ) i x f x N N N 6 4 7 48
- 13. activation/influx ratefrom switch to 2 from 1 per phagocytized switch to 1 from 2 1 1 2 1 2 1 1 2 ( , , 1, )1 ( 1, ) 2 ( , , 1, ) ( 2) RM M M AN M M mr M m m anm i m m mr M M s R P N M ANdM k k ANf M N k M dt u R P N M AN R M 6 4 4 4 4 4 44 7 4 4 4 4 4 4 48 6 4 4 44 7 4 4 4 48 64 7 8 decay 1 activation/influx rate from switch rate from M1 per phagocytized switch from 2 to 2 1 2 1 2 1 1 2 1 ( 2)2 ( 1, ) 2 ( , , 1, ) ( 2) R m M AN M mr M m m anm i m m mr M M M s R MdM k k ANf M N k M dt u R P N M AN R M 4 6 7 8 6 4 4 4 4 4 44 7 4 4 4 4 4 4 48 6 4 4 44 7 4 4 4 48 } } decay1 2 activation by 1s and their cytokinesactivation by activation by byproducts of 1 1 1 1 1 activation by necrotic 1 activati 2 2 2 2 1 2 M m MP N M m p m n m m AN m an an M m m M R k P k N k M k AN R k M 64 7 48 64 7 48 64 7 48 6 4 7 4 8 }on by 2s and their cytokines background anti-inflammatory cytokines 2 ( , ) 1 ( ) M c i k x f x N N N 64 7 48 Macrophages
- 14. Amouse model of peritonitis was used to gather immune cell influx data using flow cytometry
- 18. Check sensitivity of outputs to parameters
- 20. 1 Cov( )T F S S F p Brun et al. https://github.com/gabora/visid
- 24. Simulations varying kan Hypothesis: efferocytosis is a key driver of macrophage phenotype change (M1 to M2) and it requires a sufficiently sized population of apoptotic cells Dysregulation of neutrophil population level and turnover is known to contribute to human inflammatory and autoimmune diseases Macrophages themselves are known to modulate neutrophil lifespan by releasing cytokines that can delay apoptosis and some microbial pathogens delay or accelerate neutrophil apoptosis to promote their own growth.
- 25. Simulations varying kan Case kan=0 (no population of apoptotic neutrophils available to eat): neutrophils remain the dominant immune cell. Case kan low: sustained inflammation due to too many inflammatory neutrophil byproducts and low M2 population levels. Case kan fit to data: “normal response” Case kan high: faster resolution of inflammation
- 27. References Cooper, Racheal L., et al. "Modeling the effects of systemic mediators on the inflammatory phase of wound healing." Journal of theoretical biology 367 (2015): 86-99. Reynolds, Angela, et al. "A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation." Journal of theoretical biology 242.1 (2006): 220-236. Porcheray, F., S. Viaud, A.-C. Rimaniol, C. Léone, B. Samah, N. Dereuddre- Bosquet, D. Dormont, and G. Gras. 2005. “Macrophage Activation Switching: An Asset for the Resolution of Inflammation.” Clinical & Experimental Immunology 142 (3): 481–89. doi:10.1111/j.1365-2249.2005.02934.x. Brown, Bryan N., Brian M. Sicari, and Stephen F. Badylak. 2014. “Rethinking Regenerative Medicine: A Macrophage-Centered Approach.” Frontiers in Immunology 5 (November). doi:10.3389/fimmu.2014.00510. Stout, R. D. 2010. “Editorial: Macrophage Functional Phenotypes: No Alternatives in Dermal Wound Healing?” Journal of Leukocyte Biology 87 (1): 19–21. doi:10.1189/jlb.0509311. Novak, Margaret L., and Timothy J. Koh. 2013. “Macrophage Phenotypes during Tissue Repair.” Journal of Leukocyte Biology 93 (6): 875–81. doi:10.1189/jlb.1012512. Duffield, Jeremy S. 2003.“The Inflammatory Macrophage: A Story of Jekyll and Hyde.” Clinical Science 104(1):27–38. DiStefano III, Joseph. Dynamic systems biology modeling and simulation. Academic Press, 2015. Eisenberg, M. C., & Hayashi, M. A. L. (2014). Determining identifiable parameter combinations using subset profiling. Mathematical Biosciences, 256, 116–126. https://doi.org/10.1016/j.mbs.2014.08.008 Gábor, A., Villaverde, A. F., & Banga, J. R. (2017). Parameter identifiability analysis and visualization in large-scale kinetic models of biosystems. BMC Systems Biology, 11(1). https://doi.org/10.1186/s12918-017-0428-y Hengl, S., Kreutz, C., Timmer, J., & Maiwald, T. (2007). Data-based identifiability analysis of non-linear dynamical models. Bioinformatics, 23(19), 2612–2618. https://doi.org/10.1093/bioinformatics/btm382 Maiwald, T., & Timmer, J. (2008). Dynamical modeling and multi-experiment fitting with PottersWheel. Bioinformatics, 24(18), 2037–2043. https://doi.org/10.1093/bioinformatics/btn350
Hinweis der Redaktion
- We developed a model for the inflammatory response that includes macrophage polarization between M1 and M2 phenotypes since it’s dysfunction of this phenotypic switch can disrupt the timely influx and egress of immune cells during the healing process and lead to chronic wounds or disease. The modulation of macrophage population has been suggested as a strategy to dampen inflammation in diseases that feature chronic inflammation, such as diabetes and atherosclerosis. Our model is able to reproduce the expected timing of sequential influx of neutrophils and macrophages in response to an inflammatory stimulus. Model parameters were estimated with in vivo experimental data from a mouse model of peritonitis. I’ll give a brief overview of the biology, then model development, next the parameter selection process, and finally with what we call the identifiable parameter set we perform sensitivity analysis do some simulations
- We start with some kind of injury or infection in the body…
- This attracts immune cells to the cite
- This is a simplification – there is a whole spectrum
- But we are interested in whether the macrophages are “mostly” PRO inflammatory or mostly “ANTI” inflammatory
- Too many M1s – chronic inflammation seen in everything from wounds that don’t heal, to autoimmune, to diabetes and heart disease. Too many M2s can also lead to problems like keloid scars on organs or even cancer. We want just enough. We want M1s to come in and kill pathogen and clean things up first, followed by M2s to calm things down and rebuild, and then we want them to mostly go away so that healing is complete
- How does the population shift over time given certain assumptions about what makes macrophages more M1 or M2?
- Here all variables represent local levels. To create this model, previous models of immune response to a wound [14, 20, 23] have been adapted. What’s new is this polarization of macrophages between phenotypes M1 and M2, transition of neutrophils to the apoptotic state, and the injection of nutrient broth to induce growth of pathogen and stimulate immune response. Initial conditions: the healthy peritoneal cavity is impermeable and is assumed to be nearly sterile prior to inflammatory stimulus, with very low levels of pathogen, and so has no immune cell influx. So, all of our immune cell variables have an initial condition of zero
- The injection of nutrient broth drives a spike in pathogen growth that quickly dies down as broth is consumed and pathogen is removed by macrophages and neutrophils.
- Immune cells are assumed to activate and influx into the local environment rapidly compared to other dynamics, so the quasi-steady state assumption is used. This gives rise to Michaelis-Menten type activation and influx terms in Eqs 1–3. In addition, we do not explicitly model cytokines because we did not have in vivo measurements. We let the production of immune cells to act as an indicator of associated cytokine level. Resting neutrophils become activated by pathogen and the debris formed by apoptotic neutrophils at the rate RN. As neutrophils become laden with bacteria, they undergo apoptosis at rate kan. Apoptotic neutrophils are then removed by M1s at rate kanm1, M2s at rate kanm2, and by active neutrophils at rate kann. We have chosen kann to be much smaller than both kanm1 and kanm2 as appropriate for the case when both macrophages and neutrophils are present, but in the absence of macrophages, the clearance of apoptotic cells by neutrophils may take on greater importance [44, 45]. Apoptotic neutrophils that are not cleared undergo secondary necrosis at rate μan, contributing to the positive feedback described in the neutrophil activation term RN. The inhibition term fi(x, N) models the inhibition of macrophage activity by neutrophils due to oxidation of the environment. The same parameter, n∞, is used to determine the level at which the presence of neutrophils inhibit the macrophages regardless of phenotype (M1 or M2) and what they are phagocytosing (pathogen or apoptotic neutrophils). This is due to the simplifying assumption that all macrophages are inhibited the same by the oxidative stress in the local environment.
- Resting monocytes (MR) arrive next. The majority of these first monocytes differentiate to an inflammatory M1 phenotype in response to pathogen, byproducts of neutrophils, M1s and their cytokines, and cytokines spilled by necrotic apoptotic neutrophils at rate RM1(P, N, M1, AN). Background levels of anti-inflammatory cytokines, kc cause a small portion of monocytes to differentiate to an M2 phenotype. Intrinsic decay is assumed to occur at rate μm1 in M1s and at rate μm2 in M2s. M1s are assumed to be able to switch to M2s at rate km1m2, and this switch is assumed to be promoted by the phagocytosis of apoptotic cells [1, 3, 42, 46]. we also allow for the possibility of a transition from M2 to M1 in Eq 1 at rate km2m1 as well. Late arriving monocytes are assumed to be able to activate to the M2 phenotype in response to anti-inflammatory cytokines produced by M2s at rate RM2(M2).
- Cell count data is given in units of 107 cells. The model given by Eqs 1–6 was fit to experimental data using the trust region method within PottersWheel, a MATLAB toolbox for parameter estimation [47]. The trust region approach uses the lsqnonlin algorithm of MATLAB’s optimization toolbox, which allows for the specification of bounds on the parameter space to be searched. Bounds for each parameter are given in Table 1.
- Estimate all parameters. Use the fitted model to generate the discretized sensitivity matrix S. Fix insensitive parameters. Use S to rank parameters by sensitivity. Set a threshold such that parameters with sensitivity below the threshold (insensitive) are fixed and parameters with sensitivity above the threshold (sensitive) are analyzed in Step 3. Select low collinearity group of parameters as identifiable (ID) subset. Check for pairwise correlations between parameters by deriving an approximate correlation matrix from S. Check for collinearity between groups of parameters with a collinearity index (CI) measure. Set a threshold such that groups of parameters with CI above the threshold are considered collinear. Groups of parameters with CI below the threshold are considered identifiable subsets. Estimate identifiable (ID) subset of parameters. One identifiable subset of parameters is selected to be estimated. The remaining parameters are fixed.
- For all three observable model outputs (N, M1, and M2) sampled at 10 time points with 24 model parameters, a 30 × 24 discretized sensitivity matrix S is produced. To test structural identifiability of the model a posteriori, we generated these matrices at a variety of locations in parameter space within the bounds given in Table 1 and found the rank and the singular values for each. Since each of these matrices was determined to have full column rank and no zero singular values, we concluded that the model is locally SI [51] within the bounds we had set for parameter estimation. Has some nice properties: We had determined that all singular values were greater than zero, indicating SI, but only 6 of the 24 singular values obtained had values with order of magnitude greater than zero. If we consider the very small singular values essentially zero for the purpose of rank calculation (in order to reduce problems with numerical identifiability) this gives rank(S) = 6, and since rank(S) can be used to identify the number of parameters that can be included in an identifiable subset [36, 50], a subset of size 6 is suggested. The parameter estimation problem was therefore reduced to finding identifiable subsets of size 6 out of the 16 sensitive parameters.
- Next, we ranked the impact of each parameter on all three observable model outputs (N, M1, and M2) by calculating a root mean square sensitivity measure, as defined in Brun et al. [34], for each column j of the normalized sensitivity matrix. Parameter j is deemed insensitive if RMSj is less than 5% of the value of the maximum RMS value calculated over all parameters. By this measure, 8 parameters were deemed insensitive, as shown in Fig 4, and fixed at their nominal values.
- First we look at pairwise correlations using the Fisher Information Matrix for the sensitive subset of parameters. There are many significant linear correlations (greater than 0.7) between sensitive parameters that appear as black or white squares on the off diagonal. Linearly dependent parameters compensate for changes in any 1 of the parameters so the impact of these variations on output isn’t clear and they cannot be reliably estimated together. The same is true of groups of parameters. We were looking for a minimally correlated group of 6 parameters and to do these we used a collinearity index measure CI defined by Brun et al. [34] that is similar to condition number. We construct submatrices of the sensitivity matrix with 6 sensitivity vectors, across all combinations of 6 parameters, and then compute λk which is the smallest eigenvalue of the Fisher Information Matrix of each of the sensitivity sub matrices. We used CI = 20 as the cutoff for parameter subset selection since a change in output caused by a change in any parameter in the subset can be compensated for up to 5% by other parameters in the subset) [34]. To do this we used the VisId toolbox in MATLAB.
- The most influential parameters on M1 behavior are snr and smr (availability of resting neutrophils and monocytes), kpg (behavior of inflammatory stimulus), km1p and knp (response of M1s and neutrophils to inflammatory stimulus), and uan (rate of secondary necrosis of neutrophils). In the present context, M1s are primarily activated by initial inflammatory stimulus and necrosis of apoptotic neutrophils that have not been phagocytosed. This supports the hypothesis that effective clearance of apoptotic cells is important in the resolution of inflammation [1, 40, 46, 54–59]. If our parameter estimates had been obtained by fitting to data from chronic inflammation, feedback from existing M1s and the pro-inflammatory byproducts of existing neutrophils would likely be greater contributors to M1 response. Negatively related to magnitude of M1 response are parameters μm1 (decay or efflux rate of M1s) and n∞ (the level of neutrophils required to inhibit macrophage activity by 50%). As the threshold for inhibition of M1s increases, the magnitude of the M1 population decreases because less M1s are required to mount an adequate response.
- The importance of neutrophils and neutrophil apoptosis in mounting a timely and sufficient M2 response is evidenced by the high sensitivity of M2 peak timing and amplitude to neutrophil-associated parameters snr, uan, kan, knp, unr, n∞, kanm1, kanm2, and knan. The magnitude of the M2 population peak is also strongly positively associated with km1m2 (switch rate from M1s) and km2m2 (feedback from existing M2s). Increasing rates of decay or efflux for resting monocytes (μmr) and resting neutrophils (unr) diminishes M2 population magnitude, as does reduced M1 activation by pathogen (km1p), indicating M2 dependence on the population size of other immune cells.