A Nonconventional Analysis of CD4^{+} and CD8^{+} T Cell Responses During and After Acute Lymphocytic Choriomeningitis Virus Infection
AA Nonconventional Analysis of CD4 + and CD8 + T Cell ResponsesDuring and After Acute Lymphocytic Choriomeningitis Virus Infection by Dwayne John
March 23, 2018Computational Science ProgramDepartment of ChemistryMiddle Tennessee State University301 E Main StMurfreesboro, TN 37132
A mathematical model from a previous work was re-fitted and analyzed for experimental data regarding the cellularimmune response to the lymphocytic choriomeningitis virus. Specifically, the CD + T cell response to six MHC classI-restricted epitopes (GP* and NP*) and CD + T cell responses to two MHC class II-restricted epitopes[2]. In thiswork, we use calibration through log likelihood maximization to investigate if different parameters can produce amore accurate fit of the model presented previously in the paper titled
Different Dynamics of CD + and CD + TCell Responses During and After Acute Lymphocytic Choriomeningitis Virus Infection [2]. a r X i v : . [ q - b i o . T O ] M a y I. HISTORICAL BACKGROUND
The leading cause of death among people under the age of 45 is trauma. A main cause of death after trauma isinternal bleeding of the abdominal organs. Of those abdominal organs, the spleen is one of the most often injureddue to blunt trauma[8], so much so that it is affected in 32% of patients that have traumatic abdominal injuries[7].Every year, about 39,000 patients are admitted in hospitals throughout the United States of America for the treat-ment of blunt splenic trauma (BSI). Of those admitted patients, 28,000 will undergo nonoperative management care[6].It was once believed that the spleen could not heal spontaneously and would also rupture later during patientrecovery. These ideas started to be re-examined in the 1970s when data about post-operative infections regardinglaparotomies and removing the spleen was published[8]. Due to this new knowledge, more doctors are now perform-ing procedures such as splenic artery embolization (SAE) which preserve the spleen and also hold a high success rate[7].Like the appendix, the spleen’s function in the human body was unknown to many. In 1919, two individuals(Morris and Bullock) showed that dogs with their spleen that were infected with rat plague bacillus had a highersurvival rate as opposed to those that were infected without their spleen. Due to limitations in diagnostic medicineat the time, splenectomy was a main standard of care. Until about 65 years ago, no one even questioned the idea ofsplenectomy as a treatment en masse when there were a plethora of overwhelming polstsplenectomy infection (OPSI)cases. Things started to change roughly fifty years ago when care shifted to splenorraphies and then again about adecade later when clinicians started expectant management of patients became mainstream. Contrary to older ideasregarding the spleen’s function, we now know the spleen assists with immune function. It can help filter specificantigens and microorganisms and act in cell regeneration[3].
II. INTRODUCTION
The lymphocytic choriomeningitis (LCM) virus is a rodent disease that an be passed in many different ways fromrodent to rodent, human to human, and rodent to human. It can stay dormant in mice for as long as 35 years[4]. Amember of the arenaviridae family of viruses, lymphocytic choriomeningitis virus (LCMV) is a human pathogen thatcan infect a large amount of the human population[1].
III. MATERIALS AND METHODS
In the previous study by De Boer et al.[2], data was collected by first injecting six to eight week old mice, male andfemale, with LCMV. At different time intervals, spleen cell samples were extracted and measured from an average ofthree to four mice per data point. The number of specific T cells per spleen was measured. A set of linear differentialequations was used to describe the data. Due to the linear nature of the model, a solution to the differential equationscan be obtained. De Boer, et al. estimated the parameters of the model using the DNLS1 subroutine from theCommon Los Alamos Software Library and based on the Levenberg-Marquardt algorithm[5].We employ a different, more efficient strategy to arrive at the same destination: parameter optimization through loglikelihood maximization.In order to calculate the model results with the necessary parameters, we first calculate the values of the populationof activated cells, A and the memory cells, M. The death rate of memory cells and activated cells is given by δ M and δ A , respectively.There are three phases in the model: rapid expansion, rapid contraction, and slower contraction of the T cellpopulation. The three phases for modeling the amount of T cells per spleen correspond to the following time inter-vals: t < T, t between T and T+∆, and T > T+∆.The population of A cells in the initial phase of rapid cellular growth is given by the following differential equa-tion: dAdt = ρA, FIG. 1. Cubic spline interpolation of specific CD + T cells per spleen (two responses GP61 and NP309) with respect to daysafter LCMV infection[2].FIG. 2. Cubic spline interpolation of specific CD + T cells per spleen (two responses GP61 and NP309) with respect to daysafter LCMV infection[2]. the parameter ρ is the net expansion rate.The population of A and M cells during the contraction phase is given by the following equations: dAdt = − ( r + α + δ A ) AdMdt = rA − δ M M FIG. 3. Cubic spline interpolation of specific CD + T cells per spleen (six responses GP33, GP118, NP205, NP396, GP276,GP92) with respect to days after LCMV infection[2].
The parameter α is the cell death rate, also known as rapid apoptosis and r is the contraction phase rate. Attime T, A reaches its peak value and is calculated by A ( T ) = A (0) exp[ ρT ].We then calculate the value of A at T + ∆. The ∆ term is the duration of the phase of rapid contraction of T cells pop-ulation. That phase is followed by a phase of slower contraction of the population. Now, A ( T +∆) = A ( T ) exp[ − δ (∆)].Next, we calculate the value of M at T+∆, M ( T + ∆) = exp[ − δ M ( T + ∆)]( rA ( T )(1 − exp[ − ( δ A − δ M )( T + ∆)])) δ − δ M For t < T , A = A (0) exp[ ρt ] and M=0.For t > T , A = A ( T ) exp[ − δ ( t − T )] and M = exp[ − δ M ( t )] rA ( T )(1 − exp[ − ( δ − δ M ) t ]) δ − δ M , where δ = r + δ A + α .For T < t < T + ∆, A = A ( T + ∆) exp[ − δ (cid:48) ( t − T − ∆)] and M ( t ) = exp[ − δ M t ]( rA ( T + ∆)(1 − exp[ − ( δ (cid:48) − δ M ) t ])) + M ( T + ∆)( δ (cid:48) − δ M ) δ (cid:48) − δ M , and δ (cid:48) = r + δ A .Data was collected from citation III.1. Sensitivity Analysis
Sobol method was used first to see which parameters have the most influence on the fluctuations of the results. Theresults of these calculations were inconclusive and we were not able to ascertain which parameter had the greatesteffect. We suspect that all parameters have a similar influence on the results of the differential equations.
IV. DISCUSSION
Our sensitivity analysis on the parameters of the model did not show that any particular parameter is significantlymore important than the others.It was found that the optimized parameters calculated in this work, using a simple log likelihood maximizationmethod, are noticeably different from those calculated in the paper[2], using the Levenberg-Marquard algorithm.A few factors that influenced the results are the following. The data was acquired directly from the graphs inthe published paper[2] and not provided via a database from the corresponding author due to time constraints of theproject. The use of this method is subject to errors that are difficult to quantify due to human-machine interaction.Another influential factor is that our study used only the data corresponding to the first 70 days, while the originalarticle used the full data set up to 921 days. Despite these issues, in most cases the newly estimated parameters fallwithin the 95% combined confidence intervals reported in the paper.
Parameter Log Likelihood Maximization Method 95% Combined CI[2] Units ρ d − δ A d − δ M d − r d − α d − T A (0) 19.3 0.4-518.8 CellsTABLE I. Results for implementation of log-likelihood maximization, compared to De Boer et al.’s results from the paper[2]. V. CONCLUSION
We can conclude that, in cases where a quick and simple verification is in order, “data thieving” from publishedgraphs and the log likelihood maximization method can be useful tools. In this case, we were able to obtain resultsconsistent with previously published[2] work using the calibration likelihood estimation parameter optimization.
LIST OF FIGURES CD + T cells per spleen (two responses GP61 and NP309) withrespect to days after LCMV infection[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Cubic spline interpolation of specific CD + T cells per spleen (two responses GP61 and NP309) withrespect to days after LCMV infection[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Cubic spline interpolation of specific CD + T cells per spleen (six responses GP33, GP118, NP205,NP396, GP276, GP92) with respect to days after LCMV infection[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES
I Results for implementation of log-likelihood maximization, compared to De Boer et al.’s results fromthe paper[2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 [1] Bonthius, D. J., in
Seminars in pediatric neurology , Vol. 19 (Elsevier, 2012) pp. 89–95.[2] De Boer, R. J., Homann, D., and Perelson, A. S., The Journal of Immunology , 3928 (2003).[3] Kaseje, N., Agarwal, S., Burch, M., Glantz, A., Emhoff, T., Burke, P., and Hirsch, E., The American Journal of Surgery , 213 (2008).[4] Lehmann-Grube, F., in
Lymphocytic Choriomeningitis Virus (Springer, 1971) pp. 1–173.[5] Mor´e, J. J., in
Numerical analysis (Springer, 1978) pp. 105–116.[6] Requarth, J. A., Journal of Trauma and Acute Care Surgery , 1423 (2010).[7] van der Vlies, C. H., Hoekstra, J., Ponsen, K. J., Reekers, J. A., van Delden, O. M., and Goslings, J. C., Cardiovascularand interventional radiology , 76 (2012).[8] van der Vlies, C. H., Olthof, D. C., Gaakeer, M., Ponsen, K. J., van Delden, O. M., and Goslings, J. C., InternationalJournal of Emergency Medicine4