key: cord-1044335-ox7t41iu authors: Blanco-Rodríguez, Rodolfo; Du, Xin; Hernández-Vargas, Esteban title: Computational Simulations to Dissect the Cell Immune Response Dynamics for Severe and Critical Cases of SARS-CoV-2 Infection date: 2021-09-20 journal: Comput Methods Programs Biomed DOI: 10.1016/j.cmpb.2021.106412 sha: bdedb280a4fdde66b84b63a2aba0cc77da6761ff doc_id: 1044335 cord_uid: ox7t41iu Background: COVID-19 is a global pandemic leading high death tolls worldwide day by day. Clinical evidence suggests that COVID-19 patients can be classified as non-severe, severe and critical cases. In particular, studies have highlighted the relationship between the lymphopenia and the severity of the illness, where CD8 [Formula: see text] T cells have the lowest levels in critical cases. However, quantitative understanding of the immune responses in COVID-19 patients is still missing. Objectives: In this work, we aim to elucidate the key parameters that define the course of the disease deviating from severe to critical cases. Dynamics of different immune cells are taken into account in mechanistic models to elucidate those that contribute to the worsening of the disease. Methods: Several mathematical models based on ordinary differential equations are proposed to represent data sets of different immune response cells dynamics such as CD8 [Formula: see text] T cells, NK cells, and also CD4 [Formula: see text] T cells in patients with SARS-CoV-2 infection. Parameter fitting is performed using the differential evolution algorithm. Non-parametric bootstrap approach is introduced to abstract the stochastic environment of the infection. Results: The mathematical model that represents the data more appropriately is considering CD8 [Formula: see text] T cell dynamics. This model had a good fit to reported experimental data, and in accordance with values found in the literature. The NK cells and CD4 [Formula: see text] T cells did not contribute enough to explain the dynamics of the immune responses. Conclusions: Our computational results highlight that a low viral clearance rate by CD8 [Formula: see text] T cells could lead to the severity of the disease. This deregulated clearance suggests that it is necessary immunomodulatory strategies during the course of the infection to avoid critical states in COVID-19 patients. tionship between the lymphopenia and the severity of the illness, where CD8 + T cells 23 have the lowest levels in critical cases. However, quantitative understanding of the 24 immune responses in COVID-19 patients is still missing. Objectives: In this work, we aim to elucidate the key parameters that define the 26 course of the disease deviating from severe to critical cases. Dynamics of different 27 immune cells are taken into account in mechanistic models to elucidate those that 28 contribute to the worsening of the disease. Methods: Several mathematical models based on ordinary differential equations 30 are proposed to represent data sets of different immune response cells dynamics such 31 as CD8 + T cells, NK cells, and also CD4 + T cells in patients with SARS-CoV-2 infec-32 tion. Parameter fitting is performed using the differential evolution algorithm. Non-33 confinement of their population, travel restrictions, forced use of mask in public spaces, and even a night-It is still controversial whether virus persistence can increase the severity of the disease. SARS-Cov- 80 2 viral dynamics have shown remarked differences between severe patients and non-severe patients. It 81 has been reported that the viral load peak is higher in non-severe patients (∼ 10 8 copies/mL) than severe 82 patients (∼ 10 7 copies/mL), and viral shedding time has been longer in severe patients [55] , even the 83 virus is detectable until death [68] . On the other hand, it also has been reported the mean viral load of 84 severe cases around 60 times higher then mild cases [33] . Hasanoglu et al. [21] conducted a study to 85 evaluate six different types of samples from 60 COVID-19 patients with different age and clinical history, 86 including asymptomatic patients. They showed that asymptomatic patients could have higher viral load 87 than symptomatic patients, and observed a significant decrease in viral load as disease severity increases. environment, which would lead to multi-system inflammatory syndrome in children. 99 There are several studies around the dynamic changes of lymphocytes [35, 32] interaction with the immune response, there has not been any study to quantify the differences between 131 severe and critical patients with COVID-19. As far as we know, there are no models fitted to T cells 132 data in order to examine the relation between T cell dynamic and severity of the illness. In this work, we 133 contribute to the mathematical study of SARS-CoV-2 dynamic and the T cell dynamics to elucidate the 134 principal role of lymphocytes in the develop of the disease between severe ad critical patients. severe and critical groups. The moderate cases were those with fever, typical symptoms and pneumonia. 140 206 severe cases had respiratory distress, blood oxygen saturation less than 93%, or arterial partial pressure 141 of O 2 to fraction of inspired oxygen ratio less than 300 mmHg. 91 critical cases had respiratory failure, 142 shock or multiple organ dysfunction needing intensive care unit treatment. where is the virus level, the number of CD8 + T cells, the viral replication rate with maximum , follows a log-sigmoidal form with half saturation constant . The parameter = (0) represents lack of data before illness onset, we assumed the initial level of T cells equal to the median of the CD8 + 172 T cells in the day 3 ( (0) = (3)) from the reported data for each case. Infection time was assumed at −3 173 days after illness onset (daio). This infection time was also varied in −10, −5, −3 and 0 daio in order to 174 find the best value but we found no significant difference between the results. A stability analysis of this 175 model can be found in [2] . 176 In our model, we included CD8 + T cells in the peripheral blood of patients described above, who have 177 a wide range of comorbidities. We only considered several and critical cases, since data from moderate 178 cases showed no marked changes during the disease course. suggested that CD4/CD8 ratio is significantly higher in critical patients than non-critical patients [43] . Be-182 cause of that, we modified the Model 1. We now assumed that CD4 + helps to the proliferation of CD8 + T 183 cells which occurs at rate 4 where 4 is CD4 + T cell level and is a free parameter to be estimated. We The fixed parameters were the same as the Model 1, and , , , and were estimated as the same way 187 by using CD8 + T cell data. We also explored two different ways to integrate CD4 + T cell data, which is shown in Model 3 and 4. The results can be found in Figure S1 in the Supplemental Material. Here, and 4 parameters are estimated. In this model, the dynamics of the virus was not appreciated 190 for severe cases, in critical cases the virus was not cleared. The results can be found in Figure where and 4 parameters are estimated. In this model the viral load did not drop to zero. The results can 193 be found in Figure S3 in the Supplemental Material. The parameters , , and were estimated using CD8 + T cell data. The results shows that tends to not decrease enough to be below detectable levels. These results are showed in Figure S4 in Supplemental Material. Model 6. CD8 + T cell, CD4 + T cell and NK cell responses: We also explored combining the above 205 two models to use CD4 + T cells and NK cells data without improving the AIC value. This model is given The results of this model are very similar to Model 2, so the NK cell response is not supported by this 208 data set. These results can be found in Figure S5 in Supplemental Material. The ordinary differential equations of the model were solved using a Dormand-Prince fifth-order Runge-Kutta algorithm. The estimation of the free parameters was performed by minimize the Root Mean Square Error (RMSE) using the difference between the experimental measurement ( ) and the predictive output (̄ ) as follows: where is the corresponding sample and is the total number of measurement. To minimize the RMSE In order to compare between different models, we used the Akaike information criterion (AIC) defined by: where is the number of data points and is the number of unknown parameters. A lower AIC values 225 means a better description of the data. A mathematical model is said to be identifiable when the parameter set can be uniquely determined. Here we used the profile likelihood method proposed by [47] . In this method one by one the parame- The experimental data for CD8 + T cells and their respective fit of the Model 1 are displayed in Fig. 3a . The experimental data were reproduced from [64]. For severe cases, the CD8+ T cell response starts about 252 10 to 20 daio reaching its peak between 35 to 45 daio, while for critical the CD8+ T cell response starts 253 late, around 30-40 daio with a peak between 40 to 50 daio. Note that critical cases begin with a lower level 254 of CD8 + T cells than severe cases (half of them), however, both reaches approximately the same level of 255 the moderate cases at the end of the disease course. The total count of cells, that is the area under the 256 curve (AUC), between both cases are in the same order of magnitude although is lower for critical cases 257 (1.3 × 10 7 cells days/mL) than severe cases (2.2 × 10 7 cells days/mL). The viral load obtained using the Model 1 with the parameter fitted to CD8 + T cells experimental 259 data is displayed in Fig. 3b . The viral load peaks around 40 daio for critical cases and 20 daio for severe cases. There is a delay in the peak of the viral load for critical cases compared to severe cases, also critical 261 viral load peak is around a order of magnitude lower than severe viral load peak, and even the total viral 262 count (the AUC) is higher for severe case (1.2 × 10 8 copies days/mL) than critical cases (4.5 × 10 6 copies 263 days/mL). This outcome is debatable since high viral loads have been reported from patients who develop noted that the medians of the data were used in this analysis; however, there is a large variability among 276 them, so the parameters could be practically non-identifiable. The best fitted parameters are presented in Table 1 for CD8 + T cells and the two cases of illness 278 severity. The viral replication rate for critical cases is a half of that for severe cases. The viral clearance 279 for critical cases is approximately one third of that for severe cases, while the CD8 + T cell proliferation 280 rate is higher for critical cases. These results suggest that the rapid proliferation of CD8 + T cells may 281 compensate the low clearance rate, which could be a key aspect in the development of the disease. Due to high variability of the data, we performed bootstrap fits in order to obtained the confidence value for the best fit. This discrepancy could due to high variability of the experimental data used. In order to explore the dependencies of the parameters, we displayed scatter plots in Figs 4g, 4h and 292 4i. These plots reveal that there are no strong correlation between , and . However, we can notice a 293 slight inter-dependence between and parameters for critical cases; and and parameters for severe 294 cases. In the former, increasing decrease ; and in the latter, increasing increase . 295 We explored the modification of the Model 1 by adding CD4 + T cells and NK responses as mentioned 296 above. Fitting these models to the data revealed that including CD4 + T cells as a helper of the proliferation 297 of CD8 + T cells does not improve the fits, similarly for NK response. In the Table 2 The role of the immune system during SARS-CoV-2 infection is fragmented. The T cell kinetics seem 309 to be decisive in the resolution of severe or non-severe patients [32] . CD8 + T cells are relevant for killing On the other hand, fitting results show a viral clearance rate for severe cases is higher than that for 331 critical cases. The viral replication rate for severe cases is also higher than that for critical cases which translates to a higher viral peak. Therefore, although the severe cases have a low production of CD8 + T The authors declare that the research was conducted in the absence of any commercial or financial 377 relationships that could be construed as a potential conflict of interest. [10] Cartocci, A., Cevenini, G., Barbini, P., 2021. 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