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Immune‐viral dynamics modeling for SARS‐CoV‐2 drug development

Abstract Coronavirus disease 2019 (COVID‐19) global pandemic is caused by severe acute respiratory syndrome‐coronavirus 2 (SARS‐CoV‐2) viral infection, which can lead to pneumonia, lung injury, and death in susceptible populations. Understanding viral dynamics of SARS‐CoV‐2 is critical for development of effective treatments. An Immune‐Viral Dynamics Model (IVDM) is developed to describe SARS‐CoV‐2 viral dynamics and COVID‐19 disease progression. A dataset of 60 individual patients with COVID‐19 with clinical viral load (VL) and reported disease severity were assembled from literature. Viral infection and replication mechanisms of SARS‐CoV‐2, viral‐induced cell death, and time‐dependent immune response are incorporated in the model to describe the dynamics of viruses and immune response. Disease severity are tested as a covariate to model parameters. The IVDM was fitted to the data and parameters were estimated using the nonlinear mixed‐effect model. The model can adequately describe individual viral dynamics profiles, with disease severity identified as a covariate on infected cell death rate. The modeling suggested that it takes about 32.6 days to reach 50% of maximum cell‐based immunity. Simulations based on virtual populations suggested a typical mild case reaches VL limit of detection (LOD) by 13 days with no treatment, a moderate case by 17 days, and a severe case by 41 days. Simulations were used to explore hypothetical treatments with different initiation time, disease severity, and drug effects to demonstrate the usefulness of such modeling in informing decisions. Overall, the IVDM modeling and simulation platform enables simulations for viral dynamics and treatment efficacy and can be used to aid in clinical pharmacokinetic/pharmacodynamic (PK/PD) and dose‐efficacy response analysis for COVID‐19 drug development.


Methods
Model parameters with variability are assumed to be log-normal distributed with a mean at the population parameter and a standard deviation of . The model was fitted to the log10transformed viral load data. The residual error between the log10-transformed viral load and model predictions is assumed to follow a Gaussian distribution with standard deviation : log 10 ( ) − log 10 ( )~(0,1), in which was estimated as the residual error parameter. In this study, the initial condition for the model in fitting and simulations was set to 0 = , 0 = 1 / , 0 = 10 / , where is the estimated initial size of target cells.
We evaluated the adequacy of the final model using visual predictive check (VPC), in which 500 replicates of the analysis dataset were simulated with each replicate has the same number of participants as the analysis dataset.
A virtual population (N=1000) for each disease severity level was generated and simulated from the IVDM using the mrgsolve R package [8] based on the final model and estimated parameter values. Model parameter of a virtual subject is generated by sampling a Gaussian random number using the log population parameter log( ) as the mean and the IIV as the standard deviation ~(log( ) , 2 ), and the actual parameter is generated as = exp( ). Multiple sets of simulations are performed to explore the hypothetical treatment efficacy in different scenarios in all disease severity levels. These explorations include varying the drug effect parameter ( = 0, 0.80, 0.85, 0.90, 0.95), start time of treatment (day 0, 1, …, 20 post symptom onset), and duration of treatments (long-term treatment or 10-day treatment). Two key endpoints, positive VL (+VL) duration days and total VL AUC are extracted from the simulated results to explore the usefulness of the model to inform decisions in hypothetical scenarios in clinical outcomes.

Results
The inter-individual variabilities (IIVs) are estimated for four key parameters: , , , and 50 , and are shown in CV% in Table 1. The IIV for viral production rate and viral clearance rate was estimated to be very high at 427% and 188%, suggesting the model attributes the high variability in viral load peak and set points to these two parameters. IIVs for other parameters are all smaller than 100%. The estimated residual error is high at 1.65, which might be driven by different VL assays and sampling approaches across these studies.
The cell death rate in mild cases is = , in moderate cases as = exp ( ), and in severe cases as = exp ( ). The covariate parameters are estimated as = −0.294 and = −0.942. Moderate and severe disease population was generated from mild virtual population by multiplying the estimated disease severity covariate parameter on the infected cell death rate.
IVDM model parameters were estimated from the VL dataset using nonlinear mixed effects (NLME) approach. These parameter values can help understand the dynamics of SARS-CoV-2 viral infection and time-dependent adaptive immune response. The density of target cell and viral production rate are not identifiable, so it is not feasible to estimate both parameters with variabilities. In the final model we chose to estimate without variability and with variability based on the performance of the model. In biology, both and may be varying across different populations. Therefore, the estimated high variability of (CV%=427%) may be partly coming from actual variability of the size of susceptible target cells. The typical viral clearance rate was estimated as = 25.4 per day, which is much faster than the estimated infected cell death rate = 0.994, suggesting that the rate of first phase VL decline is mainly determined by the rate of infected cell death, not by viral clearance, which is similar to the mechanisms in other viruses such as HIV [9]. The Hill coefficient was estimated as = 8.18, and the time-dependent immune response parameters are estimated as 50 = 32.6 and = 175 −1 , the estimated variability (%CV) around 50 is 77.5%.. These values suggest a steep adaptive immune response occurring around 4 weeks after infection with a large variability between individuals (between 11 days and 97 days), and the SARS-CoV-2-specific adaptive immunity may significantly increase the infected cell death rate (estimated 175-fold) at its maximum strength. However, these estimates were based on the available viral load data only, immunology data from future studies are needed to validate these estimates. There are some emerging studies showing that both SARS-CoV-2-specific CD4 + T-cell and CD8 + T-cell responses occur within 1-2 weeks after symptom onset in most patients [10], but more detailed data is yet to be seen.
The SARS-CoV-2 IVDM can be readily used to generate virtual populations of different COVID-19 disease severities to simulate clinical trials and inform clinical decisions. Consistent with the viral load data, the peak viral load in clinical trial simulations exhibits large variabilities between individuals ranging from 10 3 to >10 7 c/mL with N=1000 subjects across mild, moderate, and severe diseases. This high variability in VL is originated from the data and may be caused by different viral load sampling approaches and assays across five different studies. But on the other hand, the high VL variability could also be reflecting the truth as shown in other studies [11]. Another significant difference to note between different disease severity levels is that the duration of positive VL (+VL) is significantly prolonged in severe cases, while the mean peak VL does not differentiate among disease severity levels. The large difference in VL AUC between different disease severities are mainly driven by the difference in VL duration controlled by the infected cell death rate. The difference in peak VL between different severities are much smaller, only about 8% higher in moderate cases than in mild cases, and 20% higher in severe cases than in mild cases (Table S4).  Figure S1 Representative VL profiles and fits for mild, moderate, and severe diseases. Red dashed lines are the VL LOQ level.
Figure S2 Viral load profiles simulated from a virtual population of N=100 with no treatment. Figure S3 Goodness of fits plot of the IVDM model to the viral load data. Blue dots: the observed above LOQ viral load data, and red dots: the simulated below LOQ data from Monolix. Figure S4. Visual predictive check (VPC) for the model fitting. Prediction intervals and empirical percentiles are calculated based on 500 simulated replicates of the dataset. Blue lines are the 10th, 50th and 90th percentiles calculated over bins for the original data set. The blue and pink bands represent the 90% prediction interval for these same percentiles calculated using the simulated data sets. Red circles highlight when the blue lines are not within the prediction interval.