Comparative assessment of viral dynamic models for SARS‐CoV‐2 for pharmacodynamic assessment in early treatment trials

Pharmacometric analyses of time series viral load data may detect drug effects with greater power than approaches using single time points. Because SARS‐CoV‐2 viral load rapidly rises and then falls, viral dynamic models have been used. We compared different modelling approaches when analysing Phase II‐type viral dynamic data. Using two SARS‐CoV‐2 datasets of viral load starting within 7 days of symptoms, we fitted the slope‐intercept exponential decay (SI), reduced target cell limited (rTCL), target cell limited (TCL) and TCL with eclipse phase (TCLE) models using nlmixr. Model performance was assessed via Bayesian information criterion (BIC), visual predictive checks (VPCs), goodness‐of‐fit plots, and parameter precision. The most complex (TCLE) model had the highest BIC for both datasets. The estimated viral decline rate was similar for all models except the TCL model for dataset A with a higher rate (median [range] day−1: dataset A; 0.63 [0.56–1.84]; dataset B: 0.81 [0.74–0.85]). Our findings suggest simple models should be considered during pharmacodynamic model development.


| INTRODUCTION
The COVID-19 pandemic continues to threaten public health largely now due to new variants of concern with increasing ability to evade antibody responses. Most importantly, these variants challenge vaccination efforts to halt the pandemic, thereby necessitating efforts to develop new antivirals as well as repurposing of existing antiviral therapies. 1 So far, the ongoing development of novel antivirals is promising, albeit drug development processes are time-consuming. Drug repurposing is a time-saving approach as clinical efficacy and safety data are already known for other therapeutic indications. 2 Nonetheless, the push for repurposing therapies for SAR-CoV-2 has been hampered by clinical inefficiencies such as non-randomised placebo-controlled trials and an overemphasis on hospitalised patients. 2,3 As with other respiratory viral infections, an understanding of SARS-CoV-2 viral dynamics could shape the future of potential treatment options to identify antivirals which can disrupt viral replication.
The target cell limited (TCL) model has previously been used to support antiviral development for respiratory infections. 4,5 For SARS-CoV-2, extensions and simplifications of the TC model have been described in recent studies. [6][7][8][9][10] During pharmacokinetic model building, common practice involves starting with the simplest model (often one-compartment) and then adding complexity (further compartments) where data supports this. The goal is to find a model that adequately describes the data and from which important secondary parameters such as area under the curve (AUC) or highest observed concentration (C max ) can be derived. SARS-CoV-2 viral pharmacodynamic modelling has so far often not taken this approach, in that only one model is often considered. As Phase II type trials of repurposed and novel antivirals read out, it is important to consider a model-building approach that is sufficient to characterise viral decline rate as the clinical endpoint of interest. And for most pharmacodynamic models, characterisation of the infected cells death rate (i.e., δ) is the main driver of viral decline rate since often virus clearance (e.g., c in the TCL model) is much faster than viral production rate. 7

What this study adds
• This study compared the simplified and extended forms of the TCL model and found no advantage of the more complex (TCL, TCLE) models over simplified forms (SI, rTCL), which could inform the selection of a suitable modelling approach for SARS-CoV-2 viral dynamics.

| Viral dynamic models for SARS-CoV-2
Schematic diagrams of the slope-intercept exponential decay (SI), respectively. For the SI and rTCL models, the assumption of quasi-steady state between I and V due to the typically faster c than δ translates δ as the overall viral elimination rate as previously at a rate of k. I 2 subsequently release viruses at a rate of ρ with a viral clearance rate of c. Productively infected cells die at a rate of δ. # For models (A) and (B), the assumption of quasi-steady state between I and V due to the typically faster c than δ translates δ as the overall viral elimination rate as previously described. 6,8 duration of virus production (L) was also derived for all models using Equation (8). 10 2.3 | Model-fitting assessment  All four models yielded goodness-of-fit plots that were in satisfactory agreement with trends observed with both datasets ( Figure S3 in the Supporting Information). VPC plots were adequate for all models for dataset B. However, VPC plots for the TCL and TCLE models displayed poor predictive performance on the 5th percentile below the limit of detection (LOD) at early time points for dataset A ( Figure S4 in the Supporting Information).

| DISCUSSION
In the present study, the model performance of the TCL model includ- Likewise, for SARS-CoV-2, having more complex models may be useful for hypothesis testing but particularly challenging for fitting data where strong prior information on required parameters may be lacking. Thus, in the proposed rTCL model to characterise SARS-CoV-2 viral dynamics, Kim et al., 8  The estimates for δ across the different models for both datasets were also largely consistent with those previously reported for SARS-CoV-2 (range: 0.27-2.29 day À1 ). 7,8,14 Of note, an alternative approach known as model averaging has been described for viral dynamic models where different models yielding similarly good fits are simultaneously utilised to account for model uncertainty. 17 Although this approach may be reasonable, such complexity may not be required as the primary focus of viral dynamic models is the estimation of δ, which can equally be well characterised by simpler models, as seen here.
There are some limitations worth noting in this study. Firstly, only two datasets were evaluated and therefore our results may not be universally representative. Secondly, R 0 was poorly estimated with the datasets employed in this study and as such the results should be interpreted with caution. Thirdly, participants in the two datasets were recruited prior to the emergence of SARS-CoV-2 variants of concern and, therefore, the results here ought to be interpreted within this context. Further studies should explore the performance of these models with SARS-CoV-2 emerging variants. In addition, our analysis was restricted to models proposed to describe antiviral effects in clinical trials and we did not test viral dynamic models from epidemiological studies, 18,19 which would be interesting to address in future work. Future studies may also consider a joint pharmacometric and epidemiological modelling approach to broaden the understanding of SARS-CoV-2 viral dynamics. Finally, we did not compare the performance of the different models in addressing other potential goals in viral dynamics modelling such as detecting antiviral effects and the impact of timing of therapeutic interventions on treatment outcomes.
Such evaluations may therefore necessitate the use of more complex models and a minimalist model may not be the best choice. In such context, complex models may be considered, particularly where their structural identifiability could be improved without compromising the intended modelling goal.
In conclusion, as shown in the present study, we found no advantage of the complex models over simplified forms. This emphasises the need to explore both simplified and extended models to ascertain the most appropriate pharmacodynamic model development for

SARS-CoV-2 viral dynamics.
ACKNOWLEDGEMENT This paper is dedicated to our co-author Dr. Tabitha Mahungu who sadly passed away in April 2022.