## 2 Materials and Methods
### CytoSolve In Silico Modeling Protocol
#### 2.2.1 Control Conditions  
Under control conditions, apigenin and hesperidin concentrations are kept at zero. For the IL-1\(\beta\)-induced COX-2 synthesis model, chondrocyte was presumed to be in an inflammatory state. Under these conditions, IL-1\(\beta\) levels were found to be 0.0032 nM [20]. With this as an initial condition (IL-1\(\beta\) = 0.0032 nM), COX-2 concentrations were estimated for the control condition by simulating a COX-2 synthesis model for a 2-day period. For the arachidonic acid metabolism model, chondrocyte was assumed to be under inflammatory conditions. Under such inflammatory conditions, arachidonic acid levels were found to be 35 mM [21; 22]. Arachidonic acid upregulates the production of PGE2, which is a pro-inflammatory biomarker. Using the initial condition of arachidonic acid = 35 mM, PGE2 concentrations were estimated for the control condition by simulating an arachidonic acid metabolism model for a 2-day period. The predicted steady-state concentration of PGE2 from the arachidonic acid metabolism model was used as an initial condition for the PGE2 signaling model. TRPV1 and CGRP concentrations were estimated for the control condition by simulating PGE2 signaling model for a 2-day period. Under an inflammatory state, NADPH oxidase activity, which leads to production of ROS, doubled in endothelial cells [23]. The oxidative stress model is simulated for a 2-day period to obtain the control value of ROS using doubled NADPH oxidase activities as an initial condition. 2.2 Computer Simulations to Study the Effect of Apigenin and Hesperidin on Integrated Model of Joint Pain  
The joint-pain integrated model includes four (4) individual models: (1) arachidonic acid metabolism model, (2) PGE2 signaling model, (3) COX-2 synthesis model, and (4) oxidative stress model. The effect of apigenin and hesperidin, individually and in combination, was analyzed by estimating the concentration levels of PGE-2, TRPV1, CGRP, COX-2, and ROS in the presence of apigenin and hesperidin. The integrated model of joint pain was simulated using standardized the CytoSolve(r) protocol detailed in Supplementary Materials Section S1.8. Initial simulations indicated that all five biomarkers reached a steady-state value at a simulation period of 2 days, and increasing the simulation time beyond 2 days did not change results. Based on these data, a decision was made to assign the 2 days as the duration of simulations. The administration of apigenin and hesperidin occurred at the start of simulations (\(t\) = 0 s), and their levels were kept constant throughout the simulation period. The following computer simulations were performed:  
* Individual and combination effects of apigenin and hesperidin on COX-2 production;
* Individual and combination effects of apigenin and hesperidin on PGE-2 production;
* Individual and combination effects of apigenin and hesperidin on TRPV1 production;
* Individual and combination effects of apigenin and hesperidin on CGRP production;
* Individual and combination effects of apigenin and hesperidin on ROS production. The output from the above simulations included the time-dependent concentration profiles of the five (5) biomarkers of joint pain--COX-2, PGE2, TRPV1, CGRP, and ROS. 2.3 Computer Simulations to Determine Synergistic Effects of Apigenin and Hesperidin on an Integrated Model of Joint Pain  
A multi-combination cocktail may have greater therapeutic benefits if they can act in synergy versus in an additive manner. To determine whether the specific combination in this study acts synergistically or in an additive manner to modulate joint pain, it is necessary to quantify the synergistic effect versus the additive effect. This calculation is adapted from an earlier work applied to in vitro and in vivo studies [24]. In such quantification, the _additive effect_ range--two endpoints--must first be determined for a particular biomarker. Determining the first endpoint of this range involves a sequential addition of compounds at their specific dosages.