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22 April 2023 Inhalation tion indicated by a ratio P 3 /P 2 that is smaller than 0.5 (an approximation of the sonic flow condition predicted by adiabatic ideal gas behavior; e.g., equa- tions 2.18 and 2.19 of Baker [5]). is flow condi- tion is often called "critical" flow. When the sonic flow condition is obtained, the mass flow rate of air throughout the system (inhaler, Dose Uniformity Sampling Apparatus (DUSA), vacuum source) is proportional to the gas density at the flow control valve (equation 1): Ṁ s = K s MW * P 2,s RT (1) Here, Ṁ is the mass flow rate, R is the universal gas constant, T is the absolute temperature just upstream of the control valve, and MW is the molecular weight of air (29 g/mole). e subscript "S" on the mass flow rate, on the proportionality constant K, and on the pressure P 2 indicates that the values are those derived when "setting" the conditions along the flow path before dispensing a dose from the inhaler. We have depicted these steps in Figures 1a, b and c to enable a nomenclature for the relevant parameters that we will need for analyzing the system in detail. Derivation of independent equations Perhaps surprisingly there are ten independent equa- tions and ten unknowns describing these three steps. We begin by deriving these ten equations, according to the three major steps of the testing procedure for total dose (TD) content from a DPI, in the context of determining DDU. When reading this detailed section, it is important to understand that our pur- pose is to enable the practicing user to evaluate when bias might be important and how to address it, if desired and needed. Step 1: Set-up at a device pressure drop of 4 kPa e pharmacopeial procedure [1, 2] aims to draw a specified volume of air (two liters) through the inhal- er-on-test at a target flow rate that is chosen to impart a pressure drop of 4 kPa across the device (Figure 1a). e vacuum equipment for fixing the flow rate at this target value must be sufficient to achieve sonic air flow velocity across the flow control valve, a condi- Nomenclature K s – setting of the flow control valve Ṁ s – mass flow rate into the total dose tube during the set-up step (Figure 1a) Ṁ m – mass flow rate into the total dose tube during the measurement step (Figure 1b) Ṁ a – mass flow rate into the total dose tube during the dose dispensing step (Figure 1c) MW - molecular weight of air (29 g/mole) P 1 – pressure inside the total dose tube P 2 – pressure upstream of the flow control valve P 3 – pressure downstream of the flow control valve ΔP – pressure difference between the atmosphere and the inside of the total dose tube ΔP fm,m – pressure drop across the flow meter at the time of the flow measurement ΔP f,s – pressure drop across the filter at the time of the set-up ΔP f,m – pressure drop across the filter at the time of the measurement ΔP f,a – pressure drop across the filter at the time of the actual test ΔP dev,a – device pressure drop at the time of the actual test Q s – volumetric flow rate entering the total dose tube during the set-up Q m – measured volumetric flow rate Q a – volumetric flow rate entering the total dose tube during the actual test T TD – run time of the total dose test V TD,a – actual volume of air that flows out of the device during a test Y – dimensionless form of the measured volumetric flow rate; equal to Q m /Q s Z – dimensionless form of the volumetric flow rate during the actual test; equal to Q a /Q s