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I was skimming through a study [mRNA-based COVID-19 vaccine boosters induce neutralizing immunity against SARS-CoV-2 Omicron variant, page 6, figure 3(A)] on vaccine and I came across the international unit ($\text{IU}$). From what I read, $\text{IU}$ measures the amount of substance adjusted according to biological activity observed. [I studied maths in uni. I only learnt a bit of physics/chemistry/biology in secondary school.]

In dimensional analysis, all units are assigned a physical dimension which is expressed in term of length ($\text{L}$), mass ($\text{M}$) and time ($\text{T}$). For examples, gram has dimension $[\text{g}] = \text{M}$, mole is dimensionless, millilitre has dimension $[\text{mL}] = \text{L}^3$ and molar has dimension $[\text{M}] = \text{L}^{-3}$.

What is the physical dimension of $\text{IU}$?

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    $\begingroup$ @ARogueAnt. The $y$-axis of the graph has the label Pseudovirus neutralization (IU/mL). I don't know that it varies according to the substance measured. It's something new to me. $\endgroup$
    – user69081
    Commented Jan 15, 2022 at 23:02
  • $\begingroup$ Please link the study $\endgroup$
    – acvill
    Commented Jan 15, 2022 at 23:04
  • $\begingroup$ @acvill mRNA-based COVID-19 vaccine boosters induce neutralizing immunity against SARS-CoV-2 Omicron variant, page 6, figure 3(A) $\endgroup$
    – user69081
    Commented Jan 15, 2022 at 23:13
  • $\begingroup$ It depends totally on the substance. IUs are decided upon by committee in order to be able to compare formulations. $\endgroup$ Commented Jan 16, 2022 at 0:01
  • $\begingroup$ As you have identified it yourself, the $\rm IU$ is a measure for the "effective" amount of substance, which is one of the seven base quantities in the International System of Quantities. Therefore, in your notation, $[{\rm IU}] = {\mathsf N}$. $\endgroup$
    – Domen
    Commented Jan 16, 2022 at 0:41

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Based on the WHO mandates, the international units ($\text{IU}$) are arbitrary units corresponding to neutralization. The reason for choosing arbitrary units was that the serum used to test immune resistance is made of too many molecules for there to be a single standardized piece to measure. As a result, the units are arbitrary and that means it is based on the number of particles neutralized, so the $\text{IU}$ has a dimension of $\text{[IU]}$ = $\text{N}$, $\text{N}$ being the number of neutralized particles.

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