In the Hesselholt-Madsen paper "On the K-theory of finite algebras over Witt vectors of perfect fields", the authors develop some results concerning the Topological Hochschild homology (THH) of functors with smash products.

They use the language of equivariant spectra in the sense of Lewis-May-Steinberger.

~~My question is if there is a more modern reformulation of this work using equivariant orthogonal spectra (or other (one-categorical) variants)~~.

I am quite certain that using the results from Mandell, May, Schwede and Shipley, one can translate the results from Hesselholt-Madsen to orthogonal spectra. My Question then is if this has already been done. At this moment, I am not looking for an infinity categorical treatment.

To be more precise, I would like to see the construction of THH and topological cyclic homology using orthogonal spectra. Preferably I would like to see the construction of the Tate spectral sequence as well, where for a finite cyclic group $C$ and $T$ an equivariant $C$ spectrum, the Tate spectral sequence is $$E^2_{r,s}=\hat{H}^{-r}(C,\pi_sT)\Rightarrow \pi_{r+s}\hat{\mathbb{H}}(C,T),$$

where the $E^2$ page is the ordinary Tate cohomology of $C$ and $E^{\infty}$-page is the homotopy groups of the Tate spectrum.