The units for tpi are long tons (2240 lb) per square inch. It was used (only?) by the British gun industry. As TB said, the crusher was a copper ring placed at the base of the cartridge. The case was well oiled on the outside so that the case could more readily slide back, acting as a piston. The crusher ideally measured the thrust of the case backwards, and this thrust divided by the OD area of the case was used to estimate the pressure driving it back.

In the US, the "radial crusher" is used. The piston is on the side of the case. The cylinder is fixed in diameter, and the crusher deformation is calibrated with psi by the manufacturer of the crushers.

Under the CIP, I believe the radial crusher is placed in front of the case mouth. This would eliminate the problem of having to blow the case wall out to move the crusher, but the pressure is obviously a bit lower at the mouth than at the base. In comparing CIP crusher data (converted to psi) to SAAMI CUP/psi, the CIP numbers will be found to be about 1000 psi higher. I don't know if this means the European stuff is hotter or if the two methods of estimating pressure are typically that much off.

All three systems are mechanical and cannot respond fast enough to the rapidly changing pressures inside the chamber. Both indicate pressures well below true in cartridges like the .30-06.

The tpi and CUP cannot be "directly" compared because the amount that indicated is below true is different with each. I don't think anyone has identified all the various reasons each is low. The assumption there is no case "cling" with the oiled case in the base crusher seems iffy to me.

Anyway, to answer your question, you probably can convert between the two, but I don't have enough data on hand estimate how tpi relates to psi. CUP can be converted to psi with reasonable accuracy. Grab a SAAMI reference manual and plot CPU vs psi for the various cartridges and you can see a definite trend. A mathematically inclined fellow who works with RSI, the maker of the strain gauge rig for handloaders, believes much of the scatter in the plot of CUP vs psi is attributable to the random errors in measurement that both CUP and piezo suffer. If you can find enough data where tpi and psi were measured on the same loads, you could make a such a plot to estimate the conversion.

This fellow went on to show that the CIP actually uses a simple, straight line conversion formula to convert their crusher ratings to piezo ratings. The conversion is not exact, of course, but it was close enough to satisfy the CIP. My guess is the conversion is accurate to about 10%, well within the margin of safety of the typical gun.

An article in the 4/1967 American Rifleman gives details on the base crusher method. It was stated that the "psi" indicated by base crushers are lower than with the radial crushers. In other words, 15 tpi would be 33,600 psi indicated, but the CUP method might indicate 36,000 psi for the same load, and piezo might read 39,000 psi. (These numbers are made up; they did not come from the article.)

So many words spilled, yet no answer...