How close can our domain name be to the trademark of another company in the same industry?

In TMs/brand names, the test of infringement or unfair competition is the likelihood of confusion in the marketplace. So, obviously, if names are very similar, and they are in the same field or market, you may be looking at infringement or other types of legal troubles that you obviously want to avoid. Still further, the likelihood of confusion also depends upon the strength of the mark or brand name - if Google, for example, started to manufacture automobiles - there has been a lot of discussion lately of driverless cars - the public would naturally associate the cars with Google because the TM or brand is so strong. Similarly, if some other entity started making (driverless) automobiles and marketed them under the Google TM or brand name, they would probably be held to infringe because people would be confused - they would assume that the cars were being manufactured by the real/original Google, which they are not. Still further, therefore, if a mark or brand is not that strong, and you are operating in a different market, one could probably get away with using the same or a similar name or mark. For the examples you posted, my guess would be that the two names could possibly coexist, even in the same market, because if they are medical products or devices, the prefix "bio" is not at all strong or unique and may even be held to be descriptive, which essentially means anyone has a right to use it, therefore, the distinction between the marks would hinge upon "metrix" and "generix" - they're close, it would be tough to definitively state that yes they could coexist or definitely not coexist - my advice would be to choose a mark that you feel comfortable with that would not be confusingly similar.