Can LM35 temperature sensor be used to measure temperatures in bone in an in vitro setting? are they comparable to K type thermocouples?
An LM35 is an electronic component. To see the characteristics we refer to a datasheet. Towards the end you'll find the mechanical drawings. The smallest available package is the epoxy TO-92 package, which will fit in a 5.5mm ID protection tube. If you look at small or hypodermic tubing you'll get some idea of the minimum OD (outer diameter) of tubing required, somewhere around 5.75mm, way more than 1mm. 

Thermocouples are made in an enormous range of sizes and shapes, and you can readily find 1.0mm or even 0.8mm diameter probes. See manufacturer's data for specifications. You may have trouble finding accuracy numbers because it depends a lot on the application. 

Aside from the mechanical characteristics, we have thermal characteristics. A &lt;1mm probe will minimally affect the material being measured. A small thermocouple will have a very fast response time so there will be not much lag between a change in temperature and the resulting electrical signal. The LM35 being huge and encased in plastic, will be very sluggish in comparison. 

So the LM35, as shown in the datasheet, can't possibly meet your size requirement. 



On the other hand, the LM35 is trivially easy to interface with accuracy (measurement of the junction of the device, not necessarily measurement of the desired external material) that will meet the data sheet specifications. The premium version has a guaranteed accuracy of +/-0.5°C at 25°C, which deteriorates as you deviate from that reference temperature. Pretty good by semiconductor sensor standards but definitely not 0.1°C. If you calibrate them individually at a reference temperature they'll probably stay put, at least until the first time you autoclave them. But just about anyone can get them to work. 

A thermocouple, to provide similar accuracy will require well-designed signal conditioning circuitry that has microvolt stability, and a cold junction sensor that provides significantly higher accuracy and stability than the desired system accuracy (you can't blow your entire error budget on the cold junction sensing). 0.1°C at body temperature represents about 4uV, which is not trivial to deal with, even with modern components. The cold junction sensor also has to be isothermal with the Chromel-copper and Alumel-copper junctions to significantly higher accuracy and stability than the desired system accuracy. And any connectors have to avoid introducing thermal EMFs that would degrade the reading too much. All in all, non-trivial even for a very experienced engineer.