I want to preface by saying I'm not a physicist, I'm a hardware designer at an R&D center trying to wrap my head around temperature measurements, so please forgive me if my questions are stupid or don't make sense.

The task we have is the following: we need to measure extremely fast temperature changes in human body tissue caused by RF current. The heating time varies but is usually in the range of 50-200ms. Ideally, we would like to create a graph (T = f(t)) that shows us how the tissue heats up during this time. The most critical parameter is the maximum temperature, but if we could graph the whole process that would be ideal.

The problem, however, is that temperature measurement sensors are usually very slow in comparison to the process we are trying to measure. The fastest (non-specialized) sensor we have tested is an exposed thermocouple that had a time constant of ~5ms. Obviously we would like to measure more often - currently our goal is to measure every 0.1ms.

With all that in mind, here are my questions:

  1. From what I've researched so far, I came to the conclusion that Newton's law of cooling is what can be used to estimate a temperature at any given moment. From wikipedia: $${\displaystyle T(t)=T_{\text{env}}+(T(0)-T_{\text{env}})e^{-rt}.}$$ My current idea is the following:

    • Pick a period of time for which we assume $T_{\text{env}}$ (or in my case - the tissue temperature) is constant (say 0.1ms);
    • Record the thermocouple temperature N number of times (say 10) during this interval. We end up with N points;
    • Fit the points to the equation above. I'm not entirely sure how this works, but I should be able to figure it out with some more research;
    • Calculate $T_{\text{env}}$ and record this as the "real" tissue temperature. Repeat this whole process every 0.1ms.

    Would that work? Is there anything I'm missing?

  1. Assuming the above is correct, I would need to know $r$ - the coefficient of heat transfer. From the equations in Wikipedia, I assume $r = 1/\tau$, where $\tau$ is the thermal time constant of the thermocouple. My plan is to measure this directly as follows:

    • Get 2 containers of water at known temperatures.
    • Submerge the thermocouple into one of them and then quickly transfer to the other.
    • Measure the time it takes the thermocouple to reach 63.2% of the final measurement.
    • Do this N number of times and take the average - this should be $\tau$.

    Would that work? Am I right in assuming that $\tau$ is constant no matter the amplitude of the temperature change (e.g. it's the same when the thermocouple is transferred from 0°C -> 50°C as when it's transferred from 0°C -> 100°C)? Is there anything I'm missing?

I am open to any other notes or ideas you might have. If anyone has another idea how we can solve our problem I would be super happy to hear it.

  • $\begingroup$ As I understand it, with this method you will get interpolated body "temperature to be", but it has nothing to do with real sensor response which simply has better time resolution. Imagine you have 2 temperature spikes in these 10 points,- 1 up spike and second,- down spike. Your technics will average these spikes to 0 temperature change gradient, which is not what happened truly in a body at these peaks in time. So I suppose it's better to look for better temperature sensors with response time of 0.1ms. $\endgroup$ Commented Feb 12 at 16:32
  • $\begingroup$ @Agnius We are absolutely looking for faster temperature sensors, however to my knowledge there is only one option (coaxial thermocouple with tau≈3us) and it's very expensive. The way I understand it, sensor response is determined by the mass of the sensing element - meaning that for an equal amount of time, a faster sensor will have a larger output signal compared to a slower one - like this (Does this sound correct?). Spikes being averaged out sounds OK, if they are that quick they are either noise or we have big problems with our other equipment. $\endgroup$ Commented Feb 13 at 8:05
  • $\begingroup$ If you are assuming that every signal in the sensor response resolution of $0.1ms$ is just noise, then it means that you don't need such expensive temperature sensor. Choose these where your research assumes meaningful (at least 1 signal per resolution range) signals in sensor response duration range. So you must really decide what are your real project requirements. You either want high-resolution sensors or not, but not both. $\endgroup$ Commented Feb 13 at 8:52
  • $\begingroup$ @Agnius What I failed to clarify is that thermocouple output is analog, meaning "accurate" data for its temperature can be gathered at any moment in time. Response time just gives information about the time it takes the thermocouple to reach 62.3% of it's maximum output signal after a step change in temperature. So a "fast" thermocouple just generates a larger output signal during the same time period. The thermocouple is just the sensing element, the rest of the sensor I will design myself. I am just wondering if the method I described can be used to reconstruct the actual tissue temperature. $\endgroup$ Commented Feb 13 at 9:17

2 Answers 2


Unfortunately, back-calculating peak temperatures using measurements in the present is mathematically ill-defined- that is, there are many earlier temperatures which will result in the present temperature i.e., no unique solution- unless you have special knowledge like the exact time that the heat input began. You have that here but I am not sure if it's enough to let you converge on a solution that represents the physics.

In cases like this it is better to build a finite element 3D heat conduction model of a volume of tissue and run it on a computer. If your knowledge of the heat transfer coefficient and heat capacity of the tissue are accurate, and you can adequately model the heat exchange with the blood flowing through the control volume, then you can get close to reality.

However, to have confidence in the model you will need to experimentally verify it which puts us back in the realm of ultrafast thermometry. Can you use temperature measurements of the surface of the sample? if so, you can use very speedy infrared thermometers which actually furnish an IR image of the test sample while it is being heated.

  • $\begingroup$ Unfortunately, measuring the surface is of no use to us, we are only interested in internal temperatures. So, just to clarify, are you saying that the issue is I'm assuming the temperature is constant during the time it takes to gather the 10 points I intend to use for estimation? Would the change in temperature during that time really add such significant error to my estimation? Or is there another problem with my method? (Small clarification - we are okay with missing temp changes faster than 0.1ms, as any change that fast would have negligible effects on the tissue anyways) $\endgroup$ Commented Feb 13 at 8:31
  • $\begingroup$ @IliyanAntov, I do not know. good luck with your project. -NN $\endgroup$ Commented Feb 13 at 18:16

It sounds like $r = 5$ ms for your thermocouple.

You are on the right track, except that it might not be practical. For example, quickly transferring a thermocouple from one container to another in $< 5$ ms sounds hard.

You might imagine using an RF signal modulated with a 200 Hz sine wave. The intensity would get bigger and smaller every $5$ ms. The temperature of your sample will rise linearly with a sinsusoidal component on top of it. You might expect your thermocouple to respond on that time scale and measure the sinusiodal component.

If you double the frequency, you might expect a smaller but measurable response. There should be a response at $10$ kHz, but it will be tiny. You might be able to amplify it enough to detect it. But will you be able to pick it out from noise?

You should be able to calculate r by measuring the amplitude of the sinusoidal response as a function of frequency. But it will tell you if your thermocouple is usable even without calculating.

  • $\begingroup$ You are right, on second thought that sounds very impractical. If my assumption is correct though (that τ is independent of the amplitude of the step change), wouldn't it be easier to just hold the thermocouple in the air until it reaches room temperature and then dip it in hot water quickly? I would get 2 stable levels - one at room temperature and then, later, one at the water temperature. Wouldn't I be able to just measure the time it takes the thermocouple to reach 62.3% of its maximum output signal? Your method sounds logical as well, I'm just trying to minimize the work I have to do. $\endgroup$ Commented Feb 13 at 8:17

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