# Concentration-dependent Pore Diffusion

Reviving this thread due to topic relevance, as originally mentioned by @Flynn. I am also interested in having a concentration-dependent pore diffusion term implemented into CADET. I think this would be useful for describing behavior in chromatographic systems that cannot be explained otherwise, such as shape of the uptake in the breakthrough curve.

Here are two proposed functional forms that I came up with, please let me know your thoughts.

Both of these functions yield Dp,0 when q = 0 and this should be viewed as the “native” pore diffusion – when there is no binding and diffusion is unhindered. Also, a linear relationship can be used if Dp,2 is set to unity. I am not sure if whether the power law or exponential relationship would be appropriate.

I think these functions could also be imposed on the surface diffusion parameter, but seeing as that in some cases (e.g. very strong binding in small pores) concentration-dependent binding is relevant and surface diffusion is essentially negligible. In such a case, this dependency would useful when imposed on pore diffusion. I think this scenario is not uncommon - one example is during the loading phase in protein A chromatography. Also, it would become complicated if imposed on both Dp and Ds because Ds may also be dependent on salt concentration and/or pH. So, for now, I think it makes sense to implement these functions for Dp and possibly for Ds later on.

@s.leweke It would be great to hear your thoughts on this.

Another possible empirical expression could look like:

D_p = D_{p,max} \left[ S_1+(1-S_1) \left(1- \frac{q}{q_{max}} \right)^{s_2} \right] \tag{1}
where 0 \leq S_1 \leq 1 and S_2 >0. It was originally used by Ng et al. [1] to modify the k_f in transport dispersive model.

[1] C.K.S. Ng, H. Osuna-Sanchez, E. Valéry, E. Sørensen, D.G. Bracewell, Design of high productivity antibody capture by protein A chromatography using an integrated experimental and modeling approach, J. Chromatogr. B. 899 (2012) 116–126. Redirecting.

@Flynn I actually don’t love this expression because it is a function of qMax. This implies that if qMax changes (e.g. lower pH) that the diffusion rate may be the same or at least not scale realistically. Also, isotherms that do not have an explicit qMax term will need to have workarounds developed which complicates things.

Even though I didn’t create this expression so I can’t speak for the authors, their functional dependence makes sense to me because

It can scale realistically if their D_{p,max} depends on pH. In addition, the ‘native’ D_{p,0} in your expression could also be a a function of pH.

Personally I haven’t seen reports on hindered diffusion in modes of chromatography other than affinity and in affinity chromatography there is usually a q_{max}.

This is true, however I think it would be beneficial to try to reduce the number of model parameters - in this case particularly because we can already use surface diffusion as a function of eluent concentration. Also, thinking of Dp as the diffusion rate of the protein in the unbound state - it doesn’t make sense to me why it would be a function of pH, unless the pH is affecting viscosity and hence the molecular diffusion rate. Mechanistically, I think it would make sense for Dp to be a function of protein concentration in the adsorbed state (in the case of hindered diffusion) while Ds is a function of the protein affinity with the adsorbent. This way we can be consistent with the hypothetical mechanisms of this parallel diffusion phenomena while also minimizing the total number of model parameters.

Actually, we are working on modeling protein impurity removal in frontal-mode using a novel multimodal (AEX/RPC) chromatography resin under prominent diffusion control. There is an atypical residence time insensitivity that we cannot model using the GRM, so I would really like to see if this can be amended with a hindered diffusion model. That said, we are also using Langmuir in this model. Still, I would be in favor of more generalizable expressions where this could be applied to, for instance, the colloidal model where Bpp is used instead of qMax. Another example would be Bi-Langmuir which represents multiple binding configurations which are each assigned a qMax - using this in conjunction with a qMax-dependent hindered diffusion framework could become overly complex.

Thanks a lot for this discussion of alternative modeling options for hindered pore diffusion. The CADET philosophy would be to implement several of them for the user to choose from.

I am in agreement with that philosophy – I think it is important to be open minded since there are many problems that may benefit from different constructions.

What would the timeframe be like for implementing some of these dependences into CADET? I’d be happy to test it out since it is definitely relevant for a few of the problems I am working on.

Currently, we’re not working on this topic. But as I said, we’re happy to assist if you’re up to the task. Could also be a fun master thesis?

Unfortunately I am very limited on time since I am wrapping up my PhD work very soon…