Concentration vs tissue concentration

[plaresmedima]

I would propose in the language and notation to distinguish explicitly between two different types of indicator concentration, which you could call "concentration" and "tissue concentration". Both are in units of M and express the number of indicator particles per unit of volume, but which volume?

1. "tissue concentration": number of indicator particles (mM) divided by the total tissue volume (mL). I would propose to use capital letters "C" for tissue concentrations.
2. "concentration": number of indicator particles (mM) divided by the volume of the distribution space (mL). I would propose to use lower-case letters "c" for concentrations.

Tissue concentration and concentration are related by the volume of distribution v (dimensionless and 0<v<1), defined as the volume of the distribution space (mL) divided by the total tissue volume (mL). The relationship is:

C = vc

So C is always smaller than c because the same number of particles is distributed over a larger volume.

Why does this matter? If you don't make this distinction between the two concentrations, you get very ambiguous statements.

Consider for instance a two-compartment exchange model with compartments p and e. Using the proposed definitions and notations, we can write 3 unambiguous statements, which are all correct (v=vp+ve):

C = vp * cp+ve * ce
v * c = vp * cp + ve * ce
C = Cp + Ce

Still using the same notations, the following statements are all wrong:

C = vp * Cp + ve * Ce
v * C = vp * Cp + ve * Ce
c = cp+ce

Now lets assume we are not distinguishing explicitly between these two types of concentrations, and use the same lower-case notation for both. Then we have said that the following statements are correct:

c = vp * cp+ve * ce
v * c = vp * cp + ve * ce
c = cp + ce

and that these are wrong:

c = vp * cp + ve * ce
v * c = vp * cp + ve * ce
c = cp+ce

Very confusing, yet that is how it is commonly done - we are supposed to know from the context and definitions in the text which definition of concentration is used in each case. Depending on which definition or notation is used, any of these statements can be either true or false. What is extra confusing is that the definition is often implicit, in which case we need to try and work out from the math which concentration is which.

The distinction is relevant when looking at indicator flux as well. For instance, still in the 2-compartment exchange model, which is the correct expression for the backflux from extravascular compartment (e) to plasma compartment (p)?

PS * ce

or

kep * ce

Answer: It depends on how you define the concentration. Using the proposed notations and definitions, this is correct and unambiguous:

PS * ce = kep * Ce

You can check that this also leads to the correct relation between PS and kep. Because Ce = ve * ce we find PS = kep * ve. The mean transit time of e is ve/PS or also 1/kep.

As a note, traditional pharmacokinetics only uses the "concentration" (c, relative to distribution volume) which is why I propose to refer to this quantity as simply "concentration". It is only in an imaging context that the tissue concentration C becomes a relevant quantity, because that is what is ultimately measured. The "concentration" c can only be derived from imaging data by modelling or making assumptions on certain distribution volumes.

[mjt320]

I also came across this issue while writing code for the OSIPI package.
Agree it would be good to make the distinction.
For example, if we give the AIF as arterial blood plasma concentration, it should be lower case c_a,p according to the convention above.
It will become even more pertinent once we look at coding non-FXL situations.
I think total tissue concentration (i.e. C_t) may also be missing from the Lexicon?

[MRdep]

We need to decide on how concentration in arterial plasma is defined. My opinion is to use lower case (ca,p), as per the definition used to tissue concentration above, that is because it is multiplied by a distribution volume, not total tissue volume.

We measure arterial concentration in an artery, but as we know CA resides only in plasma, so the true plasma concentration is higher. Using lower case for arterial plasma conc, we would write: Ca = ca,p*(1-Hct), where 1-Hct acts in a similar way to vp acts on cp.

[plaresmedima]

For a measurement over a ROI or voxel taken in pure blood you have vb=1 And so tissue concentration and concentration are the same: Cb = vb cb = cb If you look in the plasma (p) of that blood you have Cp = Cb And cb = (1-Hct) cp And also vp = 1-Hct And of course Cp = vp cp If you now look at blood in tissue, or an arterial ROI with partial volume, then what is different is that vb is no longer equal to 1, and vp is no longer equal to 1-Hct. But it is still true that Cb = Cp cb = (1-Hct) cp But vb = vp/(1-Hct) These are general relations that are true no matter what the specific region is where these measurements are taken. So I would take the principle that the first index reflects the tissue component that the volumes and conentrations are taken in. Then if you want to introduce specific notations for blood or plasma measurements in an artery (a), vein (v), portal vein (pv), tissue (t), or anything else, you can do this by adding an additional index. cp,a vp,a cp,t Etc.. Maybe these notations do not need to all be defined explicitly. You can just define the p and b, and then say that a second index can be added to indicate the ROI, if needed. Sometimes this is notational overkill though, such as in situations where cp,a = cp,t and vp,a = 1. You have to avoid compulsory notations that make eqs and code look more bureaucratic than is necessary as this will reduce uptake.

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On Wed, 19 Jun 2024, 15:42 Ben Dickie, ***@***.***> wrote: We need to decide on how concentration in arterial plasma is defined. My opinion is to use lower case (ca,p), as per the definition used to tissue concentration above, that is because it is multiplied by a distribution volume, not total tissue volume. We measure arterial concentration in an artery, but as we know CA resides only in plasma, so the true plasma concentration is higher. Using lower case for arterial plasma conc, we would write: Ca = ca,p*(1-Hct), where 1-Hct acts in a similar way to vp acts on cp. — Reply to this email directly, view it on GitHub <#68 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ABOFKAY6OFEZCROBJB6Q5S3ZIGKGRAVCNFSM6AAAAABEQEDLK2VHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZDCNZYHA4DKMJSGU> . You are receiving this because you authored the thread.Message ID: ***@***.***>

[mjt320]

Hi all,

As I understand it, the suggestion is that we will replace lexicon entry C_j with two distinct quantities: c_j and C_j, as defined above. The definitions of the compartments j are discussed under a separate issue.
We would then need to review and amend other lexicon entries depending on/referring to concentration quantities. For example, model definitions, C-R relationships.

Do you agree with this proposal?

We also need to represent the total tissue concentration, which is the most widely used concentration quantity in the field.
Should this be a separate lexicon quantity or should we define "t" as a (combined) tissue "compartment", together with other composite compartments such as "b" for blood? Thoughts welcome.

[mjt320]

Hi all,

I've modified the C_j definition and added one for c_j. See:
https://github.com/mjt320/OSIPI_CAPLEX/blob/Issue-68/docs/quantities.md

Any comments, @plaresmedima @MRdep , others?

If you're happy with the change I'll work on the (many) other entries affected by this. Tracer kinetic models, for example.

[MRdep]

Hi Michael, these modifications seem clear. I think we need to add page describing what subscripts refer to. We have this information recorded in the paper, but I don't think it was ever migrated to the website. I'll try to sort that asap, then we can start to edit the other entries.