## On differential localization of tumors using relative concentrations of ctDNA. Part 2.

Part 1 of this series introduced the idea of ctDNA and its use for detecting cancers or their resurgence, and proposed a scheme whereby relative concentrations of ctDNA at two or more sites after controlled disturbance might be used to constrain the location of source tumors. In this installment, many of the technical details for doing this will be examined.

#### Time Constants, Decay Curves, Differential Erosion, and Joint Densities

There are two principal processes which affect this model of these phenomena. One is the decay time or physiological clean-up time for ctDNA of interest in the blood. That time will be denoted $\tau_{d}$ here. The other time is the time required for a deposit of a single quantity of material in a single node of the $M = 33$ to be distributed among the $M$ nodes in the equilibrium concentrations due to CVS pumping. That time will be denoted $\tau_{b}$. For the purposes of this technique, it is assumed $\tau_{d} >> \tau_{b}$ and that concentrations of the original material are significant enough to be detected any time before $\tau_{d}$.

The technique also assumes that differential erosion is negligible. By this I mean that there are not substantial differences in the relative concentration of ctDNA depending upon the paths taken through the CVS. There are concentration differences based upon diffusion and flow, but it assumes there aren’t special processes along certain paths which attenuate ctDNA concentrations in addition.

In the model considered here, a unit contribution of ctDNA is made to the blood stream in an organ, here modelled as a nde in a network. In order to use differential localization, ctDNA needs to be sampled at sites before concentrations come to equilibrium throughout the body. That means in the clinical setting, samples of ctDNA need to be taken concurrently at two or more sites a few dozen heartbeats after a generating event, such as vigorous exercise. Localization depends upon two factors, both time dependent. One is, of course, the ctDNA being present at all. The second is the time dependent non-equilibrium contributions of differing paths to the concentration at a point of observation. As the observation time, $\tau_{o}$ increases, and exceeds $\tau_{b}$, although ctDNA might be detected, the path-dependent part is washed out by mixing. Accordingly, in the case of an isolated unit contribution, there is a window of opportunity for observing path effects and, therefore, localizing the source. The assessment below will examine that window.

In actuality, the contribution of ctDNA from a site after activity is probably going to continue for some time, separately from the decay time $\tau_{d}$. To the degree that is the case, the need to remain within a window of opportunity is lessened.

#### Transition Matrix

Recall the schematic CVS model from Shandas course which was shown in the first installment:

Each node in this network is identified in this diagram by a number. There are $M = 33$ nodes. A transition matrix $\mathbf{T}$ is a quantitative depiction of this network, being a square matrix having $M$ cells on a side. The connectivity of a node to others is indicated by a non-zero positive entry in this matrix such that a flow out of node $i$ to node $j$ is indicated by having that value recorded in cell $T_{i,j}$ of that matrix. Nodes having no direct connections with one another, say, node $u$ and $v$ so blood does not flow between them have $T_{u,v} = 0$. Note the asymmetry of flow is indicated because flow is unidirectional in healthy operation: Either $T_{i,j} > 0$ or $T_{j,i} > 0$ or $T_{i,j} = 0$ in mutual exclusion. Given a node $i$ when blood flows from it to, say, three other nodes, $j$, $k$, and $l$, and just those nodes, then $T_{i,j} > 0$, $T_{i,k} > 0$, and $T_{i,l} > 0$, and, moreover, $T_{i,j} + T_{i,k} + T_{i,l} = 1$. This is because the positive numbers in $\mathbf{T}$ denote the fraction of total flow that leaves node $i$ to any other node, and each element of flow must go some place.

In general, long term equilibrium concentrations of material deposited in nodes of any such network after $\tau_{b}$ will not be uniform, and will depend upon the settings in $\mathbf{T}$. As noted, however, at equilibrium, path dependent information that’s needed for localization is lost.

In addition to the approximate nature of the Shandas network, in instances where flows split like this, the author made educated guesses regarding the proportion of flow leaving such a node for others. The reason why more care was not taken with this determination is that:

• The additional detail was not judged to be crucial for such this explanation of the technique.
• Proportions of flow vary from patient to patient.
• Proportions of flow vary from time to time within the same patient.
• The additional detail is available from the more comprehensive studies and models described in the first installment, such as in the material by Olufsen.
• Providing more details would suggest the long term stable equilibrium of such flow was all that mattered, or the expected value of such equilibria, and that is both untrue and misleading.
• A sensitivity analysis of the results of this installment to such variations will be presented in an upcoming installment.

Nevertheless, for the purposes of this description, it is assumed the transition matrix does faithfully represent the node to node flow allocations, and that, therefore, the equilibrium concentrations of any material deposited among the nodes after the time for CVS redistribution discussed just above is exceeded applies. It’s assumed that sampling of concentrations by node is arranged after such a time, and before appreciable ctDNA has decayed.

It’s presumed that material does not pool in the nodes. If it did, the entries $T_{i,i}$ would be positive in some instances. This may be a limitation of the present illustration, but, in any case, discussion of that is deferred until the installment on sensitivity analysis.

So, what about $\mathbf{T}$? Given that $\mathbf{T}$ is sparse, just its non-zero entries are listed below:

 FromNode ToNode Proportion 33 1 1.0000 31 2 0.6667 1 3 1.0000 32 4 0.0625 15 5 0.5000 15 6 0.5000 18 7 0.5000 19 8 0.1250 22 8 1.0000 21 9 0.5000 21 10 0.5000 12 11 1.0000 26 12 0.2500 27 13 0.2000 27 14 0.8000 16 15 1.0000 17 16 0.2000 31 16 0.3333 32 17 0.9375 17 18 0.8000 18 19 0.5000 19 20 0.8750 20 21 0.1250 9 22 1.0000 10 22 1.0000 8 23 1.0000 24 23 1.0000 11 24 1.0000 25 24 1.0000 13 25 1.0000 14 25 1.0000 20 26 0.8750 26 27 0.7500 7 28 1.0000 23 28 1.0000 28 29 1.0000 30 29 1.0000 5 30 1.0000 6 30 1.0000 3 31 1.0000 2 32 1.0000 4 33 1.0000 29 33 1.0000 

#### One Heartbeat

Note that if the relative proportions of a set of concentrations of materials are represented as a column vector $\mathbf{x}$ having $M$ rows, and the initial proportions are denoted $\mathbf{x}_{0}$, then the concentrations at tick 1 are given by

$\mathbf{x}_{1} = \mathbf{T} \mathbf{x}_{0}$

Accordingly after the $k$-th step they are

$\mathbf{x}_{k} = \mathbf{T}^{k} \mathbf{x}_{0}$

and the analysis sketched in the first installment can be used to describe concentrations as $k \rightarrow \infty$.

The steady-state or equilibrium concentration of any material can be described by a differential equation known as the diffusion equation, as described by G. Strang in his textbook Linear Algebra and Its Applications (2nd edition, 1980), sections 5.2-5.4.

#### Towards Tumor Localization

In particular, if one were to posit that a tumor was located in the $j$-th row of $\mathbf{x}_{0}$, and nowhere else, then the concentration of ctDNA eminating from it at the $k$-th step would be

$\mathbf{x}_{k} = \mathbf{T}^{k} \left[0, 0, \dots, 1, \dots, 0 \right]^{\top}_{0}$

where there are $j-1$ zeros before the $1$ appears, and $M-j-1$ zeros after the $1$ appears. This simply selects a column from $\mathbf{S} \boldsymbol\Lambda^{k} \mathbf{S}^{-1}$. Accordingly, at the $k$-th step, the matrix $\mathbf{T}^{k}$ records in its columns the distribution of ctDNA assuming a unit amount was introduced in the first state. And, in a clinical setting, if sampling from pairs of particular nodes is more readily done than others, say, nodes 15 and 14 for arms and legs, the ratios of the 15th and 14th rows of this matrix would give the relative concentrations of ctDNA for that locale. Similarly, if there’s uncertainty about $\mathbf{T}$ settings, any given $\mathbf{T}$ can be perturbed randomly, diagonalized, and these relative concentrations recalculated for the perturbed matrix.

#### So What Does It Look Like? And About That Window of Opportunity …

If sampling is done too early, ctDNA would not be sufficiently circulated to obtain a good sample. If it is done too late, then equilibrium concentrations dominate, and it information about the original location of the tumor would be lost. It is best to see this this graphically. The following sequence of images show the relative concentrations of hypothetical ctDNA relative to a unit quantity deposited in one of the $M = 33$ nodes or organs. The columns of these images correspond to the node number of the tumor, and the rows correspond to the concentrations at the sampling moment among the various nodes. To see a larger version of any of these images, click on the image. After doing so, to return to the posting, just use the Back button on your browser.

#### Relative Concentrations

Since absolute concentrations don’t mean anything here because the original concentration of ctDNA is not known, what’s of diagnostic interest is the relative concentration of ctDNA at two sites. For illustration, relative concentrations at various sampling times were calculated for the ratio of the concentrations at node 14 to node 15, roughly, the legs to the arms, and at node 25 to node 14, roughly a pelvic vein to the legs. In the following displays the rows correspond to the location of the tumor. If the ratios differ between putative tumor sites, there is diagnostic power in the ratio. If they are comparable, discrimination is poor or none.

The following shows sampling times across the columns, in roughly, heartbeats, and locations of tumors in the rows. This display is the ratio of legs to arms.

 tumor site R20 R25 R30 R35 R40 R45 R50 R70 R100 1 Inf 1.6276 1.8371 0.5326 0.753 1.353 0.675 0.852 0.980 2 0.0000 0.0000 2.0312 0.2274 1.378 1.349 1.555 1.058 0.976 3 72.0000 7.2187 2.2213 0.8340 1.153 0.626 0.976 1.016 0.973 4 NaN Inf 1.6276 1.8371 0.533 0.753 1.353 1.216 1.018 5 0.0000 0.6667 1.3952 1.9287 0.414 0.914 1.353 1.126 1.021 6 0.0000 0.6667 1.3952 1.9287 0.414 0.914 1.353 1.126 1.021 7 NaN 8.0000 0.1008 1.2984 1.078 1.617 0.810 1.054 1.000 8 0.0000 0.0000 2.1333 0.5967 0.514 1.260 1.475 0.953 0.992 9 NaN NaN Inf 1.6276 1.837 0.533 0.753 1.052 0.967 10 NaN NaN Inf 1.6276 1.837 0.533 0.753 1.052 0.967 11 NaN 0.1250 72.0000 7.2187 2.221 0.834 1.153 0.899 0.992 12 NaN NaN Inf 1.6276 1.837 0.533 0.753 1.052 0.967 13 NaN 0.1250 72.0000 7.2187 2.221 0.834 1.153 0.899 0.992 14 NaN 0.1250 72.0000 7.2187 2.221 0.834 1.153 0.899 0.992 15 1.3333 2.4176 0.1116 0.9070 1.955 0.662 0.930 1.080 1.012 16 0.0625 0.6180 2.2635 0.3946 1.098 1.292 1.259 1.025 0.993 17 NaN Inf 1.6276 1.8371 0.533 0.753 1.353 1.216 1.018 18 0.1250 72.0000 7.2187 2.2213 0.834 1.153 0.626 0.943 1.038 19 NaN 8.0000 0.1008 1.2984 1.078 1.617 0.810 1.054 1.000 20 0.0000 0.0000 0.0000 2.0312 0.227 1.378 1.349 0.881 1.028 21 NaN 1.3333 0.0625 0.1249 0.725 2.139 0.495 0.785 0.967 22 NaN 0.1250 72.0000 7.2187 2.221 0.834 1.153 0.899 0.992 23 Inf 0.9517 1.0379 0.0469 0.775 0.635 1.677 1.158 1.012 24 Inf 0.0000 0.6154 0.0554 1.881 0.308 1.429 1.104 1.002 25 NaN NaN 8.0000 0.1008 1.298 1.078 1.617 1.042 0.952 26 NaN 1.3333 0.0625 0.1249 0.725 2.139 0.495 0.785 0.967 27 NaN NaN Inf 1.6276 1.837 0.533 0.753 1.052 0.967 28 0.0000 0.0000 0.6926 1.9688 0.729 1.284 1.154 0.972 1.028 29 0.0000 2.0156 0.1601 1.3671 1.412 1.591 0.725 0.994 0.992 30 0.1250 2.6182 3.6406 1.2681 0.824 1.510 0.581 0.893 1.018 31 8.0000 0.1008 1.2984 1.0779 1.617 0.810 1.331 0.963 0.998 32 1.3333 0.0625 0.1249 0.7248 2.139 0.495 1.007 1.093 1.017 33 0.0625 0.1249 0.7248 2.1387 0.495 1.007 1.011 0.957 1.010 

This display is the ratio of pelvic to legs.

 tumor site S20 S25 S30 S35 S40 S45 S50 S70 S100 1 0.0556 0.0378 0.5840 3.8387 1.065 0.592 1.102 0.941 1.059 2 Inf Inf 0.4927 1.4802 0.956 1.621 0.796 0.951 0.992 3 0.2500 3.6638 0.7708 0.3123 0.814 2.699 0.930 1.045 0.970 4 Inf 0.0556 0.0378 0.5840 3.839 1.065 0.592 0.855 0.991 5 Inf 3.0556 0.0756 0.5387 3.333 1.023 0.755 0.987 0.978 6 Inf 3.0556 0.0756 0.5387 3.333 1.023 0.755 0.987 0.978 7 Inf 0.5000 2.7070 0.9991 2.970 0.759 0.457 0.765 1.029 8 NaN Inf 0.6270 0.7708 1.155 1.629 1.176 1.224 0.963 9 NaN Inf 0.0556 0.0378 0.584 3.839 1.065 1.107 0.945 10 NaN Inf 0.0556 0.0378 0.584 3.839 1.065 1.107 0.945 11 NaN 0.0000 0.2500 3.6638 0.771 0.312 0.814 1.005 1.066 12 NaN Inf 0.0556 0.0378 0.584 3.839 1.065 1.107 0.945 13 NaN 0.0000 0.2500 3.6638 0.771 0.312 0.814 1.005 1.066 14 NaN 0.0000 0.2500 3.6638 0.771 0.312 0.814 1.005 1.066 15 0.0469 0.4909 9.4785 0.9625 0.726 1.070 0.980 0.900 1.001 16 16.0000 0.8426 0.9456 1.1454 0.705 1.413 1.097 1.029 0.987 17 Inf 0.0556 0.0378 0.5840 3.839 1.065 0.592 0.855 0.991 18 0.0000 0.2500 3.6638 0.7708 0.312 0.814 2.699 1.240 0.972 19 Inf 0.5000 2.7070 0.9991 2.970 0.759 0.457 0.765 1.029 20 NaN Inf Inf 0.4927 1.480 0.956 1.621 1.483 0.946 21 NaN 0.0000 0.0000 16.2500 0.932 0.489 1.246 0.859 1.094 22 NaN 0.0000 0.2500 3.6638 0.771 0.312 0.814 1.005 1.066 23 1.9091 0.0000 0.3924 12.8378 2.057 0.851 0.618 0.865 1.002 24 0.0000 Inf 3.5000 5.7070 0.564 1.973 0.908 0.937 0.993 25 NaN Inf 0.5000 2.7070 0.999 2.970 0.759 0.954 0.986 26 NaN 0.0000 0.0000 16.2500 0.932 0.489 1.246 0.859 1.094 27 NaN Inf 0.0556 0.0378 0.584 3.839 1.065 1.107 0.945 28 Inf Inf 2.1063 0.3858 0.555 0.979 1.588 1.267 0.957 29 Inf 0.4961 2.1852 0.9673 1.556 0.790 0.745 0.837 1.029 30 0.0000 0.1250 3.6323 1.4639 0.364 0.647 2.279 1.134 1.004 31 0.5000 2.7070 0.9991 2.9703 0.759 0.457 0.808 0.993 1.032 32 0.0000 0.0000 16.2500 0.9320 0.489 1.246 1.248 0.962 0.991 33 0.0000 16.2500 0.9320 0.4892 1.246 1.248 1.344 1.247 0.963 

It’s clear that the most discrimination appears at sampling times 30 through 50. Accordingly, in this example, at the 40 tick point, a legs to arms concentration ratio of 0.5 suggests node 20, or a liver source. In practice, these ratios will have highest density intervals (“HDI“) which may overlap and, so, there may be ambiguity as to source. Such ambiguity might be resolved using other clinical knowledge, repeating the sampling slightly later, or using another site to develop additional ratios.

#### Sampling Plans

In addition to discriminating among locations, the same scheme can be used to plan detection. For instance, if there is a strong prior on certain locations for a tumor, this prior in combination with these kinds of studies can be used to plan where and when samples can be taken.

#### Revising Use in a Clinical Setting

Recall the proposed use in a clinical setting:

1. Patient is identified with a tumor having specific ctDNA.
2. Patient undergoes exercise or vigorous activity as tolerated.
3. Patient returns to a resting posture.
4. Blood samples are quickly drawn from members which study suggests will best constrain location of tumor. Roughly speaking, and depending upon further details from models of the CVS, samples should be taken without a minute of stopping exercise.
5. Relative concentrations of ctDNA are obtained from blood samples.
6. Relative concentrations are combined with results from present study and CVS model to estimate probable location of sources.
7. As needed, additional samples are drawn to further constrain location of tumor.
8. Relative concentrations of ctDNA are obtained from these.
9. Bayes Rule is used to update the estimate of location of the tumor.
10. Imaging is applied to locations to confirm and further specify locations.

Note that the relaxation time for ctDNA concentrations depends upon finer details of the CVS model, and upon a sensitivity analysis of this calculation to variations in the network transition matrix, $\mathbf{T}$. Such an analysis along with suggestions on how to develop HDIs for concentration ratios will be presented in the next installment, which will be the next-to-last one.

The final installment will look at extensions, including how to combine two of these ratio measurements in an updated, Bayesian estimate of tumor location, and also how tracking these ratios over time might improve accuracy by using a state-space filter.