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:
CVS_biomechanics_2015-09-13_163921_Shandas2004_markup_virtual

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.

20_steps_2015-10-26_112040

25_steps_2015-10-26_112813

30_steps_2015-10-26_111906

35_steps_2015-10-26_112844

40_steps_2015-10-26_111948

45_steps_2015-10-26_112916

50_steps_2015-10-26_112130

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.

About hypergeometric

See http://www.linkedin.com/in/deepdevelopment/ and http://667-per-cm.net
This entry was posted in Bayes, Bayesian, Bayesian inversion, cancer research, ctDNA, differential equations, diffusion, diffusion processes, engineering, linear algebra. Bookmark the permalink.

One Response to On differential localization of tumors using relative concentrations of ctDNA. Part 2.

  1. Pingback: On differential localization of tumors using relative concentrations of ctDNA. Part 1. | Hypergeometric

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