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

Posted on 26 October 2015 by ecoquant

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 (2^nd 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 15^th and 14^th 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:

Patient is identified with a tumor having specific ctDNA.
Patient undergoes exercise or vigorous activity as tolerated.
Patient returns to a resting posture.
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.
Relative concentrations of ctDNA are obtained from blood samples.
Relative concentrations are combined with results from present study and CVS model to estimate probable location of sources.
As needed, additional samples are drawn to further constrain location of tumor.
Relative concentrations of ctDNA are obtained from these.
Bayes Rule is used to update the estimate of location of the tumor.
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 ecoquant

See https://wordpress.com/view/667-per-cm.net/ Retired data scientist and statistician. Now working projects in quantitative ecology and, specifically, phenology of Bryophyta and technical methods for their study.

View all posts by ecoquant →

This entry was posted in Bayes, Bayesian, Bayesian inversion, cancer research, ctDNA, differential equations, diffusion, diffusion processes, engineering, linear algebra. Bookmark the permalink.

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

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