Like most mammalian tissue, tumors often produce shards of DNA as a byproduct of cell death and fracture. This *circulating tumor DNA* is being studied as a means of detecting tumors or their resurgence after treatment. (See also a Q&A with Victor Velculescu in *Nature*.) These shards have a natural half-life of hours to days. ctDNA is known to increase after exercise, and temper with rest. Understanding the relationship between ctDNA and its host tumors and their lifecycles is an exciting, relatively benign way of ascertaining tumor status.

This and following related posts propose use of relative concentrations of ctDNA associated with a specific tumor in blood draws taken concurrently as a means of aiding tumor localization. The idea is that while such a ratio is likely to be noisy, with the defined network that is the human cardiovascular system (“CVS”), even a noisy ratio may yield a few bits of information regarding location of a tumor. Constraints on location can then be used to inform imaging campaigns to better identify the source.

In addition to new developments pertaining to ctDNA, this is possible in part because of several projects during the last 15 years which sought to build models of the fluid dynamics and hemodynamics of the human CVS, notably the works of Olufsen, Quarteroni, Formaggia, Müller, and Toro. Several links are provided below.

- J. T. Ottesen, M. S. Olufsen, J. K. Larsen,
*Applied Mathematical Models in Human Physiology*,*SIAM*, 2004 - M. S. Olufsen, “Modeling blood flow in the cardiovascular system”, MA325, Spring 2013, North Carolina State University
- L. O. Müller & E. F. Toro, “A global multi-scale mathematical model for the human circulation with emphasis on the venous system”, 2013. Details from their tables and lumped system models are shown below.
- M. Olufsen, A. Nadim, “On deriving lumped models for blood flow and systemic pressure in the systemic arteries”,
*Mathematical Biosciences and Engineering*, June 2004 - A. Quarteroni, L. Formaggia, “Mathematical modelling and numerical simulation of the cardiovascular system”, 2002
- H. Tamaddon, M. Behnia, M. Behnia, “A numerical model for simulation of blood flow in vascular networks”,
*International Science Index*, 8(6), 2014.

While I *hope* use of ctDNA for differential localization won’t demand the full features of these models, those being devoted primarily to dynamics, the sophistication of the lumped system and compartment models for representing blood flow is key to constructing the *network transition matrix* which will be seen to be key in the forthcoming analysis. These are not considered difficult: Olufsen in her MA325 course assigns solving for variables of Volume, Pressure, and Flow from the model parameters, Resistence, Compliance, and Heart-Pumping as a homework. It is, after all, but algebra.

The specifics of the technique are not at all new, at least to engineers and statisticians. Applications to this field are new. In particular, these papers address the general question of locating a point source given information regarding the concentrations of its materials in a geometry once diffused.

- P. C. Pinto, P. Thiran, M. Vetterli, “Locating the source of diffusion in large-scale networks”, 2012
- E. Yee, “Bayesian inversion of concentration data for an unknown number of contaminant sources”, 2007

The key idea is to calculate the equibriuum concentration for the *diffusion equation* with a discrete approximation corresponding to an idealized (or detailed!) model of the CVS. This can be constructed from an eigenanalysis of the state transition matrix using standard methods, such as those described by G. Strang in Chapter 5 of his *Linear Algebra and its Applications*, 2^{nd} edition, 1980. The premise is that ctDNA in a quiescent patient will rapidly achieve this equilibriuum, affected thereafter only by the decay constant associated with metabolic cleanup. While the references give a much more detailed presentation, the model I’ll use for the CVS is shown below, adapted from Shandas course on cardiovascular biomechanics (I, II, and III).

How would this be used 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.
- These locations are hypothesized to be location of tumors.
- Imaging is applied to locations to confirm and further specify locations.

#### Details of Some CVS Model Data and Examples

Müller and Totter offer the following tables of CVS data, examples taken from their Tables, 3, 6, and 8.

Müller and Toro also offer lumped model electrical circuit equivalents to CVS parts, such as the following for feeding arteries and collecting veins:

These are described by the following parameters:

The use of such networks for modeling dynamics is taught by many, including via the tutorials of Professor John Baez of University of California, Riverside. They are more general than it may seem, being both equivalent to linear differential equations and, therefore, having analogies with mechanical systems. These analogies are being used in these CVS models. See work by John Baez and Brendan Fong for a deeper analysis.

#### Prospectus

Subsequent posts will describe the details of a study of the technique, including the analysis of the technique in terms of a network transition matrix, illustrative applications of it to isolating hypothetical sources of ctDNA, and a sensitivity analysis of the illustrations to errors in the transition matrix, whether because of approximation, imperfections in measurement, or dynamics.

The second installment is now available.

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Cool stuff! By the way, I’m at U. C. Riverside, not U. C. Davis.

Sorry, John. Fixed the location.

And thanks!