Physiological random processes in precision cancer therapy

<div><p>Many different physiological processes affect the growth of malignant lesions and their response to therapy. Each of these processes is spatially and genetically heterogeneous; dynamically evolving in time; controlled by many other physiological processes, and intrinsically random and unpredictable. The objective of this paper is to show that all of these properties of cancer physiology can be treated in a unified, mathematically rigorous way via the theory of random processes. We treat each physiological process as a random function of position and time within a tumor, defining the joint statistics of such functions via the infinite-dimensional characteristic functional. The theory is illustrated by analyzing several models of drug delivery and response of a tumor to therapy. To apply the methodology to precision cancer therapy, we use maximum-likelihood estimation with Emission Computed Tomography (ECT) data to estimate unknown patient-specific physiological parameters, ultimately demonstrating how to predict the probability of tumor control for an individual patient undergoing a proposed therapeutic regimen.</p></div

Tumor growth curves.

<p>A comparison of several differential growth curves of the form <i>N</i>Φ(<i>N</i>). Note that in each case except unbounded exponential growth, the differential growth rate goes to zero as the tumor approaches its maximum size.</p

Precision medicine trials.

<p>Our proposed clinical trial framework for precision medicine consists of selecting a data collection, processing and model-based decision-making protocol, then testing the entire joint protocol in a classical trial setting.</p

Tumor growth under uncertainty.

<p>A simulation of Gompertzian growth of the spatial random field <i>n</i>(<i><b>r</b></i>, <i>t</i>), as modeled in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199823#pone.0199823.e011" target="_blank">Eq (9)</a> with <i>c</i>(<i><b>r</b></i>, <i>t</i>)≡0, is displayed on the left. In this simulation, both the local growth constant <i>μ</i>(<i><b>r</b></i>) and the local carrying capacity <i>n</i>_max(<i><b>r</b></i>) are spatially constant but random; the initial condition <i>n</i>(<i><b>r</b></i>, 0) is taken as a so-called <i>lumpy background</i> random process (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199823#pone.0199823.s002" target="_blank">S2 Appendix</a>). Three realizations (rows) are shown at three times. The total cell number <i>N</i>(<i>t</i>) as defined in (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0199823#pone.0199823.e007" target="_blank">6</a>) is shown on the right for 16 realizations of the process.</p

TOC curves.

<p>An illustration of a Therapy Operating Characteristic (TOC) curve. (a) Schematic plot of Tumor-Control Probability (TCP) and Normal-Tissue-Complication Probability (NTCP) vs. injected mass of drug, <i>M</i> (arbitrary units); (b) The TOC curve: Plot of TCP vs. NTCP as <i>M</i> is varied.</p

Modeling uncertainty <i>in-silico</i>.

<p>The paradigm of using <i>in-silico</i> modeling to make predictions about treatment effect under uncertainty about the physiological processes involved. Relevant physiological processes <i><b>f</b></i>₁, …, <i><b>f</b></i>_<i>n</i> are identified, and a model predicting response is selected. Because of uncertainty in the processes, they are modeled as random; hence the response <i>Y</i> is random, having PDF pr(<i>y</i>). A desired effect is identified, for instance response less than a threshold <i>Y</i>_<i>c</i>. Its probability is the area under pr(<i>y</i>) left of <i>Y</i>_<i>c</i>.</p

Information flow in precision medicine.

<p>The information pipeline for precision medicine involves collecting patient-specific data, using this data to personalize validated mathematical and computational models, then using the output of such models to perform treatment selection, optimization, and assess efficacy.</p