Characteristic functionals

While the statistical properties of random variables and random vectors are easily specified by probability density functions (PDFs), describing the statistical properties of a random process requires the application of more sophisticated tools. A random process consists of a set of functions {f(r, t)}, each of which can be regarded as an element of an infinite-dimensional vector space, so it is not usually convenient to define conventional PDFs for them (see S1 Appendix for a discussion of this issue). Nevertheless, all the statistical properties of a random process can be derived from its characteristic functional, a powerful mathematical tool which to our knowledge has not previously been applied to oncology or precision medicine. A functional accepts a function as its input and returns a single scalar value, and we define the characteristic functional for a physiological random process consisting of functions {f(r, t)} by (1) where ϕ(r) is an arbitrary real-valued test function and the integral is over the volume V of a tumor or some other region of interest. The angle brackets denote an average over all possible realizations of the random process for some ensemble of patients or subjects. The functional (1) contains a complete description of the spatial statistics of the process at any given time; the time variable t in Eq (1) allows these spatial statistics to evolve as the disease progresses or therapeutic interventions take effect, but it is not intended to capture joint statistical variations across multiple times, so the test function ϕ(r) is not a function of time. An analogous characteristic functional can be defined when temporal correlations are of interest; see S1 Appendix.

For a survey of the mathematical properties of random processes and characteristic functionals, see [10, 11] and S1 Appendix. One of the key advantages of the characteristic functional is that there are many general classes of random processes for which Ψ_f is known analytically, which makes it possible to compute any of the conventional statistical properties, such as moments or marginal probability density functions. This includes many classes of non-Gaussian processes which are typically challenging to manipulate—see [10, 11], and also S2 Appendix. Moreover, the characteristic functional can be applied to personalized medicine by collecting molecular images or other data from a particular patient and using them to estimate any parameters that appear in a patient-specific characteristic functional [12].

Emission computed tomography

A general method for measuring physiological random processes is Emission Computed Tomography (ECT), defined broadly as three-dimensional (3D) imaging of molecules or cells that have been labeled so that they emit light, high-energy photons or charged particles without significant alteration of their biological function. The labeling can use radionuclides or light-emitting molecules, so the emissions can be nuclear decay products, including electrons, positrons and high-energy photons, or visible or near-infrared photons.

The most familiar forms of ECT are SPECT (Single-Photon Emission Computed Tomography) and PET (Positron Emission Tomography), in which molecules labeled with radionuclides are imaged. SPECT and PET are widely used in clinical medicine and in preclinical studies with small animals. There is considerable interest in pushing the spatial resolution of radionuclide imaging ever finer, especially in SPECT where intrinsic detector resolutions around 50 μm have been reported from several labs, and reconstructed resolutions for small-animal SPECT are approaching 100 μm [13].

Much finer resolution in radionuclide imaging can be obtained with autoradiography, which uses the charged particles emitted by a radionuclide (alpha particles, beta particles or positrons) rather than the high-energy photons. Classical autoradiography is carried out by placing a thin slab of excised tissue on film or an electronic detector and taking a long exposure. More modern approaches use particle detectors operating at video rates [14]. Images with spatial resolution around 5 μm and sub-nanoCurie sensitivity can be obtained ex vivo with excised slices, and 3D images can be synthesized by stacking the slice images in a computer.

In-vivo charged-particle autoradiography can be performed at high resolution with endoscopic devices, in mouse window chambers or for superficial lesions clinically, but to date only without tomography. Recent work has shown the possibility of true ECT with charged particles in vivo [15]. We refer to the general method as CPET (Charged-Particle Emission Tomography), with the special cases of BET (Beta-particle Emission Tomography) and αET (Alpha-particle Emission Tomography).

Finally, there is great interest in OpET (Optical Emission Tomography), where the radionuclide label is replaced by a fluorescent or bioluminescent molecule. Fluorescence imaging of cells in culture has been the mainstay of molecular biology, and immunohistochemistry (IHC) with fluorescent antibodies is the foundation of modern pathology.

There are numerous ways to produce 3D images with optical emitters. At the clinical scale, fluorescence-enhanced optical imaging of the breast or lymphatic vessels [16, 17] yields 3D reconstructed images with a resolution around a millimeter. Several commercial preclinical imaging systems collect light emerging at different angles from a mouse or other small animal and reconstruct OpET images. There is also a burgeoning interest in various forms of 3D fluorescence imaging at the microscopic scale; these include deconvolution methods, scanning confocal microscopes, optical projection tomography [18] and 3D superresolution methods derived from the Nobel-prize-winning work of Betzig, Hell and Moerner [19, 20].

For all of these ECT modalities and for almost any chosen physiological random process, many different labeling agents (tracers) have been developed [21, 22].

A physiological random process f = f(r, t) imaged via molecular imaging will give rise to a random data set g, usually taken to be a finite-dimensional random vector. In the population ensemble picture, we view the pair {f, g} as statistical covariates. As patient j gives rise to a realization f_j(r, t), the conditional random vector g|f_j represents the random data vector for patient j. The statistics of g|f_j are related to the physics of imaging, which we will not discuss here; see e.g. [10]. A major goal of this paper is to show how ECT data and characteristic functionals can be used to study the physiological random processes that influence outcomes in cancer chemotherapy.

We first discuss how classical models of tumor growth, drug delivery and response to therapy can be modified into spatiotemporal models involving Physiological Random Processes (PRPs). The goal of this exercise is to demonstrate that under the PRP hypothesis, it is possible to derive clinically relevant parameters that are related to the prediction of treatment response for individual patients. Later, we discuss how ECT data can be used to perform statistical estimation of unknown parameters in these models and demonstrate how the theory of maximum likelihood estimation allows for the quantification of uncertainties in these estimates.

Classical models

There is an extensive literature on mathematical models for tumor growth; common mathematical forms are exponential, logistic, Gompertz and von Bertalanffy [23–30]. There is another extensive literature on the response of cancer cells to chemotherapy drugs, radiation or other interventions [31–37]. For simplicity we will consider only avascular and premetastatic growth; incorporating both angiogenesis and metastasis requires more complex mathematical models.

Many of the papers cited present ordinary differential equations for the time dependence of the total number of clonogenic cells in a tumor, denoted N(t). It is common to regard N(t) as a deterministic quantity that can be found by solving the differential equations, with no spatial inhomogeneity, uncertainty or stochastic effects considered. The differential equations always involve a small number of free parameters, which can be fixed by fitting the predicted behavior of N(t) to measurements on tumors in humans or animals.

These investigations often result in equations having the generic form (2) where the first term describes the growth of the tumor in the absence of a therapeutic intervention and the second accounts for a chemotherapy drug.

The simplest form for the growth term is dN(t)/dt|_growth = βN(t) (where β is a positive constant), which leads to exponential growth, N(t) ∝ exp(βt). It is widely observed, however, that the growth rates of solid tumors decrease as the tumors get larger, and hence several common models take the form dN(t)/dt|_growth = N(t)Φ[N(t)], where Φ(⋅) is a monotonically decreasing function. The resulting differential growth rates for several common models are shown in Fig 3. For example, the commonly used Gompertzian model assumes that the tumor size reaches a “carrying capacity” N_max limited by blood supply and nutrients. The Gompertz model enforces this condition by taking Φ[N(t)] ∝ −ln(N(t)/N_max). Mechanistic explanations for Gompertzian tumor growth have been given by Gyllenberg [38, 39], Hahnfeldt [40] and Norton [24].

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Tumor growth curves.

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

https://doi.org/10.1371/journal.pone.0199823.g003

A common form for the drug effect in (2), derived from the classical law of mass action, assumes that dN(t)/dt|_drug = −αC(t)N(t), where C(t) is the drug concentration and α is positive. Equivalently, (3) This form, often called linear log-kill, says that the fractional rate of cell killing is a linear function of the drug concentration and is usually attributed to Skipper [41]. Linear log-kill is a reasonable model for certain cell-cycle-independent drugs such as doxorubicin, at least in vitro [42]. More generally, cytotoxic drugs can affect cell populations nonlinearly with respect to C(t), for instance due to saturation effects and cell cycle inhomogeneities. We discuss several nonlinear response models later; refer to Eqs (32) and (37).

With the general growth model dN(t)/dt|_growth = N(t)Φ[N(t)] and the mass-action assumption for the drug effect, the overall model (2) becomes (4) With the choice of Gompertzian growth kinetics, (5) Eq (5), paired with an initial condition of N(0) = N₀ ≥ 0 total cells, describes the deterministic evolution of N(t). Again, neither spatial effects nor uncertainty are addressed in such a model. We will address both by replacing N(t) with a corresponding physiological random process n(r, t).

Tumor growth and response as random processes

We can define a continuum density of cells, denoted n(r, t), which has units of number of cells per unit volume. In the viewpoint of this paper, n(r, t) is a spatial random process that evolves in time, and N(t) is now viewed as a time-dependent random variable. The two random entities are related by (6) where V(t) is the evolving tumor volume. If we are interested in the means of these random quantities (statistical expectations with respect to some ensemble of tumors), we can indicate the means by overbars and write (7) This result depends only on the linearity of statistical expectations, not on any particular statistical properties of n(r, t) or N(t).

We should not necessarily expect to be spatially uniform to any reasonable approximation: malignant cells may be fairly widely separated in a tumor, spaced by the extracellular matrix and interspersed with fibroblasts, lymphocytes and other cells in the tumor microenvironment. Moreover, tumor cells often tend to grow in clusters called nests or islets, surrounded by a collagenous shell.

We postulate that a reasonable random-process counterpart of Eq (4) is (8) where now n(r, t), α(r, t) and c(r, t) are treated as random processes. The growth nonlinearity Φ may also depend on some auxiliary random process parameters; for example, with a Gompertzian model, we would have (9) where the growth rate μ(r), carrying capacity n_max(r) and drug sensitivity α(r) are all allowed to be spatial random processes. We assume further that the initial condition n(r, 0) is strictly positive at all points within the potential tumor volume: in mechanistic terms, we assume tumor growth proceeds from pre-existing clonogenic cells rather than spontaneous mutations, so it cannot begin in a region where there is zero density of malignant cells at t = 0.

Similarly, to be consistent with the idea of linear log-kill, we assume for the remainder of this section that the drug sensitivity α(r) is independent of time and statistically independent of the drug distribution c(r, t). More general response models can also be considered [43].

Another class of spatial growth models, which we will not discuss here, allows for diffusive and chemotactic migration of tumor cells; see e.g. [44–47].

Because each function in (9) is treated as a random quantity, we must specify the sense in which the equality holds. We will assume throughout that a differential equation such as (9) holds in the classical (or weak, in the sense of generalized functions) sense for each fixed realization of the random parameters. This assumption allows us to solve (9) using standard differential equation techniques, then derive statistical distributions for the resulting solution and any quantities of interest related to it. Such an equation is frequently called a random differential equation, to distinguish from stochastic differential equations which require more sophisticated solution strategies such as the Itô formalism (see discussion in [48], section 4.7). See Fig 4 for an illustration.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Tumor growth under uncertainty.

A simulation of Gompertzian growth of the spatial random field n(r, t), as modeled in Eq (9) with c(r, t)≡0, is displayed on the left. In this simulation, both the local growth constant μ(r) and the local carrying capacity n_max(r) are spatially constant but random; the initial condition n(r, 0) is taken as a so-called lumpy background random process (see S2 Appendix). Three realizations (rows) are shown at three times. The total cell number N(t) as defined in (6) is shown on the right for 16 realizations of the process.

https://doi.org/10.1371/journal.pone.0199823.g004

In S3 Appendix, we derive an expression for the evolution of the characteristic functional for ln n(r, t) under the hypothesis that the density of tumor cells is a lognormal random process (see S2 Appendix).

Other response models

The formulas for tumor response developed above are all linear, in the sense that the log-kill is a linear function of the drug concentration, and local, in the sense that the response of the tumor at some point r depends on the drug concentration only at that point. There are situations in clinical oncology, however, where the response can be nonlinear, nonlocal or both. In this subsection we look at how these effects can be incorporated into a general theory of physiological random processes in oncology. The models presented can be used in combination with patient-specific imaging data to extract detailed spatiotemporal physiological information, which can subsequently be used to better predict treatment response.

Approach

To this point we have derived characteristic functionals for a number of physiological random processes related to cancer therapy, and we have used the results to obtain finite-dimensional characteristic functions and probability density functions for clinically relevant random variables such as the integrated log kill, which then permits us to compute probability of tumor control as in Eqs (18)–(20). All of these quantities are defined in terms of hypothetical infinite ensembles of patients, so their use for individuals is so far somewhat limited. In precision medicine, the objective is to make use of patient-specific data and mathematical models to enhance decision making; refer again to Fig 1. This process begins with identifying clinically relevant treatment efficacy biomarkers such as the integrated log-kill Y; because Y may not be directly observable, we must also identify relevant clinically observable physiological processes; we have discussed several spatiotemporal processes which are accessible via molecular imaging. Then, by combining imaging data and the mathematical models presented here, we can compute an estimate of Y which is specific to a particular patient. Even with patient-specific imaging, uncertainties remain, and thus the patient-specific Y is still a random variable, but these uncertainties are almost certainly lower than if no imaging were performed. We discuss the statistical properties of patient-specific estimates of Y in a later section (Eqs (39)–(43) and Discussion).

In several previous papers [9, 52, 61–63], we have considered the problem of optimizing a diagnostic imaging study or therapeutic regimen in order to get the best outcome for an individual patient. This would seem to be fundamentally impossible since both diagnostic and therapeutic efficacy are typically defined in terms of large groups of patients and evaluated by clinical trials, but we have developed rigorous mathematical methods for patient-specific optimization of SPECT imaging [9], radiation therapy [61, 62] and chemotherapy [52, 63]. None of these papers, however, makes use of random processes and characteristic functionals as in this paper.

In the paradigm of personalized medicine it is reasonable to expect most cancer patients to undergo genomic analysis and in many cases to have sensitivity-resistance assays to aid in selection of therapeutic drugs. In addition, there are attempts to use databases to correlate genomic mutations with drug efficacy in the hope that genomically similar patients will have similar drug responses. However this effort is problematic because there is no consensus on how to define genomic similarity and because a genomic mutation leads to an abnormal protein only after transcription and translation. For this reason there is considerable interest in the transcriptome and the proteome, but the databases are not as extensive at these levels.

The paradigm we consider for personalized cancer therapy is thus [52, 64]:

1. Obtain images, genomic data and other data for a specific patient;
2. Optimize the therapy regimen using this information and supporting mathematical models;
3. Evaluate the resulting patient-specific protocol with multiple patients as in conventional clinical trials.

The first step requires reconstruction of molecular images for each patient, which requires the application of statistical image science methodologies [10]; the spatial statistics of reconstructed images are discussed in [10, 65] and elsewhere.

The second step requires the application of mathematical or heuristic models for estimating the probability of tumor control and probability of normal tissue complication for each patient, and all of the expressions for Pr(TC) given in this paper are potentially applicable. To optimize a treatment, we advocate using a curve-based analysis such as the Therapy Operating Characteristic (TOC) curve discussed below and elsewhere [52], which models the trade-off between tumor control and undesired side effects.

The third step requires recruiting multiple patients and applying the joint data collection and adaptive therapy protocol independently to each subject in the trial.

This paradigm provides a logical structure for clinical trials of personalized therapy, yielding a joint evaluation of the quality of the patient-specific data; the strategy for adapting the therapy regimen to the patient, and the models that connect the data and the strategy. As we stressed in [52], the only thing one can ever evaluate in a clinical trial for personalized medicine is the combination of the patient-specific data, the model of tumor response and the personalization strategy (see Fig 5).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Precision medicine trials.

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.

https://doi.org/10.1371/journal.pone.0199823.g005

Therapeutic efficacy

Any optimization procedure requires the selection of a scalar figure of merit to maximize or minimize, so an important question in the second step of the precision medicine paradigm proposed above is the selection of an appropriate figure of merit. One could imagine maximizing probability of tumor control, Pr(TC), over an admissible set of therapy parameters such as administered mass, but tumor control by itself does not constitute therapeutic success; we must also be concerned with adverse side effects. Both the tumor-control probability (TCP) and the probability of some specified normal-tissue complication (NTCP) generally increase with drug dosage. If the total mass of the drug administered to patient j is denoted M, then with suitable models such as those given in Eqs (18)–(20), we can calculate TCP_j(M) and NTCP_j(M) for patient j and plot both as a function of M as shown in Fig 6(a). By plotting TCP_j(M) vs. NTCP_j(M) as the drug mass is varied, we thus construct the patient-specific Therapy Operating Characteristic (TOC) curve, as shown in Fig 6(b), which is the therapeutic analog of the familiar Receiver Operating Characteristic (ROC) curve, used routinely in evaluating diagnostic imaging methods and laboratory tests. The TOC curve was originally developed in the context of radiation therapy [61, 62] and recently extended to chemotherapy [52]. Ultimately, we expect the TOC curve to be a useful, simplified graphical decision-making tool in clinical practice; the clinician and patient can together select treatment parameters that balance the chance of success and chance of side effects according to the patient’s level of risk aversion.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. TOC curves.

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, M (arbitrary units); (b) The TOC curve: Plot of TCP vs. NTCP as M is varied.

https://doi.org/10.1371/journal.pone.0199823.g006

As with ROC curves, we can use the area under the TOC curve, denoted AUTOC, as a metric of therapeutic efficacy and hence as a measure of both treatment quality and image quality for situations where the images are used for therapy planning. This approach is particularly attractive for radiation therapy where established radiobiological models can be used to compute TCP and NTCP, and the theory developed in this paper facilitates the estimation of TCP in chemotherapy. Estimation of NTCP in chemotherapy is more problematic, in large part because there is very little literature on the whole-body distributions of common drugs and their patient-to-patient variability. One can find publications on maximum tolerated dose (MTD) and the corresponding dose-limiting side effect, but very little on the variation of NTCP with injected mass of the drug. Moreover, the injected mass is always reported as milligrams per square meter of body surface area (possibly also per week).

Estimation tasks in precision medicine