Multifocality and recurrence risk: a quantitative model of field cancerization

Abstract

Primary tumors often emerge within genetically altered fields of premalignant cells that appear histologically normal but have a high chance of progression to malignancy. Clinical observations have suggested that these premalignant fields pose high risks for emergence of recurrent tumors if left behind after surgical removal of the primary tumor. In this work, we develop a spatio-temporal stochastic model of epithelial carcinogenesis, combining cellular dynamics with a general framework for multi-stage genetic progression to cancer. Using the model, we investigate how various properties of the premalignant fields depend on microscopic cellular properties of the tissue. In particular, we provide analytic results for the size-distribution of the histologically undetectable premalignant fields at the time of diagnosis, and investigate how the extent and the geometry of these fields depend upon key groups of parameters associated with the tissue and genetic pathways. We also derive analytical results for the relative risks of local vs. distant secondary tumors for different parameter regimes, a critical aspect for the optimal choice of post-operative therapy in carcinoma patients. This study contributes to a growing literature seeking to obtain a quantitative understanding of the spatial dynamics in cancer initiation.

Keywords: Cancer initiation; Evolution; Stochastic spatial models.

Figure 1. Lattice dynamics

(A) Schematic of spatial Moran model in d = 2: each cell divides at rate according to its fitness and replaces one of its 2d neighbors: if the light blue cell divides, its offspring replaces one of the dark blue neighbors, chosen uniformly at random. Every lattice site is occupied at all times (not shown). (B) Simulation example of the model: growth of an advantageous clone (light blue) starting from one cell with fitness advantage s = 0.2 over the surrounding field (dark blue).

Figure 2. Geometry of squamous epithelium

A Basal layer (vertical perspective) before initiation with local field (left), and after initiation where the tumor is growing within the local field (right). B Sideways view of the fields before and after initiation, along the dashed lines in panel A. The proliferative cells inhabiting the two-dimensional lattice in the model reside in the basal layer of the epithelium.

Figure 3. Simulations of clonal expansion rate for large s

Dependence of the growth rate c₂ on the fitness advantage s. Statistics performed on M = 100 samples for each s-value. The error bars represent 95% confidence intervals.

Figure 4. The three dynamic regimes

Regime 1: first successful type-2 cell (arrow) arises in the first premalignant clone, Γ = 0.055. Regime 2: several premalignant clones are present at the time of the first successful type-2 cell, Γ = 54.47. Regime 3: a large number of small premalignant clones are present by the time of the first successful type-2 cell, Γ = 5.45 × 10⁴. Simulations obtained with parameter values as in Figure 5.

Figure 5. Waiting time until first successful type-2

Cumulative distribution function (cdf) of σ₂, the waiting time until the first successful type-2 mutation, for increasing N (see (4)). Regime 1: u₁ = 7.5 · 10⁻⁸, Regime 2: u₁ = 7.5 · 10⁻⁷, Regime 3: u₁ = 7.5 · 10⁻⁶. All other parameters are fixed: d = 2, N = 2 · 10⁵, s₁ = s₂ = 0.1, u₂ = 2 · 10⁻⁵, c₂(s₁) = 0.16.

Figure 6. Local and distant recurrences

Local (blue) and distant (green) premalignant fields give rise to second field tumors and second primary tumors (both red), respectively. In scenario A, there is only one premalignant field (the local field) present at time of cancer initiation (middle panel), and the recurrence occurs inside the local field. In scenario B, two unrelated precancerous fields are present at time of initiation (middle panel), and the recurrence may occur as a second primary tumor in the distant field.

Figure 7. Size-distribution of local field

The size-distribution (8) of the local field is shown for different scenarios, corresponding to different Γ-values and regimes R1, R2 and R3 as explained in Section 2.4. A For varying arrival times t; B for varying type-1 mutation rates u₁; C for varying type-2 mutation rates u₂; (D) for varying type-1 fitness advantages s₁. The non-varying parameters are held constant at d = 2, N = 2 · 10⁵, u₁ = 7.5 · 10⁻⁷, u₂ = 2 · 10⁻⁵, s₁ = s₂ = 0.1 and c₂(s₁) = 0.16.

Figure 8

The distribution of the total size of the distant field is shown for different scenarios, corresponding to the three regimes R1, R2 and R3 illustrated in Figure 4 for varying type-1 mutation rates u₁. The non-varying parameters are held constant at d = 2, N = 2 · 10⁵, u₂ = 2 · 10⁻⁵, s₁ = s₂ = 0.1 and c₂(s₁) = 0.16.

Figure 9. Dynamic clone-size distribution

For each of the three regimes in Figure 5, the expected number of type-1 clones of sizes comprised in the corresponding intervals I_j are shown as functions of time up to E(σ₂) (expectations are conditioned on {t = E(σ₂)}). The intervals are defined as I₁ = [0, 1500), I₂ = [1500, 3000), I₃ = [3000, 4500) and I₄ = [4500, +∞). Parameter values as in Figure 5.

Figure 10. Time to local recurrence

A The cumulative distribution function of the time to recurrence of a second field tumor is shown for three different scenarios, corresponding to u₂ = 2 · 10⁻³ (Regime 1), u₂ = 2 · 10⁻⁵ (Regime 2) and u₂ = 2 · 10⁻³ (Regime 2/3), respectively. The remaining parameters are d = 2, N = 2·10⁵, u₁ = 7.5·10⁻⁷, s₁ = s₂ = 0.1, t = E(σ₂). B Schematic of the relative initiation times of the primary tumor (yellow) and sizes of the local fields (blue), for the three scenarios in panel A. The numerical values for expected initiation time and local field size are: (a) (σ₂) = 123, Ê(R_l) = 8; (b) (σ₂) = 281, Ê(R_l) = 31; (c) (σ₂) = 474, Ê(R_l) = 55.

Figure 11. Local vs. distant recurrence

A For each of the three regimes in Figure 5, we show: the distribution of time to local recurrence $\widehat{P}(T_R^f\in d\tau )$, and the distribution of time to distant recurrence $\widehat{P}({\stackrel{\sim }{T}}_R^p\in d\tau )$. The distribution of $T_R^f$ is given in Corollary 4.1 and we set ${\stackrel{\sim }{T}}_R^p=min\{T_R^p,{\sigma }_2\}$ to account both for contributions from type-1 clones already existing at σ₂ as well as contributions from type-1 clones born after σ₂ (for which time to recurrence is distributed as σ₂). Expected times to recurrence: $\widehat{E}(T_R^f)=81$ and $\widehat{E}({\stackrel{\sim }{T}}_R^p)=733$ (Regime 1); $\widehat{E}(T_R^f)=98$ and $\widehat{E}({\stackrel{\sim }{T}}_R^p)=86$ (Regime 2); $\widehat{E}(T_R^f)=149$ and $\widehat{E}({\stackrel{\sim }{T}}_R^p)=34$ (Regime 3). The parameter values are as in Figure 5.