A Joint Model of Breast Tumour Size at Diagnosis and Time to Recurrence

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.title[
# A Joint Model of Breast Tumour Size at Diagnosis and Time to Recurrence
]
.author[
### <u>Alessandro Gasparini</u> & Keith Humphreys
]
.date[
### <a href="mailto:alessandro.gasparini@ki.se" class="email">alessandro.gasparini@ki.se</a> · 2022-05-03
]

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# Breast Cancer

* Breast cancer is the fifth leading cause of cancer mortality worldwide, with 685000 deaths;

* Among women, breast cancer accounts for 1 in 4 cancer cases and for 1 in 6 cancer deaths.

> Source: _Global Cancer Statistics 2020: GLOBOCAN Estimates of Incidence and Mortality Worldwide for 36 Cancers in 185 Countries_.

Prognosis for Stage IV breast cancer is much poorer ([SEER data](https://www.cancer.org/cancer/breast-cancer/understanding-a-breast-cancer-diagnosis/breast-cancer-survival-rates.html)):

* 5-year relative survival of 99% for localised tumours,
* 86% for tumours that have spread to the lymph nodes only,
* 29% for tumours that have metastasised elsewhere in the body.

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## Developing models for understanding distant metastatic spread and linking it to the natural history of cancer is fundamentally important.

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## The model introduced here is aimed at being used on breast cancer <u>incident</u> cases collected in a population where screening is offered.

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## We are currently working on extending this to other settings.

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# Our Approach

The natural history of breast cancer has traditionally been modelled using:

1. Discrete multi-state models,

1. Natural history models.

The former can be easily extended, but the complexity grows quickly (e.g., if adding intermediate states).

The latter offers flexibility, parsimony, and biologically-motivated assumptions.

> For our modelling, we build upon the latter approach.
> We know the current model is _naïve_ in some sense, but hopefully, with further developments, it could provide a useful tool.

???

Our understanding is that CISNET uses a mixture of the two approaches, with the different working groups.

---

# Natural History Models

These models combine a growth model with a continuous function of screening sensitivity.

Features:

* Random effects can be accommodated to account for between-subjects heterogeneity;

* Components can be added on top of the basic framework (e.g., models for detection and spread);

* For our aims, we specify a model for the dynamic process of metastatic spread from the onset of the primary tumour.

> I will now introduce the mathematical details of our modelling approach.

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# Modelling Tumour Growth

We assume exponential growth:

`$$V(t | r) = V_{\text{Cell}} \exp(t / r)$$`

where `$V_{\text{Cell}}$` is the volume of a single, spherical cell of diameter `$d_{\text{Cell}}$` = 0.01 mm.

Then, we accommodate between-subjects heterogeneity by assuming a random effect on the inverse growth rates `$R$`, following a Gamma distribution with shape `$\tau_1$` and rate `$\tau_2$`:

`$$f_R(r) = \frac{\tau_2^{\tau_1}}{\Gamma(\tau_1)} r^{\tau_1 - 1} \exp(-\tau_2 r), r \ge 0$$`

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# Symptomatic Detection

We assume tumours are detectable with non-zero probability starting from size `$V_0$` and onwards:

`$$V_0 = V_{\text{Cell}} \exp(t_0 / r)$$`

Here, `$t_0$` is the time it takes for a tumour to grow from a single cell to be detectable.

We assume that, in the absence of screening, the rate of symptomatic detection at time `$T_{\text{de}}$` is proportional to the size of the tumour:

`$$P(T_{\text{det}} \in [t, t + dt) | T_{\text{det}} \ge t, R = r) = \eta V(t, r) dt + o(dt), t \ge t_0$$`

Finally, from the above we can define the density of tumour size at symptomatic detection as:

`$$f_{V_\text{det}}(v_\text{det}) = \eta \tau_1 \frac{\tau_2^{\tau_1}}{(\tau_2 + \eta (v_{\text{det}} - V_0))^{\tau_1 + 1}}, v_{\text{det}} > V_0$$`

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# Metastasis Seeding

We start by defining a model for successful distant metastatic seeding.

First, we define a non-homogeneous Poisson process with intensity function proportional to the number and rate of cell divisions:

`$$\lambda(t, r) = \sigma^* D(t, r)^k D'(t, r)$$`

The exponent `$k \ge -1$` adds additional flexibility to the model.

The cumulative intensity function of the Poisson process is defined as:

`$$\Lambda(t, r) = \int_0^t \lambda(u, r) du = \sigma \left[ \log \frac{V(t, r)}{V_{\text{Cell}}} \right]^{k + 1} = \sigma \left[ \frac{t}{r} \right]^{k + 1}$$`

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# Metastasis Seeding

The non-homogenous Poisson process leads to the following probability of having `$u$` successful seeded metastases at detection time `$t_{\text{det}}$`:

`$$P(U = u | T_{\text{det}} = t_{\text{det}}, R = r) = \frac{\Lambda(t_{\text{det}}, r)^u}{u!} \exp(-\Lambda(t_{\text{det}}, r))$$`

The corresponding survival probability is
`$$S(t | R = r) = P(U = 0 | T_{\text{det}} = t, R = r) = \exp(-\Lambda(t, r)) = \exp\left(-\frac{\sigma}{r^{k + 1}} t^{k + 1} \right),$$`

which is the survival function of a Weibull distribution with scale parameter `$\sigma / r^{k + 1}$` and shape parameter `$k + 1$`.
We can therefore derive the hazard function too:

`$$h(t | R = r) = \frac{\sigma}{r^{k + 1}} (k + 1) t^k, t \ge 0$$`

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# Detection of Metastasis

The model that we just defined is stated in terms of _unobservable_ quantities, the seeding of metastases.

We need to reformulate the problem in terms of observable quantities such as the presence of distant metastases at diagnosis and the times of distant metastases from diagnosis of the primary tumour.

This requires the following, additional assumptions:

* It takes time `$t_0$` for a metastasis to grow from seeding to detection;

* Breast cancers are only diagnosed through the primary tumour;

* Metastases growing to a detectable size before the diagnosis of the primary tumour will be visible at diagnosis of the primary tumour, but not detected beforehand.

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# Detection of Metastasis

We define a random variable W counting time to diagnosis of metastasis from detection of the primary tumour:

`$$w = t + t_0 - t_{\text{det}}$$`

With this, we can reformulate hazard and survival functions for time to detection of distant metastasis as:

`$$S(w | V_{\text{det}} = v_{\text{det}}, R = r) = \exp\left[-\sigma \left( \frac{w}{r} + \log \frac{v_{\text{det}}}{V_0}\right)^{k + 1} \right]$$`

`$$h(w | V_{\text{det}} = v_{\text{det}}, R = r) = \frac{\sigma}{r} (k + 1) \left( \frac{w}{r} + \log \frac{v_{\text{det}}}{V_0}\right)^k$$`
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# Detection of Metastasis

We now introduce additional assumptions:

* Metastatic seeding completely stops at diagnosis of the primary tumour;

* Already seeded, successful colonies are not affected at diagnosis.

These assumptions constrain the hazard (and density) functions to be null for values of `$w > t_0$`, and equivalently, the survival function to be constant beyond time `$t_0$`.

> Note that alternative, less stringent assumptions concerning the spread/seeding and growth/detection of distant metastases, as well as treatment effects, could in principle be incorporated.

???

We are currently working on some of these extensions.

One thing we did was estimate `$t_0$` from data, which resulted in a `$d_0$` close to the assumed value of 0.5 mm.

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# Summary of Key Time Points

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# We have now defined all key components of our model.

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# In the next few slides we will introduce the estimation procedure for incident cases.

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# Likelihood in the Absence of Screening

We work in a frequentist framework.
The likelihood function is defined as the joint probability of tumour size and metastasis:

`$$f_{W, V_{\text{det}}}(w, v_{\text{det}}) = f_W(w | V_{\text{det}} = v_{\text{det}}) f_{V_{\text{det}}}(v_{\text{det}})$$`

The density of `$W$`, `$f_W(w | V_{\text{det}})$`, will change depending on whether a woman was left-censored, right-censored, or had an observed event.

More details and formulae are included in the [SMMR paper](https://doi.org/10.1177%2F09622802211072496), more on this later.

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# Likelihood for a Screened Population

The likelihood function from the previous slide needs to be adjusted in the settings of a screened population.

First, we define a logistic screening sensitivity function:

`$$\text{Screening Sensitivity} = \frac{\exp(\beta_1 + \beta_2 d)}{1 + \exp(\beta_1 + \beta_2 d)},$$`

with `$d$` the diameter of the tumour. 
We also assume that sensitivity is zero for `$d$` < 0.5 mm.

This extra component, together with the models introduced before, is used to define the likelihood function for a screened population.

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# Likelihood for a Screened Population

For screen-detected tumours the likelihood contribution can be written as:

`$$L_{v,w} \propto P(B_0|V = v) P(V = v, W = w | A) P(B^c| A, V = v, W = w)$$`
where `$A$` denotes the presence of a tumour in the breast, `$B_0$` that a tumour is screen-detected, and `$B^c$` the history of negative screens.

Thanks to some theoretical results, and after some algebra, it can be shown that the likelihood contribution can be re-written as:

`$$\begin{aligned}
L_{v,w} \propto &P(B_0|V = v) \times P(V_{\text{det}} = v) \times \\ 
                & \quad \int_R \left[ \prod_{q = 1}^p P(B^c_q | R = r, V = v) \right] f_W(w | R = r, V = v) f_R(r | V_{\text{det}} = v) dr
\end{aligned}$$`

As in the absence of screening, `$f_W$` will have a different formulation for left- and right-censored subjects and observed events.

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# Likelihood for a Screened Population

Conversely, for tumours that are detected through symptoms, the likelihood contribution can be written as:

`$$L_{v,w} \propto P(V = v, W = w | A) P(B^c| A, V = v, W = w)$$`

and then re-written as

`$$\begin{aligned}
L_{v,w} \propto & P(V_{\text{det}} = v) \times \\ 
                & \quad \int_R \left[ \prod_{q = 1}^p P(B^c_q | R = r, V = v) \right] f_W(w | R = r, V = v) f_R(r | V_{\text{det}} = v) dr
\end{aligned}$$`

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# Estimation Details

1. In practice, we discretise the distribution of tumour size: `$f_{V_{\text{det}}}(v_{\text{det}}) \approx P(V_{\text{det}} \in I_i)$`;

1. The probability of `$p$` negative screens is calculated by backwards projection, as described in Abrahamsson and Humphreys (2016);

1. Discretising time makes it straightforward to calculate the likelihood when it is known only to a proportionality constant, without being too computationally demanding.

1. We still require numerical integration to marginalise over the distribution of growth rates given size, `$f_R(r | V_{\text{det}} = v)$`.

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# Model-Based Predictions

After fitting the natural history model that we just introduced, we can obtain a variety of model-based predictions.

_Standard_ quantities:

* Growth rates distribution,
* Doubling time,
* Etc.

_Novel_ quantities:

* Survival probabilities of distant metastasis over time,
* Probability of having latent, already seeded but not yet manifest distant metastases at diagnosis,
* Probability of having detected distant metastasis at diagnosis of the primary.

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# Probability of Distant Metastasis

The vast majority of breast cancer patients do not have a detected distant metastasis at diagnosis: for these, we can estimate _conditional survival probabilities_ using this model.

For instance, for symptomatically-detected tumours, the survival probability at time `$w^*$` is:

`$$P(W > w^∗|V = v, W > 0, B^c) = \frac{P(W > w^∗|V = v, B^c)}{P(W > 0|V = v)}$$`

Note that the numerator of the equation above needs to be normalised as in the likelihood calculations.

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# Finally, we illustrate our modelling approach in practice using Swedish data on postmenopausal breast cancer patients

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# Data Resource

* Data from a case-control study of postmenopausal breast cancer in Sweden (CAHRES; Cancer And Hormone REplacement Study);

* The study included women born and residing in Sweden, aged 50-74, diagnosed with an incident primary invasive breast cancer in 1993-1995;

* We included 1614 breast cancer patients, of whom 1035 (64.13%) detected via screening and 579 (35.87%) detected symptomatically;

* Median tumour diameter at detection was 15 mm (interquartile interval [IQI]: 10–22 mm);

* Four (0.25%) women had metastases at diagnosis, and an additional 293 (18.20%) were diagnosed with metastases during follow-up;

* Median follow-up was 5.49 years (IQI: 5.41–5.58 years).

???

Note that this is old data, which has some advantages for our purposes.

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# Picking k

Remember that `$k$` is the parameter denoting _genomic instability_, i.e. the primary tumour mutating at an accelerating or decelerating rate.

Fitting `$k$` from data, we obtained `$k$` = 3.887 with a log-likelihood value of −4559.10.
Conversely, log-likelihood values for models with different values of `$k$`:

| `$k = 1$` | `$k = 2$` | `$k = 3$` | `$k = 4$` | `$k = 5$` |
| :-----: | :-----: | :-----: | :-----: | :-----: |
| -4,669.44 | -4,596.21 | -4,567.58 | -4,559.18 | -4,565.68 |

> We decided to go with `$k$` = 4 since integer values are attractive for interpretability.

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# Model Estimates

Maximum likelihood parameter estimates, including standard errors computed by inverting the Hessian matrix at the optimum:

|  | Estimate | SE | 95% C.I. | |
|-:|----------:|----:|:----------:|:-|
| `$\log(\tau_1)$` | 0.746 | 0.101 | 0.549 to 0.943 | Shape of `$f_R$` |
| `$\log(\tau_2)$` | 0.910 | 0.162 | 0.592 to 1.227 | Rate of `$f_R$` |
| `$\beta_1$` | -4.654 | 0.133 | -4.914 to -4.394 | Screen-detection, intercept |
| `$\beta_2$` | 0.439 | 0.022 | 0.396 to 0.483 | Screen-detection, size |
| `$-\log(\eta)$` | 9.078 | 0.081 | 8.920 to 9.237 | Symptomatic detection |
| `$\log(\sigma)$` | -16.399 | 0.082 | -16.559 to -16.239 | Seeding of metastasis |

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# Model-Based Predictions

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# Model-Based Predictions

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# Model-Based Predictions

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# Thanks for listening. Any questions?