Accurate and Flexible Bayesian Mutation Call from Multi-regional Tumor Samples

Abstract

We propose a Bayesian method termed MultiMuC for accurate detection of somatic mutations (mutation call) from multi-regional tumor sequence data sets. To improve detection performance, our method is based on the assumption of mutation sharing: if we can predict at least one tumor region has the mutation, then we can be more confident to detect a mutation in more tumor regions by lowering the original threshold of detection. We find two drawbacks in existing methods for leveraging the assumption of mutation sharing. First, existing methods do not consider the probability of the “No-TP (True Positive)” case: we could expect mutation candidates in multiple regions, but actually, no true mutations exist. Second, existing methods cannot leverage scores from other state-of-the-art mutation calling methods for a single-regional tumor. We overcome the first drawback through evaluation of the probability of the No-TP case. Next, we solve the second drawback by the idea of Bayes-factor-based model construction that enables flexible integration of probability-based mutation call scores as building blocks of a Bayesian statistical model. We empirically evaluate that our method steadily improves results from mutation calling methods for a single-regional tumor, e.g., Strelka2 and NeuSomatic, and outperforms existing methods for multi-regional tumors through a real-data-based simulation study. Our implementation of MultiMuC is available at https://github.com/takumorizo/MultiMuC.

A Appendix

1.1 A.1 Increasing Posterior Odds Score of Mutation Call Given $C=1$

In the main text, we mentioned the assumption that lowering the threshold of scores given $C=1$ leads to performance improvement at Sect. 2.1. Here, we show that the assumption is based on an increase of posterior odds for mutation call. We assume that each score is represented by posterior odds form $V_i := Pr(X_i = 1|D_i)/Pr(X_i = 0|D_i)$. If we observe $C = 1$ in addition to the observed sequence data set, the true posterior odds can be represented as follows.

$$\begin{aligned} V_i^{\prime }&:= \frac{Pr(X_i = 1|C=1, D_i)}{Pr(X_i = 0|C=1, D_i)} = \frac{Pr(X_i = 1, C=1, D_i)}{Pr(X_i = 0, C=1, D_i)} \\&= \frac{ Pr( X_i = 1|C= 1 )}{Pr( X_i = 0| C = 1)} \frac{Pr( D_i|X_i = 1, C=1 )}{ Pr( D_i|X_i = 0, C=1 ) } \end{aligned}$$

The true posterior odds is greater than the original posterior odds as shown in the following lemma.

Lemma 2 (Increasing posterior odds)

[Increasing posterior odds]

$$\begin{aligned}&\text{ If } Pr(D_i|X_i, C) = Pr(D_i|X_i),\, 0< Pr(C = 0) < 1,\text{ and } V_i, V_i^{\prime } \in \mathbb {R},\,\text{ then } V_i^{\prime } > V_i\,. \end{aligned}$$

Proof

It is sufficient to show that the following condition holds true.

$$\begin{aligned}&\cdot \frac{ Pr(X_i = 1|C= 1)}{Pr(X_i = 0| C = 1)} > \frac{ Pr(X_i = 1)}{Pr(X_i = 0)} \end{aligned}$$

The condition can be proved by evaluating $Pr(X_i = 1)$ and $Pr(X_i = 0)$ as follows.

$$\begin{aligned}&Pr(X_i = 1) \\&= Pr(X_i = 1|C = 1) Pr(C = 1) + Pr(X_i = 1|C = 0) Pr(C = 0) \\&= Pr(X_i = 1|C = 1) Pr(C = 1) \,(\because \,Pr(X_i = 1|C = 0) = 0) \\&Pr(X_i = 0) \\&= Pr(X_i = 0|C = 1) Pr(C = 1) + Pr(X_i = 0|C = 0) Pr(C = 0) \\&= Pr(X_i = 0|C = 1) Pr(C = 1) + Pr(C = 0) \,(\because \,Pr(X_i = 0|C = 0) = 1) \\&> Pr(X_i = 0|C = 1) Pr(C = 1) \,(\because \,0< Pr(C = 0) < 1) \end{aligned}$$

By using the above evaluations, we can show $\frac{ Pr(X_i = 1|C= 1)}{Pr(X_i = 0| C = 1)} > \frac{ Pr(X_i = 1)}{Pr(X_i = 0)}$.

From this condition and the given hypothesis,

$$\begin{aligned}&V_i^{\prime } = \frac{ Pr( X_i = 1|C= 1 )}{Pr( X_i = 0| C = 1)} \frac{Pr( D_i|X_i = 1, C=1 )}{ Pr( D_i|X_i = 0, C=1 ) } > \frac{ Pr( X_i = 1 )}{Pr( X_i = 0)} \frac{Pr( D_i|X_i = 1 )}{ Pr( D_i|X_i = 0) } = V_i \end{aligned}$$

   $\square$

1.2 A.2 The Probability of No-TP Case in General

Here, we evaluate the probability of the No-TP case when $M < N$. For simplicity, we define variables and relational operators between vector and scalar. For Eq. (10), we also define similar relational operators for $\ge , <, \le , =$ between vector and scalar.

$$\begin{aligned}&\varvec{V} :=(V_{1},\cdots ,V_{M}),\, \widetilde{\varvec{V}} :=(V_{M+1},\cdots ,V_{N}),\varvec{X} :=(X_{1},\cdots ,X_{M}),\, \widetilde{\varvec{X}} :=(X_{M+1},\cdots ,X_{N}), \nonumber \\&\varvec{u}> v \iff u_{i} > v \,\,(\forall i ) \\&\varvec{u} \ne v \iff u_{i} \ne v \,\,(\exists i ) \nonumber \end{aligned}$$

(10)

The probability of the No-TP case can be represented as follows.

$$\begin{aligned}&Pr( \varvec{X} = 0, \widetilde{\varvec{X}} = 0|\varvec{V} > v, \widetilde{\varvec{V}} \le v ) \end{aligned}$$

(11)

For $Pr( \varvec{V} > v, \widetilde{\varvec{V}} \le v )$, we can obtain a lower bound as follows.

$$\begin{aligned}&Pr(\varvec{V}> v, \widetilde{\varvec{V}} \le v) \nonumber \\&\qquad = \sum _{\varvec{X}, \widetilde{\varvec{X}}} Pr( \varvec{X},\widetilde{\varvec{X}} ) Pr(\varvec{V}> v, \widetilde{\varvec{V}} \le v| \varvec{X}, \widetilde{\varvec{X}}) \nonumber \\&\qquad = \sum _{\varvec{X}, \widetilde{\varvec{X}}} Pr( \varvec{X},\widetilde{\varvec{X}} ) \prod _{i=1}^{M} Pr(V_{i} > v|X_{i}) \prod _{k=M+1}^{N} Pr(V_{k} \le v|X_{k}) \,(\because \, Eq.\, (2)) \nonumber \\&\qquad = \sum _{\varvec{X}, \widetilde{\varvec{X}}} Pr( \varvec{X}, \widetilde{\varvec{X}} ) \prod _{i=1}^{M} (1-s_{i}(v))^{1-X_{i}} R_{i}(v)^{X_{i}} \prod _{k=M+1}^{N} s_{k}(v)^{1-X_{k}} (1-R_{k}(v))^{X_{k}} \nonumber \\&\qquad \ge Pr( \varvec{X} = 1, \widetilde{\varvec{X}} = 0 ) \prod _{i=1}^{M} R_{i}(v) \prod _{k=M+1}^{N} s_{k}(v) =: A, \end{aligned}$$

(12)

where $R_{i}(v) := Pr(V_{i} > v|X_{i} = 1)$ corresponds to recall.

From Eq. (12), if $A > 0$, we can derive an upper bound for Eq. (11) as follows.

$$\begin{aligned}&Pr( \varvec{X} = 0, \widetilde{\varvec{X}} = 0|\varvec{V}> v, \widetilde{\varvec{V}} \le v ) = \frac{ Pr( \varvec{X} = 0, \widetilde{\varvec{X}} = 0, \varvec{V}> v, \widetilde{\varvec{V}} \le v ) }{ Pr( \varvec{V} > v, \widetilde{\varvec{V}} \le v ) } \nonumber \nonumber \\&\le min\left( 1, \frac{ Pr( \varvec{X} = 0, \widetilde{\varvec{X}} = 0 ) \prod _{i=1}^{M} (1-s_{i}(v)) \prod _{k=M+1}^{N} s_{k}(v) }{ Pr( \varvec{X} = 1, \widetilde{\varvec{X}} = 0 ) \prod _{i=1}^{M} R_{i}(v) \prod _{k=M+1}^{N} s_{k}(v) } \right) \nonumber \\&= min \left( 1, \frac{Pr( \varvec{X} = 0, \widetilde{\varvec{X}} = 0 )}{ Pr( \varvec{X} = 1, \widetilde{\varvec{X}} = 0 ) } \prod _{i=1}^{M} \frac{ 1-s_{i}(v) }{ R_{i}(v)} \right) \end{aligned}$$

(13)

From Eq. (13), as the specificity increases, the probability of the No-TP case also decreases when $M < N$.