Appendix1.1 Integral solutions

Integral 2.4 with \(n=1\) can be partitioned over the domain of f,

$$\begin&\mathbb [}] = \int _\mathbb [(x')}] \ f(x') \ \text x'\nonumber \\&= 2 \int _0^m \big [ 1 \big ] \ \Big [ \frac} e^(\frac)^2} \Big ] \ \text x'\nonumber \\&\quad + 2\int _m^ \Big [ \frac (2r + m - |x'|)^2 (x'^2 \nonumber \\&\quad - 3 m^2 + 2|x'|(m+2r)) \Big ] \ \Big [ \frac} e^(\frac)^2} \Big ] \ \text x'\end$$

(A.1)

$$\begin&\mathbb [}] = 2 \Bigg [ \frac \,}}\Big ( \frac}\Big ) \Bigg ] \nonumber \\&\quad +2 \Biggr [ \bigg [ \frac} \bigg ] \bigg [ e^} \sigma ^2 \bigg ( 5 m^2 + 8 m r + 8 r^2 \nonumber \\&\quad - 2 \sigma ^2 - e^} \Big ( 5 m^2 + 12 m r + 12 r^2 - 2 \sigma ^2 \Big ) \bigg ) \nonumber \\&\quad + \frac \bigg (m + 2 r \bigg ) \bigg ( -8 \sigma \sqrt \Big ( m^2 + m r + r^2 \Big ) \,}}\Big (\frac} \Big ) \nonumber \\&\quad + 8 \sigma \sqrt \Big ( m^2 + m r + r^2 \Big ) \,}}\Big ( \frac} \Big ) +3 m^2 \big (\text \Big ( \frac \Big ) \nonumber \\&\quad - \text \Big ( \frac \Big ) \big ) \Big (m + 2 r \Big )^2 \bigg ) \biggr ] \Biggr ] \ , \end$$

(A.2)

where \(\,}}(\cdot )\) is the error function

$$\begin \,}}(x) = \frac} \int _0^x e^ \text t \ , \end$$

(A.3)

and \(\text (\cdot )\) is the exponential integral

$$\begin \text (x) = \int _^x \frac \text t \ . \end$$

(A.4)

For comparison with the \(V_\) numerical results in Fig. 9, we are interested in the limit \(m \rightarrow 0\),

$$\begin&\lim _ \Big [ \mathbb [}] \Big ] = \frac}} \bigg (r^2 \sigma \Big (4 - 6 e^}\Big ) + \sigma ^3\Big (e^} - 1\Big )\bigg ) \nonumber \\&\quad + \,}}\Big (\frac}\Big ) \ . \end$$

(A.5)

Similarly, the integral 2.4 with \(n=2\) can be separated

$$\begin \mathbb [^2}]&= \int _\mathbb [(x')}]^2 \ f(x') \ \text x' \nonumber \\&= 2 \int _0^m \big [ 1 \big ]^2 \ \Big [ \frac} e^(\frac)^2} \Big ] \ \text x'\nonumber \\&+ 2\int _m^ \Big [ \frac (2r + m - |x'|)^2 (x'^2 \nonumber \\ &\quad - 3 m^2 + 2|x'|(m+2r)) \Big ]^2 \ \Big [ \frac} e^(\frac)^2} \Big ] \ \text x' \ , \end$$

(A.6)

which evaluates to

$$\begin&\mathbb [^2}] = 2 \Bigg [ \frac \nonumber \\&\quad erf \Big ( \frac}\Big ) \Bigg ] \nonumber \\&\quad +2 \Bigg [ \bigg [ \frac r^6 \sigma }\bigg ] \bigg [ e^} \Big [ m + 2 r \Big ] \Big [ -9 m^4 (m + 2 r)^2 \nonumber \\&\quad + (61 m^4 + 152 m^3 r + 216 m^2 r^2 + 128 m r^3 \nonumber \\&\quad + 64 r^4) \sigma ^2 + (-m^2 + 20 m r + 20 r^2) \sigma ^4 - 15 \sigma ^6 \Big ] \nonumber \\&\quad + e^} \Big [ 9 m^3 (m + 2 r)^4 - (61 m^5 + 336 m^4 r \nonumber \\&\quad + 768 m^3 r^2 + 1024 m^2 r^3 + 816 m r^4 + 384 r^5) \sigma ^2 \nonumber \\&\qquad + (m^3 + 24 m^2 r + 24 m r^2 + 64 r^3) \sigma ^4 + 15 m \sigma ^6 \Big ] \nonumber \\&\quad + \frac} \,}}\Big ( \frac \sigma )} \Big ) \Big [9 m^4 (m + 2 r)^4 \nonumber \\&\quad - 4 (m + 2 r)^2 (5 m^2 + 2 m r + 2 r^2) (5 m^2 + 8 m r + 8 r^2) \sigma ^2 \nonumber \\&\qquad - 6 (5 m^4 + 20 m^3 r + 44 m^2 r^2 + 48 m r^3 + 24 r^4) \sigma ^4 \nonumber \\&\quad + 36 (m^2 + 2 m r + 2 r^2) \sigma ^6 - 15 \sigma ^8 \Big ] \nonumber \\&\quad + \,}}\Big ( \frac \sigma } \Big ) \Big [ -9 m^4 (m + 2 r)^4 + 4 (m \nonumber \\ &\quad + 2 r)^2 (5 m^2 + 2 m r + 2 r^2) (5 m^2 + 8 m r \nonumber \\&\quad +8 r^2) \sigma ^2 \nonumber \\&\quad + 6 (5 m^4 + 20 m^3 r + 44 m^2 r^2 + 48 m r^3 + 24 r^4) \sigma ^4 \nonumber \\&\quad -36 (m^2 + 2 m r + 2 r^2) \sigma ^6 + 15 \sigma ^8 \Big ] \nonumber \\&\quad + 48 m^2 \sigma \big ( \text \Big ( \frac \Big ) \nonumber \\&\quad - \text \Big ( \frac \Big ) \big ) (m + 2 r)^3 (m^2 + m r + r^2) \bigg ] \Bigg ] \ . \end$$

(A.7)

The variance of \(V_\) is then \(\hbox (V_) = \mathbb [^2}] - \mathbb [}]^2\), or, in the limit \(m \rightarrow 0\),

$$\begin&\lim _ \Big [ \hbox (V_) \Big ] = \frac \bigg [\nonumber \\&\quad -8 \sigma \big (-6 r^2 + \sigma ^2\big ) \big (-8 \sqrt r^3 - 6 r^2 s + \sigma ^3\big ) - 8 e^} \big (\sigma ^3 \nonumber \\&\quad - 4 r^2 \sigma \big )^2 \nonumber \\&\quad + 2\sigma e^} \big (64 r^5 \sqrt + 192 r^4 \sigma + 20 r^3 \sigma ^2 \sqrt \nonumber \\ &\quad - 80 r^2 \sigma ^3 - 15 r \sigma ^4 \sqrt + 8 \sigma ^5\big ) \nonumber \\&\quad + 64 r^3 \sigma e^} \sqrt \big (\sigma ^2 - 4 r^2\big ) \,}}\Big (\frac}\Big ) \nonumber \\ &\quad + \big (64 r^3 \sigma \sqrt (6 r^2 - \sigma ^2) \nonumber \\ &\quad + \pi (256 r^6 + 144 r^4 \sigma ^2 - 72 r^2 \sigma ^4 + 15 \sigma ^6)\big ) \,}}\Big (\frac}\Big ) \nonumber \\ &\quad - 256 \pi r^6 \,}}\Big (\frac}\Big )^2\bigg ] \ . \end$$

(A.8)

Lastly, the integral 2.5

$$\begin p(V_ > V^*)&= p(|x| < x^*) \nonumber \\&= 2 \int _^ f(x') \ \text x' \ , \end$$

(A.9)

requires finding a critical \(x^*\) for which if \(|x| > x^*\), \(V_ < V^*\). The expression

$$\begin x^*\Bigg [ \frac \big (2r + m - x^*\big )^2 \big (^2 - 3 m^2 + 2x^*(m+2r)\big ) - V^* = 0 \Bigg ] \end$$

(A.10)

is quartic in \(x^*\) and has a root within the interval \((m, 2r + m)\) at

$$\begin x^* = \frac (\alpha - \beta ) \ , \end$$

(A.11)

withs

$$\begin \alpha&= \Bigg [\bigg [ 4 (m^2+2 m r+2 r^2) \bigg ] \nonumber \\&\quad + \frac}}\Big [-432 \Big (m^2+2 m r+2 r^2\Big )^3 \nonumber \\&\quad -1296 \Big (m^4+4 m^3 r\nonumber \\ &\quad +4 m^2 r^2\Big ) \Big (m^2+2 m r+2 r^2\Big ) \nonumber \\&\quad +1728 \Big (m^3+3 m^2 r+3 m r^2-2 r^3 V+2 r^3\Big )^2 \nonumber \\ &\quad +\Big (\big (-432 (m^2+2 m r+2 r^2)^3+1728 (m^3 \nonumber \\ &\quad +3 m^2 r+3 m r^2-2 r^3 V+2 r^3)^2 \nonumber \\&\quad -1296 (m^4+4 m^3 r+4 m^2 r^2) (m^2+2 m r+2 r^2)\big )^2 \nonumber \\ &\quad -4 \big (144 m^2 r^2+288 m r^3 +144 r^4\big )^3\Big )^\bigg ]^ \nonumber \\&\quad + \bigg [ 48 \root 3 \of \Big (m^2 r^2+2 m r^3+r^4\Big )\bigg ]\bigg [-432 \Big (m^2+2 m r+2 r^2\Big )^3 \nonumber \\ &\quad +1728 \Big (m^3+3 m^2 r+3 m r^2-2 r^3 V+2 r^3\Big )^2 \nonumber \\&\quad - 1296 \Big (m^4+4 m^3 r+4 m^2 r^2\Big )\nonumber \\&\quad \Big (m^2+2 m r+2 r^2\Big )\nonumber \\ &\quad + \Big (\big (-432 (m^2+2 m r+2 r^2)^3+1728 (m^3+3 m^2 r \nonumber \\ &\quad +3 m r^2-2 r^3 V+2 r^3)^2 \nonumber \\&\quad -1296 (m^4+4 m^3 r+4 m^2 r^2) (m^2+2 m r+2 r^2)\big )^2\nonumber \\ &\quad -4 \big (144 m^2 r^2+288 m r^3+144 r^4\big )^3\Big )^\bigg ]^\Bigg ]^ \ , \end$$

(A.12)

and

$$\begin \beta&= \Bigg [ \bigg [ 8 (m^2+2 r m+2 r^2) \bigg ]\nonumber \\&\quad - \frac}\bigg [\bigg (-432 \Big (m^2+2 r m+2 r^2\Big )^3 \nonumber \\ &\quad -1296 \Big (m^4 +4 r m^3 + 4 r^2 m^2\Big ) \Big (m^2 +2 r m+2 r^2\Big )\nonumber \\ &\quad +1728 \Big (m^3+3 r m^2+3 r^2 m+2 r^3-2 r^3 V\Big )^2\nonumber \\&\quad +\Big (\big (-432(m^2+2 r m+2 r^2)^3-1296 (m^4+4 r m^3 \nonumber \\ &\quad +4 r^2 m^2) (m^2+2 r m+2 r^2)+1728 (m^3+3 r m^2\nonumber \\ &\quad +3 r^2 m+2 r^3-2 r^3 V)^2\big )^2 \nonumber \\&\quad -4 \big (144 r^4+288 m r^3+144 m^2 r^2\big )^3\Big )^\bigg )^\bigg ] \nonumber \\&\quad -\bigg [48\ 2^} (r^4+2 m r^3+m^2 r^2)\bigg ]\bigg [-432 (m^2+2 r m+2 r^2)^3 \nonumber \\ &\quad -1296 (m^4 +4 r m^3+4 r^2 m^2) (m^2+2 r m+2 r^2) \nonumber \\&\quad +1728 (m^3+3 r m^2+3 r^2 m+2 r^3-2 r^3V)^2\nonumber \\&\quad +((-432 (m^2+2 r m+2 r^2)^3-1296 (m^4+4 r m^3 \nonumber \\ &\quad +4 r^2 m^2) (m^2+2 r m+2 r^2) +1728 (m^3+3 r m^2+3 r^2 m\nonumber \\ &\quad +2 r^3-2 r^3 V)^2)^2-4 (144 r^4+288 m r^3+144 m^2 r^2)^3)^\bigg ]^ \nonumber \\&\quad -\bigg [16 (m^3+3 r m^2+3 r^2 m+2 r^3-2 r^3 V)\bigg ]\bigg [4 (m^2+2 r m\nonumber \\&\quad +2r^2)\nonumber \\ &\quad +\frac}\Big [-432 (m^2+2 r m+2 r^2)^3 \nonumber \\&\quad -1296 (m^4+4 r m^3+4 r^2 m^2) (m^2+2 r m+2 r^2)+1728 (m^3\nonumber \\ &\quad +3 r m^2+3 r^2 m+2 r^3-2 r^3 V)^2+((-432 (m^2+2 r m+2 r^2)^3 \nonumber \\&\quad -1296 (m^4+4 r m^3+4 r^2 m^2) (m^2+2 r m+2 r^2)+1728 (m^3\nonumber \\ &\quad +3 r m^2+3 r^2 m\nonumber \\&\quad +2 r^3-2 r^3 V)^2)^2-4 (144 r^4+288 m r^3+144 m^2 r^2)^3)^\Big ]^ \nonumber \\&\quad + \Big [48 \root 3 \of (r^4+2 m r^3+m^2 r^2)\Big ]\Big [-432 (m^2+2 r m+2 r^2)^3\nonumber \\&\quad -1296 (m^4+4 r m^3+4 r^2 m^2) (m^2+2 r m+2 r^2) \nonumber \\&\quad +1728 (m^3+3 r m^2+3 r^2 m+2 r^3-2 r^3V)^2\nonumber \\&+((-432 (m^2+2 r m+2 r^2)^3\nonumber \\&\quad -1296 (m^4+4 r m^3+4 r^2 m^2) (m^2+2 r m+2 r^2)\nonumber \\&\quad +1728 (m^3+3 r m^2+3 r^2 m+2 r^3-2 r^3 V)^2)^2\nonumber \\&-4 (144 r^4+288 m r^3+144 m^2 r^2)^3)^\Big ]^\bigg ]^\Bigg ]^ \end$$

(A.13)

Having found \(x^*\), the equation 2.5 integral is trivial,

$$\begin p(V_ > V^*)&= 2 \int _^ f(x') \ \text x' \nonumber \\&= \,}}\Big (\frac} \Big ) \ . \end$$

(A.14)

Fig. 1The alternative text for this image may have been generated using AI.

\(V_\) is expressed as the intersection volume of spheres having radii r and \(r + m\). The misalignment distance between sphere centers is denoted x

Fig. 2The alternative text for this image may have been generated using AI.

Visualization of \(V_(x)\) (equation 2.3) for example target radii r and margins m. Notice a more rapid coverage fall-off for smaller targets utilizing similar margins

Fig. 3The alternative text for this image may have been generated using AI.

The expected value of \(V_\) for geometric errors x with probability density \(\mathcal (0, \sigma ^2)\) according to equation A.2. The variance of \(V_\) is shown by the boundaries of the shaded region

Fig. 4The alternative text for this image may have been generated using AI.

Probability of meeting coverage objectives \(V_ > 50\%\) and \(V_ > 90\%\), and \(V_ > 95\%\) for various r and m as a function of \(\sigma _x\) according to equation A.14

Fig. 5The alternative text for this image may have been generated using AI.

Clinical single-isocenter multi-target treatment plans were created using an in-house planning script with four arcs and systematically generated optimization structures

Fig. 6The alternative text for this image may have been generated using AI.

Relative DVH perturbations for 8 targets for 1.2 mm translation uncertainties in the x, y, and z directions. Nominal DVHs are shown with solid lines and six perturbed DVHs are shown with dashed lines in each subplot

Fig. 7The alternative text for this image may have been generated using AI.

Sensitivity of \(V_\) to translation errors, stratified by shift direction (left) and number of targets (right). First and third quartiles are represented by the bottoms and tops of the error bars. Translations of 1 mm can reduce \(V_\) by as much as 10%

Fig. 8The alternative text for this image may have been generated using AI.

Measured \(V_\) compared to samples from the SVM ensemble. Error bar caps and the shaded region’s boundaries are representative of the first and third quartiles of the coverage distribution for a given shift. Notice an increased variance in the ensemble’s description of coverage for larger |x|

Fig. 9The alternative text for this image may have been generated using AI.

The SVM ensembles are used to describe coverage distributions for various \(\sigma _\). Left: The center line shows the \(V_\) median and the shaded region shows the first and third quartiles. Right: The center line shows the expected value \(\mathbb [V_]\) and the shaded region shows \(\pm \sigma _}\)

Fig. 10The alternative text for this image may have been generated using AI.

The probability of achieving \(V_ > 95\%\) is evaluated for error models \(\sigma _ \in (0 \hbox , 3 \hbox )\). For the analytical model we used parameters \((r = 7.9\) mm, \(m= 0\) mm) for \(p(V_ > V^*)\) and \((r = 7.9\) mm, \(m= 0.4\) mm) for \(p(V_ > V^*)\)

Table 1 Notation used throughout the manuscriptTable 2 Tabulated tolerances on maximum translational uncertainty required to achieve coverage with at least some probability