# The Multilinear Reference Tissue Model

### Description

The multilinear reference tissue model (MRTM) estimates the binding potential from an observed time activity curve without the need for arterial sampling. Instead, a second time activity curve must be provided from a suitable reference region where there is negligible binding.

### Usage

1 | ```
multilinearReferenceTissueModel(tac, ref, time, tstar, MRTM2 = TRUE, k2prime = NULL)
``` |

### Arguments

`tac` |
a vector corresponding to the time activity curve from the tissue (in Bq/mL). |

`ref` |
a vector corresponding to the time activity curve from the reference region (in Bq/mL). |

`time` |
a vector of average frame times (in minutes). |

`tstar` |
the time (in minutes) where the linear relationship between the response and covariates may be assumed to be true. |

`MRTM2` |
a logical value that selects the three-parameter model (MRTM) or the two-parameter model (MRTM2), where k2prime is fixed. |

`k2prime` |
the value of k2prime that has been fixed. |

### Details

See the references.

The numeric integration required to construct the design matrix is
performed by interpolating the time activity curves, both for the
tissue and reference region, to one-second resolution and then
performing the `cumsum`

operation on them.

Given the nonlinear relationship between binding potential and the
regression parameters, the `deltamethod`

is used to approximate
its standard error.

### Value

`BP` |
Binding potential |

`BP.error` |
Approximate standard error of the binding potential |

`R1` |
Ratio of the volumes of distrubution for the tissue and reference region (assumes a one-tissue model is valid) |

`R1.error` |
Approximate standard error for the ratio |

`k2` |
Clearance rate constant from the tissue to plasma (assumes a one-tissue model is valid) |

`k2.error` |
Approximate standard error for k2 |

`X` |
Design matrix used in the linear regression |

`beta` |
Regression coefficients |

### Author(s)

Brandon Whitcher bwhitcher@gmail.com

### References

Ichise, M., Ballinger, J.R., Golan, H., Vines, D., Luong, A., Tsai,
S. and Kung, H.F. (1996) Noninvasive quantification of dopamine D2
receptors with iodine-123-IBF SPECT, *Journal of Nuclear
Medicine*, **37**, 513-520.

Ichise, M., Liow, J.-S., Lu, J.-Q., Takano, A., Model, K., Toyama, H.,
Suhara, T., Suzuki, K., Innis, R.B., Carson, R.E. (2003) Linearized
reference tissue parametric imaging methods: Application to [11C]DASB
positron emission tomography studies of the serotonin transporter in
human brain, *Journal of Cerebral Blood Flow \& Metabolism*,
**23**, 1096-1112.

### See Also

`cumsum`

, `deltamethod`