In this thesis, we will show how certain classes of quantum many-body Hamiltonians with $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on $k$. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of $k$ equations and diagonalizing $k\times k$ matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions. The SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have $\su{2}$ spectrum generating algebras. Next, we extend the technique to the non-compact $\su{1,1}$ algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a $k$-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ spectrum generating algebra. An attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of $10\times 10$ matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. The thesis concludes by exploring possible future developments of the SHA.

In this thesis, we will show how certain classes of quantum many-body Hamiltonians with $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on $k$. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of $k$ equations and diagonalizing $k\times k$ matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions. The SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have $\su{2}$ spectrum generating algebras. Next, we extend the technique to the non-compact $\su{1,1}$ algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a $k$-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a $\su{2}_1 \oplus \su{2}_2 \oplus \ldots \oplus \su{2}_k$ spectrum generating algebra. An attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of $10\times 10$ matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. The thesis concludes by exploring possible future developments of the SHA.

The overexpression of the human epidermal growth factor receptor-2 (HER-2) in 20-25% of human breast cancers was investigated as a target for development of a gold nanoparticle (AuNP) based radiosensitizer for improving the efficacy of neoadjuvant X-radiation therapy of the disease. HER-2 targeted AuNPs were developed by covalently conjugating trastuzumab, a Health Canada approved monoclonal antibody for the treatment of HER-2-overexpressing breast cancer, to 30 nm AuNPs. Trastuzumab conjugated AuNPs were efficiently internalized by HER-2-overexpressing breast cancer cells (as assessed by darkfield microscopy and transmission electron microscopy) and increased DNA damage from X-radiation in these cells by more than 5-fold. To optimize delivery of AuNPs to HER-2-overexpressing tumors, high resolution microSPECT/CT imaging was used to track the in vivo fate of 111In-labelled non-targeted and HER-2 targeted AuNPs following intravenous (i.v.) or intratumoral (i.t.) injection. For i.v. injection, the effects of GdCl3 (for deactivation of macrophages) and non-specific (anti-CD20) antibody rituximab (for blocking of Fc mediated liver and spleen uptake) were studied. It was found that HER-2 targeting via attachment of trastuzumab paradoxically decreased tumor uptake as a result of faster elimination of the targeted AuNPs from the blood while improving internalization in HER-2-positive tumor cells as compared to non-targeted AuNPs. This phenomenon could be attributed to Fc-mediated recognition and subsequent sequestration of trastuzumab conjugated AuNP by the reticuloendothelial system (RES). Blocking of the RES did not increase tumor uptake of either HER-2 targeted or non-targeted AuNPs. Following i.t. injection, our results suggest that Au-NTs redistribute over time and traffick to the liver via the ipsilateral axillary lymph node leading to comparable exposure as seen with i.v. administration. In contrast, targeted AuNPs are bound and internalized by HER-2-overexpressing tumor cells following i.t. injection, with a lower proportion of AuNPs redistributing to normal tissues. In vivo, the combination of HER-2 targeted AuNPs injected i.t. and X-radiation (11 Gy) yielded a 46% decrease in tumor size over a 4 month period in contrast to an 11.5% increase in tumor size for X-radiation treatment alone. Toxicology studies (evaluated through complete blood cell counts, by serum transaminase and creatinine measurements and by monitoring the body weight) demonstrated no apparent normal organ toxicity from the combination of HER-2 targeted AuNPs and X-radiation. These results are promising for the clinical translation of HER-2-targeted AuNPs for radiosensitization of tumors to X-radiation.

The overexpression of the human epidermal growth factor receptor-2 (HER-2) in 20-25% of human breast cancers was investigated as a target for development of a gold nanoparticle (AuNP) based radiosensitizer for improving the efficacy of neoadjuvant X-radiation therapy of the disease. HER-2 targeted AuNPs were developed by covalently conjugating trastuzumab, a Health Canada approved monoclonal antibody for the treatment of HER-2-overexpressing breast cancer, to 30 nm AuNPs. Trastuzumab conjugated AuNPs were efficiently internalized by HER-2-overexpressing breast cancer cells (as assessed by darkfield microscopy and transmission electron microscopy) and increased DNA damage from X-radiation in these cells by more than 5-fold. To optimize delivery of AuNPs to HER-2-overexpressing tumors, high resolution microSPECT/CT imaging was used to track the in vivo fate of 111In-labelled non-targeted and HER-2 targeted AuNPs following intravenous (i.v.) or intratumoral (i.t.) injection. For i.v. injection, the effects of GdCl3 (for deactivation of macrophages) and non-specific (anti-CD20) antibody rituximab (for blocking of Fc mediated liver and spleen uptake) were studied. It was found that HER-2 targeting via attachment of trastuzumab paradoxically decreased tumor uptake as a result of faster elimination of the targeted AuNPs from the blood while improving internalization in HER-2-positive tumor cells as compared to non-targeted AuNPs. This phenomenon could be attributed to Fc-mediated recognition and subsequent sequestration of trastuzumab conjugated AuNP by the reticuloendothelial system (RES). Blocking of the RES did not increase tumor uptake of either HER-2 targeted or non-targeted AuNPs. Following i.t. injection, our results suggest that Au-NTs redistribute over time and traffick to the liver via the ipsilateral axillary lymph node leading to comparable exposure as seen with i.v. administration. In contrast, targeted AuNPs are bound and internalized by HER-2-overexpressing tumor cells following i.t. injection, with a lower proportion of AuNPs redistributing to normal tissues. In vivo, the combination of HER-2 targeted AuNPs injected i.t. and X-radiation (11 Gy) yielded a 46% decrease in tumor size over a 4 month period in contrast to an 11.5% increase in tumor size for X-radiation treatment alone. Toxicology studies (evaluated through complete blood cell counts, by serum transaminase and creatinine measurements and by monitoring the body weight) demonstrated no apparent normal organ toxicity from the combination of HER-2 targeted AuNPs and X-radiation. These results are promising for the clinical translation of HER-2-targeted AuNPs for radiosensitization of tumors to X-radiation.