<p>This paper aims to investigate the mathematical modeling of Casson hybrid nanofluid flow, which uses pure blood as the base fluid and incorporates the impacts of titanium dioxide (TiO₂) and silver (Ag) nanoparticles on a Riga plate that helps to stabilize and disperse drug molecules efficiently through a drug-delivery system. The bottom plate is assumed to be implemented with thermal-source effects where the fluid flow has time-dependent attributes. The squeezing characteristics are considered to be induced on the surface of the upper Riga plate that is moving with some speed. A set of suitable variables are incorporated to convert modeled equations to dimensionless form. The problem was initially solved through homotopy analysis method (HAM) and then the artificial neural network (ANN) is used on the basis of HAM. Medical diagnostics could benefit from this model, particularly in the process of drug delivery and the flow dynamics of the microcirculatory mechanism. It has been observed in this study that, with growth in the modified Hartman number, as well as the volumetric fraction of titanium dioxide nanoparticles, the velocity distribution was retarded for both Ag/blood nanofluid and Ag+TiO_2/ blood hybrid nanofluid. For an increase in the volumetric fraction of silver nanoparticles and thermal-source factor there is a corresponding progression in thermal distribution both for Ag/blood nanofluid and Ag+TiO_2/ blood hybrid nanofluid. The heat-transfer rate determines the sustainability of drug delivery by ensuring its safe administration. It is observed that using the 5% nanoparticle volume fraction the obtained results show that a 10.06% increase has been achieved using the hybrid nanofluid in comparison with Ag nanofluid that has increased the heat-transfer rate up to 7.79%. With an increase in the squeezing factor <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>S</mi> </math></EquationSource> <EquationSource Format="TEX">$S$</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="304" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mi>S</mi> <mo>=</mo> <mn>0.0</mn> <mo>,</mo> <mo>−</mo> <mn>0.2</mn> <mo>,</mo> <mo>−</mo> <mn>0.4</mn> <mo>,</mo> <mo>−</mo> <mn>0.6</mn> <mo>,</mo> <mo>−</mo> <mn>0.8</mn> <mo>,</mo> <mo>−</mo> <mn>1.0</mn> <mo>,</mo> <mo>−</mo> <mn>1.2</mn> </math></EquationSource> <EquationSource Format="TEX">$S = 0.0, - 0.2, - 0.4, - 0.6, - 0.8, - 1.0, - 1.2$</EquationSource> </InlineEquation> there is a reduction in the thermal distribution. The optimal model performance is observed at epochs 211, 179, 115, 181, and 168, as indicated in the data displayed at these stated epochs throughout the training. For all five scenarios gradient values are linked at <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>9.94</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$9.94 \times 10^{ - 8}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>9.88</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>9</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$9.88 \times 10^{ - 9}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>9.90</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$9.90 \times 10^{ - 8}$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>9.90</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$9.90 \times 10^{ - 8}$</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11043_2025_9785_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mn>9.93</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>8</mn> </mrow> </msup> </math></EquationSource> <EquationSource Format="TEX">$9.93 \times 10^{ - 8}$</EquationSource> </InlineEquation>. Medical diagnostics could benefit from this model, particularly in the process of drug delivery and the flow dynamics of the microcirculatory mechanism.</p>